diff --git a/changes-set.txt b/changes-set.txt index f4cd30a4e..5b500283d 100644 --- a/changes-set.txt +++ b/changes-set.txt @@ -85,6 +85,43 @@ make a github issue.) DONE: Date Old New Notes +24-Feb-25 naddid1 naddrid +24-Feb-25 naddid2 naddlid +24-Feb-25 mulid1 mulrid +24-Feb-25 mulid2 mullid +24-Feb-25 mulid1i mulridi +24-Feb-25 mulid2i mullidi +24-Feb-25 mulid1d mulridd +24-Feb-25 mulid2d mullidd +24-Feb-25 addid1 addrid +24-Feb-25 addid2 addlid +24-Feb-25 addid1i addridi +24-Feb-25 addid2i addlidi +24-Feb-25 addid1d addridd +24-Feb-25 addid2d addlidd +24-Feb-25 xaddid1 xaddrid +24-Feb-25 xaddid2 xaddlid +24-Feb-25 xaddid1d xaddridd +24-Feb-25 xmulid1 xmulrid +24-Feb-25 xmulid2 xmullid +24-Feb-25 mulrid mulridx +24-Feb-25 dchrmulid2 dchrmullid +24-Feb-25 hvaddid2 hvaddlid +24-Feb-25 hvaddid2i hvaddlidi +24-Feb-25 hoaddid1i hoaddridi +24-Feb-25 hoaddid1 hoaddrid +24-Feb-25 homulid2 homullid +24-Feb-25 readdid1addid2d readdridaddlidd +24-Feb-25 reneg0addid2 reneg0addlid +24-Feb-25 resubidaddid1lem resubidaddlidlem +24-Feb-25 resubidaddid1 resubidaddlid +24-Feb-25 readdid2 readdlid +24-Feb-25 sn-addid2 sn-addlid +24-Feb-25 readdid1 readdrid +24-Feb-25 sn-addid1 sn-addrid +24-Feb-25 remulid2 remullid +24-Feb-25 sn-mulid2 sn-mullid +24-Feb-25 xaddid2d xaddlidd 24-Feb-25 abbi1 abbi reverse direction of abbib 23-Feb-25 abbi abbib flip sides of bi-conditional 23-Feb-25 addsid1 addsrid diff --git a/discouraged b/discouraged index ad05af074..e1d6de335 100644 --- a/discouraged +++ b/discouraged @@ -1272,7 +1272,7 @@ "ax-hv0cl" is used by "hon0". "ax-hv0cl" is used by "hsn0elch". "ax-hv0cl" is used by "hv2neg". -"ax-hv0cl" is used by "hvaddid2". +"ax-hv0cl" is used by "hvaddlid". "ax-hv0cl" is used by "hvmul0". "ax-hv0cl" is used by "hvsub0". "ax-hv0cl" is used by "ifhvhv0". @@ -1302,9 +1302,9 @@ "ax-hvaddid" is used by "3oalem2". "ax-hvaddid" is used by "5oalem1". "ax-hvaddid" is used by "5oalem2". -"ax-hvaddid" is used by "hoaddid1i". +"ax-hvaddid" is used by "hoaddridi". "ax-hvaddid" is used by "hst1h". -"ax-hvaddid" is used by "hvaddid2". +"ax-hvaddid" is used by "hvaddlid". "ax-hvaddid" is used by "hvaddsub4". "ax-hvaddid" is used by "hvpncan". "ax-hvaddid" is used by "hvsub0". @@ -1344,7 +1344,7 @@ "ax-hvcom" is used by "hvadd12". "ax-hvcom" is used by "hvadd32". "ax-hvcom" is used by "hvaddcan2". -"ax-hvcom" is used by "hvaddid2". +"ax-hvcom" is used by "hvaddlid". "ax-hvcom" is used by "hvcomi". "ax-hvcom" is used by "hvpncan2". "ax-hvcom" is used by "hvsub32". @@ -1402,7 +1402,7 @@ "ax-hvmulid" is used by "h1de2bi". "ax-hvmulid" is used by "hhssnv". "ax-hvmulid" is used by "hilvc". -"ax-hvmulid" is used by "homulid2". +"ax-hvmulid" is used by "homullid". "ax-hvmulid" is used by "hv2times". "ax-hvmulid" is used by "hvaddsubval". "ax-hvmulid" is used by "hvmul0or". @@ -4924,7 +4924,7 @@ "df-mul" is used by "axmulf". "df-mul" is used by "mulcnsr". "df-mulr" is used by "hlhilsmulOLD". -"df-mulr" is used by "mulrid". +"df-mulr" is used by "mulridx". "df-mulr" is used by "mulrndx". "df-mulr" is used by "opsrmulrOLD". "df-mulr" is used by "resvmulrOLD". @@ -7564,8 +7564,8 @@ "ho0f" is used by "ho0subi". "ho0f" is used by "hoaddass". "ho0f" is used by "hoaddcom". -"ho0f" is used by "hoaddid1". -"ho0f" is used by "hoaddid1i". +"ho0f" is used by "hoaddrid". +"ho0f" is used by "hoaddridi". "ho0f" is used by "hoaddsubass". "ho0f" is used by "hocsubdir". "ho0f" is used by "hoddi". @@ -7588,7 +7588,7 @@ "ho0val" is used by "adj0". "ho0val" is used by "df0op2". "ho0val" is used by "ho0coi". -"ho0val" is used by "hoaddid1i". +"ho0val" is used by "hoaddridi". "ho0val" is used by "idleop". "ho0val" is used by "leop3". "ho0val" is used by "leoprf". @@ -7628,7 +7628,7 @@ "hoaddcli" is used by "hoaddassi". "hoaddcli" is used by "hoaddcomi". "hoaddcli" is used by "hoaddfni". -"hoaddcli" is used by "hoaddid1i". +"hoaddcli" is used by "hoaddridi". "hoaddcli" is used by "hocadddiri". "hoaddcli" is used by "hodsi". "hoaddcli" is used by "honegsubi". @@ -7653,14 +7653,14 @@ "hoadddi" is used by "hosubdi". "hoadddi" is used by "opsqrlem6". "hoadddir" is used by "ho2times". -"hoaddid1" is used by "hosubid1". -"hoaddid1i" is used by "ho0subi". -"hoaddid1i" is used by "hoaddid1". -"hoaddid1i" is used by "hodidi". -"hoaddid1i" is used by "hopncani". -"hoaddid1i" is used by "hosd1i". -"hoaddid1i" is used by "hosubeq0i". -"hoaddid1i" is used by "pjclem1". +"hoaddrid" is used by "hosubid1". +"hoaddridi" is used by "ho0subi". +"hoaddridi" is used by "hoaddrid". +"hoaddridi" is used by "hodidi". +"hoaddridi" is used by "hopncani". +"hoaddridi" is used by "hosd1i". +"hoaddridi" is used by "hosubeq0i". +"hoaddridi" is used by "pjclem1". "hoaddsub" is used by "hosubsub". "hoaddsubass" is used by "hoaddsub". "hoaddsubass" is used by "hoaddsubassi". @@ -7766,7 +7766,7 @@ "hoeq" is used by "homco1". "hoeq" is used by "homco2". "hoeq" is used by "homulass". -"hoeq" is used by "homulid2". +"hoeq" is used by "homullid". "hoeq1" is used by "adjadj". "hoeq1" is used by "adjmo". "hoeq1" is used by "hoeq2". @@ -7774,7 +7774,7 @@ "hoeqi" is used by "ho0coi". "hoeqi" is used by "hoaddassi". "hoeqi" is used by "hoaddcomi". -"hoeqi" is used by "hoaddid1i". +"hoeqi" is used by "hoaddridi". "hoeqi" is used by "hocadddiri". "hoeqi" is used by "hocsubdiri". "hoeqi" is used by "hoddii". @@ -7840,7 +7840,7 @@ "homulcl" is used by "homco1". "homulcl" is used by "homco2". "homulcl" is used by "homulass". -"homulcl" is used by "homulid2". +"homulcl" is used by "homullid". "homulcl" is used by "honegsubdi". "homulcl" is used by "honegsubdi2". "homulcl" is used by "honegsubi". @@ -7854,12 +7854,12 @@ "homulcl" is used by "nmopnegi". "homulcl" is used by "opsqrlem1". "homulcl" is used by "opsqrlem6". -"homulid2" is used by "ho2times". -"homulid2" is used by "honegneg". -"homulid2" is used by "leopmul". -"homulid2" is used by "nmopleid". -"homulid2" is used by "opsqrlem1". -"homulid2" is used by "opsqrlem6". +"homullid" is used by "ho2times". +"homullid" is used by "honegneg". +"homullid" is used by "leopmul". +"homullid" is used by "nmopleid". +"homullid" is used by "opsqrlem1". +"homullid" is used by "opsqrlem6". "homval" is used by "adjmul". "homval" is used by "hmopm". "homval" is used by "hoadddi". @@ -7868,7 +7868,7 @@ "homval" is used by "homco1". "homval" is used by "homco2". "homval" is used by "homulass". -"homval" is used by "homulid2". +"homval" is used by "homullid". "homval" is used by "honegsubi". "homval" is used by "leopmuli". "homval" is used by "leopnmid". @@ -7941,7 +7941,7 @@ "hosval" is used by "hoaddcomi". "hosval" is used by "hoadddi". "hosval" is used by "hoadddir". -"hosval" is used by "hoaddid1i". +"hosval" is used by "hoaddridi". "hosval" is used by "hocadddiri". "hosval" is used by "hodsi". "hosval" is used by "honegsubi". @@ -8084,20 +8084,20 @@ "hvaddcli" is used by "normpythi". "hvaddcli" is used by "polidi". "hvaddeq0" is used by "superpos". -"hvaddid2" is used by "3oalem2". -"hvaddid2" is used by "5oalem2". -"hvaddid2" is used by "hilablo". -"hvaddid2" is used by "hilid". -"hvaddid2" is used by "hv2neg". -"hvaddid2" is used by "hvaddid2i". -"hvaddid2" is used by "hvaddsub4". -"hvaddid2" is used by "shunssi". -"hvaddid2" is used by "spanunsni". -"hvaddid2i" is used by "hhssnv". -"hvaddid2i" is used by "hsn0elch". -"hvaddid2i" is used by "hvaddcani". -"hvaddid2i" is used by "hvsubeq0i". -"hvaddid2i" is used by "shscli". +"hvaddlid" is used by "3oalem2". +"hvaddlid" is used by "5oalem2". +"hvaddlid" is used by "hilablo". +"hvaddlid" is used by "hilid". +"hvaddlid" is used by "hv2neg". +"hvaddlid" is used by "hvaddlidi". +"hvaddlid" is used by "hvaddsub4". +"hvaddlid" is used by "shunssi". +"hvaddlid" is used by "spanunsni". +"hvaddlidi" is used by "hhssnv". +"hvaddlidi" is used by "hsn0elch". +"hvaddlidi" is used by "hvaddcani". +"hvaddlidi" is used by "hvsubeq0i". +"hvaddlidi" is used by "shscli". "hvaddsub12" is used by "3oalem2". "hvaddsub12" is used by "5oalem1". "hvaddsub12" is used by "pj3si". @@ -17106,8 +17106,8 @@ New usage of "hoaddcomi" is discouraged (6 uses). New usage of "hoadddi" is discouraged (4 uses). New usage of "hoadddir" is discouraged (1 uses). New usage of "hoaddfni" is discouraged (0 uses). -New usage of "hoaddid1" is discouraged (1 uses). -New usage of "hoaddid1i" is discouraged (7 uses). +New usage of "hoaddrid" is discouraged (1 uses). +New usage of "hoaddridi" is discouraged (7 uses). New usage of "hoaddsub" is discouraged (1 uses). New usage of "hoaddsubass" is discouraged (4 uses). New usage of "hoaddsubassi" is discouraged (4 uses). @@ -17147,7 +17147,7 @@ New usage of "homndx" is discouraged (13 uses). New usage of "homul12" is discouraged (1 uses). New usage of "homulass" is discouraged (6 uses). New usage of "homulcl" is discouraged (21 uses). -New usage of "homulid2" is discouraged (6 uses). +New usage of "homullid" is discouraged (6 uses). New usage of "homval" is discouraged (15 uses). New usage of "hon0" is discouraged (1 uses). New usage of "honegdi" is discouraged (2 uses). @@ -17223,8 +17223,8 @@ New usage of "hvaddcani" is discouraged (2 uses). New usage of "hvaddcl" is discouraged (37 uses). New usage of "hvaddcli" is discouraged (13 uses). New usage of "hvaddeq0" is discouraged (1 uses). -New usage of "hvaddid2" is discouraged (9 uses). -New usage of "hvaddid2i" is discouraged (5 uses). +New usage of "hvaddlid" is discouraged (9 uses). +New usage of "hvaddlidi" is discouraged (5 uses). New usage of "hvaddsub12" is discouraged (5 uses). New usage of "hvaddsub4" is discouraged (2 uses). New usage of "hvaddsubass" is discouraged (2 uses). diff --git a/set.mm b/set.mm index 3edc79174..99b266539 100644 --- a/set.mm +++ b/set.mm @@ -89185,7 +89185,7 @@ general ordinal versions of these theorems (in this case ~ oa0r ) so $d A a b c x $. $( Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.) $) - naddid1 $p |- ( A e. On -> ( A +no (/) ) = A ) $= + naddrid $p |- ( A e. On -> ( A +no (/) ) = A ) $= ( va vb vc vx cv c0 cnadd co wceq oveq1 id eqeq12d con0 wcel wral wa crab cint adantr weq wss 0elon naddov2 mpan2 ral0 biantrur wel wb eleq1 ralimi ralbi syl adantl dfss3 bitr4di bitr3id rabbidv inteqd intmin 3eqtrd tfis3 @@ -89198,8 +89198,8 @@ general ordinal versions of these theorems (in this case ~ oa0r ) so $( Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 20-Feb-2025.) $) - naddid2 $p |- ( A e. On -> ( (/) +no A ) = A ) $= - ( con0 wcel c0 cnadd co wceq 0elon naddcom mpan2 naddid1 eqtr3d ) ABCZADEFZ + naddlid $p |- ( A e. On -> ( (/) +no A ) = A ) $= + ( con0 wcel c0 cnadd co wceq 0elon naddcom mpan2 naddrid eqtr3d ) ABCZADEFZ DAEFZAMDBCNOGHADIJAKL $. ${ @@ -89278,7 +89278,7 @@ general ordinal versions of these theorems (in this case ~ oa0r ) so $( Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 21-Jan-2025.) $) naddword1 $p |- ( ( A e. On /\ B e. On ) -> A C_ ( A +no B ) ) $= - ( con0 wcel wa c0 cnadd co wceq naddid1 adantr wss 0ss 0elon naddss2 mp3an1 + ( con0 wcel wa c0 cnadd co wceq naddrid adantr wss 0ss 0elon naddss2 mp3an1 wb ancoms mpbii eqsstrrd ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBMUB UAUFUGQZFCDUBUAUHNFBAOPRST $. @@ -126625,7 +126625,7 @@ of the other axioms (see ~ cnexALT ), but the proof requires the axiom of $d A x y $. $( ` 1 ` is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, derived from ZF set theory. Weakened from - the original axiom in the form of statement in ~ mulid1 , based on ideas + the original axiom in the form of statement in ~ mulrid , based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ~ ax-1rid . (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) $) @@ -126950,7 +126950,7 @@ of the other axioms (see ~ cnexALT ), but the proof requires the axiom of $( ` 1 ` is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by Theorem ~ ax1rid . Weakened from - the original axiom in the form of statement in ~ mulid1 , based on ideas + the original axiom in the form of statement in ~ mulrid , based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.) $) ax-1rid $a |- ( A e. RR -> ( A x. 1 ) = A ) $. $( $j restatement 'ax-1rid' of 'ax1rid'; $) @@ -127194,7 +127194,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a $d A x y $. $( The number 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) $) - mulid1 $p |- ( A e. CC -> ( A x. 1 ) = A ) $= + mulrid $p |- ( A e. CC -> ( A x. 1 ) = A ) $= ( vx vy cc wcel cv ci cmul co caddc wceq cr wrex c1 ax-icn ax-1cn ax-1rid recn syl eqtrd cnre wa sylancr adddir mp3an3 syl2an mulass mp3an13 oveq2d mulcl oveqan12d oveq1 id eqeq12d syl5ibrcom rexlimivv ) ADEABFZGCFZHIZJIZ @@ -127204,10 +127204,10 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a QUITUKTVAVBVGAUTAUTNHULVAUMUNUOUPS $. $} - $( Identity law for multiplication. See ~ mulid1 for commuted version. + $( Identity law for multiplication. See ~ mulrid for commuted version. (Contributed by NM, 8-Oct-1999.) $) - mulid2 $p |- ( A e. CC -> ( 1 x. A ) = A ) $= - ( cc wcel c1 cmul co wceq ax-1cn mulcom mpan mulid1 eqtrd ) ABCZDAEFZADEFZA + mullid $p |- ( A e. CC -> ( 1 x. A ) = A ) $= + ( cc wcel c1 cmul co wceq ax-1cn mulcom mpan mulrid eqtrd ) ABCZDAEFZADEFZA DBCMNOGHDAIJAKL $. ${ @@ -127264,12 +127264,12 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a ${ axi.1 $e |- A e. CC $. $( Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) $) - mulid1i $p |- ( A x. 1 ) = A $= - ( cc wcel c1 cmul co wceq mulid1 ax-mp ) ACDAEFGAHBAIJ $. + mulridi $p |- ( A x. 1 ) = A $= + ( cc wcel c1 cmul co wceq mulrid ax-mp ) ACDAEFGAHBAIJ $. $( Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) $) - mulid2i $p |- ( 1 x. A ) = A $= - ( cc wcel c1 cmul co wceq mulid2 ax-mp ) ACDEAFGAHBAIJ $. + mullidi $p |- ( 1 x. A ) = A $= + ( cc wcel c1 cmul co wceq mullid ax-mp ) ACDEAFGAHBAIJ $. axi.2 $e |- B e. CC $. $( Closure law for addition. (Contributed by NM, 23-Nov-1994.) $) @@ -127338,13 +127338,13 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a addcld.1 $e |- ( ph -> A e. CC ) $. $( Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) $) - mulid1d $p |- ( ph -> ( A x. 1 ) = A ) $= - ( cc wcel c1 cmul co wceq mulid1 syl ) ABDEBFGHBICBJK $. + mulridd $p |- ( ph -> ( A x. 1 ) = A ) $= + ( cc wcel c1 cmul co wceq mulrid syl ) ABDEBFGHBICBJK $. $( Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) $) - mulid2d $p |- ( ph -> ( 1 x. A ) = A ) $= - ( cc wcel c1 cmul co wceq mulid2 syl ) ABDEFBGHBICBJK $. + mullidd $p |- ( ph -> ( 1 x. A ) = A ) $= + ( cc wcel c1 cmul co wceq mullid syl ) ABDEFBGHBICBJK $. addcld.2 $e |- ( ph -> B e. CC ) $. $( Closure law for addition. (Contributed by Mario Carneiro, @@ -127394,7 +127394,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a $( Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) adddirp1d $p |- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + B ) ) $= - ( c1 caddc co cmul 1cnd adddird mulid2d oveq2d eqtrd ) ABFGHCIHBCIHZFCIHZ + ( c1 caddc co cmul 1cnd adddird mullidd oveq2d eqtrd ) ABFGHCIHBCIHZFCIHZ GHOCGHABFCDAJEKAPCOGACELMN $. $} @@ -128480,7 +128480,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a muladd11 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) ) $= ( cc wcel wa c1 caddc cmul wceq ax-1cn addcl mpan adddi mp3an2 sylan adantr - co mulid1d adddir eqtrd mp3an1 mulid2 adantl oveq1d oveq12d ) ACDZBCDZEZFAG + co mulridd adddir eqtrd mp3an1 mullid adantl oveq1d oveq12d ) ACDZBCDZEZFAG QZFBGQHQZUIFHQZUIBHQZGQZUIBABHQZGQZGQUFUICDZUGUJUMIZFCDZUFUPJFAKLZUPURUGUQJ UIFBMNOUHUKUIULUOGUFUKUIIUGUFUIUSRPUHULFBHQZUNGQZUOURUFUGULVAIJFABSUAUHUTBU NGUGUTBIUFBUBUCUDTUET $. @@ -128488,7 +128488,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a $( Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) $) 1p1times $p |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) ) $= - ( cc wcel c1 caddc co 1cnd id cmul mulid2 oveq12d joinlmuladdmuld ) ABCZDAD + ( cc wcel c1 caddc co 1cnd id cmul mullid oveq12d joinlmuladdmuld ) ABCZDAD AAEFMGZMHNMDAIFZAOAEAJZPKL $. $( A theorem for complex numbers analogous the second Peano postulate @@ -128517,7 +128517,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a 00id $p |- ( 0 + 0 ) = 0 $= ( vc vy cc0 cr wcel cv caddc co wceq 0re wa cmul c1 cc 0cn syl2anc eqtr3d recnd oveq1d remulcl wrex ax-rnegex oveq2 eqeq1d biimpd adantld ax-rrecex - wi wne adantlr simplll simprl mulass mp3an3 oveq1 mulid2i eqtrdi ad2antll + wi wne adantlr simplll simprl mulass mp3an3 oveq1 mullidi eqtrdi ad2antll simpllr mpan2 adddir mp3an2i sylancr addass eqtr2d sylan2 readdcl sylancl ad2antrl readdcan mp3an2 mpbid rexlimddv expcom pm2.61ine rexlimiva mp2b wb ) CDEZCAFZGHZCIZADUACCGHZCIZJACUBWBWDADVTDEZWBKZWDUHVTCVTCIZWBWDWEWGWB @@ -128540,7 +128540,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a B = ( B + B ) ) $= ( vy cr wcel cc0 cmul co wne wa cc c1 wceq caddc recnd mulcld 0cn syl2anc mp3an3 eqtrd cv wrex 0re remulcl mpan ax-rrecex adantr 00id oveq2i eqcomi - sylan simprl simplll simplr mul32 mul31 simprr oveq1d mulid2 ad2antlr syl + sylan simprl simplll simplr mul32 mul31 simprr oveq1d mullid ad2antlr syl adddi mp3an23 oveq12d 3eqtr3a rexlimddv ) ADEZFAGHZFIZJZBKEZJZVHCUAZGHZLM ZBBBNHZMCDVJVOCDUBZVKVGVHDEZVIVQFDEVGVRUCFAUDUECVHUFUKUGVLVMDEZVOJZJZVMAG HZBGHZFGHZWCFFNHZGHZBVPWFWDWEFWCGUHUIUJWAWDWBFGHZBGHZBWAWBKEZVKWDWHMZWAVM @@ -128588,13 +128588,13 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.) $) - addid1 $p |- ( A e. CC -> ( A + 0 ) = A ) $= + addrid $p |- ( A e. CC -> ( A + 0 ) = A ) $= ( vc vx c1 cr wcel caddc co cc0 wceq cc 1re wa ci ax-1cn mul01 eqtr3i 0re cmul a1i cv wrex ax-rnegex wne ax-1ne0 oveq2 eqeq1d biimpcd ax-icn mulcli - wi 0cn adddii mulid1i ax-mp ax-i2m1 eqtr4i oveq12i eqeq12i addassi oveq2i + wi 0cn adddii mulridi ax-mp ax-i2m1 eqtr4i oveq12i eqeq12i addassi oveq2i eqtri oveq1i 00id eqcomi wb readdcan mp3an 3bitri sylib necon3d ax-rrecex syl6 mpi sylan2 simpr simplrl recnd mulcld simplll adddid addassd simpllr - oveq1d eqtr3d 3eqtr4a readdcld syl3anc mpbid oveq2d mul31 simplrr mulid2d + oveq1d eqtr3d 3eqtr4a readdcld syl3anc mpbid oveq2d mul31 simplrr mullidd 3eqtrd syl oveq12d 3eqtr3d exp42 rexlimdv mpd rexlimiva mp2b ) DEFZDBUAZG HZIJZBEUBAKFZAIGHZAJZUKZLBDUCXFXJBEXDEFZXFMZXDCUAZSHZDJZCEUBZXJXFXKXDIUDZ XPXFDIUDXQUEXFXDIDIXFXDIJZDIGHZIJZDIJZXRXFXTXRXEXSIXDIDGUFUGUHXTNNSHZYBSH @@ -128620,7 +128620,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a cnegex $p |- ( A e. CC -> E. x e. CC ( A + x ) = 0 ) $= ( va vb vc vd cc wcel cv ci cmul co caddc wceq cr wrex cc0 recnd eqtr3d wa cnre ax-rnegex anim12i reeanv sylibr ax-icn a1i simplrr mulcld simplrl - addcld simplll simpllr adddid simprr oveq2d mul01 ax-mp eqtrdi addid1 syl + addcld simplll simpllr adddid simprr oveq2d mul01 ax-mp eqtrdi addrid syl addassd 3eqtrd oveq1d simprl oveq2 eqeq1d rspcev syl2anc rexlimdvva oveq1 ex mpd rexbidv syl5ibrcom rexlimivv ) BGHBCIZJDIZKLZMLZNZDOPCOPBAIZMLZQNZ AGPZCDBUAWAWECDOOVQOHZVROHZTZWEWAVTWBMLZQNZAGPZWHVQEIZMLZQNZVRFIZMLZQNZTZ @@ -128638,7 +128638,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a $( Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.) $) cnegex2 $p |- ( A e. CC -> E. x e. CC ( x + A ) = 0 ) $= - ( cc wcel ci cmul co caddc cc0 wceq cv wrex ax-icn mulcli mulcl c1 mulid2 + ( cc wcel ci cmul co caddc cc0 wceq cv wrex ax-icn mulcli mulcl c1 mullid mpan oveq2d ax-i2m1 oveq1i ax-1cn adddir mul02 eqtr3d oveq1 eqeq1d rspcev mp3an12 3eqtr3a syl2anc ) BCDZEEFGZBFGZCDZUNBHGZIJZAKZBHGZIJZACLUMCDZULUO EEMMNZUMBORULUNPBFGZHGZUPIULVCBUNHBQSULUMPHGZBFGZIBFGVDIVEIBFTUAVAPCDULVF @@ -128651,10 +128651,10 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) $) - addid2 $p |- ( A e. CC -> ( 0 + A ) = A ) $= + addlid $p |- ( A e. CC -> ( 0 + A ) = A ) $= ( vx vy cc wcel cv caddc co cc0 wceq cnegex wrex ad2antrl w3a 0cn mp3an12 wa addass oveq2d 3eqtr3rd adantr 3ad2ant3 00id oveq1i simp1 simp2l simp3l - addassd simp2r oveq1d simp3r addid1 3ad2ant1 eqtr3d eqtrid expd rexlimddv + addassd simp2r oveq1d simp3r addrid 3ad2ant1 eqtr3d eqtrid expd rexlimddv 3expia rexlimdv mpd ) ADEZABFZGHZIJZIAGHZAJZBDBAKVAVBDEZVDQZQZVBCFZGHZIJZ CDLZVFVGVMVAVDCVBKMVIVLVFCDVIVJDEZVLVFVAVHVNVLQZVFVAVHVONZIIGHZVJGHZIIVJG HZGHZAVEVOVAVRVTJZVHVNWAVLIDEZWBVNWAOOIIVJRPUAUBVPVRVSAVQIVJGUCUDVPAIGHZV @@ -128666,7 +128666,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a 27-May-2016.) $) addcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) ) $= - ( vx cc wcel w3a cv caddc co cc0 wceq wb wa oveq1d addassd addid2 3eqtr3d + ( vx cc wcel w3a cv caddc co cc0 wceq wb wa oveq1d addassd addlid 3eqtr3d oveq2 syl wrex cnegex2 simprr simprl simpl1 simpl2 simpl3 eqeq12d impbid1 3ad2ant1 imbitrid rexlimddv ) AEFZBEFZCEFZGZDHZAIJZKLZABIJZACIJZLZBCLZMDE UMUNUSDEUAUODAUBUJUPUQEFZUSNZNZVBVCVBUQUTIJZUQVAIJZLVFVCUTVAUQISVFVGBVHCV @@ -128678,7 +128678,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a (Revised by Scott Fenton, 3-Jan-2013.) $) addcan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) = ( B + C ) <-> A = B ) ) $= - ( vx cc wcel w3a cv caddc co cc0 wb wa oveq1 addassd oveq2d addid1 3eqtrd + ( vx cc wcel w3a cv caddc co cc0 wb wa oveq1 addassd oveq2d addrid 3eqtrd wceq syl wrex cnegex 3ad2ant3 simpl1 simpl3 simprl simprr simpl2 imbitrid eqeq12d impbid1 rexlimddv ) AEFZBEFZCEFZGZCDHZIJZKSZACIJZBCIJZSZABSZLDEUO UMUSDEUAUNDCUBUCUPUQEFZUSMZMZVBVCVBUTUQIJZVAUQIJZSVFVCUTVAUQINVFVGAVHBVFV @@ -128703,13 +128703,13 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a mul.1 $e |- A e. CC $. $( ` 0 ` is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.) $) - addid1i $p |- ( A + 0 ) = A $= - ( cc wcel cc0 caddc co wceq addid1 ax-mp ) ACDAEFGAHBAIJ $. + addridi $p |- ( A + 0 ) = A $= + ( cc wcel cc0 caddc co wceq addrid ax-mp ) ACDAEFGAHBAIJ $. $( ` 0 ` is a left identity for addition. (Contributed by NM, 3-Jan-2013.) $) - addid2i $p |- ( 0 + A ) = A $= - ( cc wcel cc0 caddc co wceq addid2 ax-mp ) ACDEAFGAHBAIJ $. + addlidi $p |- ( 0 + A ) = A $= + ( cc wcel cc0 caddc co wceq addlid ax-mp ) ACDEAFGAHBAIJ $. $( Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) $) @@ -128782,13 +128782,13 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a $( ` 0 ` is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.) $) - addid1d $p |- ( ph -> ( A + 0 ) = A ) $= - ( cc wcel cc0 caddc co wceq addid1 syl ) ABDEBFGHBICBJK $. + addridd $p |- ( ph -> ( A + 0 ) = A ) $= + ( cc wcel cc0 caddc co wceq addrid syl ) ABDEBFGHBICBJK $. $( ` 0 ` is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.) $) - addid2d $p |- ( ph -> ( 0 + A ) = A ) $= - ( cc wcel cc0 caddc co wceq addid2 syl ) ABDEFBGHBICBJK $. + addlidd $p |- ( ph -> ( 0 + A ) = A ) $= + ( cc wcel cc0 caddc co wceq addlid syl ) ABDEFBGHBICBJK $. addcomd.2 $e |- ( ph -> B e. CC ) $. $( Addition commutes. Based on ideas by Eric Schmidt. (Contributed by @@ -128902,7 +128902,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a $( Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) ltaddneg $p |- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < B ) ) $= - ( cr wcel wa cc0 clt wbr caddc co wb ltadd2 mp3an2 wceq recn addid1d adantl + ( cr wcel wa cc0 clt wbr caddc co wb ltadd2 mp3an2 wceq recn addridd adantl 0re breq2d bitrd ) ACDZBCDZEZAFGHZBAIJZBFIJZGHZUEBGHUAFCDUBUDUGKRAFBLMUCUFB UEGUBUFBNUAUBBBOPQST $. @@ -129100,7 +129100,7 @@ it could represent the (meaningless) operation of negeu $p |- ( ( A e. CC /\ B e. CC ) -> E! x e. CC ( A + x ) = B ) $= ( vy cc wcel wa cv caddc co wceq wreu wrex cnegex adantr wral simpl simpr cc0 wb addcl syl2anr simplrr oveq1d simplll simplrl simpllr eqeq2d addcld - addassd addid2d 3eqtr3rd addcand bitrd ralrimiva reu6i syl2anc rexlimddv + addassd addlidd 3eqtr3rd addcand bitrd ralrimiva reu6i syl2anc rexlimddv ) BEFZCEFZGZBDHZIJZSKZBAHZIJZCKZAELZDEUSVDDEMUTDBNOVAVBEFZVDGZGZVBCIJZEFZ VGVEVLKZTZAEPVHVJVIUTVMVAVIVDQUSUTRVBCUAUBVKVOAEVKVEEFZGZVGVFBVLIJZKVNVQC VRVFVQVCCIJSCIJVRCVQVCSCIVAVIVDVPUCUDVQBVBCUSUTVJVPUEZVAVIVDVPUFZUSUTVJVP @@ -129335,13 +129335,13 @@ it could represent the (meaningless) operation of $( Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) $) subid $p |- ( A e. CC -> ( A - A ) = 0 ) $= - ( cc wcel cc0 caddc co cmin addid1 oveq1d wceq 0cn pncan2 mpan2 eqtr3d ) AB + ( cc wcel cc0 caddc co cmin addrid oveq1d wceq 0cn pncan2 mpan2 eqtr3d ) AB CZADEFZAGFZAAGFDOPAAGAHIODBCQDJKADLMN $. $( Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) $) subid1 $p |- ( A e. CC -> ( A - 0 ) = A ) $= - ( cc wcel cc0 caddc co cmin addid1 oveq1d wceq 0cn pncan mpan2 eqtr3d ) ABC + ( cc wcel cc0 caddc co cmin addrid oveq1d wceq 0cn pncan mpan2 eqtr3d ) ABC ZADEFZDGFZADGFAOPADGAHIODBCQAJKADLMN $. $( Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) $) @@ -129398,7 +129398,7 @@ it could represent the (meaningless) operation of subsub2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( A + ( C - B ) ) ) $= ( cc wcel w3a cmin co caddc wceq cc0 subcl 3adant1 simp1 simp3 simp2 add12d - syl2anc npncan2 oveq2d addid1d 3eqtrd wb addcld subadd syl3anc mpbird ) ADE + syl2anc npncan2 oveq2d addridd 3eqtrd wb addcld subadd syl3anc mpbird ) ADE ZBDEZCDEZFZABCGHZGHACBGHZIHZJZULUNIHZAJZUKUPAULUMIHZIHAKIHAUKULAUMUIUJULDEZ UHBCLMZUHUIUJNZUKUJUIUMDEUHUIUJOUHUIUJPCBLRZQUKURKAIUIUJURKJUHBCSMTUKAVAUAU BUKUHUSUNDEUOUQUCVAUTUKAUMVAVBUDAULUNUEUFUG $. @@ -129551,7 +129551,7 @@ it could represent the (meaningless) operation of Carneiro, 27-May-2016.) $) negsub $p |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) ) $= ( cc wcel wa cneg caddc co cc0 cmin wceq df-neg oveq2i a1i addsubass mp3an2 - 0cn simpl addid1d oveq1d 3eqtr2d ) ACDZBCDZEZABFZGHZAIBJHZGHZAIGHZBJHZABJHU + 0cn simpl addridd oveq1d 3eqtr2d ) ACDZBCDZEZABFZGHZAIBJHZGHZAIGHZBJHZABJHU FUHKUDUEUGAGBLMNUBICDUCUJUHKQAIBOPUDUIABJUDAUBUCRSTUA $. $( Relationship between subtraction and negative. (Contributed by NM, @@ -129565,7 +129565,7 @@ it could represent the (meaningless) operation of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.) $) negneg $p |- ( A e. CC -> -u -u A = A ) $= - ( cc wcel cneg cc0 caddc co cmin df-neg wceq 0cn subneg eqtrid addid2 eqtrd + ( cc wcel cneg cc0 caddc co cmin df-neg wceq 0cn subneg eqtrid addlid eqtrd mpan ) ABCZADZDZEAFGZAQSERHGZTRIEBCQUATJKEALPMANO $. $( Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by @@ -130385,7 +130385,7 @@ used to convert hypothesis of the inference (deduction) form of this identity of the complex numbers. (Contributed by AV, 17-Jan-2021.) $) addid0 $p |- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X <-> Y = 0 ) ) $= ( cc wcel wa caddc co wceq cc0 simpl simpr subaddd eqcom subid adantr eqtrd - cmin wi ex biimtrid sylbird oveq2 addid1 sylan9eqr impbid ) ACDZBCDZEZABFGZ + cmin wi ex biimtrid sylbird oveq2 addrid sylan9eqr impbid ) ACDZBCDZEZABFGZ AHZBIHZUHUJAAQGZBHZUKUHAABUFUGJZUNUFUGKLUFUMUKRUGUMBULHZUFUKULBMUFUOUKUFUOE BULIUFUOKUFULIHUOANOPSTOUAUFUKUJRUGUFUKUJUKUFUIAIFGABIAFUBAUCUDSOUE $. @@ -130539,7 +130539,7 @@ used to convert hypothesis of the inference (deduction) form of this 6-May-1999.) $) ine0 $p |- _i =/= 0 $= ( ci cc0 wceq c1 ax-1ne0 neii caddc cmul oveq2 ax-icn mul01i eqtr2di oveq1d - co ax-1cn addid2i ax-i2m1 3eqtr3g mto neir ) ABABCZDBCDBEFUABDGNAAHNZDGNDBU + co ax-1cn addlidi ax-i2m1 3eqtr3g mto neir ) ABABCZDBCDBEFUABDGNAAHNZDGNDBU ABUBDGUAUBABHNBABAHIAJKLMDOPQRST $. $( Product with negative is negative of product. Theorem I.12 of [Apostol] @@ -130581,7 +130581,7 @@ used to convert hypothesis of the inference (deduction) form of this $( Product with minus one is negative. (Contributed by NM, 16-Nov-1999.) $) mulm1 $p |- ( A e. CC -> ( -u 1 x. A ) = -u A ) $= - ( cc wcel c1 cneg cmul co wceq ax-1cn mulneg1 mpan mulid2 negeqd eqtrd ) AB + ( cc wcel c1 cneg cmul co wceq ax-1cn mulneg1 mpan mullid negeqd eqtrd ) AB CZDEAFGZDAFGZEZAEDBCOPRHIDAJKOQAALMN $. $( Addition with product with minus one is a subtraction. (Contributed by @@ -130717,13 +130717,13 @@ used to convert hypothesis of the inference (deduction) form of this $( Multiplication by one minus a number. (Contributed by Scott Fenton, 23-Dec-2017.) $) muls1d $p |- ( ph -> ( A x. ( B - 1 ) ) = ( ( A x. B ) - A ) ) $= - ( c1 cmin co cmul 1cnd subdid mulid1d oveq2d eqtrd ) ABCFGHIHBCIHZBFIHZGH + ( c1 cmin co cmul 1cnd subdid mulridd oveq2d eqtrd ) ABCFGHIHBCIHZBFIHZGH OBGHABCFDEAJKAPBOGABDLMN $. $( Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.) $) mulsubfacd $p |- ( ph -> ( ( A x. B ) - B ) = ( ( A - 1 ) x. B ) ) $= - ( c1 cmin co cmul 1cnd subdird mulid2d oveq2d eqtr2d ) ABFGHCIHBCIHZFCIHZ + ( c1 cmin co cmul 1cnd subdird mullidd oveq2d eqtr2d ) ABFGHCIHBCIHZFCIHZ GHOCGHABFCDAJEKAPCOGACELMN $. $} @@ -130968,7 +130968,7 @@ Ordering on reals (cont.) $( Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.) $) ltaddpos $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( B + A ) ) ) $= - ( cr wcel wa cc0 clt wbr caddc co wb ltadd2 mp3an1 wceq recn addid1d adantl + ( cr wcel wa cc0 clt wbr caddc co wb ltadd2 mp3an1 wceq recn addridd adantl 0re breq1d bitrd ) ACDZBCDZEZFAGHZBFIJZBAIJZGHZBUFGHFCDUAUBUDUGKRFABLMUCUEB UFGUBUEBNUAUBBBOPQST $. @@ -131121,7 +131121,7 @@ Ordering on reals (cont.) $( A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.) $) addge01 $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> A <_ ( A + B ) ) ) $= - ( cr wcel wa cc0 cle wbr caddc co wb leadd2 mp3an1 ancoms wceq recn addid1d + ( cr wcel wa cc0 cle wbr caddc co wb leadd2 mp3an1 ancoms wceq recn addridd 0re adantr breq1d bitrd ) ACDZBCDZEZFBGHZAFIJZABIJZGHZAUGGHUCUBUEUHKZFCDUCU BUIRFBALMNUDUFAUGGUBUFAOUCUBAAPQSTUA $. @@ -131137,7 +131137,7 @@ Ordering on reals (cont.) add20 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) ) $= ( cr wcel cc0 cle wbr wa caddc wceq simpllr simplrl simplll addge02 syl2anc - co wb mpbid simpr breqtrd simplrr letri3d mpbir2and oveq2d addid1d 3eqtr3rd + co wb mpbid simpr breqtrd simplrr letri3d mpbir2and oveq2d addridd 3eqtr3rd 0red recnd jca ex oveq12 00id eqtrdi impbid1 ) ACDZEAFGZHZBCDZEBFGZHZHZABIP ZEJZAEJZBEJZHZVAVCVFVAVCHZVDVEVGVBAEIPEAVGBEAIVGVEBEFGUSVGBVBEFVGUPBVBFGZUO UPUTVCKVGURUOUPVHQUQURUSVCLZUOUPUTVCMZBANORVAVCSZTUQURUSVCUAVGBEVIVGUGUBUCZ @@ -131148,7 +131148,7 @@ Ordering on reals (cont.) subge0 $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ ( A - B ) <-> B <_ A ) ) $= ( cr wcel wa cc0 caddc co cle wbr cmin wb 0red simpr simpl leaddsub syl3anc - recnd addid2d breq1d bitr3d ) ACDZBCDZEZFBGHZAIJZFABKHIJZBAIJUDFCDUCUBUFUGL + recnd addlidd breq1d bitr3d ) ACDZBCDZEZFBGHZAIJZFABKHIJZBAIJUDFCDUCUBUFUGL UDMUBUCNZUBUCOFBAPQUDUEBAIUDBUDBUHRSTUA $. $( Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof @@ -131224,7 +131224,7 @@ Ordering on reals (cont.) $( 0 is less than 1. Theorem I.21 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.) $) 0lt1 $p |- 0 < 1 $= - ( cc0 c1 cmul co clt wcel wne wbr 1re ax-1ne0 msqgt0 ax-1cn mulid1i breqtri + ( cc0 c1 cmul co clt wcel wne wbr 1re ax-1ne0 msqgt0 ax-1cn mulridi breqtri cr mp2an ) ABBCDZBEBOFBAGAQEHIJBKPBLMN $. $( 0 is less than or equal to 1. (Contributed by Mario Carneiro, @@ -131997,7 +131997,7 @@ Ordering on reals (cont.) $( When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) sublt0d $p |- ( ph -> ( ( A - B ) < 0 <-> A < B ) ) $= - ( cmin co cc0 clt wbr caddc 0red ltsubaddd recnd addid2d breq2d bitrd ) A + ( cmin co cc0 clt wbr caddc 0red ltsubaddd recnd addlidd breq2d bitrd ) A BCFGHIJBHCKGZIJBCIJABCHDEALMARCBIACACENOPQ $. $} @@ -132084,7 +132084,7 @@ Ordering on reals (cont.) 27-May-2016.) $) mulcand $p |- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) ) $= ( vx cmul co wceq c1 cc wcel wa oveq2 adantr oveq1d mulassd cv wi cc0 wne - wrex recex syl2anc simprl mulcomd simprr mulid2d 3eqtr3d eqeq12d imbitrid + wrex recex syl2anc simprl mulcomd simprr mullidd 3eqtr3d eqeq12d imbitrid eqtrd rexlimddv impbid1 ) ADBJKZDCJKZLZBCLZADIUAZJKZMLZUTVAUBINADNOZDUCUD VDINUEGHIDUFUGUTVBURJKZVBUSJKZLAVBNOZVDPZPZVAURUSVBJQVJVFBVGCVJVBDJKZBJKM BJKVFBVJVKMBJVJVKVCMVJVBDAVHVDUHZAVEVIGRZUIAVHVDUJUOZSVJVBDBVLVMABNOVIERZ @@ -132190,9 +132190,9 @@ Ordering on reals (cont.) [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.) $) muleqadd $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = ( A + B ) <-> ( ( A - 1 ) x. ( B - 1 ) ) = 1 ) ) $= - ( cc wcel wa c1 cmin co cmul wceq caddc ax-1cn mulsub mpanr2 mpanl2 mulid1i - cc0 oveq2i a1i mulid1 oveqan12d oveq12d addsub mp3an2 syl2anc 3eqtrd eqeq1d - mulcl addcl subcld 0cn addcan2 mp3an23 syl addid2i eqeq2i subeq0ad 3bitr2rd + ( cc wcel wa c1 cmin co cmul wceq caddc ax-1cn mulsub mpanr2 mpanl2 mulridi + cc0 oveq2i a1i mulrid oveqan12d oveq12d addsub mp3an2 syl2anc 3eqtrd eqeq1d + mulcl addcl subcld 0cn addcan2 mp3an23 syl addlidi eqeq2i subeq0ad 3bitr2rd wb bitr3di ) ACDZBCDZEZAFGHBFGHIHZFJABIHZABKHZGHZFKHZFJZVGQJZVEVFJVCVDVHFVC VDVEFFIHZKHZAFIHZBFIHZKHZGHZVEFKHZVFGHZVHVAFCDZVBVDVPJZLVAVSEVBVSVTLAFBFMNO VCVLVQVOVFGVLVQJVCVKFVEKFLPRSVAVBVMAVNBKATBTUAUBVCVECDZVFCDZVRVHJZABUHZABUI @@ -132207,7 +132207,7 @@ Ordering on reals (cont.) receu $p |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> E! x e. CC ( B x. x ) = A ) $= ( vy cc wcel cc0 w3a cv cmul co wceq wrex wa wi wral 3adant1 oveq2 eqeq1d - c1 wreu recex simprl simpll mulcld oveq1 ad2antll mulassd mulid2d 3eqtr3d + c1 wreu recex simprl simpll mulcld oveq1 ad2antll mulassd mullidd 3eqtr3d wne simplr rspcev syl2anc rexlimdvaa 3adant3 eqtr3 mulcan imbitrid expcom mpd 3expa ralrimivv reu4 sylanbrc ) BEFZCEFZCGUKZHZCAIZJKZBLZAEMZVLCDIZJK ZBLZNZVJVNLZOZDEPAEPVLAEUAVIVOTLZDEMZVMVGVHWAVFDCUBQVFVGWAVMOVHVFVGNZVTVM @@ -132466,7 +132466,7 @@ Ordering on reals (cont.) divrec $p |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) ) $= ( cc wcel cc0 wne w3a cdiv co c1 cmul wceq simp2 simp1 reccl 3adant1 mul12d - recid oveq2d mulid1d 3eqtrd wa wb mulcld 3simpc divmul syl3anc mpbird ) ACD + recid oveq2d mulridd 3eqtrd wa wb mulcld 3simpc divmul syl3anc mpbird ) ACD ZBCDZBEFZGZABHIAJBHIZKIZLZBUNKIZALZULUPABUMKIZKIAJKIAULBAUMUIUJUKMUIUJUKNZU JUKUMCDUIBOPZQULURJAKUJUKURJLUIBRPSULAUSTUAULUIUNCDUJUKUBUOUQUCUSULAUMUSUTU DUIUJUKUEAUNBUFUGUH $. @@ -132591,7 +132591,7 @@ Ordering on reals (cont.) $( A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.) $) div1 $p |- ( A e. CC -> ( A / 1 ) = A ) $= - ( cc wcel c1 cdiv co wceq mulid2 wb cc0 wne wa ax-1cn ax-1ne0 pm3.2i divmul + ( cc wcel c1 cdiv co wceq mullid wb cc0 wne wa ax-1cn ax-1ne0 pm3.2i divmul cmul mp3an3 anidms mpbird ) ABCZADEFAGZDAQFAGZAHUAUBUCIZUAUADBCZDJKZLUDUEUF MNOAADPRST $. @@ -132604,7 +132604,7 @@ Ordering on reals (cont.) diveq1 $p |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( A / B ) = 1 <-> A = B ) ) $= ( cc wcel cc0 wne w3a cdiv co c1 wceq cmul wb wa ax-1cn divmul2 3impb simp2 - mp3an2 mulid1d eqeq2d bitrd ) ACDZBCDZBEFZGZABHIJKZABJLIZKZABKUCUDUEUGUIMZU + mp3an2 mulridd eqeq2d bitrd ) ACDZBCDZBEFZGZABHIJKZABJLIZKZABKUCUDUEUGUIMZU CJCDUDUENUJOAJBPSQUFUHBAUFBUCUDUERTUAUB $. $( Move negative sign inside of a division. (Contributed by NM, @@ -132651,7 +132651,7 @@ Ordering on reals (cont.) ( ( A - ( C x. B ) ) / C ) = ( ( A / C ) - B ) ) $= ( cc wcel cc0 wne wa w3a cmul co cmin cdiv simp3l simp2 mulcld divsubdir c1 wceq eqtrd syld3an2 simprl simpl simpr div23 syl3anc divid oveq1d sylan9eqr - mulid2 3adant1 oveq2d ) ADEZBDEZCDEZCFGZHZIZACBJKZLKCMKZACMKZUSCMKZLKZVABLK + mullid 3adant1 oveq2d ) ADEZBDEZCDEZCFGZHZIZACBJKZLKCMKZACMKZUSCMKZLKZVABLK UMUSDEUNUQUTVCSURCBUMUNUOUPNUMUNUQOPAUSCQUAURVBBVALUNUQVBBSUMUNUQHZVBCCMKZB JKZBVDUOUNUQVBVFSUNUOUPUBUNUQUCUNUQUDCBCUEUFUQUNVFRBJKBUQVERBJCUGUHBUJUITUK ULT $. @@ -132703,7 +132703,7 @@ Ordering on reals (cont.) -> ( ( A / B ) / ( C / D ) ) = ( ( A x. D ) / ( B x. C ) ) ) $= ( cc wcel cc0 wa cdiv co cmul wceq divcl syl3anc mulcomd divmuldiv eqtrd c1 wne mulcld simprrl simprll simprlr simplrl simplrr syl22anc simprrr mulne0d - simpll simplr simprl oveq2d simprr divid syl2anc mulassd mulid2d 3eqtr3d wb + simpll simplr simprl oveq2d simprr divid syl2anc mulassd mullidd 3eqtr3d wb oveq1d eqtr3d mulne0 ad2ant2lr divne0 adantl divmul syl112anc mpbird ) AEFZ BEFZBGSZHZHZCEFZCGSZHZDEFZDGSZHZHZHZABIJZCDIJZIJADKJZBCKJZIJZLZWCWFKJZWBLZW AWCDCIJZWBKJZKJZWHWBWAWKWFWCKWAWKWBWJKJZWFWAWJWBWAVQVNVOWJEFVMVPVQVRUAZVMVN @@ -132719,7 +132719,7 @@ Ordering on reals (cont.) divcan5 $p |- ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) ) $= ( cc wcel cc0 wne wa w3a cdiv co cmul c1 divid oveq1d 3ad2ant3 simp3l simp1 - wceq simp3 simp2 divmuldiv syl22anc divcl 3expb mulid2d 3adant3 3eqtr3d ) A + wceq simp3 simp2 divmuldiv syl22anc divcl 3expb mullidd 3adant3 3eqtr3d ) A DEZBDEZBFGZHZCDEZCFGZHZIZCCJKZABJKZLKZMURLKZCALKCBLKJKZURUOUIUSUTSULUOUQMUR LCNOPUPUMUIUOULUSVASUIULUMUNQUIULUORUIULUOTUIULUOUACACBUBUCUIULUTURSUOUIULH URUIUJUKURDEABUDUEUFUGUH $. @@ -132761,7 +132761,7 @@ Ordering on reals (cont.) recdiv $p |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( 1 / ( A / B ) ) = ( B / A ) ) $= ( cc wcel cc0 wne wa c1 cdiv cmul 1div1e1 oveq1i wceq ax-1cn ax-1ne0 pm3.2i - co divdivdiv mpanl12 mulid2 eqtr3id oveqan12rd ad2ant2r eqtrd ) ACDZAEFZGBC + co divdivdiv mpanl12 mullid eqtr3id oveqan12rd ad2ant2r eqtrd ) ACDZAEFZGBC DZBEFZGGZHABIQZIQZHBJQZHAJQZIQZBAIQZUIUKHHIQZUJIQZUNUPHUJIKLHCDZURHEFZGUIUQ UNMNURUSNOPHHABRSUAUEUGUNUOMUFUHUGUEULBUMAIBTATUBUCUD $. @@ -132805,7 +132805,7 @@ Ordering on reals (cont.) divdiv1 $p |- ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / B ) / C ) = ( A / ( B x. C ) ) ) $= ( cc wcel cc0 wne wa cdiv co c1 cmul ax-1cn ax-1ne0 pm3.2i divdivdiv mpanr2 - w3a wceq 3impa div1 oveq2d ad2antrl 3adant1 mulid1 oveq1d 3ad2ant1 3eqtr3d + w3a wceq 3impa div1 oveq2d ad2antrl 3adant1 mulrid oveq1d 3ad2ant1 3eqtr3d ) ADEZBDEBFGHZCDEZCFGZHZRABIJZCKIJZIJZAKLJZBCLJZIJZUNCIJZAURIJZUIUJUMUPUSSZ UIUJHUMKDEZKFGZHVBVCVDMNOABCKPQTUJUMUPUTSZUIUKVEUJULUKUOCUNICUAUBUCUDUIUJUS VASUMUIUQAURIAUEUFUGUH $. @@ -132814,7 +132814,7 @@ Ordering on reals (cont.) divdiv2 $p |- ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ ( C e. CC /\ C =/= 0 ) ) -> ( A / ( B / C ) ) = ( ( A x. C ) / B ) ) $= ( cc wcel cc0 wne wa c1 cdiv co cmul ax-1cn ax-1ne0 pm3.2i divdivdiv mpanl2 - w3a wceq 3impb div1 3ad2ant1 oveq1d mulid2 ad2antrl 3adant3 oveq2d 3eqtr3d + w3a wceq 3impb div1 3ad2ant1 oveq1d mullid ad2antrl 3adant3 oveq2d 3eqtr3d ) ADEZBDEZBFGZHZCDECFGHZRZAIJKZBCJKZJKZACLKZIBLKZJKZAUPJKURBJKUIULUMUQUTSZU IIDEZIFGZHULUMHVAVBVCMNOAIBCPQTUNUOAUPJUIULUOASUMAUAUBUCUNUSBURJUIULUSBSZUM UJVDUIUKBUDUEUFUGUH $. @@ -132869,8 +132869,8 @@ Ordering on reals (cont.) conjmul $p |- ( ( ( P e. CC /\ P =/= 0 ) /\ ( Q e. CC /\ Q =/= 0 ) ) -> ( ( ( 1 / P ) + ( 1 / Q ) ) = 1 <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) ) $= ( cc wcel cc0 wne wa cmul co cdiv caddc wceq cmin reccl adantr recid 3eqtrd - c1 adantl ad2ant2r simpll simprl mul32d oveq1d mulid2 mulassd oveq2d mulid1 - ad2antrl ad2antrr oveq12d mulcl adddid addcom 3eqtr4d mulid1d eqeq12d addcl + c1 adantl ad2ant2r simpll simprl mul32d oveq1d mullid mulassd oveq2d mulrid + ad2antrl ad2antrr oveq12d mulcl adddid addcom 3eqtr4d mulridd eqeq12d addcl syl2an mulne0 ax-1cn mulcan mp3an2 syl12anc eqcom muleqadd bitrid 3bitr3d wb ) ACDZAEFZGZBCDZBEFZGZGZABHIZRAJIZRBJIZKIZHIZVQRHIZLZABKIZVQLZVTRLZARMIB RMIHIRLZVPWAWDWBVQVPVQVRHIZVQVSHIZKIBAKIZWAWDVPWHBWIAKVPWHAVRHIZBHIZRBHIZBV @@ -133646,7 +133646,7 @@ Ordering on reals (cont.) subrec $p |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( 1 / A ) - ( 1 / B ) ) = ( ( B - A ) / ( A x. B ) ) ) $= ( cc wcel cc0 wne wa c1 cdiv cmin cmul 1cnd simpll simprl simplr divsubdivd - co simprr mulid2d oveq12d oveq1d eqtrd ) ACDZAEFZGZBCDZBEFZGZGZHAIQHBIQJQHB + co simprr mullidd oveq12d oveq1d eqtrd ) ACDZAEFZGZBCDZBEFZGZGZHAIQHBIQJQHB KQZHAKQZJQZABKQZIQBAJQZUMIQUIHAHBUILZUCUDUHMZUOUEUFUGNZUCUDUHOUEUFUGRPUIULU NUMIUIUJBUKAJUIBUQSUIAUPSTUAUB $. @@ -134188,7 +134188,7 @@ Ordering on reals (cont.) ltrec $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) ) $= ( cr wcel cc0 clt wbr wa c1 cdiv co cmul 1red simprl simpll simplr ltmuldiv - wb syl112anc recnd mulid2d breq1d gt0ne0d divrecd 3bitr3d rereccld ltdivmul + wb syl112anc recnd mullidd breq1d gt0ne0d divrecd 3bitr3d rereccld ltdivmul breq2d simprr bitr4d ) ACDZEAFGZHZBCDZEBFGZHZHZABFGZIBIAJKZLKZFGZIBJKUSFGZU QIALKZBFGZIBAJKZFGZURVAUQICDZUNUKULVDVFRUQMZUMUNUONZUKULUPOZUKULUPPZIBAQSUQ VCABFUQAUQAVJTZUAUBUQVEUTIFUQBAUQBVITVLUQAVKUCZUDUHUEUQVGUSCDUNUOVBVARVHUQA @@ -134370,7 +134370,7 @@ Ordering on reals (cont.) recp1lt1 $p |- ( ( A e. RR /\ 0 <_ A ) -> ( A / ( 1 + A ) ) < 1 ) $= ( cr wcel cc0 cle wbr wa c1 caddc co cdiv clt cmul ltp1 cc wceq recn adantr 1re recnd ax-1cn addcom sylancl breqtrd readdcl mpan addgtge0 mpanr1 mpanl1 - 0lt1 gt0ne0d divcan1d mulid2d 3brtr4d simpl redivcld ltmul1 mp3an2 syl12anc + 0lt1 gt0ne0d divcan1d mullidd 3brtr4d simpl redivcld ltmul1 mp3an2 syl12anc wb mpbird ) ABCZDAEFZGZAHAIJZKJZHLFZVFVEMJZHVEMJZLFZVDAVEVHVILVBAVELFVCVBAA HIJZVELANVBAOCZHOCVKVEPAQZUAAHUBUCUDRVDAVEVBVLVCVMRVDVEVBVEBCZVCHBCZVBVNSHA UEUFZRZTVDVEVOVBVCDVELFZSVOVBGDHLFVCVRUJHAUGUHUIZUKZULVBVIVEPVCVBVEVBVEVPTU @@ -135514,7 +135514,7 @@ Dedekind cut construction of the reals (see ~ df-mp ). (Contributed by 6-May-1999.) $) inelr $p |- -. _i e. RR $= ( ci cr wcel cc0 wceq ine0 neii cmul co clt wbr wn c1 0re 1re ltnsymi ax-mp - 0lt1 caddc cneg ixi renegcli eqeltri ltadd1i ax-1cn addid2i ax-i2m1 breq12i + 0lt1 caddc cneg ixi renegcli eqeltri ltadd1i ax-1cn addlidi ax-i2m1 breq12i bitri mtbir wne msqgt0 ex necon1bd mpi mto ) ABCZADEZADFGUQDAAHIZJKZLURUTMD JKZDMJKVALRDMNOPQUTDMSIZUSMSIZJKVADUSMNUSMTBUAMOUBUCOUDVBMVCDJMUEUFUGUHUIUJ UQUTADUQADUKUTAULUMUNUOUP $. @@ -136022,7 +136022,7 @@ subset of complex numbers ( ~ nnsscn ), in contrast to the more elementary 25-Aug-1999.) $) nnge1 $p |- ( A e. NN -> 1 <_ A ) $= ( vx vy c1 cv cle wbr caddc co breq2 wcel cr wi cc0 clt wn 0re 1re lenlt - wb 1le1 cn nnre recn addid1d breq2d 0lt1 axltadd mp3an12 wa readdcl mpan2 + wb 1le1 cn nnre recn addridd breq2d 0lt1 axltadd mp3an12 wa readdcl mpan2 mpi peano2re mp3an3 syl2anc mpand con3d sylancr 3imtr4d sylbird syl nnind lttr ) DBEZFGDDFGDCEZFGZDVFDHIZFGZDAFGBCAVEDDFJVEVFDFJVEVHDFJVEADFJUAVFUB KVFLKZVGVIMVFUCVJVGDVFNHIZFGZVIVJVKVFDFVJVFVFUDUEUFVJVKDOGZPZVHDOGZPZVLVI @@ -136070,7 +136070,7 @@ subset of complex numbers ( ~ nnsscn ), in contrast to the more elementary nnne0 $p |- ( A e. NN -> A =/= 0 ) $= ( vx vy cn wcel c1 cc0 clt wbr wne 1re 0re wa cv caddc breq1 imbi2d breq2 wi wceq wo ax-1ne0 lttri2i mpbi co w3a simp1 nnred 1red readdcld readdcli - id a1i 0red simp3 ltadd1dd ax-1cn addid2i simp2 eqbrtrid lttrd 3exp nnind + id a1i 0red simp3 ltadd1dd ax-1cn addlidi simp2 eqbrtrid lttrd 3exp nnind cr a2d imp lt0ne0d addgt0d gt0ne0d jaodan mpan2 ) ADEZFGHIZGFHIZUAZAGJZFG JVOUBFGKLUCUDVLVMVPVNVLVMMAVLVMAGHIZVMBNZGHIZSVMVMSVMCNZGHIZSVMVTFOUEZGHI ZSVMVQSBCAVRFTZVSVMVMVRFGHPQVRVTTZVSWAVMVRVTGHPQVRWBTZVSWCVMVRWBGHPQVRATZ @@ -136188,7 +136188,7 @@ subset of complex numbers ( ~ nnsscn ), in contrast to the more elementary nndivtr $p |- ( ( ( A e. NN /\ B e. NN /\ C e. CC ) /\ ( ( B / A ) e. NN /\ ( C / B ) e. NN ) ) -> ( C / A ) e. NN ) $= ( cn wcel cc w3a cdiv co wa cmul nnmulcl c1 cc0 wne wceq 3ad2ant2 nnne0 jca - nncn 3ad2ant1 divmul24 syl22anc dividd oveq1d sylan2 ancoms mulid2d 3adant2 + nncn 3ad2ant1 divmul24 syl22anc dividd oveq1d sylan2 ancoms mullidd 3adant2 simp3 divcl 3expb 3eqtrd eleq1d imbitrid imp ) ADEZBDEZCFEZGZBAHIZDECBHIZDE JZCAHIZDEZVCVAVBKIZDEUTVEVAVBLUTVFVDDUTVFBBHIZVDKIZMVDKIZVDUTBFEZUSAFEZANOZ JZVJBNOZJZVFVHPURUQVJUSBTZQUQURUSUJUQURVMUSUQVKVLATARSZUAURUQVOUSURVJVNVPBR @@ -136570,7 +136570,7 @@ of the complex number axiom system (see ~ df-0 and ~ df-1 ). $( 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.) $) 0p1e1 $p |- ( 0 + 1 ) = 1 $= - ( c1 ax-1cn addid2i ) ABC $. + ( c1 ax-1cn addlidi ) ABC $. $( Function value at ` N + 1 ` with ` N ` replaced by ` 0 ` . Technical theorem to be used to reduce the size of a significant number of proofs. @@ -136581,7 +136581,7 @@ of the complex number axiom system (see ~ df-0 and ~ df-1 ). $( 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) $) 1p0e1 $p |- ( 1 + 0 ) = 1 $= - ( c1 ax-1cn addid1i ) ABC $. + ( c1 ax-1cn addridi ) ABC $. $( 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.) $) 1p1e2 $p |- ( 1 + 1 ) = 2 $= @@ -136767,11 +136767,11 @@ ordinal natural numbers (finite integers starting at 0), so that proof $( 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.) $) 1t1e1 $p |- ( 1 x. 1 ) = 1 $= - ( c1 ax-1cn mulid1i ) ABC $. + ( c1 ax-1cn mulridi ) ABC $. $( 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.) $) 2t1e2 $p |- ( 2 x. 1 ) = 2 $= - ( c2 2cn mulid1i ) ABC $. + ( c2 2cn mulridi ) ABC $. $( 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.) $) 2t2e4 $p |- ( 2 x. 2 ) = 4 $= @@ -136779,7 +136779,7 @@ ordinal natural numbers (finite integers starting at 0), so that proof $( 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.) $) 3t1e3 $p |- ( 3 x. 1 ) = 3 $= - ( c3 3cn mulid1i ) ABC $. + ( c3 3cn mulridi ) ABC $. $( 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.) $) 3t2e6 $p |- ( 3 x. 2 ) = 6 $= @@ -137053,7 +137053,7 @@ ordinal natural numbers (finite integers starting at 0), so that proof halfpm6th $p |- ( ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 ) /\ ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 ) ) $= ( c1 c2 cdiv co c6 cmin wceq caddc cmul 3cn ax-1cn 2cn 3ne0 2ne0 divmuldivi - c3 oveq1i mulid1i 3t2e6 oveq12i dividi halfcn mulid2i eqtri 3eqtr3i cc wcel + c3 oveq1i mulridi 3t2e6 oveq12i dividi halfcn mullidi eqtri 3eqtr3i cc wcel cc0 wne wa 6cn 6re 6pos gt0ne0ii pm3.2i divsubdir mp3an 3m1e2 oveq2i reccli 3eqtr2i c4 divdiri df-4 3eqtr4ri 2t2e4 divcli ) ABCDZAECDZFDZAPCDZGVHVIHDZB PCDZGVJPECDZVIFDZPAFDZECDZVKVHVNVIFPPCDZVHIDZPAIDZPBIDZCDVHVNPPABJJKLMNOVSA @@ -137148,7 +137148,7 @@ ordinal natural numbers (finite integers starting at 0), so that proof number. (Contributed by AV, 28-Jun-2021.) $) subhalfhalf $p |- ( A e. CC -> ( A - ( A / 2 ) ) = ( A / 2 ) ) $= ( cc wcel c2 cdiv co cmin cmul 2cnd cc0 wne 2ne0 a1i divcan1d eqcomd oveq1d - id halfcl c1 3eqtrd mulcomd mulsubfacd wceq 2m1e1 mulid2d ) ABCZAADEFZGFUGD + id halfcl c1 3eqtrd mulcomd mulsubfacd wceq 2m1e1 mullidd ) ABCZAADEFZGFUGD HFZUGGFDUGHFZUGGFZUGUFAUHUGGUFUHAUFADUFQUFIZDJKUFLMNOPUFUHUIUGGUFUGDARZUKUA PUFUJDSGFZUGHFSUGHFUGUFDUGUKULUBUFUMSUGHUMSUCUFUDMPUFUGULUETT $. @@ -137333,7 +137333,7 @@ ordinal natural numbers (finite integers starting at 0), so that proof xp1d2m1eqxm1d2 $p |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) - 1 ) = ( ( X - 1 ) / 2 ) ) $= ( cc wcel c1 caddc co cdiv cmin peano2cn halfcld peano2cnm syl 2cnd cc0 wne - c2 2ne0 a1i cmul divcan1d 1cnd subdird mulid2d oveq12d wceq 2m1e1 oveq2d id + c2 2ne0 a1i cmul divcan1d 1cnd subdird mullidd oveq12d wceq 2m1e1 oveq2d id subsub3d 3eqtr2rd 3eqtrd mulcan2ad ) ABCZADEFZPGFZDHFZADHFZPGFZPUMUOBCUPBCU MUNAIZJZUOKLUMUQAKZJUMMZPNOUMQRZUMUPPSFUOPSFZDPSFZHFUNPHFZURPSFZUMUODPUTUMU AZVBUBUMVDUNVEPHUMUNPUSVBVCTUMPVBUCUDUMVGUQAPDHFZHFVFUMUQPVAVBVCTUMVIDAHVID @@ -137347,7 +137347,7 @@ ordinal natural numbers (finite integers starting at 0), so that proof ( cr wcel c6 cle wbr c4 co c1 caddc cmin c2 cmul a1i wceq adantr wb syl3anc cc cc0 wa cdiv 6re id leadd2d biimpa times2d breqtrrd 4cn 2cn addassd 4p2e6 recn oveq2i eqtrdi breq1d clt 4re wne redivcld peano2re peano2rem rehalfcld - mpbird 4ne0 syl 4pos pm3.2i lemul1 divcan1d mulid2i oveq12d joinlmuladdmuld + mpbird 4ne0 syl 4pos pm3.2i lemul1 divcan1d mullidi oveq12d joinlmuladdmuld recnd 1cnd 2t2e4 eqcomi oveq2d w3a mulass eqcomd 2ne0 oveq1d subdird 3eqtrd breq12d readdcld 2re remulcld leaddsub bicomd 3bitrd ) ABCZDAEFZUAZAGUBHZIJ HZAIKHZLUBHZEFZAGJHZLJHZALMHZEFZWOXDADJHZXCEFZWOXEAAJHZXCEWMWNXEXGEFWMDAADB @@ -137610,7 +137610,7 @@ Nonnegative integers (as a subset of complex numbers) (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.) $) nnnn0addcl $p |- ( ( M e. NN /\ N e. NN0 ) -> ( M + N ) e. NN ) $= - ( cn0 wcel cn cc0 wceq wo caddc co elnn0 nnaddcl wa oveq2 addid1d sylan9eqr + ( cn0 wcel cn cc0 wceq wo caddc co elnn0 nnaddcl wa oveq2 addridd sylan9eqr nncn simpl eqeltrd jaodan sylan2b ) BCDAEDZBEDZBFGZHABIJZEDZBKUBUCUFUDABLUB UDMUEAEUDUBUEAFIJABFAINUBAAQOPUBUDRSTUA $. @@ -137638,8 +137638,8 @@ Nonnegative integers (as a subset of complex numbers) (Contributed by Mario Carneiro, 17-Jul-2014.) $) un0addcl $p |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M + N ) e. T ) $= ( wcel caddc co cc0 wo wa eleq2i elun bitri cc sselda eqeltrd csn ssun1 - cun sseqtrri sselid expr addid2d wss a1i elsni oveq1d eleq1d syl5ibrcom - wi impancom jaodan sylan2b 0cnd snssd unssd eqsstrid addid1d simpr jaod + cun sseqtrri sselid expr addlidd wss a1i elsni oveq1d eleq1d syl5ibrcom + wi impancom jaodan sylan2b 0cnd snssd unssd eqsstrid addridd simpr jaod oveq2d biimtrid impr ) ADCIZECIZDEJKZCIZVIEBIZELUAZIZMZAVHNZVKVIEBVMUCZ IVOCVQEGOEBVMPQVPVLVKVNVHADBIZDVMIZMZVLVKUNZVHDVQIVTCVQDGODBVMPQAVRWAVS AVRVLVKAVRVLNNBCVJBVQCBVMUBGUDZHUEUFAVLVSVKAVLNZVKVSLEJKZCIWCWDECWCEABR @@ -138663,7 +138663,7 @@ nonnegative integers (cont.)". $) gtndiv $p |- ( ( A e. RR /\ B e. NN /\ B < A ) -> -. ( B / A ) e. ZZ ) $= ( cc0 cz wcel cr cn clt wbr w3a cdiv co c1 caddc wn 0z nnre 3ad2ant2 wa wb simp1 nngt0 adantl wi 0re lttr mp3an1 sylan ancoms mpand divgt0d simp3 cmul - 3impia 1re ltdivmul2 mp3an2 syl12anc recn mulid2d breq2d bitrd mpbird 0p1e1 + 3impia 1re ltdivmul2 mp3an2 syl12anc recn mullidd breq2d bitrd mpbird 0p1e1 3ad2ant1 breqtrrdi btwnnz mp3an2i ) CDEAFEZBGEZBAHIZJZCBAKLZHIVMCMNLZHIVMDE OPVLBAVJVIBFEZVKBQZRZVIVJVKUAZVJVICBHIZVKBUBZRVIVJVKCAHIZVIVJSVSVKWAVJVSVIV TUCVJVIVSVKSWAUDZVJVOVIWBVPCFEVOVIWBUECBAUFUGUHUIUJUNZUKVLVMMVNHVLVMMHIZVKV @@ -138678,7 +138678,7 @@ nonnegative integers (cont.)". $) $( Three halves is not an integer. (Contributed by AV, 2-Jun-2020.) $) 3halfnz $p |- -. ( 3 / 2 ) e. ZZ $= ( c1 cz wcel c3 c2 cdiv co clt wbr caddc wn cmul cr wb 3re 2re mp3an breq2i - c4 mpbir 1z 2cn mulid2i 2lt3 eqbrtri cc0 1re 2pos pm3.2i ltmuldiv mpbi 3lt4 + c4 mpbir 1z 2cn mullidi 2lt3 eqbrtri cc0 1re 2pos pm3.2i ltmuldiv mpbi 3lt4 wa 2t2e4 1p1e2 ltdivmul bitri btwnnz ) ABCADEFGZHIZUSAAJGZHIZUSBCKUAAELGZDH IZUTVCEDHEUBUCUDUEAMCDMCZEMCZUFEHIZUMZVDUTNUGOVFVGPUHUIZADEUJQUKVBDEELGZHIZ VKDSHIULVJSDHUNRTVBUSEHIZVKVAEUSHUORVEVFVHVLVKNOPVIDEEUPQUQTAUSURQ $. @@ -139257,8 +139257,8 @@ nonnegative integers (cont.)". $) $( 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.) $) 9p1e10 $p |- ( 9 + 1 ) = ; 1 0 $= - ( c1 cc0 cdc c9 caddc co cmul df-dec cn 9nn 1nn nnaddcl mp2an nncni mulid1i - wcel oveq1i addid1i 3eqtrri ) ABCDAEFZAGFZBEFTBEFTABHUATBETTDIPAIPTIPJKDALM + ( c1 cc0 cdc c9 caddc co cmul df-dec cn 9nn 1nn nnaddcl mp2an nncni mulridi + wcel oveq1i addridi 3eqtrri ) ABCDAEFZAGFZBEFTBEFTABHUATBETTDIPAIPTIPJKDALM NZOQTUBRS $. $( Version of the definition of the "decimal constructor" using ` ; 1 0 ` @@ -139320,13 +139320,13 @@ nonnegative integers (cont.)". $) $( Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.) $) num0u $p |- ( T x. A ) = ( ( T x. A ) + 0 ) $= - ( cmul co cc0 caddc nn0mulcli nn0cni addid1i eqcomi ) BAEFZGHFMMMBACDIJKL + ( cmul co cc0 caddc nn0mulcli nn0cni addridi eqcomi ) BAEFZGHFMMMBACDIJKL $. $( Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.) $) num0h $p |- A = ( ( T x. 0 ) + A ) $= - ( cc0 cmul co caddc nn0cni mul01i oveq1i addid2i eqtr2i ) BEFGZAHGEAHGANE + ( cc0 cmul co caddc nn0cni mul01i oveq1i addlidi eqtr2i ) BEFGZAHGEAHGANE AHBBCIJKAADILM $. numcl.2 $e |- B e. NN0 $. @@ -139408,7 +139408,7 @@ nonnegative integers (cont.)". $) $( Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.) $) numnncl2 $p |- ( ( T x. A ) + 0 ) e. NN $= - ( cmul co cc0 caddc cn nnmulcli nncni addid1i eqeltri ) BAEFZGHFNINNBACDJ + ( cmul co cc0 caddc cn nnmulcli nncni addridi eqeltri ) BAEFZGHFNINNBACDJ ZKLOM $. $} @@ -139442,7 +139442,7 @@ nonnegative integers (cont.)". $) $( Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) $) numltc $p |- ( ( T x. A ) + C ) < ( ( T x. B ) + D ) $= - ( cmul co caddc clt wbr cle nn0rei wcel numlt nnrei ax-1cn adddii mulid1i + ( cmul co caddc clt wbr cle nn0rei wcel numlt nnrei ax-1cn adddii mulridi c1 recni oveq2i eqtri breqtrri cn0 wb nn0ltp1le mp2an cc0 nngt0i peano2re mpbi cr ax-mp lemul2i remulcli readdcli ltletri nn0addge1i ) EAMNZCONZEBM NZPQZVHVHDONZRQVGVJPQVGEAUFONZMNZPQVLVHRQZVIVGVFEONZVLPACEEFGIFKUAVLVFEUF @@ -139610,7 +139610,7 @@ nonnegative integers (cont.)". $) Carneiro, 18-Feb-2014.) $) numsucc $p |- ( N + 1 ) = ( ( T x. B ) + 0 ) $= ( c1 caddc co cmul cc0 cn0 1nn0 nn0addcli nn0cni oveq2i 3eqtr4ri eqeltrri - eqeltri mulid1i ax-1cn adddii eqcomi numsuc num0u 3eqtri ) DKLMZCAKLMZNMZ + eqeltri mulridi ax-1cn adddii eqcomi numsuc num0u 3eqtri ) DKLMZCAKLMZNMZ CBNMZUNOLMCANMZCKNMZLMUOCLMUMUKUPCUOLCCCEKLMZPGEKFQRUCZSZUDTCAKUSAHSUEUFA ECCDURHFCUQGUGJUHUAULBCNITBCURULBPIAKHQRUBUIUJ $. $} @@ -139633,7 +139633,7 @@ nonnegative integers (cont.)". $) $( Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) $) dec10p $p |- ( ; 1 0 + A ) = ; 1 A $= - ( c1 cdc cc0 cmul co caddc dfdec10 10nn nncni mulid1i oveq1i eqtr2i ) BACBD + ( c1 cdc cc0 cmul co caddc dfdec10 10nn nncni mulridi oveq1i eqtr2i ) BACBD CZBEFZAGFNAGFBAHONAGNNIJKLM $. ${ @@ -139703,7 +139703,7 @@ nonnegative integers (cont.)". $) $( Add two decimal integers ` M ` and ` N ` (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) $) numadd $p |- ( M + N ) = ( ( T x. E ) + F ) $= - ( co caddc c1 cmul numcl eqeltri nn0cni mulid1i oveq1i 1nn0 eqtri numma + ( co caddc c1 cmul numcl eqeltri nn0cni mulridi oveq1i 1nn0 eqtri numma cn0 eqtr3i ) HUAUBSZITSHITSEFUBSGTSUMHITHHHEAUBSBTSUKOABEJKLUCUDUEUFUGA BCDUAEFGHIJKLMNOPUHAUAUBSZCTSACTSFUNACTAAKUEUFUGQUIBUAUBSZDTSBDTSGUOBDT BBLUEUFUGRUIUJUL $. @@ -139716,7 +139716,7 @@ nonnegative integers (cont.)". $) $( Add two decimal integers ` M ` and ` N ` (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) $) numaddc $p |- ( M + N ) = ( ( T x. E ) + F ) $= - ( co c1 cmul caddc cn0 numcl eqeltri nn0cni mulid1i oveq1i 1nn0 addassi + ( co c1 cmul caddc cn0 numcl eqeltri nn0cni mulridi oveq1i 1nn0 addassi ax-1cn 3eqtr2i eqtri nummac eqtr3i ) HUAUBTZIUCTHIUCTEFUBTGUCTUQHIUCHHH EAUBTBUCTUDOABEJKLUEUFUGUHUIABCDUAEFGUAHIJKLMNOPUJQUJAUAUBTZCUAUCTZUCTA USUCTACUCTUAUCTFURAUSUCAAKUGZUHUIACUAUTCMUGULUKRUMBUAUBTZDUCTBDUCTEUAUB @@ -139738,7 +139738,7 @@ nonnegative integers (cont.)". $) $( The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.) $) nummul1c $p |- ( N x. P ) = ( ( T x. C ) + D ) $= - ( co cc0 caddc cmul cn0 numcl eqeltri num0u num0h nn0cni addid2i oveq2i + ( co cc0 caddc cmul cn0 numcl eqeltri num0u num0h nn0cni addlidi oveq2i 0nn0 eqtri eqtr3i nummac ) HEUARZUNSTRFCUARDTREHHFAUARBTRUBMABFIKLUCUDJ UEABSSEFCDGHSIKLUJUJMSFIUJUFJNOAEUARZSGTRZTRUOGTRCUPGUOTGGOUGUHUIPUKBEU ARZUQSTRFGUARDTREBLJUEQULUMUK $. @@ -139848,7 +139848,7 @@ nonnegative integers (cont.)". $) $( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed multiplicand ` P ` (no carry). (Contributed by AV, 16-Sep-2021.) $) decrmanc $p |- ( ( M x. P ) + N ) = ; E F $= - ( cc0 0nn0 dec0h cmul co caddc nn0mulcli nn0cni addid1i eqtri decma ) A + ( cc0 0nn0 dec0h cmul co caddc nn0mulcli nn0cni addridi eqtri decma ) A BOGCDEFGHIPJKGJQLACRSZOTSUFDUFUFACHLUAUBUCMUDNUE $. $} @@ -139860,7 +139860,7 @@ nonnegative integers (cont.)". $) $( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed multiplicand ` P ` (with carry). (Contributed by AV, 16-Sep-2021.) $) decrmac $p |- ( ( M x. P ) + N ) = ; E F $= - ( cc0 co caddc 0nn0 dec0h cmul nn0cni addid2i oveq2i eqtri decmac ) ABR + ( cc0 co caddc 0nn0 dec0h cmul nn0cni addlidi oveq2i eqtri decmac ) ABR HCDEFGHIJUAKLHKUBMNOACUCSZRFTSZTSUIFTSDUJFUITFFOUDUEUFPUGQUH $. $} $} @@ -139885,7 +139885,7 @@ nonnegative integers (cont.)". $) $( Add two numerals ` M ` and ` N ` (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) $) decaddi $p |- ( M + N ) = ; A C $= - ( cc0 0nn0 dec0h nn0cni addid1i decadd ) ABKEACDEFGLHIEHMAAFNOJP $. + ( cc0 0nn0 dec0h nn0cni addridi decadd ) ABKEACDEFGLHIEHMAAFNOJP $. $} decaddci.5 $e |- ( A + 1 ) = D $. @@ -139895,7 +139895,7 @@ nonnegative integers (cont.)". $) $( Add two numerals ` M ` and ` N ` (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) $) decaddci $p |- ( M + N ) = ; D C $= - ( cc0 0nn0 dec0h caddc co c1 nn0cni addid1i oveq1i eqtri decaddc ) ABNF + ( cc0 0nn0 dec0h caddc co c1 nn0cni addridi oveq1i eqtri decaddc ) ABNF DCEFGHOIJFIPANQRZSQRASQRDUEASQAAGTUAUBKUCLMUD $. $} @@ -139933,7 +139933,7 @@ nonnegative integers (cont.)". $) AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) Remove hypothesis ` D e. NN0 ` . (Revised by Steven Nguyen, 7-Dec-2022.) $) decmul1 $p |- ( N x. P ) = ; C D $= - ( cmul co cc0 caddc cdc cn0 deccl eqtri eqeltri num0u nn0mulcli addid1i + ( cmul co cc0 caddc cdc cn0 deccl eqtri eqeltri num0u nn0mulcli addridi 0nn0 nn0cni decrmanc ) FEMNZUHOPNCDQEFFABQRJABHISUAGUBABECDFOHIUEJGKBEM NZOPNUIDUIUIBEIGUCUFUDLTUGT $. $} @@ -139968,7 +139968,7 @@ nonnegative integers (cont.)". $) $( The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.) $) decmulnc $p |- ( N x. ; A B ) = ; ( N x. A ) ( N x. B ) $= - ( cmul co cc0 cdc eqid nn0mulcli 0nn0 nn0cni addid1i dec0h decmul2c ) ABC + ( cmul co cc0 cdc eqid nn0mulcli 0nn0 nn0cni addridi dec0h decmul2c ) ABC AGHZCBGHZCIABJZDEFTKCBDFLZMRRCADELNOSUAPQ $. $} @@ -139977,7 +139977,7 @@ nonnegative integers (cont.)". $) $( The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.) $) 11multnc $p |- ( N x. ; 1 1 ) = ; N N $= - ( c1 cdc cmul co 1nn0 decmulnc nn0cni mulid1i deceq12i eqtri ) ACCDEFACEF + ( c1 cdc cmul co 1nn0 decmulnc nn0cni mulridi deceq12i eqtri ) ACCDEFACEF ZMDAADCCABGGHMAMAAABIJZNKL $. $} @@ -140143,7 +140143,7 @@ nonnegative integers (cont.)". $) $( Lemma for ~ 4t3e12 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.) $) 4t3lem $p |- ( A x. C ) = E $= - ( cmul co c1 caddc oveq2i nn0cni ax-1cn adddii mulid1i eqtri oveq12i ) AC + ( cmul co c1 caddc oveq2i nn0cni ax-1cn adddii mulridi eqtri oveq12i ) AC KLABMNLZKLZECUBAKHOUCDANLZEUCABKLZAMKLZNLUDABMAFPZBGPQRUEDUFANIAUGSUATJTT $. $} @@ -140160,7 +140160,7 @@ nonnegative integers (cont.)". $) $( 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.) $) 5t2e10 $p |- ( 5 x. 2 ) = ; 1 0 $= - ( c5 c1 c2 cc0 cdc 5nn0 1nn0 df-2 5cn mulid1i 5p5e10 4t3lem ) ABCABDEFGHAIJ + ( c5 c1 c2 cc0 cdc 5nn0 1nn0 df-2 5cn mulridi 5p5e10 4t3lem ) ABCABDEFGHAIJ KL $. $( 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) @@ -140327,7 +140327,7 @@ nonnegative integers (cont.)". $) $( 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) $) 9t11e99 $p |- ( 9 x. ; 1 1 ) = ; 9 9 $= - ( c9 c1 cc0 cdc cmul co caddc 9cn 10nn0 nn0cni ax-1cn mulcli adddii mulid1i + ( c9 c1 cc0 cdc cmul co caddc 9cn 10nn0 nn0cni ax-1cn mulcli adddii mulridi oveq2i mulcomi eqtri oveq12i dfdec10 3eqtr4i ) ABCDZBEFZBGFZEFZUAAEFZAGFZAB BDZEFAADUDAUBEFZABEFZGFUFAUBBHUABUAIJZKLKMUHUEUIAGUHAUAEFUEUBUAAEUAUJNOAUAH UJPQAHNRQUGUCAEBBSOAAST $. @@ -140948,7 +140948,7 @@ nonnegative integers (cont.)". $) ( N + K ) e. ( ZZ>= ` M ) ) $= ( vj vk cn0 wcel cuz caddc co cv wi cc0 c1 wa wceq cc eleq1d oveq2 imbi2d cfv eluzelcn ax-1cn addass mp3an3 syl2anr adantr peano2uz adantl eqeltrrd - nn0cn exp31 a2d addid1d ibir nn0indALT impcom ) AFGCBHUAZGZCAIJZURGZUSCDK + nn0cn exp31 a2d addridd ibir nn0indALT impcom ) AFGCBHUAZGZCAIJZURGZUSCDK ZIJZURGZLUSCMIJZURGZLUSCEKZIJZURGZLUSCVGNIJZIJZURGZLUSVALDEAVGFGZUSVIVLVM USVIVLVMUSOZVIOVHNIJZVKURVNVOVKPZVIUSCQGZVGQGZVPVMBCUBZVGUKVQVRNQGVPUCCVG NUDUEUFUGVIVOURGVNBVHUHUIUJULUMUSVFUSVECURUSCVSUNRUOVBMPZVDVFUSVTVCVEURVB @@ -144109,27 +144109,27 @@ Infinity and the extended real number system (cont.) BXCWRURWQVAUTVBXEBFHJVNVSVCZSXEBFHJXFTVDVNBFNQXBHXCBUOBUNRVEVFXAAHBJVMVNVGZ TXAAHBJXGSMVHVI $. - $( Extended real version of ~ addid1 . (Contributed by Mario Carneiro, + $( Extended real version of ~ addrid . (Contributed by Mario Carneiro, 20-Aug-2015.) $) - xaddid1 $p |- ( A e. RR* -> ( A +e 0 ) = A ) $= + xaddrid $p |- ( A e. RR* -> ( A +e 0 ) = A ) $= ( cxr wcel cr cpnf wceq cmnf w3o cc0 cxad co elxr 0re wne ax-mp mp2an oveq1 - 0xr id 3eqtr4a caddc rexadd mpan2 addid1d renemnf xaddpnf2 renepnf xaddmnf2 + 0xr id 3eqtr4a caddc rexadd mpan2 addridd renemnf xaddpnf2 renepnf xaddmnf2 recn eqtrd 3jaoi sylbi ) ABCADCZAEFZAGFZHAIJKZAFZALUMUQUNUOUMUPAIUAKZAUMIDC ZUPURFMAIUBUCUMAAUIUDUJUNEIJKZEUPAIBCZIGNZUTEFRUSVBMIUEOIUFPAEIJQUNSTUOGIJK ZGUPAVAIENZVCGFRUSVDMIUGOIUHPAGIJQUOSTUKUL $. - $( Extended real version of ~ addid2 . (Contributed by Mario Carneiro, + $( Extended real version of ~ addlid . (Contributed by Mario Carneiro, 20-Aug-2015.) $) - xaddid2 $p |- ( A e. RR* -> ( 0 +e A ) = A ) $= - ( cxr wcel cc0 cxad co wceq 0xr xaddcom mpan xaddid1 eqtrd ) ABCZDAEFZADEFZ + xaddlid $p |- ( A e. RR* -> ( 0 +e A ) = A ) $= + ( cxr wcel cc0 cxad co wceq 0xr xaddcom mpan xaddrid eqtrd ) ABCZDAEFZADEFZ ADBCMNOGHDAIJAKL $. ${ - xaddid1d.1 $e |- ( ph -> A e. RR* ) $. + xaddridd.1 $e |- ( ph -> A e. RR* ) $. $( ` 0 ` is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.) $) - xaddid1d $p |- ( ph -> ( A +e 0 ) = A ) $= - ( cxr wcel cc0 cxad co wceq xaddid1 syl ) ABDEBFGHBICBJK $. + xaddridd $p |- ( ph -> ( A +e 0 ) = A ) $= + ( cxr wcel cc0 cxad co wceq xaddrid syl ) ABDEBFGHBICBJK $. $} $( Extended nonnegative integer ordering relation. (Contributed by Thierry @@ -144169,7 +144169,7 @@ Infinity and the extended real number system (cont.) nn0re wne caddc rexadd cle wb nn0ge0 add20 bitrd biimpd expcom oveq2 adantr wbr jca cmnf nn0xnn0 xnn0xrnemnf xaddpnf1 3syl adantl renepnf ax-mp pm2.21i 0re nesymi syl6bi ex sylbi com12 oveq1 xaddpnf2 syl sylan9bb imp oveq12 0xr - jaoi xaddid1 eqtrdi impbid1 ) ACDZBCDZEABFGZHIZAHIBHIEZVTWAWCWDJZVTAKDZALIZ + jaoi xaddrid eqtrdi impbid1 ) ACDZBCDZEABFGZHIZAHIBHIEZVTWAWCWDJZVTAKDZALIZ MWAWEJZANWFWHWGWAWFWEWABKDZBLIZMWFWEJZBNWIWKWJWFWIWEWFWIEZWCWDWLWCABUAGZHIZ WDWLWBWMHWFAODZBODZWBWMIWIASZBSZABUBPQWFWOHAUCULZEWPHBUCULZEWNWDUDWIWFWOWSW QAUEUMWIWPWTWRBUEUMABUFPUGUHUIWJWFWEWJWFEZWCLHIZWDXAWCALFGZHIZXBWJWCXDUDWFW @@ -144256,7 +144256,7 @@ Infinity and the extended real number system (cont.) ( cxr wcel cr wa cxad co wceq adantl oveq2d cmnf ad2antlr cpnf rexr renepnf wne syl cc0 eqtrd cxne cneg rexneg renegcl xaddmnf2 syl2anc oveq1 sylan9eqr oveq1d simpr 3eqtr4d simpll renemnf xaddass syl222anc simplr rexaddd negidd - caddc recnd xaddid1 ad2antrr pm2.61dane ) ACDZBEDZFZABGHZBUAZGHVGBUBZGHZAVF + caddc recnd xaddrid ad2antrr pm2.61dane ) ACDZBEDZFZABGHZBUAZGHVGBUBZGHZAVF VHVIVGGVEVHVIIVDBUCJKVFVJAIALVFALIZFZLVIGHZLVJAVLVIEDZVMLIZVEVNVDVKBUDZMVNV ICDZVINQVOVIOZVIPVIUEUFRVLVGLVIGVKVFVGLBGHZLALBGUGVEVSLIZVDVEBCDZBNQVTBOZBP BUEUFJUHUIVFVKUJUKVFALQZFZVJABVIGHZGHZAWDVDWCWABLQZVQVILQZVJWFIVDVEWCULVFWC @@ -144337,7 +144337,7 @@ Infinity and the extended real number system (cont.) Mario Carneiro, 21-Aug-2015.) $) xaddge0 $p |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( 0 <_ A /\ 0 <_ B ) ) -> 0 <_ ( A +e B ) ) $= - ( cxr wcel wa cc0 cle wbr cxad co 0xr a1i simplr xaddcl adantr wceq xaddid2 + ( cxr wcel wa cc0 cle wbr cxad co 0xr a1i simplr xaddcl adantr wceq xaddlid simprr syl simpll simprl xleadd1a syl31anc eqbrtrrd xrletrd ) ACDZBCDZEZFAG HZFBGHZEZEZFBABIJZFCDZULKLZUFUGUKMZUHUMCDUKABNOUHUIUJRULFBIJZBUMGULUGUQBPUP BQSULUNUFUGUIUQUMGHUOUFUGUKTUPUHUIUJUAFABUBUCUDUE $. @@ -144385,7 +144385,7 @@ Infinity and the extended real number system (cont.) ( 0 <_ ( A +e -e B ) <-> B <_ A ) ) $= ( cxr wcel cpnf wceq cmnf cc0 cxne co cle wbr wb wa adantl breq2d breqtrrid cxad eqtrdi 3bitr4d cr w3o elxr 0xr rexr xnegcl xaddcl sylan2 simpr xleadd1 - mp3an2i xaddid2 syl xnpcan breq12d bitrd pnfxr xrletri3 mpan2 wne wn mnflt0 + mp3an2i xaddlid syl xnpcan breq12d bitrd pnfxr xrletri3 mpan2 wne wn mnflt0 clt mnfxr xrltnle mp2an mpbi xaddmnf1 mtbiri ex necon4ad 0le0 oveq1 impbid1 pnfaddmnf pnfge biantrurd adantr xnegeq xnegpnf oveq2d mnfaddpnf pm2.61dane breq1 0lepnf xaddpnf1 mnfle 2thd xnegmnf 3jaodan sylan2b ) BCDZACDZBUADZBEF @@ -144616,26 +144616,26 @@ Infinity and the extended real number system (cont.) UKGBCZUTUOKUKULLZMAGNOUKUSBCULDUSEPZUTGKARUKULVCASTUSUAUBUCUDUGUMUNBCZJBCUQ URUHUMUKVAVDVBMAGUEOUFUNJUIOUJ $. - $( Extended real version of ~ mulid1 . (Contributed by Mario Carneiro, + $( Extended real version of ~ mulrid . (Contributed by Mario Carneiro, 20-Aug-2015.) $) - xmulid1 $p |- ( A e. RR* -> ( A *e 1 ) = A ) $= + xmulrid $p |- ( A e. RR* -> ( A *e 1 ) = A ) $= ( cxr wcel cr cpnf wceq cmnf w3o c1 cxmu elxr cmul 1re 1xr 0lt1 mp2an oveq1 co id 3eqtr4a mpan2 ax-1rid eqtrd cc0 clt wbr xmulpnf2 xmulmnf2 3jaoi sylbi rexmul ) ABCADCZAEFZAGFZHAIJRZAFZAKULUPUMUNULUOAILRZAULIDCUOUQFMAIUKUAAUBUC UMEIJRZEUOAIBCZUDIUEUFZUREFNOIUGPAEIJQUMSTUNGIJRZGUOAUSUTVAGFNOIUHPAGIJQUNS TUIUJ $. - $( Extended real version of ~ mulid2 . (Contributed by Mario Carneiro, + $( Extended real version of ~ mullid . (Contributed by Mario Carneiro, 20-Aug-2015.) $) - xmulid2 $p |- ( A e. RR* -> ( 1 *e A ) = A ) $= - ( cxr wcel c1 cxmu co wceq 1xr xmulcom mpan xmulid1 eqtrd ) ABCZDAEFZADEFZA + xmullid $p |- ( A e. RR* -> ( 1 *e A ) = A ) $= + ( cxr wcel c1 cxmu co wceq 1xr xmulcom mpan xmulrid eqtrd ) ABCZDAEFZADEFZA DBCMNOGHDAIJAKL $. $( Extended real version of ~ mulm1 . (Contributed by Mario Carneiro, 20-Aug-2015.) $) xmulm1 $p |- ( A e. RR* -> ( -u 1 *e A ) = -e A ) $= ( cxr wcel c1 cneg cxmu co cxne cr wceq 1re rexneg oveq1i 1xr xmulneg1 mpan - ax-mp eqtr3id xmulid2 xnegeq syl eqtrd ) ABCZDEZAFGZDAFGZHZAHZUCUEDHZAFGZUG + ax-mp eqtr3id xmullid xnegeq syl eqtrd ) ABCZDEZAFGZDAFGZHZAHZUCUEDHZAFGZUG UIUDAFDICUIUDJKDLQMDBCUCUJUGJNDAOPRUCUFAJUGUHJASUFATUAUB $. $( Lemma for ~ xmulass . (Contributed by Mario Carneiro, 20-Aug-2015.) $) @@ -144815,7 +144815,7 @@ the expression make the whole thing evaluate to zero (on both sides), ( A <_ B <-> ( A *e C ) <_ ( B *e C ) ) ) $= ( cxr wcel crp w3a cle wbr cxmu co cc0 wa wi rpxr 3ad2ant3 syl2anc syl wceq c1 rpge0 jca xlemul1a ex syl3an3 cdiv simp1 simp2 rpreccl syl112anc xmulass - xmulcl syl3anc cmul rpre rpred rexmul recnd wne rpne0 recidd oveq2d xmulid1 + xmulcl syl3anc cmul rpre rpred rexmul recnd wne rpne0 recidd oveq2d xmulrid cr eqtrd 3eqtrd breq12d sylibd impbid ) ADEZBDEZCFEZGZABHIZACJKZBCJKZHIZVLV JVKCDEZLCHIZMZVNVQNVLVRVSCOZCUAUBVJVKVTGVNVQABCUCUDUEVMVQVOTCUFKZJKZVPWBJKZ HIZVNVMVODEZVPDEZWBDEZLWBHIZVQWENVMVJVRWFVJVKVLUGZVLVJVRVKWAPZACULQVMVKVRWG @@ -144900,7 +144900,7 @@ the expression make the whole thing evaluate to zero (on both sides), xadddi $p |- ( ( A e. RR /\ B e. RR* /\ C e. RR* ) -> ( A *e ( B +e C ) ) = ( ( A *e B ) +e ( A *e C ) ) ) $= ( cr wcel cxr cc0 clt wbr cxad cxmu wceq syl2anc xmul02 syl oveq1d xmulneg1 - co cxne xmulcl w3a xadddilem simpl2 simpl3 xaddcl 0xr xaddid1 ax-mp eqtr4di + co cxne xmulcl w3a xadddilem simpl2 simpl3 xaddcl 0xr xaddrid ax-mp eqtr4di wa simpr eqtr3d oveq12d 3eqtr3d simp1 adantr cneg rexneg renegcl eqeltrd wb rexrd xlt0neg1 biimpa syl31anc xnegdi eqtr4d xneg11 mpbid w3o 0re mpjao3dan lttri4 sylancr ) ADEZBFEZCFEZUAZGAHIZABCJRZKRZABKRZACKRZJRZLZGALZAGHIZABCUB @@ -144933,10 +144933,10 @@ the expression make the whole thing evaluate to zero (on both sides), ( cxr wcel cc0 cle wbr wa cxad co cxmu wceq cpnf cmnf adantr syl2an2r oveq1 sylancr sylan9eqr w3a clt simpr simp2l ad2antrr simp3l xadddi syl3anc pnfxr wne xmulcl simpl3r 0lepnf xmulge0 mpanl12 ge0nemnf syl2anc xaddpnf2 oveq12d - cr xmulpnf2 sylan oveq1d xaddcl 0xr a1i xaddid1d xleadd2a syl31anc eqbrtrrd + cr xmulpnf2 sylan oveq1d xaddcl 0xr a1i xaddridd xleadd2a syl31anc eqbrtrrd xrltletrd 3eqtr4rd mnfxr xmulneg1 xnegmnf oveq1i eqtr3di xnegpnf eqeq12d wb cxne xneg11 sylancl bitr3d necon3bid mpbid xaddmnf2 xmulmnf2 w3o elxr sylib - simpl1 mpjao3dan simp1 xaddid2 syl eqcomd xmul01 3ad2ant1 oveq2d wo xrleloe + simpl1 mpjao3dan simp1 xaddlid syl eqcomd xmul01 3ad2ant1 oveq2d wo xrleloe oveq2 simp2r mpjaodan ) ADEZBDEZFBGHZIZCDEZFCGHZIZUAZFBUBHZABCJKZLKZABLKZAC LKZJKZMZFBMZXMXNIZAUTEZXTANMZAOMZYBYCIYCXGXJXTYBYCUCXMXGXNYCXFXGXHXLUDZUEXM XJXNYCXFXIXJXKUFZUEABCUGUHYBYDINNCLKZJKZNXSXPYBYHDEZYDYHOUJZYINMYBNDEZXJYJU @@ -144968,7 +144968,7 @@ the expression make the whole thing evaluate to zero (on both sides), 20-Aug-2015.) $) x2times $p |- ( A e. RR* -> ( 2 *e A ) = ( A +e A ) ) $= ( cxr wcel c2 cxmu co c1 cxad caddc df-2 cr wceq rexadd mp2an eqtr4i oveq1i - 1re cc0 cle wbr wa 1xr pm3.2i xadddi2r mp3an12 xmulid2 oveq12d eqtrd eqtrid + 1re cc0 cle wbr wa 1xr pm3.2i xadddi2r mp3an12 xmullid oveq12d eqtrd eqtrid 0le1 ) ABCZDAEFGGHFZAEFZAAHFZDULAEDGGIFZULJGKCZUPULUOLQQGGMNOPUKUMGAEFZUQHF ZUNGBCZRGSTZUAZVAUKUMURLUSUTUBUJUCZVBGGAUDUEUKUQAUQAHAUFZVCUGUHUI $. @@ -147023,7 +147023,7 @@ the existence of its supremum (see ~ suprcl ). (Contributed by Paul ( ( A / B ) e. ( 0 [,] 1 ) <-> A <_ B ) ) $= ( cdiv co cc0 c1 cicc wcel cr cle wbr clt w3a elicc01 df-3an bitri cmul 1re wa wb ledivmul mp3an2 simpll simprl gt0ne0 adantl redivcld divge0 biantrurd - adantlr wne jca cc recn ad2antrl mulid1d breq2d 3bitr3d bitrid ) ABCDZEFGDH + adantlr wne jca cc recn ad2antrl mulridd breq2d 3bitr3d bitrid ) ABCDZEFGDH ZUTIHZEUTJKZSZUTFJKZSZAIHZEAJKZSZBIHZEBLKZSZSZABJKZVAVBVCVEMVFUTNVBVCVEOPVM VEABFQDZJKZVFVNVGVLVEVPTZVHVGFIHVLVQRAFBUAUBUJVMVDVEVMVBVCVMABVGVHVLUCVIVJV KUDVLBEUKVIBUEUFUGABUHULUIVMVOBAJVMBVJBUMHVIVKBUNUOUPUQURUS $. @@ -147033,11 +147033,11 @@ the existence of its supremum (see ~ suprcl ). (Contributed by Paul Carneiro, 8-Sep-2015.) $) lincmb01cmp $p |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) ) $= - ( cr wcel wbr cc0 c1 cicc co cmin cmul caddc wb eqid syl22anc mpbid mulid2d + ( cr wcel wbr cc0 c1 cicc co cmin cmul caddc wb eqid syl22anc mpbid mullidd cle recnd clt w3a simpr crp 0red 1red elicc01 simp1bi adantl biimp3a adantr difrp iccdil simpl2 simpl1 resubcld mul02d oveq12d remulcld iccshftr mulcld wa eleqtrd subadd23d subdid oveq1d 1re resubcl sylancr addcomd 1cnd subdird - eqtrd oveq2d 3eqtr4d addid2d npcand 3eltr3d ) ADEZBDEZABUAFZUBZCGHIJEZVBZCB + eqtrd oveq2d 3eqtr4d addlidd npcand 3eltr3d ) ADEZBDEZABUAFZUBZCGHIJEZVBZCB AKJZLJZAMJZGAMJZWEAMJZIJZHCKJZALJZCBLJZMJZABIJWDWFGWEIJZEZWGWJEZWDWFGWELJZH WELJZIJZWOWDWCWFWTEZWBWCUCWDGDEZHDEZCDEZWEUDEZWCXANWDUEZWDUFWCXDWBWCXDGCSFC HSFCUGUHUIZWBXEWCVSVTWAXEABULUJUKGHWRWSWECWROWSOUMPQWDWRGWSWEIWDWEWDWEWDBAV @@ -147061,7 +147061,7 @@ the existence of its supremum (see ~ suprcl ). (Contributed by Paul w3a cdiv wa cle elicc01 simp1bi adantl simpl2 mulcld ax-1cn subcl sylancr simpl1 addcomd lincmb01cmp eqeltrd simpr wb elicc2 biimpa simp1d iccshftl 3adant3 eqid syl22anc mpbid resubcld difrp biimp3a adantr rpne0d divcan1d - rpcnd mul02d subidd eqtr4d mulid2d oveq12d 3eltr4d 0red rerpdivcld iccdil + rpcnd mul02d subidd eqtr4d mullidd oveq12d 3eltr4d 0red rerpdivcld iccdil crp 1red mpbird eqcom adantrr divmul3d bitrid remulcld subadd2d subadd23d wne bitrdi subdid oveq1d 1cnd subdird eqtrd oveq2d eqeq2d 3bitrd f1ocnv2d 3eqtr4d ) CGHZDGHZCDUAIZUCZABJKLMZCDLMZAUBZDNMZKXMOMZCNMZPMZBUBZCOMZDCOMZ @@ -148416,7 +148416,7 @@ Finite intervals of nonnegative integers (or "finite sets of sequential $( An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.) $) fz0tp $p |- ( 0 ... 2 ) = { 0 , 1 , 2 } $= - ( cc0 c2 cfz co caddc c1 ctp 2cn addid2i eqcomi oveq2i cz wcel wceq 0z fztp + ( cc0 c2 cfz co caddc c1 ctp 2cn addlidi eqcomi oveq2i cz wcel wceq 0z fztp ax-mp eqid id a1i 0p1e1 tpeq123d 3eqtri ) ABCDAABEDZCDZAAFEDZUDGZAFBGZBUDAC UDBBHIZJKALMUEUGNOAPQAANZUGUHNARUJAAUFFUDBUJSUFFNUJUATUDBNUJUITUBQUC $. @@ -148433,9 +148433,9 @@ Finite intervals of nonnegative integers (or "finite sets of sequential (Contributed by Alexander van der Vekens, 13-Aug-2017.) $) fz0to4untppr $p |- ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } ) $= ( cc0 c4 cfz co c2 caddc c1 c3 cuz cfv wcel wceq eqtri cz cle wbr 3z ltleii - eluz1i ax-mp cun ctp cpr df-3 2cn addid2i eqcomi oveq1i 0re 3re 0z mpbir2an + eluz1i ax-mp cun ctp cpr df-3 2cn addlidi eqcomi oveq1i 0re 3re 0z mpbir2an 3pos eqeltrri 4z 2re 4re 2lt4 2z fveq2i eleqtri fzsplit2 mp2an ax-1cn eqidd - fztp cc addid2 tpeq123d oveq1d eqtr4di wa eqid df-4 pm3.2i wb fzopth sylibr + fztp cc addlid tpeq123d oveq1d eqtr4di wa eqid df-4 pm3.2i wb fzopth sylibr a1i 3lt4 fzpr eqtrd preq2i eqtrdi uneq12i ) ABCDZAAEFDZCDZWGGFDZBCDZUAZAGEU BZHBUCZUAWIAIJZKBWGIJZKWFWKLHWIWNHEGFDZWIUDEWGGFWGEEUEUFZUGZUHMHWNKHNKZAHOP QAHUIUJUMRAHUKSULUNBEIJZWOBWTKBNKZEBOPUOEBUPUQURREBUSSULEWGIWRUTVAWGABVBVCW @@ -149239,7 +149239,7 @@ Finite intervals of nonnegative integers (or "finite sets of sequential by AV, 30-Apr-2020.) $) fzo0addel $p |- ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( D ..^ ( C + D ) ) ) $= - ( cc0 cfzo co wcel cz wa caddc fzoaddel wceq cc addid2 eqcomd adantl oveq1d + ( cc0 cfzo co wcel cz wa caddc fzoaddel wceq cc addlid eqcomd adantl oveq1d zcn syl eleqtrrd ) ADBEFGZCHGZIZACJFDCJFZBCJFZEFCUEEFADBCKUCCUDUEEUBCUDLZUA UBCMGZUFCRUGUDCCNOSPQT $. @@ -149255,7 +149255,7 @@ Finite intervals of nonnegative integers (or "finite sets of sequential (Contributed by Stefan O'Rear, 15-Aug-2015.) $) fzoaddel2 $p |- ( ( A e. ( 0 ..^ ( B - C ) ) /\ B e. ZZ /\ C e. ZZ ) -> ( A + C ) e. ( C ..^ B ) ) $= - ( cc0 cmin co cfzo wcel cz w3a caddc fzoaddel 3adant2 wceq cc zcn wa addid2 + ( cc0 cmin co cfzo wcel cz w3a caddc fzoaddel 3adant2 wceq cc zcn wa addlid adantl npcan oveq12d syl2an 3adant1 eleqtrd ) ADBCEFZGFHZBIHZCIHZJACKFZDCKF ZUECKFZGFZCBGFZUFUHUIULHUGADUECLMUGUHULUMNZUFUGBOHZCOHZUNUHBPCPUOUPQUJCUKBG UPUJCNUOCRSBCTUAUBUCUD $. @@ -149306,7 +149306,7 @@ Finite intervals of nonnegative integers (or "finite sets of sequential is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.) $) fzosubel3 $p |- ( ( A e. ( B ..^ ( B + D ) ) /\ D e. ZZ ) -> ( A - B ) e. ( 0 ..^ D ) ) $= - ( caddc co cfzo wcel cz wa cmin simpl elfzoel1 adantr zcnd addid1d eleqtrrd + ( caddc co cfzo wcel cz wa cmin simpl elfzoel1 adantr zcnd addridd eleqtrrd cc0 oveq1d 0zd simpr fzosubel2 syl13anc ) ABBCDEZFEZGZCHGZIZABQDEZUCFEZGBHG ZQHGUFABJEQCFEGUGAUDUIUEUFKUGUHBUCFUGBUGBUEUJUFABUCLMZNORPUKUGSUEUFTABQCUAU B $. @@ -149551,7 +149551,7 @@ Finite intervals of nonnegative integers (or "finite sets of sequential (Contributed by Alexander van der Vekens, 9-Nov-2017.) $) fzo0to3tp $p |- ( 0 ..^ 3 ) = { 0 , 1 , 2 } $= ( cc0 c3 cfzo co c1 cmin cfz c2 caddc ctp cz wcel 3z fzoval ax-mp 3m1e2 2cn - wceq addid2i a1i eqtr4i oveq2i 0z fztp eqidd 0p1e1 tpeq123d eqtrd 3eqtri ) + wceq addlidi a1i eqtr4i oveq2i 0z fztp eqidd 0p1e1 tpeq123d eqtrd 3eqtri ) ABCDZABEFDZGDZAAHIDZGDZAEHJZBKLUJULRMABNOUKUMAGUKHUMPHQSZUAUBAKLZUNUORUCUQU NAAEIDZUMJUOAUDUQAAUREUMHUQAUEURERUQUFTUMHRUQUPTUGUHOUI $. @@ -150331,7 +150331,7 @@ Finite intervals of nonnegative integers (or "finite sets of sequential an interval. (Contributed by BJ, 29-Jun-2019.) $) ico01fl0 $p |- ( A e. ( 0 [,) 1 ) -> ( |_ ` A ) = 0 ) $= ( cc0 c1 cico co wcel cr cle wbr clt cfl cfv wceq cxr wss 0re icossre mp2an - 1xr wb sseli w3a 0xr elico1 simp2bi simp3bi wa caddc addid2d fveqeq2d cz 0z + 1xr wb sseli w3a 0xr elico1 simp2bi simp3bi wa caddc addlidd fveqeq2d cz 0z recn flbi2 mpan bitr3d biimpar syl12anc ) ABCDEZFZAGFZBAHIZACJIZAKLBMZUSGAB GFCNFZUSGOPSBCQRUAUTANFZVBVCBNFVEUTVFVBVCUBTUCSBCAUDRZUEUTVFVBVCVGUFVAVDVBV CUGZVABAUHEZKLBMZVDVHVAVIABKVAAAUMUIUJBUKFVAVJVHTULABUNUOUPUQUR $. @@ -150369,7 +150369,7 @@ Finite intervals of nonnegative integers (or "finite sets of sequential divfl0 $p |- ( ( A e. NN0 /\ B e. NN ) -> ( A < B <-> ( |_ ` ( A / B ) ) = 0 ) ) $= ( cn0 wcel cn wa cdiv co cfl cfv cc0 wceq caddc cle wbr c1 clt cc cr wb syl - nn0nndivcl recnd addid2 eqcomd fveqeq2d cz 0z flbi2 nn0ge0div biantrurd crp + nn0nndivcl recnd addlid eqcomd fveqeq2d cz 0z flbi2 nn0ge0div biantrurd crp sylancr nn0re nnrp divlt1lt syl2an bitr3d 3bitrrd ) ACDZBEDZFZABGHZIJKLKVCM HZIJKLZKVCNOZVCPQOZFZABQOZVBVCVDKIVBVCRDZVCVDLVBVCABUBZUCVJVDVCVCUDUEUAUFVB KUGDVCSDVEVHTUHVKVCKUIUMVBVGVHVIVBVFVGABUJUKUTASDBULDVGVITVAAUNBUOABUPUQURU @@ -150479,7 +150479,7 @@ Finite intervals of nonnegative integers (or "finite sets of sequential ( cr wcel co clt wbr cdiv c1 caddc wa syl 3adant3 adantr cmul recn 3ad2ant2 wb cc crp w3a cmin cfl cfv refldivcl peano2re rerpdivcl ancoms 3adant1 1red cle 3simpa fldivle leadd1dd rpre ltaddsub syl3an2 biimpar rpcn 3ad2ant1 cc0 - 1cnd wne rpne0 divcan1d wceq mulid2d oveq12d joinlmuladdmuld breq12d mpbird + 1cnd wne rpne0 divcan1d wceq mullidd oveq12d joinlmuladdmuld breq12d mpbird 3ad2ant3 simp2 ltmul1d lelttrd ex ) ADEZBUAEZCDEZUBZACBUCFGHZABIFZUDUEZJKFZ CBIFZGHWAWBLZWEWCJKFZWFWAWEDEZWBVRVSWIVTVRVSLZWDDEZWIABUFZWDUGMNOWAWHDEZWBV RVSWMVTWJWCDEZWMABUHZWCUGZMNOWAWFDEZWBVSVTWQVRVTVSWQCBUHUIUJZOWGWDWCJWAWKWB @@ -151039,7 +151039,7 @@ The modulo (remainder) operation NM, 11-Nov-2008.) $) modfrac $p |- ( A e. RR -> ( A mod 1 ) = ( A - ( |_ ` A ) ) ) $= ( cr wcel c1 cmo co cdiv cfl cfv cmul cmin crp wceq modval mpan2 recn div1d - 1rp oveq2d eqtrd fveq2d reflcl recnd mulid2d ) ABCZADEFZADADGFZHIZJFZKFZAAH + 1rp oveq2d eqtrd fveq2d reflcl recnd mullidd ) ABCZADEFZADADGFZHIZJFZKFZAAH IZKFUEDLCUFUJMRADNOUEUIUKAKUEUIDUKJFUKUEUHUKDJUEUGAHUEAAPQUASUEUKUEUKAUBUCU DTST $. @@ -151137,8 +151137,8 @@ The modulo (remainder) operation modid $p |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A ) $= ( cr wcel wa cc0 cle wbr clt co cfl cmul cmin wceq adantr c1 ad2antlr eqtrd - cfv wb crp cmo modval caddc cc rerpdivcl recnd addid2 fveq2d rpregt0 divge0 - cdiv syl sylan2 an32s adantrr simpr rpcn mulid1d breqtrrd ad2ant2l ltdivmul + cfv wb crp cmo modval caddc cc rerpdivcl recnd addlid fveq2d rpregt0 divge0 + cdiv syl sylan2 an32s adantrr simpr rpcn mulridd breqtrrd ad2ant2l ltdivmul simpll 1re mp3an2 syl2anc mpbird cz 0z flbi2 mpbir2and eqtr3d oveq2d mul01d sylancr recn subid1d ad2antrr ) ACDZBUADZEZFAGHZABIHZEZEZABUBJZABABULJZKSZL JZMJZAWAWFWJNWDABUCOWEWJAFMJZAWEWIFAMWEWIBFLJZFWEWHFBLWEFWGUDJZKSZWHFWEWGUE @@ -151369,7 +151369,7 @@ The modulo (remainder) operation negmod $p |- ( ( A e. RR /\ N e. RR+ ) -> ( -u A mod N ) = ( ( N - A ) mod N ) ) $= ( cr wcel crp wa cmin co cneg caddc c1 cmul cc wceq rpcn recn negsub oveq1d - cmo adantr syl2anr eqcomd mulid2d adantl mulcl syl2an renegcl recnd addcomd + cmo adantr syl2anr eqcomd mullidd adantl mulcl syl2an renegcl recnd addcomd 1cnd cz simpr 1zzd modcyc syl3anc eqtrd 3eqtr2rd ) ACDZBEDZFZBAGHZBSHBAIZJH ZBSHKBLHZVBJHZBSHZVBBSHZUTVAVCBSUTVCVAUSBMDZAMDVCVANURBOZAPBAQUAUBRUTVEVCBS UTVDBVBJUSVDBNURUSBVIUCUDRRUTVFVBVDJHZBSHZVGUTVEVJBSUTVDVBURKMDVHVDMDUSURUJ @@ -151400,7 +151400,7 @@ The modulo (remainder) operation addmodid $p |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( ( M + A ) mod M ) = A ) $= ( cn0 wcel cn clt wbr w3a caddc co cmo c1 cmul wceq 3ad2ant2 oveq1d cc0 cxr - rexrd 3ad2ant1 nncn mulid2d eqcomd cz crp cico 1zzd nnrp nn0re nn0ge0 simp3 + rexrd 3ad2ant1 nncn mullidd eqcomd cz crp cico 1zzd nnrp nn0re nn0ge0 simp3 cle wb 0xr nnre elico1 sylancr mpbir3and muladdmodid syl3anc eqtrd ) ACDZBE DZABFGZHZBAIJZBKJLBMJZAIJZBKJZAVEVFVHBKVEBVGAIVEVGBVCVBVGBNVDVCBBUAUBOUCPPV ELUDDBUEDZAQBUFJDZVIANVEUGVCVBVJVDBUHOVEVKARDZQAULGZVDVBVCVLVDVBAAUISTVBVCV @@ -151781,7 +151781,7 @@ The modulo (remainder) operation ( caddc co clt wbr cc0 wcel wa cr wceq w3a adantr adantl sylbi wne syl cz wi cfzo c1 cmo csn cdif crp cle cn0 cn elfzoelz zred nn0re 3ad2ant1 readdcl elfzo0 syl2anr nnrp 3ad2ant2 jca sylanb elfzo1 nnnn0 elfzonn0 nn0ge0d simpl - nn0addcl modid syl12anc simp2 syl3anbrc zcnd 0cnd addneintr2d addid2 eqcomd + nn0addcl modid syl12anc simp2 syl3anbrc zcnd 0cnd addneintr2d addlid eqcomd cc nnne0 neeqtrrd eldifsn sylanbrc eqeltrd cmin cneg elfzoel2 mulm1d oveq2d wn cmul zaddcl negsub eqtrd oveq1d syl2an resubcld nnrpd nnre simp3 3adant3 ancoms lenltd biimprd subge0d sylibrd syl3anc anim12ci simpr jca31 ltsubadd @@ -151842,7 +151842,7 @@ The modulo (remainder) operation zcnd oveq1d eqtrd 0mod zmodidfzoimp oveq12d eqeq1d cdiv wb zsubcl mod0 cv syl2an cmul syl2an2r oveq2 elfzoel2 mul01d sylan9eq eqcom subeq0 biimtrid wrex zdiv biimpd sylbid ex wn cneg subfzo0 w3o elz pm2.24 2a1d breq1 nncn - a1d mulid1d breq2d nnre 1red ltmul2d nnge1 lenltd pm2.21 syl6bi mpd com13 + a1d mulridd breq2d nnre 1red ltmul2d nnge1 lenltd pm2.21 syl6bi mpd com13 sylbird a1dd syl6bir com15 com12 breq2 remulcl possumd oveq2d recn adddid simpr 1cnd eqtr4d peano2re remulcld 0red impcom wo nnnn0 nn0ge0d biimparc addcomd subnegd renegcl suble0d eqbrtrd sylan olcd mulle0b mpbird lensymd @@ -154066,7 +154066,7 @@ seq M ( .+ , F ) ) $= ser1const $p |- ( ( A e. CC /\ N e. NN ) -> ( seq 1 ( + , ( NN X. { A } ) ) ` N ) = ( N x. A ) ) $= ( vj vk cn wcel cc caddc c1 cmul co wceq cv wi fveq2 oveq1 eqeq12d imbi2d - cfv fvconst2g csn cxp cseq 1z 1nn mpan2 mulid2 eqtr4d seq1i wa seqp1 nnuz + cfv fvconst2g csn cxp cseq 1z 1nn mpan2 mullid eqtr4d seq1i wa seqp1 nnuz cuz eleq2s adantl peano2nn sylan2 oveq2d eqtrd nncn ax-1cn adddir syl2anr id mp3an2 adantr imbitrrid expcom a2d nnind impcom ) BEFAGFZBHEAUAUBZIUCZ SZBAJKZLZVLCMZVNSZVRAJKZLZNVLIVNSZIAJKZLZNVLDMZVNSZWEAJKZLZNVLWEIHKZVNSZW @@ -154248,7 +154248,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( ( A ^ N ) x. A ) ) $= ( cn0 wcel cc cn cc0 wceq c1 caddc co cexp cmul wa cfv sylan2 oveq2d oveq1d expnnval 3eqtr4d wo elnn0 csn cxp cseq seqp1 nnuz eleq2s peano2nn fvconst2g - cuz adantl eqtrd exp1 mulid2 eqtr4d adantr simpr 0p1e1 oveq2 exp0 sylan9eqr + cuz adantl eqtrd exp1 mullid eqtr4d adantr simpr 0p1e1 oveq2 exp0 sylan9eqr eqtrdi jaodan sylan2b ) BCDAEDZBFDZBGHZUAABIJKZLKZABLKZAMKZHZBUBVFVGVMVHVFV GNZVIMFAUCUDZIUEZOZBVPOZAMKZVJVLVNVQVRVIVOOZMKZVSVGVQWAHZVFWBBIUKOFMVOIBUFU GUHULVNVTAVRMVGVFVIFDZVTAHBUIZFAVIEUJPQUMVGVFWCVJVQHWDAVISPVNVKVRAMABSRTVFV @@ -154405,7 +154405,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ m1expcl2 $p |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } ) $= ( vx vy c1 cneg wcel cc0 wne co cc ax-1cn cmul wceq eqeltri eqeltrdi jaoi cv wo syl cdiv cpr cz negex prid1 neg1ne0 wss neg1cn prssi mp2an wi elpri - sseli mulm1d negeq negneg1e1 prid2 eqeltrd oveq1 eleq1d imbitrrid mulid2d + sseli mulm1d negeq negneg1e1 prid2 eqeltrd oveq1 eleq1d imbitrrid mullidd cexp 1ex imp oveq2 ax-1ne0 divneg2 1div1e1 negeqi eqtr3i adantr expcl2lem id mp3an mp3an12 ) DEZVPDUAZFVPGHAUBFVPAVBIVQFVPDDUCUDZUEBCVPAVQVPJFDJFZV QJUFUGKVPDJUHUIZBQZVQFZCQZVQFZWAWCLIZVQFZWBWAVPMZWADMZRZWDWFUJZWAVPDUKZWG @@ -154552,7 +154552,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ expgt1 $p |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < ( A ^ N ) ) $= ( cr wcel cn c1 clt wbr w3a cexp 1re a1i simp1 cn0 reexpcl syl2anc cmul cle co cc0 simp2 nnnn0d simp3 cmin nnm1nn0 syl wi sylancr mpd expge1 syl3anc wb - ltle 0red 0lt1 lttrd lemul1 syl112anc mpbid cc recn 3ad2ant1 mulid2d eqcomd + ltle 0red 0lt1 lttrd lemul1 syl112anc mpbid cc recn 3ad2ant1 mullidd eqcomd wceq expm1t 3brtr4d ltletrd ) ACDZBEDZFAGHZIZFAABJSZFCDZVLKLZVIVJVKMZVLVIBN DVMCDVPVLBVIVJVKUAZUBABOPVIVJVKUCZVLFAQSZABFUDSZJSZAQSZAVMRVLFWARHZVSWBRHZV LVIVTNDZFARHZWCVPVLVJWEVQBUEUFZVLVKWFVRVLVNVIVKWFUGKVPFAUMUHUIAVTUJUKVLVNWA @@ -154617,7 +154617,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ expadd $p |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) $= ( vj cc wcel cn0 caddc co cexp cmul wi cc0 c1 oveq2 oveq2d eqeq12d imbi2d - wceq eqtr4d vk wa cv nn0cn addid1d adantl expcl mulid1d exp0 adantr oveq1 + wceq eqtr4d vk wa cv nn0cn addridd adantl expcl mulridd exp0 adantr oveq1 ax-1cn addass mp3an3 syl2an adantll simpll nn0addcl expp1 syl2anc adantlr eqtr3d mulassd imbitrrid expcom a2d nn0ind expdcom 3imp ) AEFZBGFZCGFZABC HIZJIZABJIZACJIZKIZSZVLVJVKVRVJVKUBZABDUCZHIZJIZVOAVTJIZKIZSZLVSABMHIZJIZ @@ -154688,7 +154688,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) $= ( vj cc wcel cn0 cmul co cexp wceq wa wi cc0 c1 caddc oveq2 oveq2d imbi2d eqeq12d vk cv nn0cn mul01d exp0 sylan9eqr expcl eqtr4d oveq1 ax-1cn adddi - mp3an3 mulid1 adantr syl2an adantll simpll nn0mulcl simplr expadd syl3anc + mp3an3 mulrid adantr syl2an adantll simpll nn0mulcl simplr expadd syl3anc syl eqtrd expp1 sylan imbitrrid expcom a2d nn0ind expdcom 3imp ) AEFZBGFZ CGFZABCHIZJIZABJIZCJIZKZVNVLVMVSVLVMLZABDUBZHIZJIZVQWAJIZKZMVTABNHIZJIZVQ NJIZKZMVTABUAUBZHIZJIZVQWJJIZKZMVTABWJOPIZHIZJIZVQWOJIZKZMVTVSMDUACWANKZW @@ -155150,7 +155150,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ltexp2a $p |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) < ( A ^ N ) ) $= ( cr wcel cz w3a c1 clt wbr wa cexp co cmul crp rpexpcl syl2anc rpred recnd - cc0 simpl1 0red 1red 0lt1 a1i simprl lttrd elrpd simpl2 mulid2d cdiv simprr + cc0 simpl1 0red 1red 0lt1 a1i simprl lttrd elrpd simpl2 mullidd cdiv simprr cmin cn simpl3 znnsub mpbid expgt1 syl3anc wne wceq gt0ne0d expsub syl22anc wb cc breqtrd ltmuldivd mpbird eqbrtrrd ) ADEZBFEZCFEZGZHAIJZBCIJZKZKZHABLM ZNMZVSACLMZIVRVSVRVSVRVSVRAOEZVLVSOEVRAVKVLVMVQUAZVRTHAVRUBVRUCZWCTHIJVRUDU @@ -155242,7 +155242,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( A ^ M ) <_ ( A ^ N ) ) $= ( cr wcel c1 cle wbr cuz cfv cexp co crp cc0 3ad2ant3 rpexpcl syl2anc rpred cz recnd w3a cmul simp1 0red 1red clt 0lt1 simp2 ltletrd elrpd eluzel2 cdiv - a1i mulid2d cmin cn0 uznn0sub expge1 syl3anc cc wceq gt0ne0d eluzelz expsub + a1i mullidd cmin cn0 uznn0sub expge1 syl3anc cc wceq gt0ne0d eluzelz expsub wne syl22anc breqtrd lemuldivd mpbird eqbrtrrd ) ADEZFAGHZCBIJEZUAZFABKLZUB LZVOACKLZGVNVOVNVOVNVOVNAMEZBSEZVOMEVNAVKVLVMUCZVNNFAVNUDVNUEZVTNFUFHVNUGUM VKVLVMUHZUIZUJZVMVKVSVLBCUKOZABPQZRTUNVNVPVQGHFVQVOULLZGHVNFACBUOLZKLZWGGVN @@ -155272,7 +155272,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( A ^ N ) <_ ( A ^ M ) ) $= ( vj cr wcel cn0 cle wbr c1 wa cexp co wceq breq1d imbi2d reexpcl syl2anc wi oveq2 vk cuz cfv w3a cc0 cv caddc adantr leidd cmul simprll 1red simpl - simprlr eluznn0 simprrl expge0 syl3anc simprrr lemul2ad cc mulid1d eqcomd + simprlr eluznn0 simprrl expge0 syl3anc simprrr lemul2ad cc mulridd eqcomd recnd expp1 3brtr4d peano2nn0 syl ad2antrl letr mpand ex a2d uzind4i expd com12 3impia imp ) AEFZBGFZCBUBUCZFZUDUEAHIZAJHIZKZACLMZABLMZHIZVSVTWBWEW HSZWBVSVTKZWIWBWJWEWHWJWEKZADUFZLMZWGHIZSWKWGWGHIZSWKAUAUFZLMZWGHIZSWKAWP @@ -155499,7 +155499,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ integers, see ~ zzlesq . (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.) $) nnlesq $p |- ( N e. NN -> N <_ ( N ^ 2 ) ) $= - ( cn wcel cmul co c2 cexp cle c1 nncn mulid1d wbr nnge1 cr cc0 wb 1red nnre + ( cn wcel cmul co c2 cexp cle c1 nncn mulridd wbr nnge1 cr cc0 wb 1red nnre clt nngt0 lemul2 syl112anc mpbid eqbrtrrd cc wceq sqval syl breqtrrd ) ABCZ AAADEZAFGEZHUJAIDEZAUKHUJAAJZKUJIAHLZUMUKHLZAMUJINCANCZUQOASLUOUPPUJQARZURA TIAAUAUBUCUDUJAUECULUKUFUNAUGUHUI $. @@ -155633,7 +155633,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ 11-May-2014.) $) binom21 $p |- ( A e. CC -> ( ( A + 1 ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. A ) ) + 1 ) ) $= - ( cc wcel c1 caddc co c2 cexp cmul wceq ax-1cn binom2 mulid1 oveq2d sq1 a1i + ( cc wcel c1 caddc co c2 cexp cmul wceq ax-1cn binom2 mulrid oveq2d sq1 a1i mpan2 oveq12d eqtrd ) ABCZADEFGHFZAGHFZGADIFZIFZEFZDGHFZEFZUBGAIFZEFZDEFTDB CUAUGJKADLQTUEUIUFDETUDUHUBETUCAGIAMNNUFDJTOPRS $. @@ -155653,7 +155653,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ 2-Aug-2021.) $) binom2sub1 $p |- ( A e. CC -> ( ( A - 1 ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. A ) ) + 1 ) ) $= - ( cc wcel c1 cmin co cexp cmul caddc wceq binom2sub mpdan mulid1 oveq2d sq1 + ( cc wcel c1 cmin co cexp cmul caddc wceq binom2sub mpdan mulrid oveq2d sq1 c2 1cnd a1i oveq12d eqtrd ) ABCZADEFPGFZAPGFZPADHFZHFZEFZDPGFZIFZUCPAHFZEFZ DIFUADBCUBUHJUAQADKLUAUFUJUGDIUAUEUIUCEUAUDAPHAMNNUGDJUAORST $. @@ -155687,7 +155687,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( cc wcel caddc co c3 cexp cmul df-3 wceq sylancl syl oveq1d adddird oveq2d c2 c1 eqtr4d oveq12d wa oveq2i cn0 addcl 2nn0 expp1 eqtrid sqcl simpl simpr adddid binom2 mulcl sylancr addcld sqval mul32d 2cnd mulassd mulcomd 3eqtrd - 2cn eqtrd mulcld 3nn0 expcl addassd eqtr3d add4d 1cnd mulid2d 1p2e3 eqtr3id + 2cn eqtrd mulcld 3nn0 expcl addassd eqtr3d add4d 1cnd mullidd 1p2e3 eqtr3id oveq1i eqtr2d ) ACDZBCDZUAZABEFZGHFZVSQHFZVSIFZAGHFZQAQHFZBIFZIFZEFZWEEFZAB QHFZIFZQWJIFZBGHFZEFZEFZEFZWCGWEIFZEFZGWJIFZWLEFZEFVRVTVSQREFZHFZWBGWTVSHJU BVRVSCDZQUCDZXAWBKABUDZUEVSQUFLUGVRWBWAAIFZWABIFZEFWGWJEFZWEWMEFZEFWOVRWAAB @@ -155711,7 +155711,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ NM, 6-Jun-2006.) $) sq01 $p |- ( A e. CC -> ( ( A ^ 2 ) = A <-> ( A = 0 \/ A = 1 ) ) ) $= ( cc wcel c2 cexp co wceq cc0 c1 wo wa wn df-ne cmul wb sqval oveq1 3eqtr4a - wne id mulid1 eqcomd eqeq12d adantr ax-1cn mulcan mp3an2 impancom biimtrrid + wne id mulrid eqcomd eqeq12d adantr ax-1cn mulcan mp3an2 impancom biimtrrid anabss5 bitrd biimpd orrd ex sq0 sq1 jaoi impbid1 ) ABCZADEFZAGZAHGZAIGZJZU SVAVDUSVAKZVBVCVBLAHSZVEVCAHMUSVFVAVCUSVFKZVAVCVGVAAANFZAINFZGZVCUSVAVJOVFU SUTVHAVIAPUSVIAAUAUBUCUDUSVFVJVCOZUSIBCVGVKUEAIAUFUGUJUKULUHUIUMUNVBVAVCVBH @@ -155753,7 +155753,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( ( A - ( _i x. B ) ) / ( ( A ^ 2 ) + ( B ^ 2 ) ) ) ) $= ( cc0 wne c1 ci cmul co caddc cdiv c2 cexp wceq cneg sqcli ax-1cn wcel cc wo cmin recni ax-icn mulcli negsubi sqmuli oveq1i mulneg1i 3eqtri negnegi - i2 negeqi mulid2i oveq2i 3eqtr3ri wa wn neorian cr wb sumsqeq0 necon3bbii + i2 negeqi mullidi oveq2i 3eqtr3ri wa wn neorian cr wb sumsqeq0 necon3bbii subsqi mp2an bitri addcli subcli divasszi sylbi divid 3eqtr3a divclzi a1i mpan crne0 biimpi divmul mp3an1 syl12anc mpbird ) AEFBEFUAZGAHBIJZKJZLJAW CUBJZAMNJZBMNJZKJZLJZOZWDWIIJZGOZWBWDWEIJZWHLJZWHWHLJZWKGWMWHWHLWFWCMNJZP @@ -155777,7 +155777,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ 1re simpl syl2anc remulcld reexpcl anidms msqge0 jca nn0ge0 mulge0 syl2an sylan nn0cn adantl mul32d breqtrd addge01d mulcld addassd muladd11 eqtr4d mpbid mulcl wb neg1rr leadd2 mp3an13 1pneg1e0 breq1i bitrdi biimpa simprr - ad2ant2r letrd adddi mp3an3 mulid1 eqtrd addass mp3an2i expp1 exp43 com12 + ad2ant2r letrd adddi mp3an3 mulrid eqtrd addass mp3an2i expp1 exp43 com12 lemul1ad 3brtr4d impd a2d nn0ind expd 3imp21 ) BUCCZADCZEUDZAFGZEABHIZJIZ EAJIZBKIZFGZXNXOXQYBXOXQLZEAUAUEZHIZJIZXTYDKIZFGZMYCEASHIZJIZXTSKIZFGZMYC EAUBUEZHIZJIZXTYMKIZFGZMYCEAYMEJIZHIZJIZXTYRKIZFGZMYCYBMUAUBBYDSNZYHYLYCU @@ -156094,7 +156094,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ wral 3eqtrd eqtr4d divcan2d mulneg2d negeqd negsubd addsubd breqtrd mpbid readdcld subge0d eqbrtrd leadd2d mpbird suble0d resubcld ledivmuld mul01d 0red wn c1 peano2re ltnegd df-neg breqtrdi ltaddsubd expr simprr redivcld - ltp1d simprl sqcl mul02d addid2d wb 0re lenlt cif pm2.65d nne sylib sq0id + ltp1d simprl sqcl mul02d addlidd wb 0re lenlt cif pm2.65d nne sylib sq0id mul01i 0m0e0 0le0 eqbrtri eqbrtrdi wo eqid discr1 leloe mpjaodan ) AJCUFK ZDLUAMZNCEOMZOMZUBMZJPKJCUGZAUUOUCZUUSNCOMZJOMZJPUVAUUSUVBQMZJPKUUSUVCPKU VAUVDUUPUVBQMZEUBMZJPUVAUVDUVEUURUVBQMZUBMUVFUVAUUPUURUVBUVAUUPUVADRSZUUP @@ -156166,7 +156166,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( wcel c2 cmul co c1 caddc wceq cexp cmin c8 cdiv c4 a1i zcnd oveq1d eqcomd cc eqtrd cz wa oveq1 2z id zmulcld binom21 syl sylan9eqr zcn sqmuld sq2 w3a mulass syl3anc 2t2e4 oveq12d 4z zsqcl addcld pncan1 adantr 4cn adddid 4t2e8 - 2cnd oveq2d cc0 wne zaddcld 2cnne0 4ne0 pm3.2i divcan5 sqvald mulid1d adddi + 2cnd oveq2d cc0 wne zaddcld 2cnne0 4ne0 pm3.2i divcan5 sqvald mulridd adddi 1cnd 3eqtrd ) BUACZADBEFZGHFZIZUBZADJFZGKFZLMFNBDJFZEFZNBEFZHFZLMFZNWGBHFZE FZLMFZBBGHFEFZDMFZWDWFWJLMWDWFWADJFZDWAEFZHFZGHFZGKFZWJWDWEWTGKWCVTWEWBDJFZ WTAWBDJUCVTWASCXBWTIVTWAVTDBDUACVTUDOVTUEZUFPWAUGUHUIQVTXAWJIWCVTXAWJGHFZGK @@ -156402,7 +156402,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ nn0opthlem1 $p |- ( A < C <-> ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) ) $= ( c1 caddc co cle wbr cmul clt c2 nn0addcli wcel wb nn0mulcli cexp ax-1cn cn0 sqvali 1nn0 nn0le2msqi nn0ltp1le nn0cni binom2i addcli oveq1i oveq12i - mp2an 2nn0 3eqtr3i mulid1i oveq2i eqtri breq1i bitr4i 3bitr4i ) AEFGZBHIZ + mp2an 2nn0 3eqtr3i mulridi oveq2i eqtri breq1i bitr4i 3bitr4i ) AEFGZBHIZ URURJGZBBJGZHIZABKIZAAJGZLAJGZFGZVAKIZURBAECUAMDUBASNBSNVCUSOCDABUCUIVGVF EFGZVAHIZVBVFSNVASNVGVIOVDVEAACCPLAUJCPMBBDDPVFVAUCUIUTVHVAHUTVDLAEJGZJGZ FGZEEJGZFGZVHURLQGALQGZVKFGZELQGZFGUTVNAEACUDZRUEURAEVRRUFTVPVLVQVMFVOVDV @@ -156527,7 +156527,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ $( The factorial of 2. (Contributed by NM, 17-Mar-2005.) $) fac2 $p |- ( ! ` 2 ) = 2 $= ( c2 cfa cfv c1 caddc df-2 fveq2i cmul cn0 wcel wceq 1nn0 facp1 ax-mp 1p1e2 - co fac1 oveq12i 2cn eqtri mulid2i ) ABCDDEPZBCZAAUBBFGUCDBCZUBHPZADIJUCUEKL + co fac1 oveq12i 2cn eqtri mullidi ) ABCDDEPZBCZAAUBBFGUCDBCZUBHPZADIJUCUEKL DMNUEDAHPAUDDUBAHQORASUATTT $. $( The factorial of 3. (Contributed by NM, 17-Mar-2005.) $) @@ -156652,14 +156652,14 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( M ^ ( N + 1 ) ) <_ ( ( M ^ M ) x. ( ! ` N ) ) ) $= ( cn0 wcel cc0 wceq c1 caddc co cexp cfa cfv cmul cle wbr wi oveq2d wa cr adantr vj vk cn wo elnn0 oveq1 fveq2 breq12d imbi2d nnre nnge1 cuz elnnuz - biimpi leexp2ad 0p1e1 oveq2i a1i fac0 nnnn0 reexpcld recnd mulid1d eqtrid + biimpi leexp2ad 0p1e1 oveq2i a1i fac0 nnnn0 reexpcld recnd mulridd eqtrid cv 3brtr4d clt ad3antrrr simpllr peano2nn0 faccld nnred remulcld peano2re syl nn0re 3syl 0re ltle mpan sylc expge0d simplr simprr lemul12ad anandis nngt0 cc nncn expp1 syl2an facp1 adantl faccl nncnd nn0cn peano2cn eqtr4d mulassd exp32 com23 wb syl2anr reexpcl ad2antrr remulcl simpr cz ad2antlr nn0ltp1le nn0zd nnz eluz syl2anc mpbird anim12i id nn0ge0 lemulge11 letrd nnge1d ex sylbid a1dd lelttric mpjaod expcom nn0ind impcom nnnn0d nn0ge0d - a2d nn0p1nn 0expd 0exp0e1 oveq1i mulid2d oveq12 anidms oveq1d imp jaoian + a2d nn0p1nn 0expd 0exp0e1 oveq1i mullidd oveq12 anidms oveq1d imp jaoian imbitrrid sylanb ) ACDZAUCDZAEFZUDBCDZABGHIZJIZAAJIZBKLZMIZNOZAUEUUFUUHUU NUUGUUHUUFUUNUUFAUAVEZGHIZJIZUUKUUOKLZMIZNOZPUUFAEGHIZJIZUUKEKLZMIZNOZPUU FAUBVEZGHIZJIZUUKUVFKLZMIZNOZPUUFAUVGGHIZJIZUUKUVGKLZMIZNOZPUUFUUNPUAUBBU @@ -156696,7 +156696,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ faclbnd2 $p |- ( N e. NN0 -> ( ( 2 ^ N ) / 2 ) <_ ( ! ` N ) ) $= ( cn0 wcel c2 cexp co cdiv c1 cle cmul c4 sq2 oveq2i cc 2cn mpan wa cc0 wbr cr caddc cfa cfv 2t2e4 eqtr4i wceq expp1 oveq1d eqtrid wne 2cnne0 divmuldiv - expcl mpanr12 sylancl 2div2e1 halfcld mulid1d 3eqtr2rd faclbnd wb peano2nn0 + expcl mpanr12 sylancl 2div2e1 halfcld mulridd 3eqtr2rd faclbnd wb peano2nn0 2nn0 2re reexpcl sylancr faccl nnred clt 4re eqeltri 4pos breqtrri ledivmul pm3.2i mp3an3 syl2anc mpbird eqbrtrd ) ABCZDAEFZDGFZDAHUAFZEFZDDEFZGFZAUBUC ZIVTWFWADJFZDDJFZGFZWBDDGFZJFZWBVTWFWDWIGFWJWEWIWDGWEKWILUDUEMVTWDWHWIGDNCZ @@ -156739,7 +156739,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ 0le1 cn0 2re nn0nnaddcl nnnn0i 2nn0 nn0expcli reexpcl nn0rei nn0ge0i exp0 1nn cc 2cn cuz 1le2 nn0uz eleqtri leexp2a mp3an eqbrtrri elnn0 nncn exp1d nnuz mp3an13 syl eqbrtrrd breq1 mpbiri jaoi sylbi lemul12a mp2 oveq1 nnzi - cz 1exp eqtrdi oveq2 nn0cni exp1 oveq12d fveq2 fac1 oveq2d mulcli mulid1i + cz 1exp eqtrdi oveq2 nn0cni exp1 oveq12d fveq2 fac1 oveq2d mulcli mulridi recni breq12d sylbir adantr a1i faccl nn0mulcli nncni expm1t oveq12i mpbi expp1 elnnnn0 simpri mulassi eqtri nn0addcli w3a 3pm3.2i wb nnltp1le df-2 breq1i bitr4i expubnd lemul1a mul4i oveq2i 3eqtr2i breqtrdi ax-1cn eqtr3i @@ -156822,7 +156822,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ wbr cfa cfv wo elnn0 0exp nnnn0 nn0sqcl nn0expcl sylancr nn0addcl nn0mulcld 2nn0 syldan sylan2 nn0ge0d eqbrtrd 0nn0 sylancl nnexpcl mpdan id oveq1 00id 1nn oveq12d 0exp0e1 eqeltrdi jaoi sylbi nnmulcl nnge1d adantr wb oveq2 sq0i - cc exp0 ax-mp breq12d mpbird jaodan sylan2b nn0cn exp0d mpan nn0cnd mulid1d + cc exp0 ax-mp breq12d mpbird jaodan sylan2b nn0cn exp0d mpan nn0cnd mulridd 2cn sylan9eq 3brtr4d fveq2 fac0 ) BDEZADEZFZCGHZFCAIJZBCIJZKJZLALIJZIJZBBAM JZIJZKJZCUAUBZKJZNTZGAIJZBGIJZKJZXDOKJZNTZWOXLWPWOXHXDXJXKNWNWMAPEZAGHZUCXH XDNTZAUDWMXMXOXNWMXMFZXHGXDNXMXHGHWMAUEQXPXDXMWMWNXDDEAUFWOXAXCWNXADEZWMWNL @@ -156846,8 +156846,8 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ cmin wo cr nnre lelttric sylancl ancli andi sylib nnge1 wb letri3 biimpar 1re anassrs mpidan 1m1e0 eqtrdi syl faclbnd4lem3 sylan2 nnsub mpan biimpa a1d rspcv adantl jaodan faclbnd4lem2 3expa syld ralrimdva biimtrid expcom - 1nn a2d nnnn0 faclbnd3 nncn exp0d cc nn0cn expcl syl2an mulid2d eqtrd sq0 - oveq2i 2cn exp0 ax-mp eqtri a1i addid1d mpancom 3brtr4d ralrimiva ralbidv + 1nn a2d nnnn0 faclbnd3 nncn exp0d cc nn0cn expcl syl2an mullidd eqtrd sq0 + oveq2i 2cn exp0 ax-mp eqtri a1i addridd mpancom 3brtr4d ralrimiva ralbidv adantr imbi2d nn0indALT imp rspcva 3impb ) CEFZAUCFZBUCFZCAGHZBCGHZIHZJAJ GHZGHZBBAKHZGHZIHZCUDUEZIHZLMZXTYANXSDUFZAGHZBYMGHZIHZYIYMUDUEZIHZLMZDEUG ZYLXTYAYTYAYMUAUFZGHZYOIHZJUUAJGHZGHZBBUUAKHZGHZIHZYQIHZLMZDEUGZUHYAYMTGH @@ -156917,7 +156917,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( ( ! ` N ) x. ( ( N + 1 ) ^ M ) ) <_ ( ! ` ( N + M ) ) ) $= ( wcel cfa cfv c1 caddc co cexp cmul cle wbr cc0 wceq oveq2 oveq2d fveq2d wa cr adantr vm vk cn0 cv breq12d weq faccl nnred leidd cc nn0cn peano2cn - syl exp0d nncnd mulid1d addid1d 3brtr4d peano2nn0 nn0red reexpcl remulcld + syl exp0d nncnd mulridd addridd 3brtr4d peano2nn0 nn0red reexpcl remulcld eqtrd sylan nnnn0 nn0ge0d simpr expge0d mulge0d jca nn0addcl faccld nn0re cn readdcl syl2an jca31 nn0ge0 adantl wb 0re leadd2 mpbid eqbrtrrd leadd1 mp3an2i mp3an3 syl2anc ax-1cn addass breqtrd anim1ci lemul12a expp1 expcl @@ -156965,8 +156965,8 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( cn0 wcel wa co cle wbr cfv cfa cmul cr syl adantr letr syl3anc cc0 adantl wi c1 caddc c2 cdiv nn0readdcl rehalfcld flle reflcl nn0re nn0addcl nn0ge0d cfl mpand wb halfnneg2 mpbid flge0nn0 syl2anc simpl facwordi 3exp sylc wceq - faccl nncnd mulid1d nnnn0d jca nnge1d 1re lemul2a mp3anl1 syl21anc eqbrtrrd - nnred faccld remulcl syl2an mpan2d 3syld simpr mulid2d lemul1a avgle mpjaod + faccl nncnd mulridd nnnn0d jca nnge1d 1re lemul2a mp3anl1 syl21anc eqbrtrrd + nnred faccld remulcl syl2an mpan2d 3syld simpr mullidd lemul1a avgle mpjaod wo ) ACDZBCDZEZABUAFZUBUCFZAGHZWJUKIZJIZAJIZBJIZKFZGHZWJBGHZWHWKWLAGHZWMWNG HZWQWHWLWJGHZWKWSWHWJLDZXAWHWIABUDZUEZWJUFMZWHWLLDZXBALDZXAWKEWSSWHXBXFXDWJ UGMZXDWFXGWGAUHZNWLWJAOPULWHWLCDZWFWSWTSWHXBQWJGHZXJXDWHQWIGHZXKWHWIABUIUJW @@ -157084,7 +157084,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) $) bcn0 $p |- ( N e. NN0 -> ( N _C 0 ) = 1 ) $= ( cn0 wcel cc0 cbc co cfa cfv cmin cmul cdiv c1 cfz wceq 0elfz bcval2 nn0cn - syl subid1d eqtrd fveq2d oveq12 sylancl faccl mulid1d oveq2d facne0 dividd + syl subid1d eqtrd fveq2d oveq12 sylancl faccl mulridd oveq2d facne0 dividd fac0 nncnd ) ABCZADEFZAGHZADIFZGHZDGHZJFZKFZLUKDDAMFCULURNAODAPRUKURUMUMKFL UKUQUMUMKUKUQUMLJFZUMUKUOUMNUPLNUQUSNUKUNAGUKAAQSUAUIUOUMUPLJUBUCUKUMUKUMAU DUJZUETUFUKUMUTAUGUHTT $. @@ -157107,7 +157107,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ bcn1 $p |- ( N e. NN0 -> ( N _C 1 ) = N ) $= ( cn0 wcel cn cc0 wceq wo c1 cbc co elnn0 cfa cfv cmin cmul cfz cuz clt wbr cdiv 1eluzge0 a1i elnnuz biimpi elfzuzb sylanbrc bcval2 facnn2 fac1 nnm1nn0 - oveq2i faccld nncnd mulid1d eqtrid oveq12d nncn nnne0d divcan3d 3eqtrd 0nn0 + oveq2i faccld nncnd mulridd eqtrid oveq12d nncn nnne0d divcan3d 3eqtrd 0nn0 syl cz 1z 0lt1 olci bcval4 mp3an wb oveq1 eqeq12 mpancom mpbiri jaoi sylbi ) ABCADCZAEFZGAHIJZAFZAKVPVSVQVPVRALMZAHNJZLMZHLMZOJZTJZWBAOJZWBTJAVPHEAPJC ZVRWEFVPHEQMCZAHQMCZWGWHVPUAUBVPWIAUCUDHEAUEUFHAUGVBVPVTWFWDWBTAUHVPWDWBHOJ @@ -157193,7 +157193,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ simplr elfzuz3 eluznn adantrr cr crp nnre nnrp ltsubrp syl2an cz nnzd nnz wb ad2antlr zsubcld zltp1le peano2zd eluz mpbird simprr nnuz eleqtrdi fvi mpbid elfzelz zcnd eqeltrd seqsplit facnn syl oveq1d 3eqtr4d simpll faccl - expr nncn 3syl mulid2d oveq2d eqtr3d fveq2 fac0 eqtrdi oveq1 0p1e1 fveq1d + expr nncn 3syl mullidd oveq2d eqtr3d fveq2 fac0 eqtrdi oveq1 0p1e1 fveq1d seqeq1d oveq12d eqeq2d syl5ibrcom fznn0sub elnn0 sylib mpjaod eqid zsubcl wo nn0z adantr eluzelcn seqf ffvelcdmd elfznn0 faccld nncnd nnne0d 3eqtrd divcan5d nnnn0 nnne0 div0d mul02 mul01 ad2antrr simpr nn0uz elfz5 subge0d @@ -157868,7 +157868,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) $= ( com wcel coa co cfv caddc wceq wi c0 oveq2 fveq2d oveq2d cc0 cn0 c1 a1i vn vz cv csuc fveq2 eqeq12d imbi2d wf1o wf hashgf1o f1of ffvelcdmi nn0cnd - ax-mp addid1d 0z om2uz0i oveq2i nna0 3eqtr4rd w3a wa nnasuc om2uzsuci syl + ax-mp addridd 0z om2uz0i oveq2i nna0 3eqtr4rd w3a wa nnasuc om2uzsuci syl nnacl eqtrd 3adant3 cc ax-1cn addass mp3an3 syl2an oveq1 3ad2ant3 3eqtr4d 3ad2ant2 3expia expcom a2d finds impcom ) CFGBFGZBCHIZDJZBDJZCDJZKIZLZWDB UBUDZHIZDJZWGWKDJZKIZLZMWDBNHIZDJZWGNDJZKIZLZMWDBUCUDZHIZDJZWGXBDJZKIZLZM @@ -158524,7 +158524,7 @@ are used instead of sets because their representation is shorter (and more 0xp fveq2i eqtri 3eqtr4ri wn wa oveq1 adantl xpundir cin xpfi inxp disjsn mpan2 biimpri xpeq1d eqtrdi eqtrid snfi hashun mp3an2 syl2an cen wbr snex mp2an elexi xpcomen vex xpsnen entri wb hashen mpbir oveq2i adantr cvv wi - hashunsng cc ax-1cn cn0 nn0cn adddir mp3an23 syl mulid2i eqtrd 3eqtr4d ex + hashunsng cc ax-1cn cn0 nn0cn adddir mp3an23 syl mullidi eqtrd 3eqtr4d ex mp2b findcard2s ) UAUDZBDZEFZXEEFZBEFZGHZIJBDZEFZJEFZXIGHZIUBUDZBDZEFZXOE FZXIGHZIZXOUCUDZUEZUFZBDZEFZYCEFZXIGHZIZABDZEFZAEFZXIGHZIUAUBUCAXEJIZXGXL XJXNYMXFXKEXEJBKLYMXHXMXIGXEJEMNOXEXOIZXGXQXJXSYNXFXPEXEXOBKLYNXHXRXIGXEX @@ -159045,7 +159045,7 @@ are used instead of sets because their representation is shorter (and more simpr sylib cen wbr cvv hasheni c1 a2d wne wex abn0 simprl simpll exlimiv necon1bi unssbd snss pncan2d cdif wf1o f1f1orn f1oen3g sylancr hashf1lem1 crn cle oveq12d frnd diffi disjdif a1i undif 3eqtr2d oveq1d eqtr3d sylan2 - eqtrd hashunsng elv ad2antrl simprll 1cnd adddid mulid1d 3eqtrd expr syl5 + eqtrd hashunsng elv ad2antrl simprll 1cnd adddid mulridd 3eqtrd expr syl5 expcom findcard2s mpcom mpi ) ACDEUDZUEZEUFZUWFJZCBUDZUGZUHZDUWDUEZEUFZKL ZDKLZCKLZUIMZUWFKLZNMZOZUWFUJUWFPQAUWGUWSRZADCUKMZUWFADPQZCPQZUXAPQGFDCUL UMAUWEEUXAUWEUWDUXAQACDUWDVACDUWDUNADCUWDPPGFUOUPUQURAUAUDZUWFJZUWDCUSZUX @@ -159158,7 +159158,7 @@ are used instead of sets because their representation is shorter (and more ( ! ` ( # ` A ) ) ) $= ( cfn wcel wf1 cab chash cfv cfa cbc cmul wf1o wceq hashf1 anidms cen wbr cv co c1 enrefg f1finf1o mpancom abbidv fveq2d cn0 hashcl bcnn syl oveq2d - wb faccld nncnd mulid1d eqtrd 3eqtr3d ) ACDZAABRZEZBFZGHZAGHZIHZVBVBJSZKS + wb faccld nncnd mulridd eqtrd 3eqtr3d ) ACDZAABRZEZBFZGHZAGHZIHZVBVBJSZKS ZAAURLZBFZGHVCUQVAVEMAABNOUQUTVGGUQUSVFBAAPQUQUSVFUKACUAAAURUBUCUDUEUQVEV CTKSVCUQVDTVCKUQVBUFDVDTMAUGZVBUHUIUJUQVCUQVCUQVBVHULUMUNUOUP $. $} @@ -160963,7 +160963,7 @@ computer programs (as last() or lastChar()), the terminology used for Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 1-May-2020.) $) ccatlid $p |- ( S e. Word B -> ( (/) ++ S ) = S ) $= ( vx cword wcel cc0 chash cfv cfzo co c0 cconcat wfn caddc wrd0 ccatvalfn - hash0 eqtrid cmin wceq mpan oveq1i nn0cnd addid2d eqcomd oveq2d fneq2d cv + hash0 eqtrid cmin wceq mpan oveq1i nn0cnd addlidd eqcomd oveq2d fneq2d cv lencl mpbird wrdfn wa a1i oveq12d eleq2d ccatval2 mp3an1 syldan oveq2i cz biimpar elfzoelz adantl zcnd subid1d fveq2d eqtrd eqfnfvd ) BADZEZCFBGHZI JZKBLJZBVJVMVLMVMFKGHZVKNJZIJZMZKVIEZVJVQAOZKBAPUAVJVLVPVMVJVKVOFIVJVOVKV @@ -160976,7 +160976,7 @@ computer programs (as last() or lastChar()), the terminology used for Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 1-May-2020.) $) ccatrid $p |- ( S e. Word B -> ( S ++ (/) ) = S ) $= ( vx cword wcel cc0 chash cfv cfzo co c0 cconcat wfn caddc wrd0 ccatvalfn - mpan2 hash0 oveq2i lencl nn0cnd addid1d eqtr2id oveq2d fneq2d mpbird wceq + mpan2 hash0 oveq2i lencl nn0cnd addridd eqtr2id oveq2d fneq2d mpbird wceq wrdfn cv ccatval1 mp3an2 eqfnfvd ) BADZEZCFBGHZIJZBKLJZBUNUQUPMUQFUOKGHZN JZIJZMZUNKUMEZVAAOZBKAPQUNUPUTUQUNUOUSFIUNUSUOFNJUOURFUONRSUNUOUNUOABTUAU BUCUDUEUFABUHUNVBCUIZUPEVDUQHVDBHUGVCAABKVDUJUKUL $. @@ -161668,7 +161668,7 @@ computer programs (as last() or lastChar()), the terminology used for /\ L e. ( 0 ... ( # ` S ) ) ) -> ( ( S substr <. F , L >. ) ` 0 ) = ( S ` F ) ) $= ( cword wcel cc0 cfzo chash cfv cfz w3a cop csubstr caddc cmin wceq elfzofz - co 3ad2ant2 3anim2i cn fzonnsub lbfzo0 sylibr syl2anc elfzoelz zcnd addid2d + co 3ad2ant2 3anim2i cn fzonnsub lbfzo0 sylibr syl2anc elfzoelz zcnd addlidd swrdfv fveq2d eqtrd ) BAEFZCGDHSFZDGBIJKSFZLZGBCDMNSJZGCOSZBJZCBJZUPUMCGDKS FZUOLGGDCPSZHSFZUQUSQUNVAUMUOCGDRUAUPVBUBFZVCUNUMVDUOCGDUCTVBUDUEABCDGUJUFU NUMUSUTQUOUNURCBUNCUNCCGDUGUHUIUKTUL $. @@ -161927,7 +161927,7 @@ computer programs (as last() or lastChar()), the terminology used for 2thd wn swrdsbslen biantrurd nn0re clt ltnle ltle sylbird cmin wsbc caddc cr cuz simpl1l simpl2l anim12i anim1i df-3an sylibr eluz2 simpl3l syl3anc jca swrdlen2 oveq2d 0zd zsubcl syl2anr adantr fzoshftral cc nn0cn oveq12d - addid2 npcan ovex csb sbceqg csbfv2g csbvarg fveq2d eqtrd bitrd mp1i 3jca + addlid npcan ovex csb sbceqg csbfv2g csbvarg fveq2d eqtrd bitrd mp1i 3jca eqeq12d swrdfv2 sylan simpl1r simpl3r ralbidva 3bitrd 3bitr2d pm2.61ian syld ) DCHIZFEUDZJZAYEJZKZCUAJZDUAJZKZDFUELHIZDAUELHIZKZUBZFCDUFZUGMZAYPU GMZNZBUCZFLZYTALZNZBCDOMZPZQYDYOKYSUUEYDYHYKYSYNYHYKKYDYSYHYKYDYSACDEFUHU @@ -161958,7 +161958,7 @@ computer programs (as last() or lastChar()), the terminology used for ( cword wcel cc0 chash cfv cfzo co c1 caddc cs1 wceq cfz adantl cz cc eqtrd cuz wa cop csubstr swrdcl cmin simpl elfzouz elfzoelz uzid peano2uz elfzuzb 3syl sylanbrc fzofzp1 swrdlen syl3anc zcnd ax-1cn sylancl eqs1 syl2an2r csn - pncan2 0z snidg ax-mp oveq2d eqtrdi eleqtrrid swrdfv syl31anc addid2 eqcomd + pncan2 0z snidg ax-mp oveq2d eqtrdi eleqtrrid swrdfv syl31anc addlid eqcomd fzo01 syl fveq2d eqtr4d s1eqd ) CADZEZBFCGHZIJEZUAZCBBKLJZUBUCJZFWEHZMZBCHZ MVTWEVSEWBWEGHZKNWEWGNACBWDUDWCWIWDBUEJZKWCVTBFWDOJEZWDFWAOJEZWIWJNVTWBUFZW CBFTHEZWDBTHZEZWKWBWNVTBFWAUGPWCBQEZBWOEWPWBWQVTBFWAUHPZBUIBBUJULBFWDUKUMZW @@ -162127,7 +162127,7 @@ computer programs (as last() or lastChar()), the terminology used for -> ( S prefix L ) = ( x e. ( 0 ..^ L ) |-> ( S ` x ) ) ) $= ( cword wcel cc0 chash cfv cfz co wa cpfx cfzo cmpt cn0 wceq adantl nn0cn syl cop csubstr cv caddc elfznn0 pfxval sylan2 simpl 0elfz simpr swrdval2 - cmin syl3anc subid1d oveq2d elfzonn0 addid1d fveq2d mpteq12dva 3eqtrd ) C + cmin syl3anc subid1d oveq2d elfzonn0 addridd fveq2d mpteq12dva 3eqtrd ) C BEZFZDGCHIZJKFZLZCDMKZCGDUAUBKZAGDGULKZNKZAUCZGUDKZCIZOZAGDNKZVJCIZOVDVBD PFZVFVGQDVCUEZCDVAUFUGVEVBGGDJKFZVDVGVMQVBVDUHVEVPVRVDVPVBVQRDUITVBVDUJAB CGDUKUMVEAVIVLVNVOVDVIVNQVBVDVHDGNVDVPVHDQVQVPDDSUNTUORVJVIFZVLVOQVEVSVKV @@ -162169,7 +162169,7 @@ computer programs (as last() or lastChar()), the terminology used for -> ( ( W prefix L ) ` I ) = ( W ` I ) ) $= ( cword wcel cc0 chash cfv cfz co cfzo w3a cpfx cop csubstr caddc wceq cn0 wi elfznn0 pfxval sylan2 3adant3 fveq1d cmin simp1 0elfz syl 3ad2ant2 simp2 - nn0cnd subid1d eqcomd oveq2d eleq2d biimpd a1i 3imp swrdfv syl31anc addid1d + nn0cnd subid1d eqcomd oveq2d eleq2d biimpd a1i 3imp swrdfv syl31anc addridd elfzoelz zcnd 3ad2ant3 fveq2d 3eqtrd ) DCEZFZBGDHIZJKFZAGBLKZFZMZADBNKZIADG BOPKZIZAGQKZDIZADIVNAVOVPVIVKVOVPRZVMVKVIBSFZVTBVJUAZDBVHUBUCUDUEVNVIGGBJKF ZVKAGBGUFKZLKZFZVQVSRVIVKVMUGVKVIWCVMVKWAWCWBBUHUIUJVIVKVMUKVIVKVMWFVKVMWFT @@ -162577,7 +162577,7 @@ computer programs (as last() or lastChar()), the terminology used for = ( W substr <. M , ( M + L ) >. ) ) ) $= ( cword wcel cc0 cfz co cop csubstr caddc wceq cvv cn0 elfznn0 3ad2ant3 syl wa chash cfv w3a cmin cpfx ovexd pfxval syl2an fznn0sub anim1i swrdswrd imp - 0elfz syldan nn0cn addid1d adantr opeq1d oveq2d 3eqtrd ex ) EDFGZCHEUAUBIJG + 0elfz syldan nn0cn addridd adantr opeq1d oveq2d 3eqtrd ex ) EDFGZCHEUAUBIJG ZBHCIJGZUCZAHCBUDJZIJZGZEBCKZLJZAUEJZEBBAMJZKZLJZNVEVHTZVKVJHAKLJZEBHMJZVLK ZLJZVNVEVJOGAPGVKVPNVHVEEVILUFAVFQVJAOUGUHVEVHHVGGZVHTZVPVSNZVEVTVHVEVFPGZV TVDVBWCVCBHCUIRVFUMSUJVEWAWBHABCDEUKULUNVOVRVMELVOVQBVLVEVQBNZVHVDVBWDVCVDB @@ -162591,7 +162591,7 @@ computer programs (as last() or lastChar()), the terminology used for = ( W substr <. K , L >. ) ) ) $= ( wcel cc0 cfz co wa cop csubstr wceq caddc adantr syl adantl oveq2d eleq2d elfzelz cword chash cfv cpfx cn0 elfznn0 anim2i pfxval w3a cmin simpl simpr - oveq1d 0elfz 3jca cz subid1d eqcomd anbi12d biimpa swrdswrd sylc cc addid2d + oveq1d 0elfz 3jca cz subid1d eqcomd anbi12d biimpa swrdswrd sylc cc addlidd zcn zcnd opeq12d 3eqtrd ex ) EDUAZFZCGEUBUCZHIFZJZAGCHIZFZBACHIZFZJZECUDIZA BKZLIZEWALIZMVNVSJZWBEGCKLIZWALIZEGANIZGBNIZKZLIZWCWDVTWEWALWDVKCUEFZJZVTWE MVNWLVSVMWKVKCVLUFZUGOECVJUHPUMWDVKVMGVOFZUIZAGCGUJIZHIZFZBAWPHIZFZJZWFWJMV @@ -162607,7 +162607,7 @@ computer programs (as last() or lastChar()), the terminology used for -> ( ( W prefix N ) prefix L ) = ( W prefix L ) ) $= ( wcel cc0 cfz co w3a cpfx cop csubstr wa wceq elfznn0 anim2i pfxval nn0cnd cn0 syl cword chash cfv caddc 3adant3 oveq1d cmin simp1 simp2 3ad2ant2 3jca - subid1d eqcomd adantl oveq2d eleq2d biimp3a pfxswrd addid2d opeq2d 3ad2ant3 + subid1d eqcomd adantl oveq2d eleq2d biimp3a pfxswrd addlidd opeq2d 3ad2ant3 0elfz sylc 3adant2 eqtr4d 3eqtrd ) DCUAZEZBFDUBUCZGHEZAFBGHZEZIZDBJHZAJHDFB KLHZAJHZDFFAUDHZKZLHZDAJHZVMVNVOAJVMVHBSEZMZVNVONVHVJWBVLVJWAVHBVIOZPUEDBVG QTUFVMVHVJFVKEZIAFBFUGHZGHZEZVPVSNVMVHVJWDVHVJVLUHVHVJVLUIVJVHWDVLVJWAWDWCB @@ -162778,7 +162778,7 @@ computer programs (as last() or lastChar()), the terminology used for cin fzodisjsn a1i fun syl21anc cv wo elun ccats1val1 adantlr wne wn simpr fzonel nelne2 sylancl necomd fvunsn eqtr4d cvv fdmd eleq2d mtbiri fsnunfv cdm fvexd syl3anc simpl adantl cn s1len 1nn eqeltri lbfzo0 mpbir ccatval3 - s1cl s1fv nn0cnd addid2d 3eqtr2rd elsni eqeq12d syl5ibrcom jaodan sylan2b + s1cl s1fv nn0cnd addlidd 3eqtr2rd elsni eqeq12d syl5ibrcom jaodan sylan2b imp eqfnfvd ) ACUAZEZBCEZFZDGAHIZJKZXPLZMZABUBZUCKZAXPBUKLZMZXOXSCYAXOGYA HIZJKZCYANZXSCYANXOYAXLEYFCABUDCYAOPXOYEXSCYAXMYEXSQXNXMYEGXPUEUFKZJKZXSX MYDYGGJCABUGUHXMXPGULIZEYHXSQXMXPTYICAUIZUMUJGXPUNPUORUPUQURXOXSCBLZMZYCX @@ -163017,7 +163017,7 @@ computer programs (as last() or lastChar()), the terminology used for -> ( K - ( L - M ) ) e. ( 0 ..^ ( N - L ) ) ) ) $= ( cc0 co wcel wa cmin cfzo cz wi w3a cle wbr adantr sylbi cc adantl 3adant1 cfz wn elfz2 zsubcl elfzonelfzo syl caddc wceq elfz2nn0 nn0cn elfzelz subcl - cn0 zcn ancoms addid1d eqcomd simprr simpl npncan3d oveq12d ex com12 syl2an + cn0 zcn ancoms addridd eqcomd simprr simpl npncan3d oveq12d ex com12 syl2an 3syl 3adant3 imp eleq2d biimpa 0zd 3adant2 3jca fzosubel2 syl2anc syld ) CF BUBGHZDBEUBGHZIZAFDCJGZKGHAFBCJGZKGHUCIZAWAVTKGZHZAWAJGFDBJGZKGHZVSWALHZWBW DMVQWGVRVQFLHZBLHZCLHZNZFCOPCBOPZIZIWGCFBUDWKWGWMWIWJWGWHBCUEUAQRQZVTAFWAUF @@ -163044,7 +163044,7 @@ computer programs (as last() or lastChar()), the terminology used for adantr syldan fzmmmeqm oveq2d fneq2d mpbird elfzmlbm nn0zd elfzmlbp sylan simplr adantl adantrl swrdvalfn syl3anc clt cif w3a simpll elfzelz zaddcl cz cv expcom elfzoelz impel df-3an sylanbrc ccatsymb wn cle elfz2 anim12i - cr zre elnn0z 0re jctl le2add syl2anc addid2d breq1d sylibd simprl lenltd + cr zre elnn0z 0re jctl le2add syl2anc addlidd breq1d sylibd simprl lenltd recn readdcl expd com12 mpan9 breq2i notbii syl6ib com23 3adant2 elfzonn0 ex adantrr iffalsed cc ad2antlr ad2antrr addsubassd oveq2 eqeq1d imbitrid zcn fveq2d 3eqtrd 3syl impcom 3jca swrdfv ccatcl biimpi eqeltrid ad2ant2r @@ -163104,7 +163104,7 @@ computer programs (as last() or lastChar()), the terminology used for ( K - ( # ` ( A substr <. M , L >. ) ) ) ) ) ) $= ( wcel wa cc0 cfz co cfv cmin wceq w3a wi adantr imp cword chash caddc wn cfzo cconcat cop csubstr pfxccatin12lem2c simprl swrdfv syl2an2r elfzoelz - cz cn0 cle wbr elfz2nn0 cc nn0cn anim12i zcn subcl ancoms anim1ci addid1d + cz cn0 cle wbr elfz2nn0 cc nn0cn anim12i zcn subcl ancoms anim1ci addridd cpfx syl simpr simplr simpll subsub3d eqtr2d syl2an eqcoms eqeq1d syl5ibr oveq2 ax-mp 3adant3 sylbi ad2antrl syl5com impcom fveq2d pfxccatin12lem2a ex adantl wb oveq1 oveq12d eleq2d sylibr df-3an sylanbrc ccatval2 elfzel2 @@ -163375,10 +163375,10 @@ computer programs (as last() or lastChar()), the terminology used for = ( A substr <. M , ( L + ( # ` B ) ) >. ) ) $= ( wcel wa cc0 co cle cop csubstr wceq wi cn0 syl c0 oveq2d adantr cfv cfz cword chash caddc wbr cmin cconcat cif lencl nn0le0eq0 biimpd hasheq0 imp - adantl eqcomi eleq1i nn0re w3a elfz2nn0 recn addid1d breq2d anim1i ancoms + adantl eqcomi eleq1i nn0re w3a elfz2nn0 recn addridd breq2d anim1i ancoms cr wb letri3 biimprd exp4b com23 sylbid com3l impcom com12 biimtrid sylbi 3adant2 elfznn0 swrd00 eqtr4i nn0cn subidd opeq1d 3eqtr4a a1i eleq1 oveq1 - opeq2d opeq1 eqeq12d 3imtr4d a1d wn swrdcl ccatrid cc addid1 eqcomd eqtrd + opeq2d opeq1 eqeq12d 3imtr4d a1d wn swrdcl ccatrid cc addrid eqcomd eqtrd syld ifeqda ex ad3antrrr oveq2 eleq2d simpr opeq2 oveq12d ifeq12d imbi12d adantll mpbird mpdan sylbir nn0red leaddle0 syl2an pm2.24 syl6bi pm2.61d ) AEUCZGZBYBGZHZDICBUDUAZUEJZUBJZGZHZYGCKUFZHYFIKUFZCDKUFZBDCUGJZYFLZMJZA @@ -164484,7 +164484,7 @@ at index N (modulo the length of the word) of the original word. -> ( ( W cyclShift N ) ` 0 ) = ( W ` ( N mod ( # ` W ) ) ) ) $= ( cword wcel c0 wne cz w3a cc0 ccsh co cfv caddc chash cmo cfzo simp1 simp3 wceq lennncl lbfzo0 sylibr 3adant3 cshwidxmod syl3anc zcn 3ad2ant3 fvoveq1d - wa cn addid2d eqtrd ) CBDEZCFGZAHEZIZJCAKLMZJANLZCOMZPLCMZAUTPLCMUQUNUPJJUT + wa cn addlidd eqtrd ) CBDEZCFGZAHEZIZJCAKLMZJANLZCOMZPLCMZAUTPLCMUQUNUPJJUT QLEZURVATUNUOUPRUNUOUPSUNUOVBUPUNUOUJUTUKEVBBCUAUTUBUCUDJABCUEUFUQUSAUTCPUP UNUSATUOUPAAUGULUHUIUM $. @@ -164755,11 +164755,11 @@ the symbol at any position is repeated at multiples of L (modulo the ( W ` I ) = ( W ` ( ( I + ( j x. L ) ) mod ( # ` W ) ) ) ) ) $= ( wcel cz wa co wceq cc0 cfv cmul caddc cmo cn0 wi oveq2d adantl adantr vx vy cword ccsh chash cfzo cv c1 oveq1 fvoveq1d eqeq2d imbi2d weq mul02d - wral zcn elfzoelz zcnd addid1d ad2antll oveq1d zmodidfzoimp eqtr2d fveq2d + wral zcn elfzoelz zcnd addridd ad2antll oveq1d zmodidfzoimp eqtr2d fveq2d eqtrd fveq1 eqcoms ad2antrl simprll simprlr cn clt wbr elfzo0 nn0z zmulcl w3a sylan ancoms zaddcl syl2an simplr jca ex 3adant3 sylbi expd com12 imp impcom zmodfzo syl cshwidxmod syl3anc cr nn0re zre remulcl readdcl sylan2 - crp nnrp simpl modaddmod cc recn recnd addassd 1cnd adddird mulid2d com13 + crp nnrp simpl modaddmod cc recn recnd addassd 1cnd adddird mullidd com13 adantld 3eqtrd biimpd a2d nn0ind ralrimiv ) EDUCFZCGFZHZECUDIZEJZBKEUELZU FIZFZHZBELZBAUGZCMIZNIZYDOIELZJZAPUOYAYGHZYMAPYIPFYNYMYNYHBUAUGZCMIZNIZYD OIELZJZQYNYHBKCMIZNIZYDOIZELZJZQYNYHBUBUGZCMIZNIZYDOIZELZJZQYNYHBUUEUHNIZ @@ -164799,7 +164799,7 @@ the symbol at any position is repeated at multiples of L (modulo the wn lencl cn clt wbr a1i wne df-ne elnnne0 simplbi2com sylbir impcom neqne 1nn0 ad2antll wb nngt1ne1 mpbird elfzo0 syl3anbrc simprr lbfzo0 biimtrrid sylbbr imp cz elfzoelz cshweqrep sylan2 syl22anc wss 0nn0 fzossnn0 ssralv - com12 mp2b eqcom zre ax-1rid oveq2d zcn addid2d eqtrd oveq1d zmodidfzoimp + com12 mp2b eqcom zre ax-1rid oveq2d zcn addlidd eqtrd oveq1d zmodidfzoimp cr fveqeq2d biimpd biimtrid ralimia impancom csn eqid fveqeq2 ralsn mpbir c0ex fzo01 pm2.61d2 pm2.61i ) CUADZEFZCBUBGZCHUCICFZJZAUDZCDECDZFZAEXNKIZ LZMXOYCXRXOYCYAANLYAAUEXOYAAYBNXOYBEEKINXNEEKUFEUGUHUIUJUKXOUNZXRYCYDXRJX @@ -165335,7 +165335,7 @@ the symbol at any position is repeated at multiples of L (modulo the (Contributed by Mario Carneiro, 26-Feb-2016.) $) cats1fvn $p |- ( X e. V -> ( T ` M ) = X ) $= ( wcel cfv cc0 cs1 chash caddc co cconcat oveq2i cn0 cvv cn cword lencl - ax-mp eqeltrri nn0cni addid2i eqtr2i fveq12i cfzo wceq s1cli c1 eqeltri + ax-mp eqeltrri nn0cni addlidi eqtr2i fveq12i cfzo wceq s1cli c1 eqeltri s1len 1nn lbfzo0 mpbir ccatval3 mp3an eqtri s1fv eqtrid ) EDICBJZKELZJZ EVCKAMJZNOZAVDPOZJZVECVGBVHFVGKCNOCVFCKNHQCCVFCRHASUAZIZVFRIGSAUBUCUDUE UFUGUHVKVDVJIKKVDMJZUIOIZVIVEUJGEUKVMVLTIVLULTEUNUOUMVLUPUQSAVDKURUSUTE @@ -167444,10 +167444,10 @@ the symbol at any position is repeated at multiples of L (modulo the syl eqtrid cn0 w3a cn wo elnn0 relexpaddnn a1d 3exp c2 cuz cfv elnn1uz2 cid wa cdm crn cun coires1 wss simpll simplr oveq12d 1p0e1 eqtrdi simprr simprl cres mpd relexp1g releqd mpbird eqeltrdi relexpnndm ssun1 relssres relexp0g - 1nn sstrdi coeq2d cc ax-1cn addid1d 3eqtr4d exp43 simp1 relexpuzrel eluz2nn + 1nn sstrdi coeq2d cc ax-1cn addridd 3eqtr4d exp43 simp1 relexpuzrel eluz2nn simp3 simp2 eluzelcn jaoi sylbi 3impd ccnv relcnv mp1i cnvco 1nn0 relexpcnv pm3.2i cvv cnvexg 3eqtrd 0nn0 coeq12d nn0addcld 0p1e1 cnveqb sylancr oveq1d - wb relco addid2d nnnn0 3syl nn0cnd addcomd eqeltrrd relexp0rel dmeqd dmresi + wb relco addlidd nnnn0 3syl nn0cnd addcomd eqeltrrd relexp0rel dmeqd dmresi uzaddcl eqimss 00id imp ) CUAEZBUAEZADEZCBFGZHIZAJZKZUBZACLGZABLGZMZAYILGZI ZYFCUCEZCNIZUDYMYRKZCUEYSUUAYTYSYGYHYLYRYGYSYHYLYRKZKZYGBUCEZBNIZUDZYSUUCKZ BUEZUUDUUGUUEYSUUDUUCYSUUDYHUUBYSUUDYHUBYRYLABCDUFUGUHOYSUUEUUCYSCHIZCUIUJU @@ -167950,7 +167950,7 @@ a function (ordinarily on a subset of ` CC ` ) and produces a new 20-Jul-2005.) $) shftval3 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift ( A - B ) ) ` A ) = ( F ` B ) ) $= - ( cc wcel wa cc0 caddc co cmin cshi cfv 0cn shftval2 mp3an3 addid1 adantr + ( cc wcel wa cc0 caddc co cmin cshi cfv 0cn shftval2 mp3an3 addrid adantr wceq fveq2d adantl 3eqtr3d ) AEFZBEFZGZAHIJZCABKJLJZMZBHIJZCMZAUGMBCMUCUD HEFUHUJSNABHCDOPUEUFAUGUCUFASUDAQRTUEUIBCUDUIBSUCBQUATUB $. @@ -168330,7 +168330,7 @@ a function (ordinarily on a subset of ` CC ` ) and produces a new $( The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) $) reim0 $p |- ( A e. RR -> ( Im ` A ) = 0 ) $= - ( cr wcel ci cc0 cmul co caddc cim cfv wceq recn it0e0 oveq2i addid1 eqtrid + ( cr wcel ci cc0 cmul co caddc cim cfv wceq recn it0e0 oveq2i addrid eqtrid cc syl fveq2d 0re crim mpan2 eqtr3d ) ABCZADEFGZHGZIJZAIJEUDUFAIUDAQCZUFAKA LUHUFAEHGAUEEAHMNAOPRSUDEBCUGEKTAEUAUBUC $. @@ -168338,7 +168338,7 @@ a function (ordinarily on a subset of ` CC ` ) and produces a new 26-Sep-2005.) $) reim0b $p |- ( A e. CC -> ( A e. RR <-> ( Im ` A ) = 0 ) ) $= ( cc wcel cr cim cfv cc0 wceq reim0 wa ci cmul co caddc replim adantr oveq2 - cre it0e0 eqtrdi oveq2d recl recnd addid1d sylan9eqr eqtrd eqeltrd impbid2 + cre it0e0 eqtrdi oveq2d recl recnd addridd sylan9eqr eqtrd eqeltrd impbid2 ex ) ABCZADCZAEFZGHZAIUJUMUKUJUMJZAARFZDUNAUOKULLMZNMZUOUJAUQHUMAOPUMUJUQUO GNMUOUMUPGUONUMUPKGLMGULGKLQSTUAUJUOUJUOAUBZUCUDUEUFUJUODCUMURPUGUIUH $. @@ -168346,7 +168346,7 @@ a function (ordinarily on a subset of ` CC ` ) and produces a new [Gleason] p. 133. (Contributed by NM, 20-Aug-2008.) $) rereb $p |- ( A e. CC -> ( A e. RR <-> ( Re ` A ) = A ) ) $= ( cc wcel cr cre cfv wceq wa ci cim co caddc cc0 replim adantr reim0 oveq2d - cmul it0e0 eqtrdi adantl recl recnd addid1d 3eqtrrd simpr eqeltrrd impbida + cmul it0e0 eqtrdi adantl recl recnd addridd 3eqtrrd simpr eqeltrrd impbida ) ABCZADCZAEFZAGZUIUJHZAUKIAJFZRKZLKZUKMLKZUKUIAUPGUJANOUMUOMUKLUJUOMGUIUJU OIMRKMUJUNMIRAPQSTUAQUIUQUKGUJUIUKUIUKAUBZUCUDOUEUIULHUKADUIULUFUIUKDCULURO UGUH $. @@ -168537,7 +168537,7 @@ a function (ordinarily on a subset of ` CC ` ) and produces a new immul2 $p |- ( ( A e. RR /\ B e. CC ) -> ( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) ) $= ( cr wcel cc wa cmul co cim cfv cre caddc wceq recn immul sylan rere oveq1d - cc0 recnd adantr reim0 recl mul02d oveq12d imcl mulcl syl2an addid1d 3eqtrd + cc0 recnd adantr reim0 recl mul02d oveq12d imcl mulcl syl2an addridd 3eqtrd sylan9eq ) ACDZBEDZFZABGHIJZAKJZBIJZGHZAIJZBKJZGHZLHZAUQGHZSLHVCULAEDZUMUOV BMANZABOPUNURVCVASLUNUPAUQGULUPAMUMAQUARULUMVASUTGHSULUSSUTGAUBRUMUTUMUTBUC TUDUKUEUNVCULVDUQEDVCEDUMVEUMUQBUFTAUQUGUHUIUJ $. @@ -168703,14 +168703,14 @@ a function (ordinarily on a subset of ` CC ` ) and produces a new $( The real part of ` _i ` . (Contributed by Scott Fenton, 9-Jun-2006.) $) rei $p |- ( Re ` _i ) = 0 $= - ( cc0 ci c1 cmul co caddc cre ax-icn ax-1cn mulcli addid2i fveq2i wcel wceq - cfv cr 0re 1re crre mp2an mulid1i 3eqtr3ri ) ABCDEZFEZGOZUCGOABGOUDUCGUCBCH + ( cc0 ci c1 cmul co caddc cre ax-icn ax-1cn mulcli addlidi fveq2i wcel wceq + cfv cr 0re 1re crre mp2an mulridi 3eqtr3ri ) ABCDEZFEZGOZUCGOABGOUDUCGUCBCH IJKLAPMCPMUEANQRACSTUCBGBHUALUB $. $( The imaginary part of ` _i ` . (Contributed by Scott Fenton, 9-Jun-2006.) $) imi $p |- ( Im ` _i ) = 1 $= - ( ci c1 cmul cim cfv cc0 ax-icn ax-1cn mulcli addid2i eqcomi fveq2i mulid1i + ( ci c1 cmul cim cfv cc0 ax-icn ax-1cn mulcli addlidi eqcomi fveq2i mulridi co caddc cr wcel wceq 0re 1re crim mp2an 3eqtr3i ) ABCNZDEFUDONZDEZADEBUDUE DUEUDUDABGHIJKLUDADAGMLFPQBPQUFBRSTFBUAUBUC $. @@ -168721,7 +168721,7 @@ a function (ordinarily on a subset of ` CC ` ) and produces a new $( The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.) $) cji $p |- ( * ` _i ) = -u _i $= - ( ci cre cfv cim cmul co cmin cc0 ccj cneg rei c1 imi oveq2i ax-icn mulid1i + ( ci cre cfv cim cmul co cmin cc0 ccj cneg rei c1 imi oveq2i ax-icn mulridi eqtri oveq12i cc wcel wceq remim ax-mp df-neg 3eqtr4i ) ABCZAADCZEFZGFZHAGF AICZAJUFHUHAGKUHALEFAUGLAEMNAOPQRASTUJUIUAOAUBUCAUDUE $. @@ -169286,7 +169286,7 @@ a function (ordinarily on a subset of ` CC ` ) and produces a new by Mario Carneiro, 8-Jul-2013.) $) rennim $p |- ( A e. RR -> ( _i x. A ) e/ RR+ ) $= ( cr wcel ci cmul co crp wn wnel cc0 wceq cre cfv cc wi ax-icn recn sylancr - mulcl rpre rereb imbitrid syl caddc addid2d fveq2d 0re eqtr3d eqeq1d sylibd + mulcl rpre rereb imbitrid syl caddc addlidd fveq2d 0re eqtr3d eqeq1d sylibd crre mpan rpne0 necon2bi eqcoms syl6 pm2.01d df-nel sylibr ) ABCZDAEFZGCZHZ VAGIUTVBUTVBJVAKZVCUTVBVALMZVAKZVDUTVANCZVBVFOUTDNCANCVGPAQDASRZVBVABCVGVFV ATVAUAUBUCUTVEJVAUTJVAUDFZLMZVEJUTVIVALUTVAVHUEUFJBCUTVJJKUGJAUKULUHUIUJVCV @@ -169348,7 +169348,7 @@ reflection about the origin (under the map ` x |-> -u x ` ). (Contributed $( Lemma for ~ 01sqrex . (Contributed by Mario Carneiro, 10-Jul-2013.) $) 01sqrexlem2 $p |- ( ( A e. RR+ /\ A <_ 1 ) -> A e. S ) $= ( crp wcel c1 cle wbr wa c2 cexp co simpl cmul cr wceq adantr cc0 wb rpre - clt rpgt0 1re lemul1 mp3an2 syl12anc biimpa rpcn sqval eqcomd syl mulid2d + clt rpgt0 1re lemul1 mp3an2 syl12anc biimpa rpcn sqval eqcomd syl mullidd cc 3brtr3d cv oveq1 breq1d elrab2 sylanbrc ) BGHZBIJKZLZVCBMNOZBJKZBDHVCV DPVEBBQOZIBQOZVFBJVCVDVHVIJKZVCBRHZVKUABUDKZVDVJUBZBUCZVNBUEVKIRHVKVLLVMU FBIBUGUHUIUJVEBUPHZVHVFSVCVOVDBUKZTVOVFVHBULUMUNVCVIBSVDVCBVPUOTUQAURZMNO @@ -169746,7 +169746,7 @@ reflection about the origin (under the map ` x |-> -u x ` ). (Contributed take here. (Contributed by Mario Carneiro, 10-Jul-2013.) $) sqrtm1 $p |- _i = ( sqrt ` -u 1 ) $= ( c1 cneg csqrt cfv ci cmul co wcel cc0 cle wbr wceq 1re 0le1 sqrtneg mp2an - cr sqrt1 oveq2i ax-icn mulid1i 3eqtrri ) ABCDZEACDZFGZEAFGEAQHIAJKUCUELMNAO + cr sqrt1 oveq2i ax-icn mulridi 3eqtrri ) ABCDZEACDZFGZEAFGEAQHIAJKUCUELMNAO PUDAEFRSETUAUB $. $( A natural number with square one is equal to one. (Contributed by Thierry @@ -170045,7 +170045,7 @@ reflection about the origin (under the map ` x |-> -u x ` ). (Contributed absimle $p |- ( A e. CC -> ( abs ` ( Im ` A ) ) <_ ( abs ` A ) ) $= ( cc wcel ci cneg cmul co cre cfv cabs cim cle wbr negicn id mulcld absrele a1i wceq c1 syl imre fveq2d absmul mpan ax-icn ax-mp absi eqtri abscl recnd - absneg oveq1i mulid2d eqtrid eqtr2d 3brtr4d ) ABCZDEZAFGZHIZJIZUTJIZAKIZJIA + absneg oveq1i mullidd eqtrid eqtr2d 3brtr4d ) ABCZDEZAFGZHIZJIZUTJIZAKIZJIA JIZLURUTBCVBVCLMURUSAUSBCZURNRUROPUTQUAURVDVAJAUBUCURVCUSJIZVEFGZVEVFURVCVH SNUSAUDUEURVHTVEFGVEVGTVEFVGDJIZTDBCVGVISUFDULUGUHUIUMURVEURVEAUJUKUNUOUPUQ $. @@ -170055,8 +170055,8 @@ reflection about the origin (under the map ` x |-> -u x ` ). (Contributed max0add $p |- ( A e. RR -> ( if ( 0 <_ A , A , 0 ) + if ( 0 <_ -u A , -u A , 0 ) ) = ( abs ` A ) ) $= ( cr wcel cc0 cle wbr cif caddc co wceq wa adantr biimpa 0re letri3 ifeq1da - wb ifid eqtrdi oveq12d cneg cabs cfv 0red id cc recn addid1d iftrue le0neg2 - adantl simpr renegcl ad2antrr sylancl mpbir2and absid 3eqtr4d addid2d mpan2 + wb ifid eqtrdi oveq12d cneg cabs cfv 0red id cc recn addridd iftrue le0neg2 + adantl simpr renegcl ad2antrr sylancl mpbir2and absid 3eqtr4d addlidd mpan2 negcld biimprd impl le0neg1 iftrued absnid lecasei ) ABCZDAEFZADGZDAUAZEFZV KDGZHIZAUBUCZJDAVHUDVHUEVHVIKZADHIAVNVOVPAVHAUFCZVIAUGZLUHVPVJAVMDHVIVJAJVH VIADUIUKVPVMVLDDGDVPVLVKDDVPVLKZVKDJZVKDEFZVLVPWAVLVHVIWAAUJMLVPVLULVSVKBCZ @@ -170330,7 +170330,7 @@ reflection about the origin (under the map ` x |-> -u x ` ). (Contributed recan $p |- ( ( A e. CC /\ B e. CC ) -> ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> A = B ) ) $= ( cc wcel cmul co cre cfv wceq c1 ci caddc fvoveq1 eqeq12d oveq2d oveq12d - wi rspcv fveq2d wa wral cneg ax-1cn ax-mp negicn cim replim mulid2 eqcomd + wi rspcv fveq2d wa wral cneg ax-1cn ax-mp negicn cim replim mullid eqcomd cv imre eqtrd eqeqan12d syl5ibr oveq2 ralrimivw impbid1 ) BDEZCDEZUAZAUKZ BFGZHIZVBCFGZHIZJZADUBZBCJZVHVIVAKBFGZHIZLLUCZBFGHIZFGZMGZKCFGZHIZLVLCFGH IZFGZMGZJVHVKVQVNVSMKDEVHVKVQJZRUDVGWAAKDVBKJVDVKVFVQVBKBHFNVBKCHFNOSUEVH @@ -173766,10 +173766,10 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $= ( cc cc0 wcel crp cmin co cabs cfv clt wbr c1 cdiv cmul cle cdif wa cv wi csn wral wrex cif c2 1rp wne simpl eldifsn sylib absrpcl rpmulcl sylancom syl ifcl sylancr rphalfcld rpmulcld eqeltrid adantr simpld simprl syl2anc - mulcld mulne0 divsubdird mulid1d oveq1d wceq 1cnd divcan5 syl3anc mulcomd + mulcld mulne0 divsubdird mulridd oveq1d wceq 1cnd divcan5 syl3anc mulcomd eqtr3d oveq12d eqtrd fveq2d subcld absdivd cr abssubd abscld eqeltrd rpre rpred ad2antlr remulcld simprr eqbrtrd 1re min2 lemul1d mpbid caddc recnd - eqbrtrid 2halvesd min1 mulid2d breqtrd ltletrd lelttrd ltsubadd2d ltadd1d + eqbrtrid 2halvesd min1 mullidd breqtrd ltletrd lelttrd ltsubadd2d ltadd1d resubcld abs2difd mpbird ltmul2dd absmuld mul32d breqtrrd lttrd ltdivmuld 1red absrpcld expr ralrimiva breq2 rspceaimv ) CGHUEUAZIZDJIZUBZEJIZBUCZC KLZMNZEOPZQYIRLZQCRLZKLZMNZDOPZUDZBYDUFYKAUCZOPZYQUDBYDUFAJUGYGEQCMNZDSLZ @@ -175005,8 +175005,8 @@ seq N ( + , F ) e. dom ~~> ) ) $= isercolllem3 $p |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( seq M ( + , F ) ` N ) = ( seq 1 ( + , H ) ` ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) $= - ( c1 cfv wcel cn cuz wa ccnv cfz co cima caddc cc cres cc0 cv wceq addid2 - adantl addid1 addcl 0cnd chash clt wss cnvimass adantr fssdm isercolllem1 + ( c1 cfv wcel cn cuz wa ccnv cfz co cima caddc cc cres cc0 cv wceq addlid + adantl addrid addcl 0cnd chash clt wss cnvimass adantr fssdm isercolllem1 wiso wf syldan wb isercolllem2 isoeq4 syl mpbird cdm cin c0 wne a1i sylib sseqin2 1nn ffvelcdm sylancl eleqtrdi simpr elfzuzb sylanbrc wfn elpreima ffn 3syl mpbir2and ne0d eqnetrd imadisj necon3bii crn wfun ffun funimacnv @@ -175557,15 +175557,15 @@ seq N ( + , F ) e. dom ~~> ) ) $= ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) $= ( wcel c1 co c2 cmul caddc vx vn cn0 cneg cexp cseq cfv cle wbr wa cv cc0 wi wceq oveq2 2t0e0 eqtrdi oveq2d fveq2d breq1d imbi2d cz wss cuz eqsstri - uzssz a1i sselda zcnd addid1d cr wne neg1ne0 reexpclz mp3an12i ffvelcdmda + uzssz a1i sselda zcnd addridd cr wne neg1ne0 reexpclz mp3an12i ffvelcdmda neg1rr remulcld eqeltrd serfre leidd eqbrtrd cmin ad2antrr ax-1cn 2timesi wf oveq2i simpr eleqtrdi adantr eluzelz syl cc nn0cn adantl mulcl sylancr 2cn mulcli addassd eqtr3id 2nn0 nn0mulcl uzaddcl syl2anc sselid 1cnd 2cnd adddid 3eqtr4d peano2nn0 eleqtrrdi ffvelcdmd peano2uz resubcld 0red fveq2 fvoveq1 breq12d wral ralrimiva rspcdva suble0d seqp1 oveq1d eqtrd oveq12d - eqeq12d neg1cn expp1zd sylancl 3eqtr3d mulassd 3eqtrrd syl22anc mulid2d + eqeq12d neg1cn expp1zd sylancl 3eqtr3d mulassd 3eqtrrd syl22anc mullidd recnd 3eqtrd eqtr4d leadd2dd reexpclzd mulneg12 renegcld peano2zd mul2neg - mpbird mulcom mulm1d mulid1d readdcld 2timesd expaddz nn0z zaddcl expmulz + mpbird mulcom mulm1d mulridd readdcld 2timesd expaddz nn0z zaddcl expmulz 2z syl2an neg1sqe1 oveq1i 1exp eqtrid negcld addcomd negsubd 3brtr3d letr syl3anc mpand expcom a2d nn0ind com12 3impia ) AGHOZEUCOZPUDZGUEQZGRESQZT QZTCFUFZUGZSQZUVRGUWAUGZSQZUHUIZUVPAUVOUJZUWFUWGUVRGRUAUKZSQZTQZUWAUGZSQZ @@ -175649,12 +175649,12 @@ seq N ( + , F ) e. dom ~~> ) ) $= cuz wf cv simpr ffvelcdmda remulcld eqeltrd serfre 3ad2ant1 eleqtrdi 2nn0 simp3 nn0mulcl sylancr uzaddcl eleqtrrdi ffvelcdmd subdid fveq2d resubcld wa syl2anc absmuld eqtr3d wceq absexpz syl3anc ax-1cn absnegi abs1 oveq1i - cc eqtri syl eqtrid eqtrd oveq1d abscld mulid2d 3eqtrd peano2uzs 3ad2ant2 + cc eqtri syl eqtrid eqtrd oveq1d abscld mullidd 3eqtrd peano2uzs 3ad2ant2 1exp seqp1 fveq2 oveq2 oveq12d eqeq12d wral rspcdva oveq2d expp1zd neg1cn ralrimiva mulcom sylancl mulm1d mulneg1d negsubd zcnd 2timesd 2z syl22anc expmulz neg1sqe1 expaddz 3eqtr3d mulassd iseraltlem2 eqbrtrrd iseraltlem1 3brtr3d nn0zd mpbid letrd absdifled mpbir2and eqtrdi nn0cnd add32d lenegd - syl3an2 1cnd mpbird negcld mulid1d subge02d eqbrtrd readdcld addge01d jca + syl3an2 1cnd mpbird negcld mulridd subge02d eqbrtrd readdcld addge01d jca simp1 ) AGHUAZEUBUAZUCZGUDEOPZUEPZUECFUFZQZGUVAQZUGPZUHQZGRUEPZDQZSUIUUTR UEPZUVAQZUVCUGPZUHQZUVGSUIUURRUKZGTPZUVBOPZUVMUVCOPZUGPZUHQZUVEUVGSUURUVQ UVMUHQZUVEOPZRUVEOPUVEUURUVMUVDOPZUHQUVQUVSUURUVTUVPUHUURUVMUVBUVCUURUVMU @@ -176098,7 +176098,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ $( Lemma for ~ sumrb . (Contributed by Mario Carneiro, 12-Aug-2013.) $) sumrblem $p |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) ) $= - ( cuz cfv cc cc0 wcel co wceq adantl adantr cz vn wss caddc addid2 0cnd + ( cuz cfv cc cc0 wcel co wceq adantl adantr cz vn wss caddc addlid 0cnd wa cv wf cif iftrue eqeltrd ex wn iffalse 0cn eqeltrdi pm2.61d1 eluzelz fmptd syl ffvelcdmd c1 cmin cfz cdif elfzelz simplr zcnd ad2antrr npcan ax-1cn sylancl fveq2d fznuz ssneldd eldifd fveqeq2 eldifi eldifn fvmpt2 @@ -176116,7 +176116,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ fsumcvg $p |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) ) $= ( caddc cfv wcel cz syl cc wa cc0 wceq vn vm cseq cvv cuz eluzelz seqex eqid a1i eluzel2 cv cif iftrue adantl eqeltrd iffalse eqeltrdi pm2.61d1 - ex wn 0cn fvmpt2 syl2anr adantr ffvelcdmd co addid1 simpr c1 cfz elfzuz + ex wn 0cn fvmpt2 syl2anr adantr ffvelcdmd co addrid simpr c1 cfz elfzuz serf cdif sseld fznuz syl6 con2d imp eldifd fveqeq2 eldifi eldifn eqtrd syl2anc vtoclga sylan2 adantlr seqid2 eqcomd climconst ) AGLEFUCZMZUAWK GUDGUEMZWMUHAGFUEMZNZGONJFGUFPWKUDNALEFUGUIAWNQGWKADEFWNWNUHAWOFONJFGUJ @@ -176199,7 +176199,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ wa cle wbr ccnv f1ocnvfv2 sylan f1ocnv ffvelcdmda elfzle2 cxr adantr cr wss fzssuz cz uzssz zssre ressxr a1i sstrdi leisorel syl122anc eqbrtrrd sstri eluzelz eluz syl2anc mpbird elfzuzb sylanbrc ex ssrdv fsumcvg cc0 - cc addid2 adantl addid1 addcl 0cnd eleqtrrd cif iftrue eqeltrd eqeltrdi + cc addlid adantl addrid addcl 0cnd eleqtrrd cif iftrue eqeltrd eqeltrdi wn 0cn csb nfv nfcsb1v nfcv nfif eleq1 csbeq1a ifbieq1d cmpt weq fvmptg wi iffalse pm2.61d1 fmptd elfzelz ffvelcdm syl2an cdif fveqeq2 elfzelzd eldifi eldifn fvmpt2 eqtrd vtoclga nfel1 nfim fvex eleq1d imbi2d vtoclf @@ -176795,7 +176795,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ ( csu caddc co wcel cc0 wss cc wceq syldan wa cv cif wral cuz cfv cfn cun wo ssun1 sseqtrrid sselda ralrimiva olcd sumss2 syl21anc oveq12d 0cn ifcl ssun2 sylancl fsumadd eleq2d elun bitrdi biimpa iftrue adantl wn noel cin - wi c0 bitr3di mtbii imnan sylibr imp iffalsed addid1d eqtrd con2d addid2d + wi c0 bitr3di mtbii imnan sylibr imp iffalsed addridd eqtrd con2d addlidd elin jaodan sumeq2dv 3eqtr2rd ) ABDFKZCDFKZLMEFUAZBNZDOUBZFKZEWICNZDOUBZF KZLMEWKWNLMZFKEDFKAWGWLWHWOLABEPDQNZFBUCEOUDUEPZEUFNZUHZWGWLRABCUGZBEBCUI HUJZAWQFBAWJWIENZWQABEWIXBUKJSZULAWSWRIUMZBEDFOUNUOACEPWQFCUCWTWHWORAXACE @@ -177242,8 +177242,8 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ ( sum_ k e. A C + sum_ k e. B C ) ) $= ( wcel cc0 cun csu cv cif caddc co wss cc wral cuz cfv cfn wceq ralrimiva wo eqimssi a1i orcd sumss2 syl21anc wa iftrue adantl elun1 sylan2 eqeltrd - wn iffalse 0cn eqeltrdi pm2.61dan adantr elun2 isumadd addid1d wi c0 noel - cin eleq2d elin bitr3di mtbii imnan sylibr imp syl oveq12d addid2d oveq1d + wn iffalse 0cn eqeltrdi pm2.61dan adantr elun2 isumadd addridd wi c0 noel + cin eleq2d elin bitr3di mtbii imnan sylibr imp syl oveq12d addlidd oveq1d 3eqtr4rd wb elun biorf bitr4id ifbid sumeq2sdv unssad unssbd eqtr4d ) ABC UAZDEUBZIEUCZXASZDTUDZEUBZBDEUBZCDEUBZUEUFZAXAIUGDUHSZEXAUIIHUJUKZUGZIULS ZUOZXBXFUMMAXJEXAPUNAXLXMXLAIXKJUPUQURZXAIDEHUSUTAIXCBSZDTUDZXCCSZDTUDZUE @@ -177627,7 +177627,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ ( vn cc0 cfz co cmin csu wcel cc wceq cv caddc wa wral fznn0sub2 ad2antll csb expr ralrimiv eleq1d cbvralvw adantrr nfcsb1v nfel1 csbeq1a rspc sylc sylibr fsum0diag mpan9 csbeq1 fsumrev2 cz elfz3nn0 ad2antlr elfzelz nn0cn - cn0 zcn subcl syl2an syl2anc addid2 syl oveq1d sumeq2dv eqtrd adantl 3syl + cn0 zcn subcl syl2an syl2anc addlid syl oveq1d sumeq2dv eqtrd adantl 3syl csbeq1d oveq2d adantr sub32 syl3an syl3anc sumeq12rdv 3eqtr4d cuz elfzuz3 fzfid cfv elfzuzb sylanbrc wb elfzel2 fzsubel syl22anc mpbid subid simpll eleqtrd wss fzss2 sselda fsumcl oveq2 oveq1 sumeq12dv eqtr4d vex a1i ovex @@ -178127,7 +178127,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ fsumrelem $p |- ( ph -> ( F ` sum_ k e. A B ) = sum_ k e. A ( F ` B ) ) $= ( vm wceq cfv wcel co wa cc0 caddc cc vf c0 csu chash cn c1 cfz cv wf1o - wex 0cn ffvelcdmi ax-mp addid1i fvoveq1 fveq2 oveq1d eqeq12d oveq2 00id + wex 0cn ffvelcdmi ax-mp addridi fvoveq1 fveq2 oveq1d eqeq12d oveq2 00id wi eqtrdi fveq2d oveq2d vtocl2ga mp2an eqtr2i addcani mpbi sum0 3eqtr4a sumeq1 a1i cmpt ccom cseq addcl adantl wf fmpttd adantr simprr f1of syl fco syl2anc ffvelcdmda simprl nnuz eleqtrdi wral simpr eqid fvmpt2 fvex @@ -178285,7 +178285,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ wb abscld adantrr rprege0 ad2antrl absid oveq2d eqtrd cmul adantlr adantr remulcld fveq2i fsumabs eqbrtrid cun ssun2 nnred flle simprr letrd fznnfl flge1nn mpbir2and syldan recnd absge0d chash cfn cn0 3syl reflcl breqtrd - clt fzsplit sseqtrrid sselda mulid2d fsumge0 jca simprd syl31anc eqbrtrrd + clt fzsplit sseqtrrid sselda mullidd fsumge0 jca simprd syl31anc eqbrtrrd lemul1a hashcl nn0re elfzuz peano2nnd eluznn sylan peano2re fllep1 eluzle cuz simpllr nfv nffv nfbr breq2 fveq2d breq1d imbi12d syl3c sylan2 fsumle nfim fsumconst syl2anc biidd 0red mpan9 nnzd uzid rspcdva ssdomg hashdomi @@ -178524,7 +178524,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ Carneiro, 28-Apr-2014.) $) abscvgcvg $p |- ( ph -> seq M ( + , G ) e. dom ~~> ) $= ( c1 cuz cfv cz wcel uzid syl eleqtrrdi cle cabs abscld eqeltrd 1red cmul - cv wa cr wbr eleq2i eqcomd eqled recnd mulid2d breqtrrd sylan2br cvgcmpce + cv wa cr wbr eleq2i eqcomd eqled recnd mullidd breqtrrd sylan2br cvgcmpce co ) ALBCDEEFGAEEMNZFAEOPEUSPHEQRGSABUFZFPZUGZUTCNZUTDNZUANZUHIVBVDJUBZUC ZJKAUDUTUSPAVAVELVCUEURZTUIFUSUTGUJVBVEVCVHTVBVEVCVFVBVCVEIUKULVBVCVBVCVG UMUNUOUPUQ $. @@ -178609,7 +178609,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.) $) hashiun $p |- ( ph -> ( # ` U_ x e. A B ) = sum_ x e. A ( # ` B ) ) $= ( vk c1 csu chash cfv cv wcel wa cmul co cfn cc wceq ciun 1cnd wral iunfi - fsumiun ralrimiva syl2anc ax-1cn fsumconst sylancl cn0 hashcl mulid1 4syl + fsumiun ralrimiva syl2anc ax-1cn fsumconst sylancl cn0 hashcl mulrid 4syl nn0cn eqtrd sumeq2dv 3eqtr3d ) ABCDUAZIHJZCDIHJZBJUSKLZCDKLZBJABCDIHEFGAB MCNZHMDNOOUBUEAUTVBIPQZVBAUSRNZISNZUTVETACRNDRNZBCUCVFEAVHBCFUFBCDUDUGZUH USIHUIUJAVFVBUKNVBSNVEVBTVIUSULVBUOVBUMUNUPACVAVCBAVDOZVAVCIPQZVCVJVHVGVA @@ -178646,7 +178646,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ = ( ( # ` A ) x. ( ( # ` A ) - 1 ) ) ) $= ( chash cfv csu c1 co cmul wcel cfn ciun cmin cv wa csn cdif diffi adantr syl eqeltrid hash2iun 2sumeq2dv cc wceq fsumconst syl2anc sumeq2dv fveq2d - 1cnd a1i hashdifsn sylan eqtrd oveq1d cn0 hashcl nn0cnd peano2cnm mulid1d + 1cnd a1i hashdifsn sylan eqtrd oveq1d cn0 hashcl nn0cnd peano2cnm mulridd sumeq2sdv 3eqtrd ) ABDCEFUAUAMNDEFMNZCOBODEPCOZBOZDMNZVOPUBQZRQZABCDEFGAB UCZDSZUDZEDVRUEZUFZTHAWBTSZVSADTSZWCGDWAUGUIUHUJZIJKUKADEVLPBCLULAVNDEMNZ PRQZBODVPPRQZBOZVQADVMWGBVTETSPUMSVMWGUNWEVTUSEPCUOUPUQADWGWHBVTWFVPPRVTW @@ -178711,7 +178711,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ wdisj xp1st elsni rgen rgenw invdisj mp1i hashiun elinel2 wf1 wss elinel1 f1of1 elpwid f1ores fvres adantl cc hashcl nn0cn 3syl fsumf1o cmul sselda ffvelcdmi hashxp hashsng hashpw f1ocnvfv2 eqtr3d oveq2d eqtrd oveq12d 2cn - c1 expcl 3eqtrd eqidd fmptco mulid2d sumeq2dv feqmptd fveq2 mptru 3eqtr4i + c1 expcl 3eqtrd eqidd fmptco mullidd sumeq2dv feqmptd fveq2 mptru 3eqtr4i mpteq2ia f1oeq1 mpbi ) GHZIUAZGJKUBZEKHZIUAZFEUCZFUCZUDZUUPHZUEZUFZLMZUGZ AUUKUULUHZAUCZUIZUGZUJZUJZNZUUKGCNZKGUULNZUUKKUVGNZUVIAUULAUKULZUUNKUVBNU UKUUNUVFNZUVLABUVBEAUUNUVABUVDBUCZUDZUVOHZUEZUFZLMEAVEZUUTUVSLUVTUUTBUUOU @@ -178842,7 +178842,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( A ^ k ) ) ) $= ( cc wcel cn0 wa c1 caddc co cexp cc0 cfz cv cbc cmin cmul csu wceq eqtrd ax-1cn binom mp3an1 cz fznn0sub adantl nn0zd 1exp syl simpl elfznn0 expcl - oveq1d syl2an mulid2d oveq2d sumeq2dv ) ADEZCFEZGZHAIJCKJZLCMJZCBNZOJZHCV + oveq1d syl2an mullidd oveq2d sumeq2dv ) ADEZCFEZGZHAIJCKJZLCMJZCBNZOJZHCV CPJZKJZAVCKJZQJZQJZBRZVBVDVGQJZBRHDEURUSVAVJSUAHABCUBUCUTVBVIVKBUTVCVBEZG ZVHVGVDQVMVHHVGQJVGVMVFHVGQVMVEUDEVFHSVMVEVLVEFEUTVCLCUEUFUGVEUHUIUMVMVGU TURVCFEVGDEVLURUSUJVCCUKAVCULUNUOTUPUQT $. @@ -178852,7 +178852,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ binom11 $p |- ( N e. NN0 -> ( 2 ^ N ) = sum_ k e. ( 0 ... N ) ( N _C k ) ) $= ( cn0 wcel c2 cexp co cc0 cfz cv cbc c1 cmul csu caddc df-2 oveq1i ax-1cn - cc wceq binom1p mpan eqtrid cz elfzelz 1exp syl bccl2 nncnd mulid1d eqtrd + cc wceq binom1p mpan eqtrid cz elfzelz 1exp syl bccl2 nncnd mulridd eqtrd oveq2d sumeq2i eqtrdi ) BCDZEBFGZHBIGZBAJZKGZLURFGZMGZANZUQUSANUOUPLLOGZB FGZVBEVCBFPQLSDUOVDVBTRLABUAUBUCUQVAUSAURUQDZVAUSLMGUSVEUTLUSMVEURUDDUTLT URHBUEURUFUGULVEUSVEUSURBUHUIUJUKUMUN $. @@ -178866,7 +178866,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ ( cc wcel cn0 wa c1 caddc co cexp cc0 cfz cbc cmul csu wceq adantl ax-1cn sylancl cv fzfid fzssp1 nn0cn npcan oveq2d sseqtrid sselda cn bccl2 nncnd simpl elfznn0 expcl syl2an mulcld syldan fsumcl addcom oveq1d binom1p cuz - cmin cfv simpr nn0uz eleqtrdi oveq2 oveq12d fsumm1 mulid2d eqtrd mvrraddd + cmin cfv simpr nn0uz eleqtrdi oveq2 oveq12d fsumm1 mullidd eqtrd mvrraddd bcnn 3eqtrd ) ADEZCFEZGZAHIJZCKJZLCHVCJZMJZCBUAZNJZAWCKJZOJZBPZACKJZVRWBW FBVRLWAUBVRWCWBEWCLCMJZEZWFDEVRWBWIWCVRLWAHIJZMJWBWILWAUCVRWKCLMVRCDEZHDE ZWKCQVQWLVPCUDRSCHUETUFUGUHVRWJGZWDWEWNWDWJWDUIEVRWCCUJRUKVRVPWCFEWEDEWJV @@ -178928,7 +178928,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ ( vb cfn wcel chash cfv cin cmin co cmul csu wceq c0 eqtrdi fveq2d oveq2d wa wss vx vy vz vt vu cv cuni cpw c1 cneg cexp cint wral cc0 csn cun uni0 unieq ineq2d in0 hash0 pweq pw0 sumeq1d eqeq12d ralbidv weq unisnv uneq2i - uniun eqtri hashcl nn0cnd mulid2d cvv 0ex eqeltrd fveq2 neg1cn exp0 ax-mp + uniun eqtri hashcl nn0cnd mullidd cvv 0ex eqeltrd fveq2 neg1cn exp0 ax-mp cc rint0 oveq12d sumsn sylancr subid1d 3eqtr4rd rgen ineq1 simpl sumeq2dv ineq1d rspcva adantll simpr inss1 ssfi sylancl in32 inass sumeq2sdv sylan wn caddc cn0 syl hashun3 syl2anc fveq2i inindi oveq2i 3eqtr4g assraddsubd @@ -179021,7 +179021,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ syl2anc disjdif a1i cun 0elpw snssi ax-mp undif mpbi eqcomi inss1 sylancl adantr fsumsplit fveq2i oveq2i subidd eqtrid incexclem syldan negsubd cvv inidm eqtr3d 0ex fveq2 hash0 eqtrdi oveq2d neg1cn exp0 rint0 fveq2d sumsn - oveq12d sylancr mulid2d eqtr2d fsumneg mulcomd mulneg1d sumeq2dv 3eqtr4rd + oveq12d sylancr mullidd eqtr2d fsumneg mulcomd mulneg1d sumeq2dv 3eqtr4rd expm1t mulm1d 3eqtrd unissd sseqin2 subeq0d ) ACDZACEZUAZAUBZFGZAUCZHUDZU EZIJZBUFZFGZIKSZLSZYPUGZFGZMSZBNZYIYJCDZYKUHDAUIZUUDYKYJOUOPZYIYNUUBBYIYL CDZYNCDYIYGUUGYGYHUJZAUKULZYLYMUMPZYIYPYNDZUAZYSUUAUULYOYRUULIUULUNUPZUUL @@ -179137,7 +179137,7 @@ Infinite sums (cont.) cz syl cseq cli cdm eluzelz wss uzss 3sstr4g sselda syldan eqeltrd iserex cv wa mpbid isumclim2 fzfid elfzuz eleqtrrdi sylan2 fsumcl ffvelcdmda cc0 serf c0 clt wbr zred ltm1d wb peano2zm fzn syl2anc2 sumeq1d adantr eqtrdi - sum0 oveq1d addid2d eqtr2d oveq2d seqeq1 fveq1d oveq12d eqeq2d syl5ibrcom + sum0 oveq1d addlidd eqtr2d oveq2d seqeq1 fveq1d oveq12d eqeq2d syl5ibrcom oveq1 addcl adantl addass simplr simpll zcnd ax-1cn sylancl eqcomd fveq2d w3a npcan eqtrid eleqtrd eluzp1m1 sylan syl2an seqsplit fsumser eqtr4d ex seqeq1d wo uzp1 mpjaod climaddc2 isumclim ) ABEFUDUEOZUFOZBCUGZGBCUGZPOCD @@ -179340,7 +179340,7 @@ Infinite sums (cont.) ( cn0 wcel c2 c1 caddc co cfv cc0 cle wceq vx vj cexp cmin cseq wbr cv wi oveq1 0p1e1 eqtrdi oveq2d cc exp1 ax-mp df-2 eqtri oveq1d ax-1cn pncan3oi 2cn fveq2d fveq2 breq12d imbi2d fvoveq1d cmul cr eleq1d ralrimiva 1nn a1i - cn rspcdva leidd recnd mulid2d breqtrrd 1z eqidd seq1i oveq2 exp0 oveq12d + cn rspcdva leidd recnd mullidd breqtrrd 1z eqidd seq1i oveq2 exp0 oveq12d 0z eqeq12d 0nn0 3brtr4d wa cfz csu fzfid cuz 2nn peano2nn0 adantl nnexpcl simpl sylancr elfzuz eluznn syl2an syl2an2r adantr simpr simplll monoord2 syl2anc breq1d rspccva fsumle chash cfn hashcl syl nn0cnd nnred cun cz 2z @@ -179699,7 +179699,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof nnre lep1d wb nngt0 peano2re peano2nn nngt0d lerec syl22anc mpbid 3brtr4d c2 cexp fveq2d oveq12d cmpt fconstmpt 2nn nnexpcl mpan oveq2d nncn recidd nnne0 eqtrd mpteq2ia eqtr4i climcnds isumrecl arch wn cfz chash cfn fzfid - cc ax-1cn fsumconst sylancl nnnn0 hashfz1 oveq1d mulid1d 3eqtrd wss ssriv + cc ax-1cn fsumconst sylancl nnnn0 hashfz1 oveq1d mulridd 3eqtrd wss ssriv elfznn a1i 0le1 adantr isumless eqbrtrrd lenlt syl2anr nrexdv pm2.65i ) C UAUDZGZHIFUEZUCUFZUGJZUCKUBZYGYHUHGZYKYGIFHIUIUJZLHUKYGULFUFZHGZYNYMMINZY GHIYNUSUMZOYGYOUNUOYGUPBIUQZYFGZUPYMLUQYFGZYGYSCYRYFEURUTYGFUCBYMYNKGZYNB @@ -179737,7 +179737,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof oveq1d cn wo cv cexp elnn0 cmin 1zzd nnz elfzelz zcnd adantl id fsumshftm 1m1e0 oveq1i sumeq1i elfznn0 bcnp1n nn0cnd ax-1cn addcom sylancl sumeq2dv wa eqtr3d 1nn0 nnm1nn0 bcxmas sylancr eqtr4d 1cnd ppncand comraddd bcp1m1 - nnnn0 sqval eqcomd mulid2 oveq12d joinlmuladdmuld 3eqtrd c0 oveq2 sumeq1d + nnnn0 sqval eqcomd mullid oveq12d joinlmuladdmuld 3eqtrd c0 oveq2 sumeq1d nncn fz10 sum0 sq0i 00id 2cn 2ne0 div0i jaoi sylbi ) BDEZBUAEZBFGZUBHBIJZ AUCZAKZBLUDJZBMJZLNJZGZBUEWPXDWQWPWTFBHUFJZIJZCUCZHMJZCKZHHMJXEMJZXEOJZXC WPWTHHUFJZXEIJZXHCKXIWPWSXHACHHBWPUGZXNBUHWSWREZWSPEWPXOWSWSHBUIUJUKWSXHG @@ -179756,7 +179756,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof ( ( ( N ^ 2 ) - N ) / 2 ) ) $= ( cn0 wcel cn cc0 wceq c1 cmin co cfz csu c2 caddc oveq1d cc eqtrd eqtrdi cdiv c0 wo cv cexp elnn0 cuz cfv nnm1nn0 nn0uz eleqtrdi wa elfznn0 adantl - nn0cnd id fsum1p 1e0p1 oveq1i sumeq1i oveq2i fzfid elfznn addid2d eqtr3id + nn0cnd id fsum1p 1e0p1 oveq1i sumeq1i oveq2i fzfid elfznn addlidd eqtr3id nncnd fsumcl arisum syl cmul nncn oveq2d sqcld subsub4d eqtr4d binom2sub1 2timesd subcld 1cnd subsubd 3eqtr4d ax-1cn subcl sylancl npcand oveq1 clt wbr cr 0re ltm1 ax-mp cz wb peano2zm mp2an mpbi sumeq1d sum0 sq0i oveq12d @@ -179784,7 +179784,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof cmul vk cseq cc0 cli wbr cv cmpt 1zzd 1cnd nnex mptex a1i oveq1 eqid ovex oveq2d fvmpt divcnvshft seqex wa peano2nn nnrecred eqeltrd cfz csu elfznn recnd nncnd peano2cn syl nnmulcld nnne0d divsubdird ax-1cn pncan2 sylancl - oveq1d mulid1d mulcomd oveq12d divcan5d eqtr3d 3eqtr3d sumeq2dv eqtrdi cz + oveq1d mulridd mulcomd oveq12d divcan5d eqtr3d 3eqtr3d sumeq2dv eqtrdi cz 1div1e1 nnz eleqtrdi telfsum eqtrd simpr fsumser 3eqtr2rd climsubc2 mptru cuz id 1m0e1 breqtri ) EBFUBZFUCGHZFUDXAXBUDUEIUCFUAAJFAUFZFEHZKHZUGZXAFL JMIUHZIFFUAXFFLJMXGIUIZXGXFLNIAJXEUJUKULUAUFZJNZXIXFOZFXIFEHZKHZPIAXIXEXM @@ -179901,7 +179901,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) ) $= ( cexp co cmin c1 csu wceq cmul wcel cn0 expcld oveq2 vj cdiv cfzo cv cfn caddc fzofi a1i cc ax-1cn subcl sylancr wa adantr cuz cfv elfzouz eluznn0 - syl2an fsummulc1 subdid mulid1d expp1d eqcomd oveq12d sumeq2dv cfz elfzuz + syl2an fsummulc1 subdid mulridd expp1d eqcomd oveq12d sumeq2dv cfz elfzuz telfsumo 3eqtrrd syl2anc subcld fsumcl cc0 wne necomd wb subeq0 necon3bid eqtrd mpbird divmul3d ) ABDJKZBEJKZLKZMBLKZUBKZDEUCKZBCUDZJKZCNZAWGWKOWEW KWFPKZOAWLWHWJWFPKZCNWHWJBWIMUFKZJKZLKZCNWEAWHWJWFCWHUEQADEUGUHZAMUIQZBUI @@ -179947,8 +179947,8 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof elfzoelz subcld npcan1 3eqtrd cfz cz nnz fzoval sumeq1d cuz nnm1nn0 nn0uz cfv eleqtrdi elfznn0 peano2nn0 nn0cnd sub32d fznn0sub eqeltrd oveq1 oveq2 1cnd fsumm1 eleq2d fzonnsub nnnn0d syl6bir imp fsum1p exp0d subid1d simp1 - mulid2d 0p1e1 fzfid elfznn cbvsumv 1m1e0 eqcomi oveq1i subsub4d sumeq12dv - eqtrid 1zzd peano2zm fsumshftm peano2cnm subidd 3ad2ant3 mulid1d comraddd + mullidd 0p1e1 fzfid elfznn cbvsumv 1m1e0 eqcomi oveq1i subsub4d sumeq12dv + eqtrid 1zzd peano2zm fsumshftm peano2cnm subidd 3ad2ant3 mulridd comraddd exp0 pnpcan2d 3eqtrrd 3exp mul01d fzo0 eqtrdi sum0 3ad2ant2 3eqtr4rd jaoi c0 sylbi 3imp ) DUBEZAUCEZBUCEZADFGZBDFGZHGZABHGZIDUDGZACUEZFGZBDUUJHGZJH GZFGZKGZCLZKGZMZUUBDUFEZDIMZUGUUCUUDUURUHUHZDUIUUSUVAUUTUUSUUCUUDUURUUSUU @@ -180007,7 +180007,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) ) $= ( cexp co c1 cmin cc0 cmul csu cz wcel wceq 1exp syl oveq2d cn0 cfz nn0zd cfzo cv eqcomd cc pwdif syl3anc fzoval wa adantr elfzoelz adantl peano2zm - 1cnd zsubcld 3syl elfzonn0 expcld mulid1d eqtrd sumeq12dv 3eqtrd ) ABDGHZ + 1cnd zsubcld 3syl elfzonn0 expcld mulridd eqtrd sumeq12dv 3eqtrd ) ABDGHZ IJHVDIDGHZJHZBIJHZKDUCHZBCUDZGHZIDVIJHZIJHZGHZLHZCMZLHZVGKDIJHUAHZVJCMZLH AIVEVDJAVEIADNOZVEIPADFUBZDQRUESADTOBUFOZIUFOVFVPPFEAUOBICDUGUHAVOVRVGLAV HVQVNVJCAVSVHVQPVTKDUIRAVIVHOZUJZVNVJILHVJWCVMIVJLWCVKNOVLNOVMIPWCDVIAVSW @@ -180114,7 +180114,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof halfcn expcl 2cnd wne divrecd elfzelz peano2zd exprecd 3eqtr2rd peano2nn0 expp1 syl div12d 3eqtr4d sumeq2dv fzfid halfcl fsummulc1 eqtr4d 1mhlfehlf 2ne0 eqtrid oveq12d simpr divrec2d nnnn0 nnrecred recnd subcl 0re halfgt0 - ax-1cn gtneii mulassd divcan1d oveq1d 3eqtr2d halfre ltneii geoser mulid2 + ax-1cn gtneii mulassd divcan1d oveq1d 3eqtr2d halfre ltneii geoser mullid halflt1 eqcomd subdird 3eqtrd ) CEFZAGFZUAZHCIJZATBUBZKJZLJZBMHHNJZCHNJZI JZATDUBZHUCJZKJZLJZDMZOXRIJZHTLJZXTKJZDMZATLJZPJZAATCKJZLJZNJZXLXPYCBDHHC XLUDZYNXJCUNFXKCUEUFZXLXNXMFZUAZAXOXJXKYPUGYQXOYQTEFZXNUHFXOEFUIYQXNYPXNE @@ -180190,7 +180190,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof wo sylan9eq oveq2d nncn mul01d simplll 0nn0 eqeltrdi expcld mul02d jaodan eqtrd sylan2b mpteq2dva eqtrid fconstmpt nn0uz xpeq1i eqtr3i eqtrdi cz 0z seqeq3d serclim0 ax-mp eqbrtrdi seqex c0ex breldm syl cdiv wral wrex 1red - wne abscl peano2re rehalfcld crp absrpcl adantlr rerpdivcld recnd mulid2d + wne abscl peano2re rehalfcld crp absrpcl adantlr rerpdivcld recnd mullidd c2 wb 1re avglt1 sylancl mpbid eqbrtrd ltmuldivd expmulnbnd syl3anc nn0cn eluznn0 ad2antrr rpne0d expdivd breq12d nn0re reexpcl sylan rpgt0d expgt0 ltmuldiv syl112anc bitr4d sylan2 anassrs ralbidva wi oveq2 fvmpt reexpcld @@ -180304,7 +180304,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof clt wbr cdc cv wne cn0 nnnn0 expcl sylancr a1i gt0ne0ii nnz expne0d mp3an2i divrec exprecd oveq2d eqtr4d sumeq2i cabs cfv rereccli cle 0re ltleii ax-mp recgt0ii absidi 1lt10 cr recgt1 mp2an mpbi eqbrtri geoisum1c mp3an divcan2i - wb divreci ax-1cn subdii mulid1i recidi oveq12i 10m1e9 3eqtrri 9re redivcli + wb divreci ax-1cn subdii mulridi recidi oveq12i 10m1e9 3eqtrri 9re redivcli eqtri subcli mulcani 9pos divgt0ii dividi 3eqtr2i ) BCDEUAZAUBZFGZHGZAIBCDW NHGZWOFGZJGZAIZDBWQWTAWOBKZWQCDWPHGZJGZWTCLKZXBWPLKZWPEUCWQXDMNXBWNLKZWOUDK XFWNOPZWOUEWNWOUFUGXBWNWOXGXBXHUHZWNEUCXBWNOQUIZUHZWOUJZUKCWPUMULXBWSXCCJXB @@ -180360,7 +180360,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof shftval syl2an 3eqtr4rd seqfeq seqshft syl2anc subidd seqeq1d oveq1d cdiv 3eqtrd recnd clt max2 sylancl absidd 0lt1 breq1 ifboth eqbrtrd geolim cvv c1 wb seqex climshft mpbird breldm eleq1d ralrimiva rspcdva abscld 2fveq3 - fveq2 wi breq12d imbi2d leidd exp0d mulid1d breqtrrd remulcld w3a lemul2a + fveq2 wi breq12d imbi2d leidd exp0d mulridd breqtrrd remulcld w3a lemul2a eqtrd ex syl112anc mul12d expp1d wral eleq2s ax-1cn addsub mp3an2 syl2anr mulcomd breq2d sylibd cbvralvw sylib peano2uzs sylan2 max1 lemul1ad letrd absge0d peano2uz letr syl3anc mpand syld expcom a2d uzind4i impcom iserex @@ -181434,7 +181434,7 @@ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) \/ $( Lemma for ~ prodrb . (Contributed by Scott Fenton, 4-Dec-2017.) $) prodrblem $p |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) |` ( ZZ>= ` N ) ) = seq N ( x. , F ) ) $= - ( cuz cfv wa cc c1 wcel co wceq adantl cz vn cmul cv mulid2 1cnd adantr + ( cuz cfv wa cc c1 wcel co wceq adantl cz vn cmul cv mullid 1cnd adantr wss iftrue adantlr eqeltrd ex wn iffalse ax-1cn eqeltrdi pm2.61d1 fmptd cif uzssz sselid ffvelcdmd cmin cdif elfzelz caddc simplr npcand fveq2d cfz zcnd sseqtrrd fznuz ssneldd eldifd fveqeq2 eldifi eldifn syl fvmpt2 @@ -181455,7 +181455,7 @@ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) \/ seq M ( x. , F ) ~~> ( seq M ( x. , F ) ` N ) ) $= ( cmul cfv wcel cz syl cc wa c1 wceq vn vm cseq cvv cuz eluzelz seqex a1i eluzel2 cv cif adantl iftrue adantlr eqeltrd ex wn iffalse ax-1cn - eqid eqeltrdi pm2.61d1 fvmpt2 syl2anc prodf ffvelcdmd co mulid1 simpr + eqid eqeltrdi pm2.61d1 fvmpt2 syl2anc prodf ffvelcdmd co mulrid simpr adantr caddc cfz elfzuz cdif sseld fznuz syl6 con2d imp eldifd eldifi fveqeq2 eldifn eqtrd vtoclga sylan2 seqid2 eqcomd climconst ) AGLEFUC ZMZUAWJGUDGUEMZWLUTAGFUEMZNZGONJFGUFPWJUDNALEFUGUHAWMQGWJADEFWMWMUTZA @@ -181557,7 +181557,7 @@ seq M ( x. , F ) ~~> ( seq 1 ( x. , G ) ` N ) ) $= cn wa cle wbr ccnv f1ocnvfv2 sylan f1ocnv ffvelcdmda elfzle2 cxr adantr wss cr fzssuz cz uzssz zssre sstri ressxr a1i sstrdi leisorel syl122anc eqbrtrrd eluzelz eluz syl2anc mpbird elfzuzb sylanbrc ex ssrdv fprodcvg - cc mulid2 adantl mulid1 mulcl 1cnd eleqtrrd cif iftrue eqeltrd eqeltrdi + cc mullid adantl mulrid mulcl 1cnd eleqtrrd cif iftrue eqeltrd eqeltrdi wn ax-1cn csb wi nfv nfcsb1v nfcv nfif csbeq1a ifbieq1d cmpt weq fvmptg eleq1 iffalse pm2.61d1 fmptd elfzelz ffvelcdm syl2an cdif eldifi eldifn fveqeq2 elfzelzd fvmpt2 eqtrd vtoclga iftrued nfim eleq1d imbi2d vtoclf @@ -182399,7 +182399,7 @@ seq n ( x. , ( cprod cmul co wcel c1 wa cc wceq adantl iffalsed cv cif iftrue prodeq2i ssun1 sseqtrrid sselda syldan eqeltrd cdif eldifn fprodss eqtr3id oveq12d cun ssun2 ax-1cn ifcl sylancl fprodmul wo eleq2d elun bitrdi biimpa c0 wn - cin disjel sylan mulid1d eqtrd ex con2d mulid2d jaodan prodeq2dv 3eqtr2rd + cin disjel sylan mulridd eqtrd ex con2d mullidd jaodan prodeq2dv 3eqtr2rd imp ) ABDFKZCDFKZLMEFUAZBNZDOUBZFKZEWBCNZDOUBZFKZLMEWDWGLMZFKEDFKAVTWEWAW HLAVTBWDFKWEBWDDFWCDOUCZUDABEWDFABCUOZBEBCUEHUFZAWCPZWDDQWCWDDRAWJSZAWCWB ENZDQNZABEWBWLUGJUHZUIWBEBUJNZWDORAWRWCDOWBEBUKTSIULUMAWACWGFKWHCWGDFWFDO @@ -183086,7 +183086,7 @@ seq n ( x. , fprodle $p |- ( ph -> prod_ k e. A B <_ prod_ k e. A C ) $= ( vj cc0 cle wbr co wcel adantr adantlr wceq wral cprod wa cmul cdiv 1red wne c1 nfra1 nfan cfn cv cr rspa adantll redivcld fprodreclf fprodge0 crp - ne0gt0d elrpd divge1 syl3anc fprodge1 lemul2ad fprodclf mulid1d fproddivf + ne0gt0d elrpd divge1 syl3anc fprodge1 lemul2ad fprodclf mulridd fproddivf recnd cc oveq2d fprodn0f divcan2d eqtrd 3brtr3d wn csb wrex rexbii rexnal nne nfcsb1v nfeq1 csbeq1a eqeq1d cbvrexw 3bitr3i nf3an 3ad2ant1 3ad2antl1 nfv w3a simp2 biimparc 3ad2antl3 fprodeq0g rexlimdv3a imp eqbrtrd sylan2b @@ -183512,7 +183512,7 @@ seq m ( x. , G ) ~~> z ) ) $. ( cc wcel cn0 wa cfallfac co cneg cexp cmul crisefac oveq2d neg1cn 3eqtr3rd c1 wceq adantl fallfaccl sylan c2 caddc nn0cn 2timesd cz nn0z m1expeven syl expadd mp3an1 anidms negneg adantr oveq1d oveq12d expcl mpan negcld mulassd - negcl mulid2d risefallfac eqtr4d ) BCDZAEDZFZBAGHZPIZAJHZVIBIZIZAGHZKHZKHZV + negcl mullidd risefallfac eqtr4d ) BCDZAEDZFZBAGHZPIZAJHZVIBIZIZAGHZKHZKHZV IVJALHZKHVFVIVIKHZVLKHPVGKHVNVGVFVPPVLVGKVEVPPQVDVEVHUAAKHZJHZVHAAUBHZJHZPV PVEVQVSVHJVEAAUCUDMVEAUEDVRPQAUFAUGUHVEVTVPQZVHCDZVEVEWANVHAAUIUJUKORVFVKBA GVDVKBQVEBULUMUNUOVFVIVIVLVEVICDZVDWBVEWCNVHAUPUQRZWDVDVKCDVEVLCDVDVJBUTZUR @@ -183592,14 +183592,14 @@ seq m ( x. , G ) ~~> z ) ) $. 5-Jan-2018.) $) risefac1 $p |- ( A e. CC -> ( A RiseFac 1 ) = A ) $= ( cc wcel c1 crisefac cc0 caddc 0p1e1 oveq2i cmul wceq 0nn0 risefacp1 mpan2 - co cn0 risefac0 addid1 oveq12d mulid2 3eqtrd eqtr3id ) ABCZADEOAFDGOZEOZAUD + co cn0 risefac0 addrid oveq12d mullid 3eqtrd eqtr3id ) ABCZADEOAFDGOZEOZAUD DAEHIUCUEAFEOZAFGOZJOZDAJOAUCFPCUEUHKLAFMNUCUFDUGAJAQARSATUAUB $. $( The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.) $) fallfac1 $p |- ( A e. CC -> ( A FallFac 1 ) = A ) $= ( cc wcel c1 cfallfac co cc0 caddc oveq2i cmin cmul cn0 wceq 0nn0 fallfacp1 - 0p1e1 mpan2 fallfac0 subid1 oveq12d mulid2 3eqtrd eqtr3id ) ABCZADEFAGDHFZE + 0p1e1 mpan2 fallfac0 subid1 oveq12d mullid 3eqtrd eqtr3id ) ABCZADEFAGDHFZE FZAUEDAEPIUDUFAGEFZAGJFZKFZDAKFAUDGLCUFUIMNAGOQUDUGDUHAKARASTAUAUBUC $. ${ $d N k $. @@ -183691,9 +183691,9 @@ seq m ( x. , G ) ~~> z ) ) $. eqtr4d uzid peano2uz fzss2 4syl sselda syldan peano2nn0 3eqtrd fallfacp1d sumeq2dv sylan9eq fzfid fsummulc1 bcpasc peano2zm adddird eqtr3d eleqtrdi syl nn0uz oveq2 fsump1 clt wbr wo nn0red ltp1d olcd bcval4 syl3anc subidd - sylancl eqeltrd syl2anc mul02d fsumcl addid1d oveq1 df-neg eqtr4di fsum1p + sylancl eqeltrd syl2anc mul02d fsumcl addridd oveq1 df-neg eqtr4di fsum1p 0nn0 cneg neg1lt0 orci mp3an23 subid1d 1zzd 0zd binomfallfaclem1 fsumshft - neg1z zcnd subsub3d npcand eqtr2d weq cbvsumv addid2d fsumadd ) ABUBZKFUC + neg1z zcnd subsub3d npcand eqtr2d weq cbvsumv addlidd fsumadd ) ABUBZKFUC LZFEUDZUELZCFUUCMLZNLZDUUCNLZOLZOLZCDPLZFMLZOLZEUGZUUBUUDCFQPLZUUCMLZNLZU UGOLZOLZUUDUUFDUUCQPLZNLZOLZOLZPLZEUGZUUJUUNNLZKUUNUCLZUUNUUCUELZUUQOLZEU GZAUUMUVDUFBAUUBUULUVCEAUUCUUBRZUBZUULUUDUUHUUKOLZOLUUDUUQUVAPLZOLUVCUVKU @@ -183948,7 +183948,7 @@ seq m ( x. , G ) ~~> z ) ) $. bpoly1 $p |- ( X e. CC -> ( 1 BernPoly X ) = ( X - ( 1 / 2 ) ) ) $= ( vk cc wcel c1 cbp cc0 cmin cfz cbc caddc cdiv cmul csu wceq 1nn0 eqtrdi co c2 oveq12d cexp cv cn0 bpolyval mpan 1m1e0 oveq2i sumeq1i cz 0z bpoly0 - exp1 oveq1d oveq2d halfcn mulid2i eqeltrdi oveq2 bcn0 ax-mp oveq1 eqtr4di + exp1 oveq1d oveq2d halfcn mullidi eqeltrdi oveq2 bcn0 ax-mp oveq1 eqtr4di 1m0e1 df-2 fsum1 sylancr eqtrd eqtrid ) ACDZEAFRZAEUARZGEEHRZIRZEBUBZJRZV NAFRZEVNHRZEKRZLRZMRZBNZHRZAESLRZHREUCDZVIVJWBOPBEAUDUEVIVKAWAWCHAULVIWAG GIRZVTBNZWCVMWEVTBVLGGIUFUGUHVIWFEGAFRZSLRZMRZWCVIGUIDWICDWFWIOUJVIWIWCCV @@ -183989,7 +183989,7 @@ seq m ( x. , G ) ~~> z ) ) $. cz elfznn0 simpr bpolycl syl2anr fznn0sub adantl nn0p1nn syl nncnd nnne0d cn divcld mulcld oveq2 oveq1 oveq1d fsumm1 bcnn adantr nn0cn subidd 0p1e1 eqtrdi div1d eqtrd bpolyval eqcomd expcl ancoms fzfid fzssp1 ax-1cn npcan - mulid2d sylancl sseqtrid sselda syldan fsumcl subaddd mpbid 3eqtrd ) BDEZ + mullidd sylancl sseqtrid sselda syldan fsumcl subaddd mpbid 3eqtrd ) BDEZ CFEZGZHBIJZBAUAZKJZWTCLJZBWTMJZNTJZOJZPJZAUBHBNMJZIJZXFAUBZBBKJZBCLJZBBMJ ZNTJZOJZPJZTJXIXKTJZCBUCJZWRXFXOAHBWRBDHUDUEWPWQUFZUGUHWRWTWSEZGZXAXEXTXA WRWPWTUMEXADEXSXRWTHBUIWTBUJUKULXTXBXDXSWTDEWQXBFEWRWTBUNWPWQUOWTCUPUQXTX @@ -184018,7 +184018,7 @@ seq m ( x. , G ) ~~> z ) ) $. cexp syl2an nn0cnd elfznn0 bpolycl syl2anr cn fzssp1 nncnd ax-1cn npcan bccl sylancl oveq2d sseqtrid sselda nn0p1nn nnne0d divcld mulcld fsumcl fznn0sub sub4d c2 bccl2 adantl expcl syldan addcom oveq1d binom1p eqtrd - cuz cfv nn0uz eleqtrdi fsumm1 bcnn mulid2d 3eqtrd mvrraddd nnm1nn0 1cnd + cuz cfv nn0uz eleqtrdi fsumm1 bcnn mullidd 3eqtrd mvrraddd nnm1nn0 1cnd oveq2 subsub4d df-2 oveq2i eqtr4di sumeq1d bcnm1 oveq1 fsum1p bpoly0 0z subid1d eqeltrd peano2nnd nnrecred recnd wss fzp1ss ax-mp sseli pnpcand sylan2 1zzd 0zd nnzd 2z zsubcl 2cnd subsubd 2m1e1 fsumshft 0p1e1 oveq1i @@ -184125,7 +184125,7 @@ seq m ( x. , G ) ~~> z ) ) $. oveq2 oveq12d cexp cv csu c6 2nn0 bpolyval mpan 2m1e1 0p1e1 eqtr4i oveq2i cn0 sumeq1i cneg cuz cfv 0nn0 nn0uz eleqtri a1i wo cpr cz 0z ax-mp eleq2i fzpr vex elpr bitri wa bcn0 oveq1 oveq1d 2cn subid1i oveq1i bpoly0 oveq2d - df-3 3cn 3ne0 reccli mulid2i sylan9eqr eqeltrdi eqeq2i ax-1cn npcan mp2an + df-3 3cn 3ne0 reccli mullidi sylan9eqr eqeltrdi eqeq2i ax-1cn npcan mp2an bcn1 sylbi bpoly1 halfcn subcl mpan2 wne divcan2 mp3an23 syl eqtrd adantr eqeltrd jaodan sylan2b fsump1 fsum1 sylancr addsub12 negsubdi2i halfthird 2ne0 mp3an13 negeqi eqtr3i 6pos gt0ne0ii negsub 3eqtrd eqtrid sqcl subsub @@ -184157,7 +184157,7 @@ seq m ( x. , G ) ~~> z ) ) $. sumeq1i c6 cuz cfv 1eluzge0 a1i w3o ctp cpr csn cun cz ax-mp 0p1e1 preq2i 0z fzpr 3eqtr3ri sneqi uneq12i df-tp fzsuc 3eqtr4ri eleq2i vex eltp bitri oveq2 bcn0 eqtrdi oveq1 subid1i oveq1i df-4 eqtr4i bpoly0 oveq2d 4cn 4ne0 - wa reccli mulid2i sylan9eqr eqeltrdi bcn1 ax-1cn npcan mp2an bpoly1 subcl + wa reccli mullidi sylan9eqr eqeltrdi bcn1 ax-1cn npcan mp2an bpoly1 subcl halfcn mpan2 wne 3ne0 divcan2 mp3an23 syl eqtrd eqeltrd bcn2 2cn divcan4i adantr 2p1e3 subaddrii bpolycl 2cnne0 div12 mp3an13 divcli mulcom sylancl 2ne0 2nn0 6cn subdi 3eqtr2d 3eqtrd mulcl sylancr eqeq2i fsump1 cneg 2t2e4 @@ -184166,7 +184166,7 @@ seq m ( x. , G ) ~~> z ) ) $. wo fsum1 eqtr3id addcl addsub12d negsubdi2d addsub12 mulcli negsub mul12i negsubdi2i 3eqtr3i divmuldivi oveq12i divdiri negeqi eqtr3i subcli negcli 3t2e6 2t1e2 subeq0i id subadd4d subdir mp3an12 divsubdir 2div2e1 3eqtr3rd - mpbir mulid2 subid1d 3eqtr3a negeqd eqtr3d negsubd eqtrid expcl subsubd ) + mpbir mullid subid1d 3eqtr3a negeqd eqtr3d negsubd eqtrid expcl subsubd ) ACDZEAFGZAEUAGZHEIJGZUBGZEBUCZUDGZUXCAFGZEUXCJGZIKGZLGZMGZBUEZJGZUWTERLGZ ARUAGZMGZIRLGZAMGZJGZJGUWTUXNJGUXPKGEUFDZUWRUWSUXKNUGBEAUHUIUWRUXJUXQUWTJ UWRUXJHIIKGZUBGZUXIBUEZUXQUXBUXTUXIBUXAUXSHUBUXARUXSUJUKULOUMUWRUYAHIUBGZ @@ -184227,16 +184227,16 @@ seq m ( x. , G ) ~~> z ) ) $. adantl bpolycl sylan ancoms 4re zred resubcld peano2re syl recnd wne 1red clt wbr eleq2i 3re elfzle2 sylbir divcld mulcld eqeq2i oveq2 eqtrdi oveq1 cr 3lt4 oveq1d 4cn 3cn ax-1cn subaddrii oveq1i eqtr4i fsump1 sseli sylan2 - df-2 0p1e1 eleqtri ax-mp 2p1e3 5cn 0re gtneii oveq2d reccli mulid2i eqtrd + df-2 0p1e1 eleqtri ax-mp 2p1e3 5cn 0re gtneii oveq2d reccli mullidi eqtrd 0nn0 3ne0 div12d 3t2e6 divmuli mpbir mulcomd eqtrid 3nn0 2ne0 expcl mpan2 eqnetri subcld subsubd addcld adddid subdid recidi mulassd 3eqtr3a add12d 6cn addassd 3eqtr2d 3eqtrd lelttrd posdifd mpbid addgt0d gt0ne0d 1eluzge0 0lt1 4bc3eq4 3p1e4 fzssp1 sseqtrri 4bc2eq6 2p2e4 nn0uz wss 3nn nnuz fzss2 cn 3sstr4i bcn1 df-4 sylbi 5pos bcn0 subid1i 4p1e5 fsum1 bpoly0 1nn0 mp1i 0z nn0cn 4ne0 divcan2d bpoly1 eqtr3id 2nn0 bpoly2 4d2e2 bpoly3 sqcl deccl - nn0cni dfdec10 10re recni mulcli addid1i 10pos mulne0i 6pos divcli mulid2 + nn0cni dfdec10 10re recni mulcli addridi 10pos mulne0i 6pos divcli mullid addcomd divcan2i eqtr3di mul32i eqeltri 3eqtr3ri addsub12d add4d subsub4d - id divreci subdird mulid2d 2txmxeqx subadd23d 3eqtr3d halfthird 5recm6rec + id divreci subdird mullidd 2txmxeqx subadd23d 3eqtr3d halfthird 5recm6rec subcli npncand eqtr3d addsubassd 3eqtr4d ) ACDZEAUAFZAEUBFZGEHIFZJFZEBUCZ UDFZUYCAUAFZEUYCIFZHKFZLFZMFZBUEZIFZUXTHUFLFZAHNLFZIFZKFZNANUBFZAIFZHOLFZ KFZMFZKFZNASUBFZSNLFZUYPMFZIFZUYMAMFZKFZMFZKFZIFZUXTNVUBMFZIFZUYPKFHSGUGZ @@ -184315,7 +184315,7 @@ seq m ( x. , G ) ~~> z ) ) $. cneg peano2cn bpoly4 cn 4nn 0exp ax-mp 3nn 2t0e0 eqtri 0m0e0 sq0 00id 0cn df-neg 3eqtr4i a1i oveq12d 4nn0 expcl mpan2 2cn mulcl sylancr subcld sqcl addcld 0nn0 deccl nn0cni nn0rei 10pos declti gt0ne0ii reccli subcl subneg - sylancl npcan 2p2e4 eqcomi df-3 expadd mp3an23 sqcld mulid1d eqcomd exp1d + sylancl npcan 2p2e4 eqcomi df-3 expadd mp3an23 sqcld mulridd eqcomd exp1d 2nn0 1nn0 eqeltrd mul12 mp3an2i subdid 3eqtr4d oveq1d ax-1cn adddi mp3an3 syl2anc eqtr4d mp3an13 2t1e2 eqtrdi addsubass 2m1e1 subsub binom21 addass eqtrd mvrraddd 3eqtr3d mulcomd 3eqtrd eqtrid eqtr3id ) AUACZDAUBEBUCFGEBU @@ -184463,7 +184463,7 @@ seq m ( x. , G ) ~~> z ) ) $. 2re 0le2 flge0nn0 syl2anc cv wa cexp cfa eftval adantl eftcl caddc adantr eqeltrd cn eluznn0 nn0p1nn nndivred reexpcld faccld syldan absge0d absexp sylan expcl nnred nngt0d divge0 syl22anc peano2nn0 nn0red flltp1 eluzp1p1 - breqtrd eluzle ltletrd recnd 2cn mulcom sylancl nncnd mulid2d 3brtr4d crp + breqtrd eluzle ltletrd recnd 2cn mulcom sylancl nncnd mullidd 3brtr4d crp 2rp 1red nnrpd lt2mul2divd mpbid wi ltle mpd lemul2ad fveq2d expp1d eqtrd nnnn0d nn0ge0d absidd oveq12d nnne0d absdivd 3eqtr4d halfcn abscld eftabs facp1 divmuldivd oveq1d cvgrat ) AEFZGHIJZUCCKHALMZNJZUAMZYIUBMZUDUEYJUFY @@ -184699,7 +184699,7 @@ seq m ( x. , G ) ~~> z ) ) $. nncnd eftcl cfz cmin cbc adantr simpr binom syl3anc oveq1d fzfid ad2antrr bccl2 fznn0sub expcld elfznn0 mulcld fsumdivc divcld eqeltrd oveq2 fveq2d fveq2 fsumrev2 oveq2d nnmulcld divrec2d divmuldivd bcval2 divdiv32d eqtrd - c1 wne dividd eqtr4d nn0cn ad2antlr addid2d nncand div23d sumeq2dv eqtrdi + c1 wne dividd eqtr4d nn0cn ad2antlr addlidd nncand div23d sumeq2dv eqtrdi cbvsumv cdm abscld recnd efcllem mertens efval breqtrrd climuni syl2anc ) AUCGUDUJZBCUCNZUEOZUFUGZYHBUEOZCUEOZPNZUFUGYJYNQAYIUHRYKABCKLUIYIDGJUKULA YHUMBMUNZSNZYOUOOZTNZMUPZUMCUAUNZSNYTUOOZTNZUAUPZPNYNUFAYRUUBMUAEFGDUMBUQ @@ -184847,7 +184847,7 @@ seq m ( x. , G ) ~~> z ) ) $. ( cc wcel wa cmul co ce cfv cexp sylan2 fveq2d cc0 c1 caddc oveq2 eqeq12d wceq adantr eqtrd vj vk cz zcn mulcom cv cneg ef0 mul01 exp0d 3eqtr4a cn0 efcl oveq1 adantl nn0cn ax-1cn adddi mp3an3 mulcl simpl efadd expp1 sylan - mulid1 oveq2d syl2anc 3eqtr4d exp31 cn cdiv nncn mulneg2 efneg syl expneg + mulrid oveq2d syl2anc 3eqtr4d exp31 cn cdiv nncn mulneg2 efneg syl expneg wi nnnn0 syl2an syl5ibr ex zindd imp eqtr3d ) ACDZBUCDZEZABFGZHIZBAFGZHIA HIZBJGZWGWHWJHWFWEBCDWHWJRBUDABUEKLWEWFWIWLRZAUAUFZFGZHIZWKWNJGZRAMFGZHIZ WKMJGZRAUBUFZFGZHIZWKXAJGZRZAXAUGZFGZHIZWKXFJGZRZAXANOGZFGZHIZWKXKJGZRZWM @@ -184867,7 +184867,7 @@ seq m ( x. , G ) ~~> z ) ) $. Equation 30 of [Rudin] p. 164. (Contributed by Steve Rodriguez, 15-Sep-2006.) (Revised by Mario Carneiro, 5-Jun-2014.) $) efzval $p |- ( N e. ZZ -> ( exp ` N ) = ( _e ^ N ) ) $= - ( cz wcel ce cfv c1 cexp co ceu cmul zcn mulid1d fveq2d cc wceq ax-1cn mpan + ( cz wcel ce cfv c1 cexp co ceu cmul zcn mulridd fveq2d cc wceq ax-1cn mpan efexp eqtr3d df-e oveq1i eqtr4di ) ABCZADEZFDEZAGHZIAGHUCAFJHZDEZUDUFUCUGAD UCAAKLMFNCUCUHUFOPFARQSIUEAGTUAUB $. @@ -184949,9 +184949,9 @@ seq m ( x. , G ) ~~> z ) ) $. wbr cz cn expge0d jca faclbnd6 mpbid divassd divdiv1d eqtrd breqtrd letrd nnne0d clt wb mpbird eqbrtrd cmin sylancr seqex sylancl breldm eqtr4d wne lemul1a syl31anc lemuldiv2d nn0z exprecd divrecd facne0 3eqtr2rd ledivmul - nnrpd nngt0d syl112anc 0z znegcld seqshft 0cn subneg mpan addid2 climshft + nnrpd nngt0d syl112anc 0z znegcld seqshft 0cn subneg mpan addlid climshft seqeq1d sumex nnge1d nnleltp1 nn0ge0d absidd breqtrrd georeclim isermulc2 - 1nn ax-1cn pncan div23d mulcomd 3eqtrd isumle fveq2 addid2d eleq2d biimpa + 1nn ax-1cn pncan div23d mulcomd 3eqtrd isumle fveq2 addlidd eleq2d biimpa isumshft sumeq1d eqtr3d isumclim 3brtr3d ) AHUBOZCUCZEOZCUDZUEOUWGUWHFOZC UDZBUEOZHULPZHUFUGPZHUHOZHQPZRPZQPZAUWJABUISZHUJSZUWJUISMAHLUKZBCDEHIUMUN UOAUWMUPSZUXAUWLUPSABMUOZUXBUWMCDFHJUQUNAUWNUWRAUWMHUXDUXBURZAUWOUWQAUWOA @@ -185071,7 +185071,7 @@ seq m ( x. , G ) ~~> z ) ) $. sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) $= ( cc wcel c1 caddc co c2 cexp cdiv c3 a1i cc0 cfv cfa efsep oveq2i ax-1cn c6 c4 df-4 3nn0 id addcl mpan sqcl halfcld df-3 2nn0 df-2 1nn0 1e0p1 0nn0 - addcld 0cnd cuz cv csu ce cn0 efval2 nn0uz sumeq1i eqtr2di oveq2d addid2d + addcld 0cnd cuz cv csu ce cn0 efval2 nn0uz sumeq1i eqtr2di oveq2d addlidd efcl eqtr2d eft0val 0p1e1 eqtrdi exp1 wceq fac1 oveq12d div1 eqtrd fac2 fac3 ) AFGZAHAIJZAKLJZKMJZIJZWGANLJZUBMJZIJZBCDNUCEUDUEWCUFZWCWDWFHFGZWCW DFGUAHAUGUHZWCWEAUIUJUQWCAWDWGBCDKNEUKULWKWMWCAHWDBCDHKEUMUNWKWLWCUAOWCAP @@ -185497,8 +185497,8 @@ seq m ( x. , G ) ~~> z ) ) $. ( cc wcel ci cmul co ce cfv caddc c2 cdiv wceq ax-icn mulcl syl 2cn 3eqtr4d 2ne0 c1 eqtr3i cneg cmin ccos csin mpan negicn addcld subcld cc0 wne pm3.2i efcl wa divdir mp3an3 syl2anc pncan3d oveq2d addassd 2timesd oveq1d divcan3 - mp3an23 eqtr2d cosval 2mulicn 2muline0 mp3an13 sinval divrec mulid2i oveq1i - div12 ine0 dividi oveq2i ax-1cn divmuldivi halfcn mulid1i eqtr4di oveq12d ) + mp3an23 eqtr2d cosval 2mulicn 2muline0 mp3an13 sinval divrec mullidi oveq1i + div12 ine0 dividi oveq2i ax-1cn divmuldivi halfcn mulridi eqtr4di oveq12d ) ABCZDAEFZGHZDUAZAEFZGHZIFZWEWHUBFZIFZJKFZWIJKFZWJJKFZIFZWEAUCHZDAUDHZEFZIFW CWIBCZWJBCZWLWOLZWCWEWHWCWDBCZWEBCZDBCZWCXBMDANUEWDULOZWCWGBCZWHBCWFBCWCXFU FWFANUEWGULOZUGWCWEWHXEXGUHZWSWTJBCZJUIUJZUMXAXIXJPRUKWIWJJUNUOUPWCWLJWEEFZ @@ -185526,7 +185526,7 @@ seq m ( x. , G ) ~~> z ) ) $. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) $= ( cc wcel ci cmul co cfv cdiv ce cneg cmin c2 c1 oveq1i ax-icn eqtri oveq1d wceq a1i eqtrd csin ixi mulass mp3an12 mulm1 3eqtr3a fveq2d mulneg1i negeqi - negneg1e1 negicn mulid2 oveq12d mulcl mpan sinval syl irec ine0 reccli efcl + negneg1e1 negicn mullid oveq12d mulcl mpan sinval syl irec ine0 reccli efcl negnegi negcl halfcld mulneg12 sylancr 2cnd cc0 wne 2ne0 divnegd negsubdi2d subcld oveq2d divrec2d divdiv1d 3eqtr2d eqtr3id 3eqtr4d divcan3d ) ABCZDAEF ZUAGZDHFDAIGZAJZIGZKFZLHFZEFZDHFWHWAWCWIDHWADWBEFZIGZDJZWBEFZIGZKFZLDEFZHFZ @@ -185545,7 +185545,7 @@ seq m ( x. , G ) ~~> z ) ) $. ( ( ( exp ` A ) + ( exp ` -u A ) ) / 2 ) ) $= ( cc wcel ci cmul co cfv ce cneg caddc cdiv wceq ax-icn syl efcl ixi oveq1i c2 c1 mulass ccos mulcl cosval negcl mp3an12 3eqtr3a fveq2d mulneg1i negeqi - mpan mulm1 negneg1e1 3eqtri negicn mulid2 oveq12d comraddd oveq1d eqtrd ) A + mpan mulm1 negneg1e1 3eqtri negicn mullid oveq12d comraddd oveq1d eqtrd ) A BCZDAEFZUAGZDVAEFZHGZDIZVAEFZHGZJFZRKFZAHGZAIZHGZJFZRKFUTVABCZVBVILDBCZUTVN MDAUBUJVAUCNUTVHVMRKUTVHVLVJUTVKBCVLBCAUDVKONAOUTVDVLVGVJJUTVCVKHUTDDEFZAEF ZSIZAEFVCVKVPVRAEPQVOVOUTVQVCLMMDDATUEAUKUFUGUTVFAHUTVEDEFZAEFZSAEFVFAVSSAE @@ -185586,7 +185586,7 @@ seq m ( x. , G ) ~~> z ) ) $. cc ctan cneg cmin caddc csin ccos ax-icn recn mulcl sylancr rpcoshcl rpne0d tanval syl2anc oveq1d sincld recoshcl ine0 divdiv32d sinhval coshval 3eqtrd oveq12d reefcl renegcl resubcld readdcld 2cnd eqnetrrd 2ne0 divne0bd mpbird - reefcld divcan7d eqtrd rpefcld ltsubrpd ltaddrpd lttrd mulid1d 1red addgt0d + reefcld divcan7d eqtrd rpefcld ltsubrpd ltaddrpd lttrd mulridd 1red addgt0d breqtrrd wb efgt0 ltdivmul syl112anc eqbrtrd ) ABCZDAEFZUAGZDHFZAIGZAUBZIGZ UCFZWMWOUDFZHFZJKWIWLWPLHFZWQLHFZHFZWRWIWLWJUEGZWJUFGZHFZDHFXBDHFZXCHFXAWIW KXDDHWIWJTCZXCMNWKXDOWIDTCZATCZXFUGAUHZDAUIUJZWIXCAUKULZWJUMUNUOWIXBXCDWIWJ @@ -185689,7 +185689,7 @@ seq m ( x. , G ) ~~> z ) ) $. ( ( tan ` A ) x. ( tan ` B ) ) =/= 1 ) ) $= ( cc wcel wa ccos cfv cc0 co ctan cmul c1 csin wceq coscl ad2antrr ad2antlr wne eqeq1d cdiv caddc mulcld sincl subeq0ad cosadd adantr ad2ant2r ad2ant2l - cmin tanval oveq12d simprl simprr divmuldivd mulne0d divmuld mulid1d 3bitrd + cmin tanval oveq12d simprl simprr divmuldivd mulne0d divmuld mulridd 3bitrd eqtrd 1cnd 3bitr4d necon3bid ) ACDZBCDZEZAFGZHRZBFGZHRZEZEZABUAIFGZHAJGZBJG ZKIZLVKVFVHKIZAMGZBMGZKIZUIIZHNVPVSNZVLHNVOLNZVKVPVSVKVFVHVCVFCDVDVJAOPZVDV HCDVCVJBOQZUBZVKVQVRVCVQCDVDVJAUCPZVDVRCDVCVJBUCQZUBZUDVKVLVTHVEVLVTNVJABUE @@ -185706,7 +185706,7 @@ seq m ( x. , G ) ~~> z ) ) $. ( cc wcel wa ccos cfv cc0 caddc co ctan csin cdiv c1 cmul cmin wceq oveq12d wne eqtrd w3a addcl adantr simpr3 tanval sinadd cosadd simpll coscld simplr syl2anc mulcld simpr1 tancld simpr2 adddid mul32d oveq2d sincld oveq1d 1cnd - divcan2d mulassd subdid mulid1d mul4d addcld ax-1cn subcl sylancr tanaddlem + divcan2d mulassd subdid mulridd mul4d addcld ax-1cn subcl sylancr tanaddlem wb 3adantr3 necomd subeq0 necon3bid mpbird mulne0d divcan5d 3eqtr2rd eqtr4d mpbid ) ACDZBCDZEZAFGZHSZBFGZHSZABIJZFGZHSZUAZEZWJKGZWJLGZWKMJZAKGZBKGZIJZN WRWSOJZPJZMJZWNWJCDZWLWOWQQWEXDWMABUBUCWEWGWIWLUDZWJUEUKWNWQALGZWHOJZWFBLGZ @@ -185916,14 +185916,14 @@ seq m ( x. , G ) ~~> z ) ) $. csu cabs caddc cfa ci cc cn0 ax-icn cle cxr w3a wb 0xr 1re elioc2 simp1bi mp2an recnd mulcl sylancr 4nn0 eftlcl sylancl abscld reexpcl 4re readdcli cn faccl ax-mp 4nn nnmulcli nndivre remulcl 6nn cmpt eqid a1i wceq absmul - absi oveq1i crp simp2bi elrpd rpre rpge0 absidd syl oveq2d eqtrid mulid2d + absi oveq1i crp simp2bi elrpd rpre rpge0 absidd syl oveq2d eqtrid mullidd 3eqtrd simp3bi eqbrtrd eftlub oveq1d breqtrd 3pos 0re 3re 5re ltadd1i 5cn - mpbi addid2i c8 cu2 5p3e8 3cn addcomi 3eqtr2ri 3brtr3i 2re 1le2 cz ltleii + mpbi addlidi c8 cu2 5p3e8 3cn addcomi 3eqtr2ri 3brtr3i 2re 1le2 cz ltleii 4z 3lt4 3z eluz1i mpbir2an leexp2a mp3an eqeltri 6re 6pos df-4 sq2 oveq2i fac3 6cn recni 3eqtr3i 2nn0 eqtr3i 8re 2nn nnexpcl nnrei ltletri remulcli nngt0i mulgt0ii ltdiv1ii df-5 fveq2i 3nn0 facp1 eqtr2i 3eqtri mulassi 2cn - 2p2e4 expadd oveq12i nncni mulid2i nnne0i dividi ax-1cn gt0ne0ii rereccli - divmuldivi mulid1i rpexpcl wa elrp ltmul2 mp3an12 sylbi mpbii wne mp3an23 + 2p2e4 expadd oveq12i nncni mullidi nnne0i dividi ax-1cn gt0ne0ii rereccli + divmuldivi mulridi rpexpcl wa elrp ltmul2 mp3an12 sylbi mpbii wne mp3an23 divrec breqtrrd lelttrd ) AUAFUBGHZIUCJBUDDJBUEZUFJZAIKGZIFUGGZIUHJZILGZM GZLGZUWENMGZUWBUWCUWBUIALGZUJHZIUKHZUWCUJHUWBUIUJHZAUJHZUWMULUWBAUWBAOHZU AAPQZAFUMQZUAUNHFOHUWBUWQUWRUWSUOUPUQURUAFAUSVAZUTZVBZUIAVCVDZVEUWLBCDIEV @@ -186097,7 +186097,7 @@ seq m ( x. , G ) ~~> z ) ) $. ( cc0 c1 co wcel c2 c3 cdiv cmul clt wbr cle cc cr wb 1re wa 3ne0 2re 3re cioc cexp cmin ccos cfv wceq cxr w3a 0xr elioc2 mp2an simp1bi resqcld recnd wne 2cn 3cn pm3.2i div12 mp3an13 syl cz 2z expgt0 mp3an2 3adant3 sylbi 2lt3 - 3pos ltdiv1ii dividi breqtri redivcli mp3an12 mpbii syl2anc mulid1d breqtrd + 3pos ltdiv1ii dividi breqtri redivcli mp3an12 mpbii syl2anc mulridd breqtrd mpbi ltmul2 eqbrtrd wi 0re ltle imdistani le2sq2 mpanr1 stoic3 sq1 breqtrdi redivcl mp3an23 remulcl sylancr ltletr mp3an3 mp2and sylancl mpbid cos01bnd mpan posdif simpld resubcl recoscld lttr mp3an2i ) ABCUADEZBCFAFUBDZGHDZIDZ @@ -186171,7 +186171,7 @@ seq m ( x. , G ) ~~> z ) ) $. absef $p |- ( A e. CC -> ( abs ` ( exp ` A ) ) = ( exp ` ( Re ` A ) ) ) $= ( cc wcel ce cfv cabs cre c1 cmul co ci fveq2d wceq recnd sylancr cr 3eqtrd syl cc0 wbr caddc replim recl ax-icn imcl mulcl efadd syl2anc eqtrd reefcld - cim efcl absmuld absefi oveq2d abscld mulid1d clt cle efgt0 wi 0re ltle mpd + cim efcl absmuld absefi oveq2d abscld mulridd clt cle efgt0 wi 0re ltle mpd absidd ) ABCZADEZFEZAGEZDEZFEZHIJZVKVJVFVHVJKAUKEZIJZDEZIJZFEVKVOFEZIJVLVFV GVPFVFVGVIVNUAJZDEZVPVFAVRDAUBLVFVIBCVNBCZVSVPMVFVIAUCZNVFKBCVMBCVTUDVFVMAU EZNKVMUFOZVIVNUGUHUILVFVJVOVFVJVFVIWAUJZNZVFVTVOBCWCVNULRUMVFVQHVKIVFVMPCVQ @@ -186202,7 +186202,7 @@ seq m ( x. , G ) ~~> z ) ) $. cabs syl cc0 cim replim recl imcl mulcl sylancr adddi mp3an2i oveq1i mulass cre ixi mp3an12i mulm1d 3eqtr3a fveq2d renegcld syl2anc eqeq1d efcl absmuld efadd absefi reefcld clt wbr cle efgt0 wi 0re ltle mpan sylc absidd oveq12d - mulid2d 3eqtrrd fveq2 sylan9eq sylbid negeq0d reim0b ef0 abs1 eqtr4i eqeq2i + mullidd 3eqtrrd fveq2 sylan9eq sylbid negeq0d reim0b ef0 abs1 eqtr4i eqeq2i ex wb reef11 sylancl bitr3id 3bitr4rd sylibd imp ) ABCZDAEFZGHZIJZAKCZWOWRA UAHZLZGHZIRHZJZWSWOWRDAUKHZEFZGHZXBEFZIJZXDWOWQXHIWOWQXFXAMFZGHZXHWOWPXJGWO WPDXEDWTEFZMFZEFZXJWOAXMDEAUBNWOXNXFDXLEFZMFZXJDBCZWOXEBCZXLBCZXNXPJOWOXEAU @@ -186240,9 +186240,9 @@ seq m ( x. , G ) ~~> z ) ) $. ( ( cos ` ( N x. A ) ) + ( _i x. ( sin ` ( N x. A ) ) ) ) ) $= ( wcel cc ccos cfv ci csin cmul caddc cexp wceq cc0 fveq2d oveq2d oveq12d co c1 ax-icn syl2anc vx vk cn0 cv wi oveq2 oveq1 imbi2d coscl sincl mulcl - eqeq12d sylancr addcl exp0 mul02 cos0 eqtrdi mul01i ax-1cn addid1i eqtr4d + eqeq12d sylancr addcl exp0 mul02 cos0 eqtrdi mul01i ax-1cn addridi eqtr4d syl sin0 wa expp1 sylan ancoms adantr adantl nn0cn sinadd sylancom mulcom - oveq1d addcom 3eqtr2d w3a adddir mulid2 3ad2ant3 syl3an1 mp3an2 cmin cneg + oveq1d addcom 3eqtr2d w3a adddir mullid 3ad2ant3 syl3an1 mp3an2 cmin cneg eqtrd mpan 3syl jca muladd syl21anc jctil mul4 oveq1i mul12 mp3an1 3eqtrd ixi adddi mulm1 negsub cosadd 3eqtr4rd exp31 a2d nn0ind impcom ) BUCCADCZ AEFZGAHFZIQZJQZBKQZBAIQZEFZGXNHFZIQZJQZLZXHXLUAUDZKQZXTAIQZEFZGYBHFZIQZJQ @@ -186326,7 +186326,7 @@ The circle constant (tau = 2 pi) eqtr3d nnne0d div12d cle nnred leidd facdiv syl3anc nnzd fsummulc2 adantr zmulcld wne facne0 divrecd eqtr4d permnn fsumzcl zsubcld wn rpmulcld cabs 0zd rpgt0d abs1 oveq1i cr wb mpbid c2 nngt0d syl112anc mpbird 1le1 eftlub - nnmulcld nndivred nnrecred mpteq2i eqbrtri rprege0d absid mulid2d 3brtr3d + nnmulcld nndivred nnrecred mpteq2i eqbrtri rprege0d absid mullidd 3brtr3d readdcld remulcld 1red nnge1d 1nn nnleltp1 sylancr ltadd2dd df-2 leadd1dd 2timesd eqbrtrid lemul1 eqbrtrrd ltletrd divcan3d 3eqtr3rd mul32d breqtrd 2re facp1 ltdivmul lelttrd ltmuldiv2d 0p1e1 breqtrrdi btwnnz pm2.65i ) AC @@ -186630,7 +186630,7 @@ seq M ( + , ( F ` A ) ) e. dom ~~> ) $= cz wbr ssdifd sstrd ssconb mpbird difssd rpnnen2lem7 syl3anc caddc cmin wb cc0 wceq rpnnen2lem9 syl cmul cc recni expp1 3cn 3ne0 divrec mp3an23 recnd eqtr4d oveq1d ax-1cn pm3.2i divsubdir mp3an 3m1e2 oveq1i 3eqtr3ri - dividi oveq2i 2cnne0 divcan7 eqtrid eqtrd oveq2d addid2d 3eqtrd breqtrd + dividi oveq2i 2cnne0 divcan7 eqtrid eqtrd oveq2d addlidd 3eqtrd breqtrd wa clt rphalflt lelttrd eluznn sylan rpnnen2lem1 syl2an2r sumeq2dv wral cif cfn wo uzid vex oveq2 eleq1d ralsn sylibr ssidd orcd syl21anc sumsn sumss2 3eqtr2d eqbrtrrd ltletrd gtned rpnnen2lem10 ex necon3ad mpd ) AG @@ -187401,7 +187401,7 @@ infinite descent (here implemented by strong induction). This is ( ( A mod N ) = ( B mod N ) <-> N || ( A - B ) ) ) $= ( wcel cz cmo co wceq wb wa cc0 caddc cr zre modadd1 3expia syl22anc oveq1d wi recnd cn cmin wbr crp nnrp adantr 0mod syl eqeq2d cneg ad2antrl ad2antll - cdvds renegcld negsubd negidd eqeq12d sylibd resubcld npcand addid2d impbid + cdvds renegcld negsubd negidd eqeq12d sylibd resubcld npcand addlidd impbid 0red zsubcl dvdsval3 sylan2 3bitr4d 3impb ) CUADZAEDZBEDZACFGZBCFGZHZCABUBG ZUMUCZIVIVJVKJZJZVOCFGZKCFGZHZVSKHZVNVPVRVTKVSVRCUDDZVTKHVIWCVQCUEUFZCUGUHU IVRVNWAVRVNABUJZLGZCFGZBWELGZCFGZHZWAVRAMDZBMDZWEMDZWCVNWJSVJWKVIVKANUKZVKW @@ -187472,13 +187472,13 @@ infinite descent (here implemented by strong induction). This is property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) $) iddvds $p |- ( N e. ZZ -> N || N ) $= - ( cz wcel c1 cmul co wceq cdvds wbr zcn mulid2d 1z dvds0lem mp3anl1 anabsan + ( cz wcel c1 cmul co wceq cdvds wbr zcn mullidd 1z dvds0lem mp3anl1 anabsan mpdan ) ABCZDAEFAGZAAHIZQAAJKQRSDBCQQRSLDAAMNOP $. $( 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) $) 1dvds $p |- ( N e. ZZ -> 1 || N ) $= - ( cz wcel c1 cmul co wceq cdvds wbr zcn mulid1d 1z dvds0lem mp3anl2 anabsan + ( cz wcel c1 cmul co wceq cdvds wbr zcn mulridd 1z dvds0lem mp3anl2 anabsan mpdan ) ABCZADEFAGZDAHIZQAAJKQRSQDBCQRSLADAMNOP $. $( Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed @@ -188248,7 +188248,7 @@ infinite descent (here implemented by strong induction). This is (Revised by AV, 8-Sep-2021.) $) 3dvdsdec $p |- ( 3 || ; A B <-> 3 || ( A + B ) ) $= ( c3 cdc cdvds wbr c9 cmul co caddc c1 eqcomi oveq1i 9cn cz wcel 3z mp2an - dfdec10 9p1e10 ax-1cn nn0cni adddiri mulid2i oveq2i 3eqtri mulcli addassi + dfdec10 9p1e10 ax-1cn nn0cni adddiri mullidi oveq2i 3eqtri mulcli addassi cc0 breq2i wa wb nn0zi zaddcl 9nn zmulcl dvdsmul1 3t3e9 3cn mulassi eqtri nnzi breqtrri pm3.2i dvdsadd2b mp3an bitr4i ) EABFZGHEIAJKZABLKZLKZGHZEVL GHZVJVMEGVJMUKFZAJKZBLKVKALKZBLKVMABUAVQVRBLVQIMLKZAJKVKMAJKZLKVRVPVSAJVS @@ -188267,7 +188267,7 @@ infinite descent (here implemented by strong induction). This is 3dvds2dec $p |- ( 3 || ; ; A B C <-> 3 || ( ( A + B ) + C ) ) $= ( c3 cdc cdvds c1 cmul co caddc c9 oveq1i nn0cni 3eqtri wcel cz mp2an wbr cc0 c2 cexp 3dec sq10e99m1 9nn0 deccl ax-1cn adddiri oveq2i 9p1e10 eqcomi - mulid2i 9cn oveq12i cc wceq mulcli wa oveq1d mp4an addcli addassi 9t11e99 + mullidi 9cn oveq12i cc wceq mulcli wa oveq1d mp4an addcli addassi 9t11e99 add4 1nn0 mulassi eqtri adddii 3t3e9 3cn breq2i wb 3z nn0zi zaddcl zmulcl dvdsmul1 pm3.2i dvdsadd2b mp3an bitr4i ) GABHCHZIUAGGGJJHZAKLZBMLZKLZKLZA BMLZCMLZMLZIUAZGWKIUAZWDWLGIWDJUBHZUCUDLZAKLZWOBKLZMLZCMLNNHZAKLZAMLZNBKL @@ -188378,7 +188378,7 @@ infinite descent (here implemented by strong induction). This is ( vy c2 cv cmul co c1 caddc wceq cz wrex wo cc0 eqeq2 rexbidv eqeq1d wcel 2cn vj vx vm orbi12d weq oveq2 oveq1d cbvrexvw bitrdi oveq1 mul02i rspcev 0z mp2an olci cn0 orcom wa cc zcn mulcom sylancl adantl eqid mpan2 eqeq2d - syl5ibcom sylbid rexlimdva peano2z mulid2i a1i oveq12d df-2 oveq2i eqtrdi + syl5ibcom sylbid rexlimdva peano2z mullidi a1i oveq12d df-2 oveq2i eqtrdi wi ax-1cn adddir mp3an23 mpan addass syl 3eqtr4d syl2an2 orim12d biimtrid mulcl nn0ind ) EBFZGHZIJHZUAFZKZBLMZAFZEGHZWMKZALMZNWLOKZBLMZWQOKZALMZNEU BFZGHZIJHZUCFZKZUBLMZDFZEGHZXGKZDLMZNZWLXGIJHZKZBLMZWQXOKZALMZNZWLCKZBLMZ @@ -188828,7 +188828,7 @@ equal to the half of the predecessor of the odd number (which is an even m1expo $p |- ( ( N e. ZZ /\ -. 2 || N ) -> ( -u 1 ^ N ) = -u 1 ) $= ( vn cz wcel c2 cdvds wbr wn c1 cneg cexp co wceq cmul caddc wrex odd2np1 cv neg1cn a1i wa oveq2 eqcoms cc cc0 neg1ne0 2z zmulcld expp1zd m1expeven - wne id oveq1d mulid2i eqtrdi eqtrd adantl sylan9eqr rexlimdva2 sylbid imp + wne id oveq1d mullidi eqtrdi eqtrd adantl sylan9eqr rexlimdva2 sylbid imp ) ACDZEAFGHZIJZAKLZVDMZVBVCEBRZNLZIOLZAMZBCPVFBAQVBVJVFBCVJVBVGCDZUAVEVDV IKLZVDVEVLMAVIAVIVDKUBUCVKVLVDMVBVKVLVDVHKLZVDNLZVDVKVDVHVDUDDVKSTVDUEUKV KUFTVKEVGECDVKUGTVKULUHUIVKVNIVDNLVDVKVMIVDNVGUJUMVDSUNUOUPUQURUSUTVA $. @@ -188839,7 +188839,7 @@ equal to the half of the predecessor of the odd number (which is an even ( vn c2 cz wcel c1 cexp co wceq wb wa cmul wrex 2z oveq2 eqcoms sylan9eqr eqtrd cc0 a1i cdvds wbr cneg cv divides mpan zcn mulcomd oveq2d m1expeven 2cnd rexlimiva syl6bi impcom simpl 2thd ax-1ne0 eqcom ax-1cn eqnegi bitri - wn nemtbir caddc odd2np1 cc neg1cn wne neg1ne0 id zmulcld expp1zd mulid2i + wn nemtbir caddc odd2np1 cc neg1cn wne neg1ne0 id zmulcld expp1zd mullidi oveq1d eqtrdi eqeq1d mtbiri 2falsed pm2.61ian ) CAUAUBZADEZFUCZAGHZFIZVTJ VTWAKWDVTWAVTWDWAVTBUDZCLHZAIZBDMZWDCDEZWAVTWHJNBCAUEUFWGWDBDWGWEDEZWCWBW FGHZFWCWKIAWFAWFWBGOPWJWKWBCWELHZGHZFWJWFWLWBGWJWECWEUGWJUKUHUIWEUJZRQULU @@ -188901,7 +188901,7 @@ equal to the half of the predecessor of the odd number (which is an even -> ( ( N - 1 ) / 2 ) e. NN ) $= ( c2 wcel c1 co cdiv cn0 cmin cn wa wi wceq clt wbr a1d cz syl adantr mpbid cr cuz cfv caddc wne eluz2b3 wo nnnn0 nn0o1gt2 sylan eqneqall nn0z peano2zm - cc0 ad2antlr cmul 2cn mulid2i nnre ltp1d 2re peano2nn nnred mp3an2i expdimp + cc0 ad2antlr cmul 2cn mullidi nnre ltp1d 2re peano2nn nnred mp3an2i expdimp lttr mpd eqbrtrid 1red crp 2rp ltmuldivd rehalfcld posdifd adantlr sylanbrc a1i elnnz wb cc nncn xp1d2m1eqxm1d2 eleq1d expcom jaoi mpcom impancom sylbi imp ) ABUAUBCZADUCEZBFEZGCZADHEBFEZICZWIAICZADUDZJWLWNKAUEWOWLWPWNADLZBAMNZ @@ -189055,13 +189055,13 @@ equal to the half of the predecessor of the odd number (which is an even ( ( -u 1 ^ k ) x. ( A ^ k ) ) ) ) $= ( vl c1 caddc co cc0 cfz cexp cmul csu wcel expcld wceq oveq2d syl neg1cn cmin cneg cv 1cnd fzfid wa cc a1i cn0 elfznn0 adantl adantr mulcld fsumcl - adddird mulcomd mulassd expp1 syl2an eqcomd 3eqtrd sumeq2dv eqtrd mulid2d + adddird mulcomd mulassd expp1 syl2an eqcomd 3eqtrd sumeq2dv eqtrd mullidd fsummulc2 oveq12d 1zzd 0zd cn nnz peano2zm peano2nn0 oveq2 oveq1 fsumshft elfzelz zcnd npcan1 cuz cfv nncnd 0p1e1 fveq2i nnuz eqtr4i 3eltr4d oveq1i cz eleq2i nnm1nn0 nnnn0 expcom elfznn syl11 imp fsumm1 pncan1 sumeq1d weq biimtrid cbvsumv eqtrdi sylib exp0 ax-mp fsum1p exp0d 1t1e1 oveq1d eleq2s elnn0uz wi impcom addcomd nnnn0d addcld addassd nncn expp1d mulm1d negidd - mul02d 3eqtr3d fsumadd wss cfn wo olcd sumz addid1d 3eqtr2d ) ABHIJKDHUBJ + mul02d 3eqtr3d fsumadd wss cfn wo olcd sumz addridd 3eqtr2d ) ABHIJKDHUBJ ZLJZHUCZCUDZMJZBYPMJZNJZCOZNJZYOYMMJZBDMJZNJZHIJZAUUABYTNJZHYTNJZIJYNYQBY PHIJZMJZNJZCOZYTIJZUUEABHYTEAUEZAYNYSCAKYMUFZAYPYNPZUGZYQYRUUPYOYPYOUHPZU UPUAUIUUOYPUJPZAYPYMUKZULZQZUUPBYPABUHPZUUOEUMZUUTQZUNZUOZUPAUUFUUKUUGYTI @@ -189105,7 +189105,7 @@ equal to the half of the predecessor of the odd number (which is an even = ( ( A + 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( ( -u 1 ^ k ) x. ( A ^ k ) ) ) ) $= ( c1 cneg cmin co cexp cmul caddc cc0 c2 cdvds wbr syl oveq1d cfz cv wceq - csu wn cz wcel nnzd oddm1even mpbid m1expe pwp1fsum nnnn0d expcld mulid2d + csu wn cz wcel nnzd oddm1even mpbid m1expe pwp1fsum nnnn0d expcld mullidd wb 3eqtr3rd ) AHIZDHJKZLKZBDLKZMKZHNKHVAMKZHNKBHNKOUSUAKURCUBZLKBVDLKMKCU DMKVAHNKAVBVCHNAUTHVAMAPUSQRZUTHUCAPDQRUEZVEGADUFUGVFVEUPADFUHDUISUJUSUKS TTABCDEFULAVCVAHNAVAABDEADFUMUNUOTUQ $. @@ -189186,7 +189186,7 @@ equal to the half of the predecessor of the odd number (which is an even cr cinf eqeltri cv wceq oveq2 crab cbvrabv eqtri elrab2 simpli nn0ge0i wn mpbi cz wne cn nnabscl mp2an nngt0i 0re cc zcn ax-mp abscli ltnlei ssrab3 cuz wss nn0uz sseqtri nn0abscl nn0sub2 mp3an12 wi nn0z c1 cmul divalglem0 - a1i 1z mpan2 recni mulid2i oveq2i breq2i syl6ib syl imp infssuzle sylancr + a1i 1z mpan2 recni mullidi oveq2i breq2i syl6ib syl imp infssuzle sylancr sylanbrc eqbrtrid wb simpld nn0red lesub mp3an3 syl2anc recnd bitrd mpbid subidd mto nn0rei mpbir pm3.2i ) LBMNBAUAUBZUCNZBBOPZADBQRZSNZBCPXOXQUDZB CUFUCUGZCJACDEFGHIUEUHADKUIZQRZSNZXQKBOCXTBUJYAXPASXTBDQUKTCADEUIZQRZSNZE @@ -189284,7 +189284,7 @@ equal to the half of the predecessor of the odd number (which is an even wi wral cr cinf divalglem2 eqeltri divalglem5 simpri breq1 rspcev mp2an cle cn0 oveq2 breq2d elrab2 simplbi nn0zd zsubcl anim12i syl2an simprbi cmul mpan dvds2sub mp3an1 sylc cc zcn caddc recni subidi oveq1i subsub2 - zrei eqtrid sub4 mpanl12 subcl ancoms addid2d 3eqtr3d mpbid wb absdvdsb + zrei eqtrid sub4 mpanl12 subcl ancoms addlidd 3eqtr3d mpbid wb absdvdsb 0cn sylancr wne cn nnabscl divides adantr divalglem8 rexlimdv mpd rgen2 nnzi ex reu4 mpbir2an ) AUCZBUDUEZUFLZADUGXKADUHZXKUAUCZXJUFLZMZXIXMNZU IZUADUJADUJCDOCXJUFLZXLCDUKUFULDKBDEFGHIJUMUNPCUTLXRBCDEFGHIJKUOUPXKXRA @@ -189708,7 +189708,7 @@ equal to the half of the predecessor of the odd number (which is an even wa wceq caddc cmul cv crab ssrab2 cr cinf wne sseqtri wrex nnssnn0 nn0red c0 2re 1lt2 expnbnd syl3anc ssrexv mpsyl rabn0 infssuzcl sylancr eqeltrid sylibr sselid nn0zd 0red zred nn0ge0d reexpcld nnred simpr breq2d cbvrabv - oveq2 elrab2 simprbi lelttrd ltexp2d elnnz sylanbrc nnm1nn0 nncnd mulid2d + oveq2 elrab2 simprbi lelttrd ltexp2d elnnz sylanbrc nnm1nn0 nncnd mullidd mpbird ltm1d ltnled mpbid wi infssuzle mpan eqbrtrid biimtrrid mpand mtod syl nltled eqbrtrd 1red crp 2rp 1z zsubcld rpexpcld lemuldivd 2cn breqtrd expm1t ltdivmuld df-2 breqtrdi rerpdivcld flbi sylancl mpbir2and syl2an2r @@ -189745,14 +189745,14 @@ equal to the half of the predecessor of the odd number (which is an even ( N e. ( 0 ..^ ( 2 ^ M ) ) <-> ( bits ` N ) C_ ( 0 ..^ M ) ) ) $= ( cz wcel cn0 wa cc0 c2 cexp co cfv cdvds wbr wn clt cr a1i c1 cle mpbird vm vn cfzo cbits wss cv cdiv cfl w3a bitsval simp32 nn0uz eleqtrdi simp1r - cuz nn0zd 2re reexpcld simp1l zred cmul recnd mulid2d simp33 crp rpexpcld + cuz nn0zd 2re reexpcld simp1l zred cmul recnd mullidd simp33 crp rpexpcld 2rp rerpdivcld 1red ltnled caddc breq2i elfzole1 3ad2ant2 divge0d wceq wb 0p1e1 0z flbi sylancl z0even id breqtrrid syl6bir mpand biimtrrid sylbird mt3d lemuldivd eqbrtrrd elfzolt2 lelttrd elfzo2 syl3anbrc 3expia biimtrid 1lt2 ltexp2d ssrdv crab cinf cneg cn cif simpr nnred simpllr max2 syl2anc nn0red simplr n2dvdsm1 2nn nnnn0d ifcld nnexpcld nndivred zcnd nncnd 2cnd simplll 2ne0 expne0d divnegd max1 uzid ax-mp bernneq3 sylancr ltled letrd - wne 2z mulid1d breqtrrd nnrpd ledivmuld eqbrtrd nngt0d lenegcon1d divgt0d + wne 2z mulridd breqtrrd nnrpd ledivmuld eqbrtrd nngt0d lenegcon1d divgt0d lt0neg1d ax-1cn neg1cn 1pneg1e0 addcomli breqtrrdi neg1z mpbir2and breq2d mtbiri bitsval2 sseldd syl mpbid pm2.65da intnand wo simpll elznn0nn eqid sylib ord bitsfzolem impbida ) BCDZAEDZFZBGHAIJZUCJDZBUDKZGAUCJZUEZUVIUVK @@ -189794,7 +189794,7 @@ equal to the half of the predecessor of the odd number (which is an even nnrpd 2ne0 expne0d dvdsval2 mpbird dvdstrd nncnd divcan2d breqtrrd nn0red zcnd dvdscmulr syl112anc pncan3d oveq1d divdird eqtr3d fveq2d wceq fladdz ltled eqtrd mvrladdd dvdssub2 syl31anc notbid 3bitr3d z0even 2rp rpexpcld - cr modcld nn0ge0d divge0d rpred modlt 1le2 leexp2ad ltletrd rpcnd mulid1d + cr modcld nn0ge0d divge0d rpred modlt 1le2 leexp2ad ltletrd rpcnd mulridd nltled 1red ltdivmuld breqtrdi rerpdivcld flbi bitr3d an12 3anass 3bitr4g 1e0p1 0z sylancl mpbir2and breqtrrid intnand 2thd con2bid bitrdi pm5.32da pm2.61dan elfzo2 elnn0uz 3anbi1i 3bitr2i anbi2i bitri bitsval elin eqrdv @@ -189904,7 +189904,7 @@ equal to the half of the predecessor of the odd number (which is an even eqbrtrid 1red ltmuldivd eqbrtrrid zlem1lt mpbird elnn0z sylanbrc eleqtrdi 0zd imp nn0uz subge02d lelttrd ltdivmuld elfzo2 syl3anbrc fzo0to2pr elpri breqtrd syl syld diveq1d oveq2d n2dvds1 breq2 mtbiri syl6 sylibrd diveq0d - ord con1d subeq0d addid1d ifbothda ) ABUAUBCZBDAEFGZUCGZUDGZBDAUCGZUDGZUV + ord con1d subeq0d addridd ifbothda ) ABUAUBCZBDAEFGZUCGZUDGZBDAUCGZUDGZUV IFGZHUVHUVJIFGZHUVHUVJUVEUVIIUEZFGZHBJCZAUFCZUGZUVIIUVIUVMHUVKUVNUVHUVIUV MUVJFUHUIIUVMHUVLUVNUVHIUVMUVJFUHUIUVQUVEUGZUVJUVHUVJKGZFGUVHUVKUVRUVJUVH UVQUVJUJCZUVEUVQUVJUVQBUVIUVOUVPUKZUVQDADUNCZUVQULLZUVOUVPUMZUOZUPUQZMUVQ @@ -190100,10 +190100,10 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + if ( ch , A , 0 ) ) ) $= ( cc cc0 cif co caddc wceq wa wb adantl ifbid iftrue oveq12d oveq1d 3eqtr4d wn iffalse wcel whad wcad c2 cmul wo 0cn ifcl mpan2 ad2antrr simpll addassd - add12d eqtr4d pm5.501 bicomd animorrl syl 2timesd eqtrd addid2d 2cnd mulcld + add12d eqtr4d pm5.501 bicomd animorrl syl 2timesd eqtrd addlidd 2cnd mulcld id 2times adantr simpl 0cnd addcomd pm2.61dan ifnot eqtr3id 3eqtr2rd hadrot nbn2 had1 bitr3id cad1 oveq2d wxo notbid df-xor bitr4di ibar simplll ifclda - biorf con1bid bitrid simpr intnanrd had0 cad0 addcld addid1d sylan9eqr ) DE + biorf con1bid bitrid simpr intnanrd had0 cad0 addcld addridd sylan9eqr ) DE UAZCABCUBZDFGZABCUCZUDDUEHZFGZIHZADFGZBDFGZIHZCDFGZIHZJWQCKZABLZDFGZABUFZXA FGZIHZXFDIHZXCXHXIAXNXOJXIAKZXEDDIHZIHZDXEIHZDIHZXNXOXPXRDXEDIHZIHXTXPXEDDW QXEEUAZCAWQFEUAZYBUGBDFEUHUIZUJZWQCAUKZYFUMXPDXEDYFYEYFULUNXPXKXEXMXQIXPXJB @@ -190225,12 +190225,12 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact f1ocnv ax-mp wceq f1oeq1 mpbir f1of ffvelcdmi syl nn0addcld nn0red 2nn0 bitsf1o adantr nn0expcld wn 0nn0 ifclda biimpa nnnn0d ifcl nn0ge0d 0red addge01d mpbid iftrue oveq1d breqtrrd addge02d oveq2d jaodan ex iffalse - clt oveq12d readdcld addid1d eqtrd fveq1i fveq2i fvresd sylancr 3eqtr3a + clt oveq12d readdcld addridd eqtrd fveq1i fveq2i fvresd sylancr 3eqtr3a f1ocnvfv2 eqsstrdi nn0zd bitsfzo syl2anc mpbird elfzolt2 eqbrtrd ltnled cz bitrd wi nn0cnd breq1 ifboth breq1d syl5ibrcom con1d bitsinvp1 ioran le2addd ad2antrl ad2antll 00id eqtrdi recnd lt2addd biimtrid impcon4bid oveqan12d lenltd con2bid biimpar ltadd1dd syl3anc mpand ltadd2d sylibrd - cad0 ltletr addid2d leidd jcad sylbid syld impbid pm2.61dan sadcp1 cmul + cad0 ltletr addlidd leidd jcad sylbid syld impbid pm2.61dan sadcp1 cmul 2cnd expp1d nncnd times2d add4d breq12d 3bitr4d ) AHBPZHCPZUAHDUBPZUCZU DHUEUFZUWBQUFZBRHUNUFZUGZGUBZCUWDUGZGUBZQUFZUVRUWBRUHZUVSUWBRUHZQUFZQUF ZSUIZUAHUJQUFZDUBPUDUWOUEUFZBRUWOUNUFZUGGUBZCUWQUGGUBZQUFZSUIAUVTUWAUWN @@ -191440,9 +191440,9 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact $( Lemma for ~ gcdaddm . (Contributed by Paul Chapman, 31-Mar-2011.) $) gcdaddmlem $p |- ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) $= ( cc0 wceq cmul co caddc wa cgcd wbr cdvds cz wcel mp2an wi ax-mp gcddvds - wn cle simpli c1 cn0 gcdcl nn0zi w3a 1z dvds2ln mpanl12 mp3an zcn mulid2i + wn cle simpli c1 cn0 gcdcl nn0zi w3a 1z dvds2ln mpanl12 mp3an zcn mullidi cc oveq2i breqtri zmulcl zaddcl dvdslegcd ex mp2ani cneg mulneg1i oveq12i - znegcl mulcli negcli negidi addcomli oveq1i addassi addid2i 3eqtr3i eqtri + znegcl mulcli negcli negidi addcomli oveq1i addassi addlidi 3eqtr3i eqtri anim12i zrei letri3i sylibr wo pm4.57 mul01i eqtrdi oveq1d eqeq1d pm5.32i oveq2 oveq12 sylbir eqtr4d sylbi jaoi pm2.61i ) BGHZABIJZCKJZGHZLZUBZWOCG HZLZUBZLZBCMJZBWQMJZHZXDXEXFUCNZXFXEUCNZLXGWTXHXCXIWTXEBONZXEWQONZXHXJXEC @@ -191476,7 +191476,7 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact (Contributed by Scott Fenton, 20-Apr-2014.) $) gcdadd $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( N + M ) ) ) $= - ( cz wcel wa cgcd co c1 cmul caddc wceq 1z gcdaddm mp3an1 zcn mulid2 oveq2d + ( cz wcel wa cgcd co c1 cmul caddc wceq 1z gcdaddm mp3an1 zcn mullid oveq2d cc syl adantr eqtrd ) ACDZBCDZEABFGZABHAIGZJGZFGZABAJGZFGZHCDUBUCUDUGKLHABM NUBUGUIKZUCUBARDZUJAOUKUFUHAFUKUEABJAPQQSTUA $. @@ -191484,7 +191484,7 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact Paul Chapman, 31-Mar-2011.) $) gcdid $p |- ( N e. ZZ -> ( N gcd N ) = ( abs ` N ) ) $= ( cz wcel cc0 cgcd co c1 cmul caddc cabs cfv wceq 1z gcdaddm mp3an13 gcdid0 - 0z cc zcn oveq2d mulid2 addid2 eqtrd syl 3eqtr3rd ) ABCZADEFZADGAHFZIFZEFZA + 0z cc zcn oveq2d mullid addlid eqtrd syl 3eqtr3rd ) ABCZADEFZADGAHFZIFZEFZA JKAAEFGBCUFDBCUGUJLMQGADNOAPUFUIAAEUFARCZUIALASUKUIDAIFAUKUHADIAUATAUBUCUDT UE $. @@ -191562,7 +191562,7 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact 3-Aug-2023.) $) gcdmultipled $p |- ( ph -> ( M gcd ( N x. M ) ) = M ) $= ( cc0 cgcd co cmul caddc cz wcel wceq nn0zd 0zd gcdaddm syl3anc nn0gcdid0 - cn0 syl zmulcld zcnd addid2d oveq2d 3eqtr3rd ) ABFGHZBFCBIHZJHZGHZBBUGGHA + cn0 syl zmulcld zcnd addlidd oveq2d 3eqtr3rd ) ABFGHZBFCBIHZJHZGHZBBUGGHA CKLBKLFKLUFUIMEABDNZAOCBFPQABSLUFBMDBRTAUHUGBGAUGAUGACBEUJUAUBUCUDUE $. $} @@ -191628,7 +191628,7 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact ( cmul co caddc wceq cz wrex cc0 wcel c1 cc cabs cfv cv wne fveq2 oveq1 cn eqeq12d rexbidv cr zre cneg 1z ax-1rid eqcomd oveq2 rspceeqv sylancr eqeq1 syl5ibrcom neg1z mulm1d neg1cn mulcom eqtr3d absor mpjaod vtoclga - recn syl wa zcnd adantr mul01d oveq2d mulcl syl2an addid1d eqtrd eqeq2d + recn syl wa zcnd adantr mul01d oveq2d mulcl syl2an addridd eqtrd eqeq2d zcn 0z mpan syl6bir reximdva mpd wi nnabscl 2rexbidv elrab2 simplbi2com ex sylsyld ) AEUAUBZEBUCZKLZFCUCZKLZMLZNZCOPZBOPZEQUDZWNUGRZWNGRZAWNWPN ZBOPZXBAEORZXGIDUCZUAUBZXIWOKLZNZBOPZXGDEOXIENZXLXFBOXNXJWNXKWPXIEUAUEX @@ -191997,7 +191997,7 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact ( K gcd M ) = 1 ) -> ( K gcd ( M x. N ) ) = ( K gcd N ) ) $= ( cn wcel w3a cgcd co c1 wceq wa cmul gcdmultiple 3adant2 nnz zmulcl syl2an cz adantr eqtrd oveq1d 3ad2ant1 3adant1 gcdass syl3anc eqtr3d nnnn0 mulgcdr - cn0 syl3an oveq1 sylan9eq cc nncn 3ad2ant3 mulid2d oveq2d ) ADEZBDEZCDEZFZA + cn0 syl3an oveq1 sylan9eq cc nncn 3ad2ant3 mullidd oveq2d ) ADEZBDEZCDEZFZA BGHZIJZKZABCLHZGHZAACLHZVEGHZGHZACGHVAVFVIJVCVAAVGGHZVEGHZVFVIVAVJAVEGURUTV JAJUSACMNUAVAAREZVGREZVEREZVKVIJURUSVLUTAOZUBURUTVMUSURVLCREZVMUTVOCOZACPQN USUTVNURUSBREZVPVNUTBOZVQBCPQUCVEVGAUDUEUFSVDVHCAGVDVHICLHZCVAVCVHVBCLHZVTU @@ -192053,7 +192053,7 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact sqgcd $p |- ( ( M e. NN /\ N e. NN ) -> ( ( M gcd N ) ^ 2 ) = ( ( M ^ 2 ) gcd ( N ^ 2 ) ) ) $= ( cn wcel wa cgcd co c2 cexp c1 cdiv wceq cz cdvds adantr adantl syl2anc cc - wbr cc0 cmul gcdnncl nnsqcld nncnd mulid1d nnsqcl nnz gcddvds syl2an simpld + wbr cc0 cmul gcdnncl nnsqcld nncnd mulridd nnsqcl nnz gcddvds syl2an simpld nnzd wi dvdssqim simprd gcddiv syl32anc nncn nnne0d sqdivd oveq12d syl31anc mpd dividd eqtr3d clt wne wb dvdsval2 syl3anc mpbid nnre nnred nngt0 nngt0d cr divgt0d elnnz sylanbrc 2nn rppwr mp3an3 3eqtr2d anim12i intnanrd gcdn0cl @@ -192854,7 +192854,7 @@ their least common multiple ( ~ df-lcmf ) are provided. Both definitions are $( The lcm of an integer and 1 is the absolute value of the integer. (Contributed by AV, 23-Aug-2020.) $) lcm1 $p |- ( M e. ZZ -> ( M lcm 1 ) = ( abs ` M ) ) $= - ( cz wcel c1 clcm co cgcd cmul cabs cfv gcd1 oveq2d cn0 lcmcl mpan2 mulid1d + ( cz wcel c1 clcm co cgcd cmul cabs cfv gcd1 oveq2d cn0 lcmcl mpan2 mulridd 1z nn0cnd eqtr2d wceq lcmgcd zcn fveq2d 3eqtrd ) ABCZADEFZUFADGFZHFZADHFZIJ ZAIJUEUHUFDHFUFUEUGDUFHAKLUEUFUEUFUEDBCZUFMCQADNORPSUEUKUHUJTQADUAOUEUIAIUE AAUBPUCUD $. @@ -193627,7 +193627,7 @@ their least common multiple ( ~ df-lcmf ) are provided. Both definitions are ( ( K || ( M x. N ) /\ ( K gcd M ) = 1 ) -> K || N ) ) $= ( cz wcel w3a cgcd co c1 wceq cmul cdvds wbr wa wb zcn breq2d bitrd adantr cc mulcom syl2an dvdsmulgcd ancoms 3adant1 gcdcom 3adant3 eqeq1d syl6bi imp - wi oveq2 mulid1d 3ad2ant3 eqtrd biimpd ex impcomd ) ADEZBDEZCDEZFZABGHZIJZA + wi oveq2 mulridd 3ad2ant3 eqtrd biimpd ex impcomd ) ADEZBDEZCDEZFZABGHZIJZA BCKHZLMZACLMZVBVDVFVGUKVBVDNZVFVGVHVFACBAGHZKHZLMZVGVBVFVKOZVDUTVAVLUSUTVAN ZVFACBKHZLMZVKVMVEVNALUTBTECTEVEVNJVABPCPZBCUAUBQVAUTVOVKOACBUCUDRUESVHVJCA LVHVJCIKHZCVBVDVJVQJZVBVDVIIJVRVBVCVIIUSUTVCVIJVAABUFUGUHVIICKULUIUJVBVQCJZ @@ -193743,7 +193743,7 @@ their least common multiple ( ~ df-lcmf ) are provided. Both definitions are wa wrex cfv c2nd cxp wi wral wreu w3a cop cdvds nnz gcddvds simpld sylan2 cq wne wb cn0 gcdcl nn0zd simpl nnne0 neneqd intnand gcdn0cl syl21anc syl wn dvdsval2 syl3anc mpbid 3adant3 clt simprd cr nnre nn0red nngt0 divgt0d - jca sylibr opelxpd gcdcld nn0cnd 1cnd cmul mulid1d cc zcn adantr divcan2d + jca sylibr opelxpd gcdcld nn0cnd 1cnd cmul mulridd cc zcn adantr divcan2d nncn mulgcd 3eqtr2rd mulcanad divcan7d eqeq2d biimp3ar ovex op1std op2ndd elnnz eqeq1d anbi12d rspcev syl12anc elxp6 simprl ad2antrr simprr simprll simprrl simprlr simprrr eqtr3d qredeq syl331anc fvex opth simplll simplrl @@ -194355,7 +194355,7 @@ being prime ( ` Prime = { p e. NN | ... ` ), but even if ` p e. NN0 ` was ( wcel c1 vn vk cn cfz co wral caddc wceq oveq2 raleqdv weq elfz1eq syl cv wb mpbiri rgen csn wa wsbc cdvds wbr c2 cmin wrex cdiv cmul peano2nn ad2antrr cuz cfv elfzuz ad2antrl eluz2nn nnne0d divcan2d clt simprr cc0 - nncnd cz wne nnzd dvdsval2 syl3anc mpbid mulid2d elfzle2 cc nncn ax-1cn + nncnd cz wne nnzd dvdsval2 syl3anc mpbid mullidd elfzle2 cc nncn ax-1cn cle pncan sylancl breqtrd nnz zleltp1 syl2anc eqbrtrd 1red nnred nngt0d ltmuldiv syl112anc eluz2b1 sylanbrc simplr fznn mpbir2and rspcdva sbcie cr vex dfsbcq bitr3id cbvralvw sylib nnrpd rpdivcld rpgt0d elnnz dividd @@ -194659,7 +194659,7 @@ being prime ( ` Prime = { p e. NN | ... ` ), but even if ` p e. NN0 ` was a1i syl3anc remulcld ltnled oveq12 anidms breq1d notbid syl5ibrcom imim2d ltmulgt11 con2 syl6 imim12d ralimdv2 wrex annim weq breq1 anbi12d ancom2s rspcev expr ad2ant2lr cdiv cz simprl wne eluzelz ad2antlr nnne0d ad2antrr - cn dvdsval2 recnd mulid2d dvdsle imp simprr neqned necomd leneltd eqbrtrd + cn dvdsval2 recnd mullidd dvdsle imp simprr neqned necomd leneltd eqbrtrd syl21anc 1red zred nnre nngt0 jca ltmuldiv eluz2b1 sylanbrc nnmulcld nnrp syl crp rpdivcl syl2an syl2anc lemul1d divmuldivd divassd divcan2d eqcomd nncnd breq12d bitr4d biimpd dvds0lem syl31anc jctird syl6an letrid mpjaod @@ -195080,7 +195080,7 @@ being prime ( ` Prime = { p e. NN | ... ` ), but even if ` p e. NN0 ` was 3lcm2e6 $p |- ( 3 lcm 2 ) = 6 $= ( c3 c2 clcm co cmul c6 cgcd c1 wceq wne 2re 2lt3 cprime wcel mp2an 3nn 2nn cn cz nnzi gtneii wb 3prm 2prm prmrp mpbir oveq2i lcmgcdnn cn0 lcmcl nn0cni - mulid1i 3eqtr3ri 3t2e6 eqtri ) ABCDZABEDZFUPABGDZEDZUPHEDUQUPURHUPEURHIZABJ + mulridi 3eqtr3ri 3t2e6 eqtri ) ABCDZABEDZFUPABGDZEDZUPHEDUQUPURHUPEURHIZABJ ZBAKLUAAMNBMNUTVAUBUCUDABUEOUFUGARNBRNUSUQIPQABUHOUPUPASNBSNUPUINAPTBQTABUJ OUKULUMUNUO $. @@ -195547,7 +195547,7 @@ reduced fraction representation (no common factors, denominator ( ( P ^ ( K - 1 ) ) x. ( P - 1 ) ) ) $= ( vx wcel cn wa co cfv c1 wceq crab chash cmin cn0 cc caddc cc0 cdiv cfl cz cprime cexp cphi cv cgcd cfz prmnn nnnn0 nnexpcl syl2an phival nnm1nn0 - syl nncnd adantr ax-1cn subdi mp3an3 syl2anc mulid1d oveq2d cdvds wbr cun + syl nncnd adantr ax-1cn subdi mp3an3 syl2anc mulridd oveq2d cdvds wbr cun cmul cfn cin c0 wss fzfi ssrab2 ssfi mp2an inrab wn wral wi elfzelz rpexp wb prmz syl3an1 3expa an32s simpr zexpcl gcdcomd eqeq1d coprm adantlr zcn 3bitr4d adantl subid1d breq2d notbid bitr4d sylan2 biimpd imnan ralrimiva @@ -195585,7 +195585,7 @@ reduced fraction representation (no common factors, denominator Carneiro, 28-Feb-2014.) $) phiprm $p |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) ) $= ( cprime wcel c1 cexp co cphi cfv cmin cmul cn wceq 1nn phiprmpw mpan2 prmz - zcnd exp1d cc0 cc fveq2d 1m1e0 oveq2i eqtrid oveq1d sylancl mulid2d 3eqtr3d + zcnd exp1d cc0 cc fveq2d 1m1e0 oveq2i eqtrid oveq1d sylancl mullidd 3eqtr3d exp0d ax-1cn subcl eqtrd ) ABCZADEFZGHZADDIFZEFZADIFZJFZAGHURUMDKCUOUSLMADN OUMUNAGUMAUMAAPQZRUAUMUSDURJFURUMUQDURJUMUQASEFDUPSAEUBUCUMAUTUIUDUEUMURUMA TCDTCURTCUTUJADUKUFUGULUH $. @@ -195751,7 +195751,7 @@ reduced fraction representation (no common factors, denominator an12s a2d syld nnind mpcom mpdan nnnn0d ccnv ccom mulcl mulcom w3a mulass ssidd f1ocnv adantrr adantrl eqeqan12d subdid breq2d 3bitr4d simp3d f1of1 zsubcld f1fveq 3bitr3d sylbid ralrimivva dff13 sylanbrc ovexi f1oen fzofi - ssfi f1finf1o f1oco f1oeq23 fvco3 fveq2d f1ocnvfv2 eqtr2d eqeltrd mulid2d + ssfi f1finf1o f1oco f1oeq23 fvco3 fveq2d f1ocnvfv2 eqtr2d eqeltrd mullidd cen seqf1o ax-1cn subdir zsubcl 3eqtr2d breqtrd mp2and mp3an3 ) ADIUCUDZU EOZIPOQIPORZIVUBQUFOZUGUHZAIVUASGQUIZUDZVUDSOZUGUHZIVUGULOZQRZVUEAIVUBVUG SOZVUASHQUIZUDZUFOZVUHUGAVULIPOZVUNIPOZRZIVUOUGUHZAVURVUKAVUAVUAUJUHZVURV @@ -195869,7 +195869,7 @@ reduced fraction representation (no common factors, denominator ( cprime wcel cz wa cdvds wbr cexp co wceq wi cgcd c1 cmul 3ad2ant1 syl2anc cmo cr oveq1d cn prmnn dvdsmodexp 3exp sylc adantr coprm prmz gcdcom eqeq1d sylan bitrd w3a cphi cfv crp cn0 simp2 phicld nnnn0d zexpcl zred 1red nnrpd - wn eulerth syl3an1 modmul1 syl221anc cmin phiprm oveq2d zcnd expm1t mulid2d + wn eulerth syl3an1 modmul1 syl221anc cmin phiprm oveq2d zcnd expm1t mullidd cc eqtr4d 3eqtr3d 3expia sylbid pm2.61d ) BCDZAEDZFZBAGHZABIJZBRJZABRJZKZWB WEWILZWCWBBUADZWKWJBUBZWLWKWKWEWIABBUCUDUEUFWDWEVEZABMJZNKZWIWDWMBAMJZNKWOB AUGWDWPWNNWBBEDWCWPWNKBUHBAUIUKUJULWBWCWOWIWBWCWOUMZABUNUOZIJZAOJZBRJZNAOJZ @@ -195966,7 +195966,7 @@ reduced fraction representation (no common factors, denominator w3a eqeq1d elrab2 cn0 elfzonn0 ad2antrl nnnn0 3ad2ant2 nn0mulcld elfzolt2 cv clt simpl1 cr wb cz elfzoelz zred nnre 3ad2ant1 nngt0 ltmuldiv syl3anc jca mpbird elfzo0 syl3anbrc cc wne nnne0 divcan1d eqcomd oveq2d nndivdvds - nncn biimp3a nnzd mulgcdr simprr oveq1d mulid2d eqtrd 3eqtrd sylanbrc cle + nncn biimp3a nnzd mulgcdr simprr oveq1d mullidd eqtrd 3eqtrd sylanbrc cle sylan2b gcddvds syl2anc eqbrtrrd nnz dvdsval2 mpbid elfzofz elfznn0 nn0re simpld cfz nn0ge0 3syl divge0 elnn0z ltdiv1 simpl2 simpl3 gcddiv syl32anc dividd 3eqtr3d simplbi anim12i ad2antll eqeq2d syl5ibrcom divcan4d impbid @@ -196108,7 +196108,7 @@ and the outer argument selects the integer or equivalence class (if you nnzd zcnd exp0d 1m1e0 eqtrdi breqtrrd syl5ibrcom impbid cdiv cfl nndivred cmul nnnn0d nn0ge0 wb nngt0d ge0div syl3anc nn0mulcld zexpcl zred expmuld 1red 1zzd odzid moddvds mpbird modexp syl221anc flcld 1exp 3eqtrd modmul1 - flge0nn0 caddc expaddd modval oveq2d nn0cnd recnd pncan3d mulid2d 3eqtr3d + flge0nn0 caddc expaddd modval oveq2d nn0cnd recnd pncan3d mullidd 3eqtr3d eqtrd eqtr3d eqeq1d sylancom 3bitr3d dvdsval3 3bitr4d ) CEFZAGFZACUAHIJZU BZBKFZUCZCABACUDUEUEZLHZMHZINHZOPZUUJQJZCABMHZINHOPZUUIBOPZUUHUUMUUNUUHUU MUUJEFZUFZUUNUUHUUMUURUCZUFUUMUUSUGUUHUUTUUIUUJUHPZUUHUUJUUIUIPZUVAUFUUHB @@ -196232,7 +196232,7 @@ and the outer argument selects the integer or equivalence class (if you ( wcel cz wa c1 cmin co cdvds wbr cmo wceq cmul simpr adantr syl3anc oveq1d syl cr zmodcld cprime c2 cexp simpll m1dvdsndvds imp w3a cfz eqid modprminv wn eqcomd modprm1div biimpar crp zre ad2antlr prmm2nn0 anim1ci zexpcl prmnn - cn0 cn nn0zd nnrpd modmulmod nn0cnd mulid2d reexpcl syl2anr modabs2 3eqtr3d + cn0 cn nn0zd nnrpd modmulmod nn0cnd mullidd reexpcl syl2anr modabs2 3eqtr3d jca eqtrd eqtr2d ex ) BUACZADCZEZBAFGHIJZABUBGHZUCHZBKHZFLVSVTEZFAWCMHBKHZW CWDVQVRBAIJUKZFWELZVQVRVTUDVSVRVTVQVRNOVSVTWFABUEUFVQVRWFUGWCFBFGHUHHCZWEFL ZEZWGABWCWCUIUJWJWEFWHWINULRPWDABKHZWCMHZBKHZFWCMHZBKHZWEWCWDWLWNBKWDWKFWCM @@ -196279,7 +196279,7 @@ a nonnegative integer (less than the given prime number) so that the sum zred nnred remulcl resubcld 3ad2ant2 nnrpd modaddmulmod syl31anc cc nncnd zcnd mulcl subdird oveq2d mulcom mulmod0 syl2anr 3adant3 mul32d modmulmod eqtrd syl3anc eqtr4d oveq12d remulcld modsubmodmod eqeq1d biimpd impancom - imp mulid2d cle wbr clt anim12ci cuz cfv elfzo2 eluz2 0red 1red 3jca 0le1 + imp mullidd cle wbr clt anim12ci cuz cfv elfzo2 eluz2 0red 1red 3jca 0le1 zre a1i anim1i letr sylc 3adant1 sylbi simp3 jca 3adant2 modid modadd2mod syl 3eqtr3d 0cnd pncan3d 0mod 3eqtrd oveq1 rspcev rexlimiva 3syl pm2.43i ex ) AUAFZDGAUBHZFZCYLFZUCZCBUDZDIHZJHZAKHZLMZBLAUBHZUEZYKYMYOUUBUFZYNYKY @@ -196381,7 +196381,7 @@ prime number there is always a second nonnegative integer (less than the 3eqtr3d oveq2d gcdass c2 cexp nn0z gcdcl syl2an 3adant2 sqvald simp13 nn0cn cc mulcomd simpl3 eqeq1d biimp3a oveq12d simp11 simp12 mulgcdr sylan ancoms wa 3adant1 cabs cfv 3ad2ant3 gcdid syl oveq1d simp2 gcdabs1 syl2anc gcdcomd - eqtrd 3eqtr4d biimpar 3adant3 mulid1d 3eqtrrd 3expia ) ADEZBFEZCDEZGZABHICH + eqtrd 3eqtr4d biimpar 3adant3 mulridd 3eqtrrd 3expia ) ADEZBFEZCDEZGZABHICH IZJKZCUAUBIZABLIZKZAACHIZUAUBIZKWCWEWHGZWJWIWILIZAWICBHIZHIZLIZAWKWIWKWIWCW EWIDEZWHVTWBWPWAVTAFEZCFEZWPWBAUCZCUCZACUDUEUFMZNUGWKAWILIZCWILIZHIZXBAWMLI ZHIZWLWOWKXCXEXBHWKCALIZCCLIZHIZACLIZWGHIZXCXEWKXGXJXHWGHWKCAWKCVTWAWBWEWHU @@ -196556,7 +196556,7 @@ prime number there is always a second nonnegative integer (less than the zaddcld zsubcld mp3an2i mpd cc subsq breqtrrd simpl2 oveq1d simpl11 nnsqcld pncand eqtr3d breqtrd adantr cprime 2prm 2nn prmdvdsexp mp3an13 mpbid mtand wb syl cneg neg1z gcdaddm pnncan 3anidm23 subcl mulm1d oveq2d addcl negsubd - 2times adantl 3eqtr4d zmulcl sylancr eqbrtrd 1z ppncan 3anidm13 mulid2d cc0 + 2times adantl 3eqtr4d zmulcl sylancr eqbrtrd 1z ppncan 3anidm13 mullidd cc0 eqtrd nnaddcl nnne0d ancoms neneqd intnand gcdn0cl syl21anc dvdsgcd syl3anc wne mp2and cn0 2nn0 mulgcd pythagtriplem3 2t1e2 eqtrdi dvdsprime orel1 sylc ) ADEZBDEZCDEZUAZAFUBGZBFUBGZHGZCFUBGZIZABJGKIZFALTZUCZMZUAZCBNGZCBHGZJGZFI @@ -197716,7 +197716,7 @@ given prime (or other positive integer) that divides the number. For gcddvds dvdsabsb syl2anc mpbid gcdn0cl sylan2 nnzd nnne0d nnabscl adantlr necon3ai dvdsval2 syl3anc nngt0 jca divgt0 syl2an elnnz sylanbrc elnn1uz2 nnre sylib simprd syl5ibcom cmul nncnd 1cnd divmuld eqeq1d bitrd absdvdsb - mulid1d 3imtr4d wrex exprmfct simprl pcdiv syl121anc cq simplll syl pcabs + mulridd 3imtr4d wrex exprmfct simprl pcdiv syl121anc cq simplll syl pcabs cmin zq oveq1d eqtrd simprr pcelnn mpbird eqeltrrd pccld cn0 simplr pczcl syl12anc znnsub nn0red ltnled simpllr nprmdvds1 ad2antrl gcdid0 oveq2d cc cif breq2d mtbird mpd biantrurd expr syl5 cxr adantl necon3bd lemin pcgcd @@ -197987,7 +197987,7 @@ given prime (or other positive integer) that divides the number. For pcadd $p |- ( ph -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) ) $= ( cdiv co wceq cn cz cpc wbr wcel wa cc0 oveq2d cmul vx vy vz vw cv caddc wrex cle cq elq sylib wi cprime cxr pcxcl syl2anc xrleidd adantr oveq2 cc - qcn syl addid1d sylan9eqr breqtrrd a1d wne reeanv ad3antrrr prmnn simplrl + qcn syl addridd sylan9eqr breqtrrd a1d wne reeanv ad3antrrr prmnn simplrl cexp cn0 simprrl wn cpnf pc0 cr simpllr pcqcl syl12anc zred ltpnf wb rexr clt pnfxr xrltnle sylancl eqnbrtrd breq1d syl5ibcom necon3bd mpd eqnetrrd mpbid simprll nncnd nnne0d div0d oveq1 eqeq1d syl5ibrcom necon3d nnexpcld @@ -198054,7 +198054,7 @@ given prime (or other positive integer) that divides the number. For pcadd2 $p |- ( ph -> ( P pCnt A ) = ( P pCnt ( A + B ) ) ) $= ( cpc co caddc wcel cq cxr pcxcl syl2anc cle wbr cc0 oveq2d cprime qaddcl xrltled pcadd cneg qnegcl syl clt wn wb xrltnle mpbid adantr pcneg breq1d - wa wceq biimpar ex qcn negcld add12d addcomd negidd eqtrd addid1d breq12d + wa wceq biimpar ex qcn negcld add12d addcomd negidd eqtrd addridd breq12d cc 3eqtrd sylibd mtod mpbird breqtrrd addassd breqtrd xrletrid ) ADBIJZDB CKJZIJZADUALZBMLZVQNLZEFDBOPZAVTVRMLZVSNLZEAWACMLZWDFGBCUBPZDVROPZABCDEFG AVQDCIJZWCAVTWFWINLZEGDCOPZHUCUDAVSDVRCUEZKJZIJVQQAVRWLDEWGAWFWLMLZGCUFUG @@ -198096,14 +198096,14 @@ given prime (or other positive integer) that divides the number. For cexp eleq1 mtbiri iffalsed 1ex fvmpt eqtri oveq2i pc1 eqtrid clt prmgt1 1z wn cr wb 1re c2 cuz prmuz2 eluzelre syl ltnle mpbid eqtr4d wa adantr sylancr wf pcmptcl simpld peano2nn ffvelcdm syl2an adantrr pccld nn0cnd - addid2d csb cvv ad2antrl ovex ifex fvmpts nfv nfcv nfcsb1v nfov nfif id + addlidd csb cvv ad2antrl ovex ifex fvmpts nfv nfcv nfcsb1v nfov nfif id csbex csbeq1a oveq12d csbief eqtrdi sylancl simprr eqeltrd eqtrd 3eqtrd ifbieq1d iftrued csbeq1d cn0 eleq1d syl2anc simpr wne nnz nnne0 syl3anc sylc jca nnred expr cdvds ad2antrr sylan9eq nfcvd csbiegf rspcv pcidlem wral oveq1 eqeq1d syl5ibrcom nnre ltp1 peano2re mpdan breq1d mtbid nnuz eqeq2d eleqtrdi seqp1 simprd pcmul prmnn leidd breqtrrd 3imtr4d simplrr ffvelcdmda necomd nfel1 prmdvdsexpr necon3ad iftrue breq2d mtbird pceq0 - rspc mpbird iffalse pm2.61dan addid1d ltlend nnleltp1 biantrud 3bitr4rd + rspc mpbird iffalse pm2.61dan addridd ltlend nnleltp1 biantrud 3bitr4rd mpd simprl biimprd pm2.61dne expcom a2d nnind mpcom ) GMNADGUCFOUDZUEZP QZDGRUFZCSUJZTZJADUAUGZUWLUEZPQZDUWRRUFZCSUJZTZUKADOUWLUEZPQZDORUFZCSUJ ZTZUKADUBUGZUWLUEZPQZDUXIRUFZCSUJZTZUKADUXIOUHQZUWLUEZPQZDUXORUFZCSUJZT @@ -198218,7 +198218,7 @@ given prime (or other positive integer) that divides the number. For sum_ k e. B if ( k e. A , 1 , 0 ) = ( # ` A ) ) $= ( cC cfn wcel wss wa c1 csu chash cfv cmul co cv cc0 cif cc wceq ax-1cn ssfi fsumconst sylancl wral cuz wo simpr a1i animorlr sumss2 syl21anc cn0 - rgenw hashcl syl nn0cnd mulid1d 3eqtr3d ) BEFZABGZHZAICJZAKLZIMNZBCOAFIPQ + rgenw hashcl syl nn0cnd mulridd 3eqtr3d ) BEFZABGZHZAICJZAKLZIMNZBCOAFIPQ CJZVCVAAEFZIRFZVBVDSBAUAZTAICUBUCVAUTVGCAUDZBDUELGZUSUFVBVESUSUTUGVIVAVGC ATUMUHUSUTVJUIABICDUJUKVAVCVAVCVAVFVCULFVHAUNUOUPUQUR $. $} @@ -198230,7 +198230,7 @@ given prime (or other positive integer) that divides the number. For if ( N || ( M + 1 ) , 1 , 0 ) ) $= ( cz wcel wa c1 caddc co wbr cdiv cfl cfv cmin cc0 wceq wb adantr cr adantl clt cdvds cif wne nnz nnne0 peano2z dvdsval2 syl2an23an biimpa flid syl cle - cn nnm1nn0 nn0red nn0ge0d nnre nngt0 divge0 syl22anc ad2antlr ltm1d mulid1d + cn nnm1nn0 nn0red nn0ge0d nnre nngt0 divge0 syl22anc ad2antlr ltm1d mulridd cmul nncn breqtrrd 1re a1i ltdivmul syl112anc nndivre mpancom flbi2 syl2anc mpbird mpbir2and eqtr4d cc ax-1cn ppncand oveq1d zcnd subcl sylancl divdird zcn eqtr3d dividd oveq2d eqtrd fveq2d sylan 1z fladdz 3eqtrrd flcld subaddd @@ -198270,7 +198270,7 @@ given prime (or other positive integer) that divides the number. For ( cn0 wcel cuz cfv cprime w3a co cc0 cle wbr c1 clt 3ad2ant1 cr wb 3ad2ant3 cexp cdiv cfl wceq caddc nn0ge0 nn0re prmnn eluznn0 3adant3 nnexpcld nngt0d cn nnred ge0div syl3anc mpbid cmul nn0red eluzle 3ad2ant2 c2 prmuz2 syl2anc - bernneq3 lelttrd nncnd mulid1d breqtrrd ltdivmul syl112anc mpbird breqtrrdi + bernneq3 lelttrd nncnd mulridd breqtrrd ltdivmul syl112anc mpbird breqtrrdi 1red 0p1e1 cz wa nndivred 0z flbi sylancl mpbir2and ) CDEZBCFGEZAHEZIZCABTJ ZUAJZUBGKUCZKWGLMZWGKNUDJZOMZWEKCLMZWIWBWCWLWDCUEPWECQEZWFQEZKWFOMZWLWIRWBW CWMWDCUFPZWEWFWEABWDWBAULEWCAUGSWBWCBDEZWDBCUHUIZUJZUMZWEWFWSUKZCWFUNUOUPWE @@ -198456,7 +198456,7 @@ given prime (or other positive integer) that divides the number. For cv ralbidv imbi2d weq breq1 breq2 cgcd wb simplrl simpll coprm syl2anc cc ad2antll prmz ad2antrl zcnd mulcomd simpl gcdcomd eqeq1d simprr coprmdvds syl3anc sylbid expdimp con1d expimpd ex vtoclga impl exp1d ad2antlr 1m1e0 - zcn cc0 oveq2i exp0d eqtrid adantl mulid1d eqtrd 3imtr4d cbvralvw zmulcld + zcn cc0 oveq2i exp0d eqtrid adantl mulridd eqtrd 3imtr4d cbvralvw zmulcld ralrimiva rspcv syl wrex cn0 nnnn0 ad2antrr zexpcl simplr divides adantll prmnn nncnd expp1d nnexpcld mulassd eqtr4d wne nnne0d dvdsmulcr syl112anc nnzd bitrd an32s syl5ibcom rexlimdva adantlr com23 a2d expm1t nncn ax-1cn @@ -198844,7 +198844,7 @@ given prime (or other positive integer) that divides the number. For ltso wne wral wrex nnssz sstri oveq1 sq1 eqtrdi breq1d 1nn a1i nnz elrabd 1dvds syl ne0d wa zsqcl id dvdsle syl2anr nnlesq adantl nnre resqcld letr adantr syl3anc mpand ralrimiva ralrab sylibr brralrspcev syl2anc suprzcl2 - mp3an2i eqeltrd sselid cbvrabv elrab2 sylib simprd nncnd mulid1d eluz2gt1 + mp3an2i eqeltrd sselid cbvrabv elrab2 sylib simprd nncnd mulridd eluz2gt1 syld cmul cc0 wb 1red eluz2nn nnred nngt0d ltmul2 syl112anc mpbid nnmulcl eqbrtrrd syl2an ltnled nnsqcld nnne0d dvdsval2 dvdscmul mpd sqmuld eqcomd simpr cc nncn divcan2d 3brtr3d suprzub breqtrrd mtand ex 3jca ) DHIZDAUBZ @@ -199822,7 +199822,7 @@ with complex numbers (gaussian integers) instead, so that we only have ( vk crn cun chash cfv clt wbr wn cin c0 wne cfn wcel cn0 cc0 c1 cmin co cfz fzfid cv c2 cexp cmo wceq cab wa wi cz cn elfzelz zsqcl cprime wrex prmnn zmodfz syl2anr eleq1a rexlimdva abssdv eqsstrid caddc prmz - syl peano2zm zcnd addid2d oveq1d adantr sselda fzrev3i ssfid zred cle + syl peano2zm zcnd addlidd oveq1d adantr sselda fzrev3i ssfid zred cle nn0red wb syl2anc mpbird cen fz01en hashen nnnn0d hashfz1 eqtrd ltp1d breqtrd cr cmul cc sylancr 2timesd 3eqtrd cmpt wf1o ex cdvds ad2antrl wf1 weq ad2antll syl3anc elfzle2 breqtrrd cabs nn0zd sylanbrc subid1d @@ -200192,7 +200192,7 @@ with complex numbers (gaussian integers) instead, so that we only have $( Lemma for ~ 4sq . Inductive step, odd prime case. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.) $) 4sqlem18 $p |- ( ph -> P e. S ) $= - ( co va vb vc vd c1 cmul cprime wcel cn prmnn syl nncnd mulid2d wceq c2 + ( co va vb vc vd c1 cmul cprime wcel cn prmnn syl nncnd mullidd wceq c2 cuz cfv wa cv cexp caddc cz wrex wn cr clt cinf wss c0 wne nnuz sseqtri ssrab3 wbr 4sqlem13 simpld infssuzcl eqeltrid oveq1 eleq1d elrab2 sylib sylancr simprd 4sqlem2 adantr wi w3a cdiv cmo simp1l cc0 simp1r simp2ll @@ -200357,9 +200357,9 @@ with complex numbers (gaussian integers) instead, so that we only have cz vx cn0 cn w3a cvdwa cfv csn cun cv wo wrex wb peano2nn0 vdwapval simp1 syl3an1 nn0cnd ax-1cn pncan sylancl eleq2d cuz nn0uz eleqtrdi elfzp12 syl bitrd anbi1d andir bitrdi exbidv df-rex 19.43 bicomi nncn 3ad2ant3 mul02d - 3bitr4g 3ad2ant2 addid1d eqtrd eqeq2d c0ex oveq1 velsn simpr 0p1e1 oveq1i + 3bitr4g 3ad2ant2 addridd eqtrd eqeq2d c0ex oveq1 velsn simpr 0p1e1 oveq1i ceqsexv 1zzd nn0zd elfzelz fzsubel syl22anc mpbid 1m1e0 zcnd 1cnd subdird - adantl mulid2d mulcld pncan3d eqtr2d subcl addassd rspceeqv syl2anc eqeq1 + adantl mullidd mulcld pncan3d eqtr2d subcl addassd rspceeqv syl2anc eqeq1 rexbidv syl5ibrcom expimpd exlimdv peano2zm fzaddel npcan eleqtrd adddird eqtr4d 0zd addcomd ovex eleq1 anbi12d spcev anbi2d impbid nnaddcl 3adant1 rexlimdva syld3an2 bitr4d orbi12d elun bitr4di eqrdv ) CUBFZAUCFZBUCFZUDZ @@ -200525,7 +200525,7 @@ with complex numbers (gaussian integers) instead, so that we only have wrex wcel cfz ffvelcdmd cun cn0 wceq nnnn0d vdwapun syl3anc cle nnred cuz eleqtrdi eluzfz1 syl nnaddcld nnrpd ltaddrpd ltled fveq2 oveq2d eleq1d wa nnuz r19.21bi cnvimass fssdm sstrd cmul cc0 cmin nnm1nn0 nn0uz ffvelcdmda - adantr nncnd mul02d addid1d eqtr2d oveq1 rspceeqv syl2anc vdwapval mpbird + adantr nncnd mul02d addridd eqtr2d oveq1 rspceeqv syl2anc vdwapval mpbird wb sseldd ralrimiva rspcdva elfzle2 letrd cz nnzd elfz5 eqidd wf fniniseg wfn ffn 3syl mpbir2and snssd oveq12d sneqd imaeq2d sseq12d sseqtrrd unssd fveq2d eqsstrd sseq1d oveq2 rspc2ev fvex sneq sseq2d spcev ovex peano2nn0 @@ -200615,7 +200615,7 @@ with complex numbers (gaussian integers) instead, so that we only have ( c1 co caddc cmul wcel cle wbr cn syl nnred cr cmin cfz elfznn nnm1nn0 c2 cn0 nn0nnaddcl syl2anc nnmulcld nnaddcld 2nn nnmulcl sylancr elfzle2 wb nnre leadd1 syl3an syl3anc mpbid nncnd 1cnd adddid nn0cnd addassd cc - ax-1cn pncan3 oveq1d eqtr3d oveq2d mulid1d 3eqtr3d breqtrrd 2timesd cc0 + ax-1cn pncan3 oveq1d eqtr3d oveq2d mulridd 3eqtr3d breqtrrd 2timesd cc0 wceq leadd1dd clt nngt0d lemul2 syl112anc cuz cfv cz nnuz eleqtrdi nnzd letrd elfz5 mpbird ) ACEBJUAKZDLKZMKZLKZJEUEDMKZMKZUBKNZWOWQOPZAWOEBDLK ZMKZWQAWOACWNACJEUBKNZCQNZICEUCRZAEWMGAWLUFNZDQNZWMQNABQNZXEABJDUBKNZXG @@ -200698,7 +200698,7 @@ with complex numbers (gaussian integers) instead, so that we only have impel subcld ax-1cn subcl sylancl npcand 3eqtr3d eqtrid addcomd 3eqtr2d 3eqtrd cnvimass fssdm vdwapid1 3sstr4d ex eqtr4d c0 wne eluzfz1 elfzuz3 ne0d nnzd uzid uzaddcl uztrn eluzle ralrimiva r19.2z rexlimiv mpbir2and - cz idd fznn peano2nnd eluzfz2 iftrue 3syl addid2d sseq12d jaod ralrimiv + cz idd fznn peano2nnd eluzfz2 iftrue 3syl addlidd sseq12d jaod ralrimiv sylbid rexeqdv rexun eqeq2d rexbidva rexsn bitrid orbi12d bitrd uneq12i abbidv rnmpt df-sn unab eqtri 3eqtr4g wfo fzfi dffn4 mpbi fofi mp2an wi cvv hashunsng sylan ralbidv rneqd fveqeq2d fveq1 vdwpc mpbird pm2.61dan @@ -200945,7 +200945,7 @@ with complex numbers (gaussian integers) instead, so that we only have a1i weq eqtr3d breq2d cn0 peano2nn0 nnm1nn0 nn0nnaddcl syl2anc cz cle nnmulcld nnaddcld nnzd sylancr nnred nncnd cr cc mpd nnmulcl leadd1dd 2nn elfzle2 2timesd breqtrrd cc0 clt nngt0d syl112anc eluz2 syl3anbrc - lemul2 nn0cnd addassd mulid1d 3eqtr3d eleqtrd fvoveq1 cbvmptv vdwlem2 + lemul2 nn0cnd addassd mulridd 3eqtr3d eleqtrd fvoveq1 cbvmptv vdwlem2 1cnd addcld adddid ax-1cn pncan3 sylbird syld expr rexlimdvva exlimdv orim2d sylbid ) AJHUHUIZKUJUKULJUMIUNUIZJUJUKULZIUHUIZUOZAJEUPZUHUIUV NEDUJMUQULZVAULZUJLUQULZVAULZHUVSHJUHURUBAUWBUWAHUSZHUWCVBABCDHILMUAS @@ -202135,7 +202135,7 @@ is defined by using the sequence builder ( ` P = seq 1 ( x. , F ) ` ), cif adantl oveq2d caddc cv peano2nn0 syl cuz nn0p1nn elnnuz sylib elfzelz cmin zcnd 1cnd ifcld eleq1 id ifbieq1d fprodm1 nn0cn pncan1 oveq1d eqcomd prodeq1d wb iftrue eqeq12d adantr mpbird fzfid elfznn 1nn fprodnncl nncnd - wn a1i mulid1d eqtr4d iffalse pm2.61ian eqtrd 3eqtrd ) ACDZAEUAFZGHZEWBIF + wn a1i mulridd eqtr4d iffalse pm2.61ian eqtrd 3eqtrd ) ACDZAEUAFZGHZEWBIF ZBUBZJDZWEERZBKZEWBEUJFZIFZWGBKZWBJDZWBERZLFZWLAGHZWBLFZWORZWAWBCDWCWHMAU CBWBNUDWAWGWMBEWBWAWBODWBEUEHDAUFWBUGUHWAWEWDDZPZWFWEEQWRWEQDWAWRWEWEEWBU IUKSWSULUMWEWBMZWFWLWEWBEWEWBJUNWTUOUPUQWAWNEAIFZWGBKZWMLFZWQWAWKXBWMLWAW @@ -202159,7 +202159,7 @@ is defined by using the sequence builder ( ` P = seq 1 ( x. , F ) ` ), $( The primorial of 2. (Contributed by AV, 28-Aug-2020.) $) prmo2 $p |- ( #p ` 2 ) = 2 $= ( c2 cprmo cfv cprime wcel c1 cmin co cmul cif cn 2nn prmonn2 ax-mp iftruei - wceq 2prm 2m1e1 fveq2i eqtri prmo1 oveq1i 2cn mulid2i ) ABCZADEZAFGHZBCZAIH + wceq 2prm 2m1e1 fveq2i eqtri prmo1 oveq1i 2cn mullidi ) ABCZADEZAFGHZBCZAIH ZUHJZAAKEUEUJPLAMNUJUIAUFUIUHQOUIFAIHAUHFAIUHFBCFUGFBRSUATUBAUCUDTTT $. $( The primorial of 3. (Contributed by AV, 28-Aug-2020.) $) @@ -202698,7 +202698,7 @@ Decimal arithmetic (cont.) Carneiro, 19-Apr-2015.) $) dec5dvds2 $p |- -. 5 || ; A C $= ( c5 cdc cdvds wbr dec5dvds caddc co cz wcel wb 5nn0 nn0zi nnnn0i dvdsadd - deccl mp2an cc0 0nn0 dec0h eqid nn0cni addid2i decadd breq2i bitri mtbi ) + deccl mp2an cc0 0nn0 dec0h eqid nn0cni addlidi decadd breq2i bitri mtbi ) HABIZJKZHACIZJKZABDEFLUOHHUNMNZJKZUQHOPUNOPUOUSQHRSUNABDBETZUBSHUNUAUCURU PHJUDHABACHUNUERDUTHRUFUNUGAADUHUIGUJUKULUM $. $} @@ -202709,7 +202709,7 @@ Decimal arithmetic (cont.) Carneiro, 19-Apr-2015.) $) dec5nprm $p |- -. ; A 5 e. Prime $= ( c2 cmul co c1 caddc c5 cdc cn wcel 2nn nnmulcli peano2nn ax-mp 5nn 1nn0 - 1lt2 nncni 5cn numlti 1lt5 cc0 mul32i 5t2e10 oveq1i eqtri mulid2i oveq12i + 1lt2 nncni 5cn numlti 1lt5 cc0 mul32i 5t2e10 oveq1i eqtri mullidi oveq12i mulcomli ax-1cn adddiri dfdec10 3eqtr4i nprmi ) CADEZFGEZHAHIZUPJKUQJKCAL BMZUPNOPAFFCLBQQRUAUBUPHDEZFHDEZGEFUCIZADEZHGEUQHDEURUTVCVAHGUTCHDEZADEVC CAHCLSZABSTUDVDVBADHCVBTVEUEUJUFUGHTUHUIUPFHUPUSSUKTULAHUMUNUO $. @@ -202860,7 +202860,7 @@ Decimal arithmetic (cont.) decexp2 $p |- ( ( 4 x. ( 2 ^ M ) ) + 0 ) = ( 2 ^ N ) $= ( c4 c2 cexp co cmul c1 caddc 2cn nn0cni wcel cn0 wceq expp1 mp2an ax-1cn 3eqtr4i 2nn0 nn0expcli mulcli cc mulcomi eqtr2i oveq1i mulcomli peano2nn0 - cc0 decbin0 ax-mp 4nn0 nn0mulcli addid1i addassi df-2 oveq2i 3eqtr2ri ) E + cc0 decbin0 ax-mp 4nn0 nn0mulcli addridi addassi df-2 oveq2i 3eqtr2ri ) E FAGHZIHZFAJKHZJKHZGHZVAUJKHFBGHFFUTIHZIHFVBGHZFIHZVAVDVEFVGFUTLUTFAUACUBZ MZUCLVEVFFIVFUTFIHZVEFUDNZAONZVFVJPLCFAQRUTFVILUEUFUGUHUTVHUKVKVBONZVDVGP LVLVMCAUIULFVBQRTVAVAEUTUMVHUNMUOBVCFGVCAJJKHZKHAFKHBAJJACMSSUPFVNAKUQURD @@ -202909,14 +202909,14 @@ Decimal arithmetic (cont.) (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.) $) decsplit0b $p |- ( ( A x. ( ; 1 0 ^ 0 ) ) + B ) = ( A + B ) $= - ( c1 cc0 cdc cexp co cmul caddc 10nn0 numexp0 oveq2i nn0cni mulid1i eqtri + ( c1 cc0 cdc cexp co cmul caddc 10nn0 numexp0 oveq2i nn0cni mulridi eqtri oveq1i ) ADEFZEGHZIHZABJTADIHASDAIRKLMAACNOPQ $. $( Split a decimal number into two parts. Base case: ` N = 0 ` . (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.) $) decsplit0 $p |- ( ( A x. ( ; 1 0 ^ 0 ) ) + 0 ) = A $= - ( c1 cc0 cdc cexp co cmul caddc decsplit0b nn0cni addid1i eqtri ) ACDEDFG + ( c1 cc0 cdc cexp co cmul caddc decsplit0b nn0cni addridi eqtri ) ACDEDFG HGDIGADIGAADBJAABKLM $. $( Split a decimal number into two parts. Base case: ` N = 1 ` . @@ -203013,8 +203013,8 @@ Decimal arithmetic (cont.) 20-Apr-2015.) $) 2exp8 $p |- ( 2 ^ 8 ) = ; ; 2 5 6 $= ( c2 c5 cdc c6 c1 c4 c8 2nn0 4nn0 nn0cni 2cn c9 1nn0 6nn0 9nn0 co 6cn caddc - cmul c3 4t2e8 mulcomli 2exp4 deccl eqid mulid1i 1p1e2 9cn addcomli decaddci - 5nn0 9p6e15 mulid2i oveq1i 6p3e9 eqtri 6t6e36 decmul1c decmul2c numexp2x + cmul c3 4t2e8 mulcomli 2exp4 deccl eqid mulridi 1p1e2 9cn addcomli decaddci + 5nn0 9p6e15 mullidi oveq1i 6p3e9 eqtri 6t6e36 decmul1c decmul2c numexp2x 3nn0 ) AABCZDCEDCZFGHIFAGFIJKUAUBUCEDVBDVCLVCEDMNUDZMNVCUEZNOEDBAVCESPLMNOV CVCVDJUFUGUKLDEBCUHQULUIUJEDLDDTVCNMNVENVAEDSPZTRPDTRPLVFDTRDQUMUNUOUPUQURU SUT $. @@ -203024,7 +203024,7 @@ Decimal arithmetic (cont.) ( c2 c1 cdc cexp co c8 c3 cmul cc0 c4 wcel 8nn0 eqtri c5 2nn0 4nn0 0nn0 8cn c6 mulcomli caddc 8p3e11 eqcomi oveq2i cn0 wceq 2cn 3nn0 expadd mp3an 2exp8 cc cu2 oveq12i 5nn0 deccl 6nn0 eqid 8t2e16 1p1e2 6p4e10 decaddci 5cn 8t5e40 - 1nn0 decmul1c 4cn addid2i decaddi 6cn 8t6e48 ) ABBCZDEZAFDEZAGDEZHEZAICZJCZ + 1nn0 decmul1c 4cn addlidi decaddi 6cn 8t6e48 ) ABBCZDEZAFDEZAGDEZHEZAICZJCZ FCZVMAFGUAEZDEZVPVLVTADVTVLUBUCUDAULKFUEKGUEKWAVPUFUGLUHAFGUIUJMVPANCZSCZFH EVSVNWCVOFHUKUMUNWBSVRFFJWCLANOUOUPUQWCURLPVQIJWBFHEJAIOQUPQPANVQIFJWBLOUOW BURQPBSIAAFHEJVEUQPFABSCRUGUSTUTQVAVBFNJICRVCVDTVFJVGVHVIFSJFCRVJVKTVFMM $. @@ -203035,7 +203035,7 @@ Decimal arithmetic (cont.) ( c2 c6 c5 cdc c3 c8 c1 2nn0 5nn0 deccl 6nn0 eqid 1nn0 caddc co cc0 decaddi 3nn0 0nn0 cmul 8nn0 8cn 2cn 8t2e16 mulcomli 2exp8 c4 4nn0 dec0h 0p1e1 1p2e3 decadd 3p1e4 8p5e13 addcomli decaddc 4p1e5 2t2e4 1p1e2 oveq12i 4p2e6 5t2e10 - 5cn eqtri addid2i decmac 6t2e12 3cn 3p2e5 oveq2i 5t5e25 decma2c 6cn decrmac + 5cn eqtri addlidi decmac 6t2e12 3cn 3p2e5 oveq2i 5t5e25 decma2c 6cn decrmac 5p3e8 6t5e30 6t6e36 decmul1c decmul2c numexp2x ) ABCDZCDZEDZBDACDZBDZFGBDZH UAFAWFUBUCUDUEUFWDBWCBWEGCDZEDZWEWDBACHIJZKJZWIKWELZKWGEGCMIJZRJACWGEWEWBEG ADZFDZWDWHHIWLRWDLZWHLWJRWMFGAMHJZUAJWDBGUGDZEAWACGWEWGWNNOWIKGUGMUHJRWKGCW @@ -203431,7 +203431,7 @@ consisting of identical symbols by at least one position (and not by as $( 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) $) 13prm $p |- ; 1 3 e. Prime $= - ( c1 c3 cdc 1nn0 3nn decnncl 1nn 3nn0 1lt10 declti c2 mulid2i df-3 dec2dvds + ( c1 c3 cdc 1nn0 3nn decnncl 1nn 3nn0 1lt10 declti c2 mullidi df-3 dec2dvds 2cn c4 4nn0 cmul co 2nn0 4cn 3cn 4t3e12 mulcomli decsuc 1lt3 ndvdsi c5 5nn0 2p1e3 3lt10 1lt2 decltc prmlem1 ) ABCZABDEFABAGHDIJAAKBDDKOLMNBUOPAEQGAKBBP RSDTUJPBAKCUAUBUCUDUEUFUGAKBUHDTHUIUKULUMUN $. @@ -203455,7 +203455,7 @@ consisting of identical symbols by at least one position (and not by as $( 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) $) 23prm $p |- ; 2 3 e. Prime $= - ( c2 c3 cdc 2nn0 3nn decnncl c1 2nn 3nn0 1nn0 1lt10 declti mulid2i dec2dvds + ( c2 c3 cdc 2nn0 3nn decnncl c1 2nn 3nn0 1nn0 1lt10 declti mullidi dec2dvds nncni df-3 c7 7nn0 cmul co 7cn 7t3e21 mulcomli 1p2e3 decaddi 2lt3 ndvdsi c5 5nn 3lt5 declt prmlem1 ) ABCZABDEFABGHIJKLAGABDJAAHOMPNBUMQAERHAGBBQSTADJDQ BAGCUABEOUBUCUDUEUFUGABUHDIUIUJUKUL $. @@ -203525,8 +203525,8 @@ consisting of identical symbols by at least one position (and not by as 37prm $p |- ; 3 7 e. Prime $= ( c3 c7 cdc 3nn0 decnncl c8 c4 c1 4nn0 7nn0 1nn0 3nn c6 c2 2nn0 cmul co cc0 caddc ndvdsi 8nn0 deccl 7lt10 3lt10 declti decltc 1lt10 3t2e6 df-7 dec2dvds - 7nn 8nn 1nn 6nn0 6p1e7 eqid 3cn mulid1i oveq1i addid1i eqtri dec0h decmul2c - 0nn0 decsuc 1lt3 2nn 2lt5 dec5dvds2 c5 5nn0 7t5e35 decaddi 2lt7 4nn mulid2i + 7nn 8nn 1nn 6nn0 6p1e7 eqid 3cn mulridi oveq1i addridi eqtri dec0h decmul2c + 0nn0 decsuc 1lt3 2nn 2lt5 dec5dvds2 c5 5nn0 7t5e35 decaddi 2lt7 4nn mullidi 5p2e7 nncni 4p3e7 addcomli decrmanc 4lt10 decmul1 2p1e3 declt 0p1e1 oveq12i 2cn decadd 7t2e14 decmac c9 9nn 9nn0 1p1e2 9p8e17 decaddc 8lt9 1lt2 prmlem2 ) ABCZABDUKEAFGCBHDFGUAIUBJKUCFGAULIDUDUEUFABHLJKUGUEAAMBDDUHUIUJAXAHNCZHLH @@ -203545,9 +203545,9 @@ consisting of identical symbols by at least one position (and not by as 43prm $p |- ; 4 3 e. Prime $= ( c4 c3 cdc 4nn0 c1 3nn0 1nn0 c2 1nn 0nn0 eqid cmul co caddc 2nn0 ndvdsi c7 cc0 c9 c5 3nn decnncl c8 8nn0 deccl 3lt10 8nn 4lt10 declti decltc 4nn 1lt10 - 2cn mulid2i dec2dvds dec0h mulid1i ax-1cn addid2i oveq12i 3p1e4 eqtri 2p1e3 + 2cn mullidi dec2dvds dec0h mulridi ax-1cn addlidi oveq12i 3p1e4 eqtri 2p1e3 df-3 3cn 4cn 4t3e12 mulcomli decsuc decma2c 1lt3 3lt5 dec5dvds c6 6nn0 1lt7 - 7t6e42 decnncl2 addid1i oveq1i 3eqtri decmac 0lt1 declt 3t3e9 9p4e13 nnnn0i + 7t6e42 decnncl2 addridi oveq1i 3eqtri decmac 0lt1 declt 3t3e9 9p4e13 nnnn0i 7nn 9nn 2p2e4 7t2e14 1p1e2 nncni addcomli 9lt10 5nn 9t2e18 8p5e13 5lt10 2nn decaddci decadd 3pos prmlem2 ) ABCZABDUAUBAUCACBEDUCAUDDUEFGUFUCAAUGDDUHUIU JABEUKFGULUIAEHBDGHUMUNZVDUOBXEEACZEUAEAGDUEIEAREBABEXGEGDJGXGKEGUPFFGBELMZ @@ -203566,8 +203566,8 @@ consisting of identical symbols by at least one position (and not by as 83prm $p |- ; 8 3 e. Prime $= ( c8 c3 cdc 8nn0 decnncl c4 c1 4nn0 3nn0 1nn0 c2 c7 2nn0 7nn0 cmul co caddc c6 eqtri c5 3nn deccl 3lt10 8nn 8lt10 declti decltc 1lt10 2cn df-3 dec2dvds - mulid2i 2nn cc0 0nn0 eqid dec0h 3t2e6 addid2i oveq12i 6p2e8 nn0cni mulcomli - 3cn 7t3e21 decaddi decma2c 2lt3 ndvdsi 3lt5 dec5dvds 7nn 6nn nnnn0i mulid1i + mullidi 2nn cc0 0nn0 eqid dec0h 3t2e6 addlidi oveq12i 6p2e8 nn0cni mulcomli + 3cn 7t3e21 decaddi decma2c 2lt3 ndvdsi 3lt5 dec5dvds 7nn 6nn nnnn0i mulridi 1p2e3 ax-1cn 7p1e8 oveq1i 7p6e13 6lt7 1nn nncni mulcomi 6lt10 5nn 6cn 1p1e2 6t3e18 8p5e13 decaddci decmac 5lt10 3p1e4 addcomli 4p4e8 7t4e28 2p1e3 declt 5lt7 c9 9nn 9nn0 9t4e36 7lt10 4nn 3t3e9 9p4e13 4lt10 1lt2 prmlem2 ) ABCZABD @@ -203590,11 +203590,11 @@ consisting of identical symbols by at least one position (and not by as 139prm $p |- ; ; 1 3 9 e. Prime $= ( c1 c3 cdc c9 1nn0 3nn0 c8 c4 9nn0 c6 6nn0 0nn0 cmul co caddc c2 2nn0 7nn0 cc0 c7 deccl 9nn decnncl 8nn0 4nn0 1lt8 3lt10 9lt10 3decltc 3nn 1lt10 4t2e8 - declti df-9 dec2dvds 1nn eqid dec0h ax-1cn addid2i oveq2i nn0cni 3cn 4t3e12 + declti df-9 dec2dvds 1nn eqid dec0h ax-1cn addlidi oveq2i nn0cni 3cn 4t3e12 2p1e3 mulcomli decsuc eqtri 8p1e9 6t3e18 decma2c 1lt3 ndvdsi 4nn 4lt5 5p4e9 - dec5dvds2 7nn 6nn 7cn mulid1i oveq12i 7p6e13 9cn 9t7e63 6p3e9 addcomli 6lt7 - decaddi 2cn nncni 1p2e3 mulid2i 00id oveq1i 7p2e9 3eqtri decmac 7lt10 10nn0 - addid1i mul01i 5nn0 8cn 5cn 8p5e13 8t7e56 6lt10 6t2e12 decrmac 2nn prmlem2 + dec5dvds2 7nn 6nn 7cn mulridi oveq12i 7p6e13 9cn 9t7e63 6p3e9 addcomli 6lt7 + decaddi 2cn nncni 1p2e3 mullidi 00id oveq1i 7p2e9 3eqtri decmac 7lt10 10nn0 + addridi mul01i 5nn0 8cn 5cn 8p5e13 8t7e56 6lt10 6t2e12 decrmac 2nn prmlem2 c5 ) ABCZDCZXNDABEFUAZUBUCAGBHDAEUDFUEIEUFUGUHUIXNDAABEUJUCZIEUKUMXNHGDXPUE ULUNUOBXOHJCZAUJHJUEKUAUPHJSABXNDAXRAUEKLEXRUQAEURFIEBHMNZSAONZONXSAONXNXTA XSOAUSUTVAAPBXSEQVEHBAPCZHUEVBVCVDVFVGVHAGDBJMNEUDVIJBAGCJKVBZVCVJVFVGZVKVL @@ -203617,10 +203617,10 @@ consisting of identical symbols by at least one position (and not by as 163prm $p |- ; ; 1 6 3 e. Prime $= ( c1 c6 cdc c3 1nn0 c8 c4 3nn0 c2 c5 cc0 0nn0 cmul co caddc eqtri 2nn0 7nn0 c7 c9 6nn0 deccl 3nn decnncl 8nn0 4nn0 6lt10 3lt10 3decltc 6nn 1lt10 declti - 1lt8 2cn mulid2i df-3 dec2dvds 5nn0 1nn eqid dec0h ax-1cn addid2i 5p1e6 5cn + 1lt8 2cn mullidi df-3 dec2dvds 5nn0 1nn eqid dec0h ax-1cn addlidi 5p1e6 5cn oveq2i 3cn 5t3e15 mulcomli decsuc 2p1e3 4cn 4t3e12 decma2c 1lt3 ndvdsi 3lt5 dec5dvds 7nn 2nn 7t2e14 4p2e6 decaddi 7t3e21 1p2e3 2lt7 9nn0 nncni addcomli - 9nn mulid1i oveq12i 4p1e5 oveq1i 9cn 9p4e13 decmac 9lt10 00id addid1i 3p3e6 + 9nn mulridi oveq12i 4p1e5 oveq1i 9cn 9p4e13 decmac 9lt10 00id addridi 3p3e6 3t2e6 7cn 6cn 7p6e13 7lt10 10nn 6p1e7 9p7e16 9t7e63 nn0cni 7pos declt 7p1e8 8cn 8p8e16 9t8e72 1lt9 decrmac 2lt10 prmlem2 ) ABCZDCZYBDABEUAUBZUCUDAFBGDA EUEUAUFHEUMUGUHUIYBDAABEUJUDHEUKULYBAIDYDEIUNUOZUPUQDYCJGCZAUCJGURUFUBUSJGK @@ -203647,10 +203647,10 @@ consisting of identical symbols by at least one position (and not by as 317prm $p |- ; ; 3 1 7 e. Prime $= ( c3 c1 cdc c7 3nn0 1nn0 c8 c4 7nn0 cc0 c5 c2 5nn0 0nn0 2nn0 caddc co eqtri cmul c9 deccl 7nn decnncl 8nn0 4nn0 1lt10 7lt10 3decltc 1nn declti c6 3t2e6 - 3lt8 df-7 dec2dvds 3nn 10nn0 2nn eqid dec0h ax-1cn addid2i 3cn mulid1i 00id - oveq12i addid1i mul01i oveq1i decma2c 5cn 5t3e15 mulcomli 5p2e7 2lt3 ndvdsi + 3lt8 df-7 dec2dvds 3nn 10nn0 2nn eqid dec0h ax-1cn addlidi 3cn mulridi 00id + oveq12i addridi mul01i oveq1i decma2c 5cn 5t3e15 mulcomli 5p2e7 2lt3 ndvdsi decaddi 2lt5 dec5dvds2 oveq2i 7t4e28 2p1e3 8p3e11 decaddci 7t5e35 2lt7 9nn0 - 9nn 9cn 2cn mulid2i 9p2e11 addcomli decmac 8cn 8p1e9 9p8e17 9lt10 5nn 4p1e5 + 9nn 9cn 2cn mullidi 9p2e11 addcomli decmac 8cn 8p1e9 9p8e17 9lt10 5nn 4p1e5 6p5e11 4cn 4t3e12 5lt10 3p1e4 decadd 1p1e2 1p2e3 7p4e11 8p5e13 6p1e7 8t7e56 7cn 6nn0 decsuc 1lt7 declt 6cn 9t6e54 4p3e7 3lt9 8nn 7p1e8 3t3e9 8lt10 1lt2 nnnn0i decltc prmlem2 ) ABCZDCZYJDABEFUAZUBUCAGBHDBEUDFUEIFUMUFUGUHYJDBABEU @@ -203685,10 +203685,10 @@ consisting of identical symbols by at least one position (and not by as 631prm $p |- ; ; 6 3 1 e. Prime $= ( c6 c3 cdc c1 3nn0 deccl c8 c4 1nn0 cc0 0nn0 c2 2nn0 caddc eqtri cmul 7nn0 co c7 c5 6nn0 1nn decnncl 8nn0 4nn0 6lt8 3lt10 1lt10 3decltc 3nn declti 2cn - mul02i 1e0p1 dec2dvds eqid dec0h 3t2e6 oveq12i 6cn addid1i 3t1e3 oveq1i 3cn + mul02i 1e0p1 dec2dvds eqid dec0h 3t2e6 oveq12i 6cn addridi 3t1e3 oveq1i 3cn 00id 3eqtri decma2c mul01i 1lt3 ndvdsi 1lt5 dec5dvds c9 7nn 9nn0 oveq2i 9cn - 0p1e1 7cn 9t7e63 mulcomli nn0cni 1lt7 5nn0 4nn 8cn addid2i 5cn 5p1e6 8p5e13 - mulid2i addcomli decmac 7p1e8 7p4e11 4lt10 1p1e2 4p2e6 4t3e12 2p1e3 decaddi + 0p1e1 7cn 9t7e63 mulcomli nn0cni 1lt7 5nn0 4nn 8cn addlidi 5cn 5p1e6 8p5e13 + mullidi addcomli decmac 7p1e8 7p4e11 4lt10 1p1e2 4p2e6 4t3e12 2p1e3 decaddi 4cn 8p3e11 8t3e24 decaddci 7lt10 2nn 1p2e3 3p3e6 7t3e21 7p5e12 7t7e49 4p1e5 9p2e11 2lt10 9nn 9t3e27 7p6e13 10nn ax-1cn 6p1e7 decadd 2t2e4 decmul1c 1lt2 7t2e14 10pos decltc prmlem2 ) ABCZDCZYJDABUAEFZUBUCAGBHDDUAUDEUEIIUFUGUHUIY @@ -203754,8 +203754,8 @@ consisting of identical symbols by at least one position (and not by as 1259lem1 $p |- ( ( 2 ^ ; 1 7 ) mod N ) = ( ; ; 1 3 6 mod N ) $= ( c2 c1 c6 cdc cc0 c8 c3 c5 1nn0 2nn0 5nn0 6nn0 8nn0 co eqid cmul caddc c4 c7 c9 cn deccl 9nn decnncl eqeltri 0z 3nn0 cexp nn0zi nn0expcli nn0cni - 2nn cmo 2cn 8t2e16 mulcomli 9nn0 4nn0 7nn0 dec0h 4cn addid2i oveq1i 4p1e5 - 0nn0 eqtri 7cn 6cn 7p6e13 addcomli decaddc 2p1e3 5cn 6p5e11 10nn0 mulid1i + 2nn cmo 2cn 8t2e16 mulcomli 9nn0 4nn0 7nn0 dec0h 4cn addlidi oveq1i 4p1e5 + 0nn0 eqtri 7cn 6cn 7p6e13 addcomli decaddc 2p1e3 5cn 6p5e11 10nn0 mulridi dec10p 1p0e1 oveq12i 5p1e6 3p2e5 3eqtri decmac 5t2e10 00id decma2c 5t5e25 3cn 2t2e4 decsuc decaddi decrmac 9cn 9t5e45 9t2e18 8p8e16 decaddci 2exp16 5p2e7 1p1e2 numexp2x 3eqtr2i mod2xi 6p1e7 mul02i decmul1c 3eqtr4i modxp1i @@ -203781,9 +203781,9 @@ consisting of identical symbols by at least one position (and not by as 1259lem2 $p |- ( ( 2 ^ ; 3 4 ) mod N ) = ( ; ; 8 7 0 mod N ) $= ( c2 c1 cdc c4 c3 c6 c8 cc0 c5 c9 1nn0 4nn0 6nn0 0nn0 cmul co caddc eqtri c7 2nn0 deccl 5nn0 9nn decnncl eqeltri 7nn0 nn0zi 3nn0 8nn0 1259lem1 eqid - cn 2nn 2cn mulid1i oveq1i 2p1e3 7t2e14 mulcomli decmul2c 9nn0 8p1e9 7p2e9 - 7cn decadd 9p7e16 3cn 3p1e4 addcomli dec0h 00id oveq12i addid1i 4cn 4p4e8 - ax-1cn 3eqtri decmac mulid2i addid2i 4t2e8 8p6e14 decma2c 5cn 9cn decaddi + cn 2nn 2cn mulridi oveq1i 2p1e3 7t2e14 mulcomli decmul2c 9nn0 8p1e9 7p2e9 + 7cn decadd 9p7e16 3cn 3p1e4 addcomli dec0h 00id oveq12i addridi 4cn 4p4e8 + ax-1cn 3eqtri decmac mullidi addlidi 4t2e8 8p6e14 decma2c 5cn 9cn decaddi 5p2e7 5t4e20 9p3e12 9t4e36 nn0cni 8cn 4p1e5 5p3e8 6cn 3t3e9 9p1e10 6t3e18 decmul1c 6p2e8 1p1e2 8p3e11 decaddci 6t6e36 eqtr4i mod2xi ) CDUAEZDFEZGFE DGEZHEZIUAEZJEZAADCEZKEZLEUNBXPLXOKDCMUBUCZUDUCZUEUFUGUODUAMUHUCXJDFMNUCZ @@ -203815,9 +203815,9 @@ consisting of identical symbols by at least one position (and not by as ( c2 c3 c8 cdc c4 c7 c6 c1 c5 c9 1nn0 2nn0 cc0 4nn0 0nn0 co cmul caddc cn deccl 5nn0 9nn decnncl eqeltri 3nn0 8nn0 4z 7nn0 nn0zi 6nn0 1259lem2 cexp 2nn cmo 2exp4 oveq1i eqid 4p4e8 decaddi 9nn0 10nn0 nn0cni dec10p addcomli - 7cn mulid2i dec0h 0p1e1 3cn ax-1cn 3p1e4 decadd 2p1e3 decsuc 5p4e9 7p5e12 - 5cn decaddci 9cn 9t11e99 mulcomli decsucc decma2c addid1i mulid1i oveq12i - decmac 9p1e10 8cn 8p5e13 eqtri 3eqtri mul02i addid2i 8t6e48 8p4e12 7t6e42 + 7cn mullidi dec0h 0p1e1 3cn ax-1cn 3p1e4 decadd 2p1e3 decsuc 5p4e9 7p5e12 + 5cn decaddci 9cn 9t11e99 mulcomli decsucc decma2c addridi mulridi oveq12i + decmac 9p1e10 8cn 8p5e13 eqtri 3eqtri mul02i addlidi 8t6e48 8p4e12 7t6e42 7p2e9 2cn 4p1e5 decmul1c 6cn decmul1 eqtr4i modxai 3t2e6 6p1e7 8t2e16 4cn decmul2c 4t2e8 8p2e10 5t4e20 9t4e36 6p5e11 7t7e49 7p7e14 decrmac mod2xi ) CDEFZGHIFHJFZKAAJCFZKFZLFUABYELYDKJCMNUBZUCUBZUDUEUFZUODEUGUHUBUIHJUJMUBZ @@ -203850,9 +203850,9 @@ consisting of identical symbols by at least one position (and not by as ( c2 c6 cdc c9 cc0 c1 co 2nn0 deccl 1nn0 c5 c8 caddc eqid c3 c4 4nn0 cmul cmin 2nn 6nn0 9nn0 0z 1nn cn0 5nn0 8nn0 nn0cni ax-1cn 8p1e9 decsuc eqtr4i mvrraddi eqeltri c7 cn 9nn decnncl 3nn0 nn0zi 7nn0 0nn0 1z 1259lem3 4p1e5 - 8nn 7cn 2cn 7t2e14 mulcomli 6cn 6t2e12 addid2i nncni mul02i oveq1i 5t5e25 - decmul2c 3eqtr4i mod2xi 2p1e3 2t2e4 eqtri 5t2e10 decmul1c modxp1i mulid1i - 5cn addid1i 3t2e6 dec0h mulid2i 1p1e2 2p2e4 decadd 5p4e9 decsucc decaddc2 + 8nn 7cn 2cn 7t2e14 mulcomli 6cn 6t2e12 addlidi nncni mul02i oveq1i 5t5e25 + decmul2c 3eqtr4i mod2xi 2p1e3 2t2e4 eqtri 5t2e10 decmul1c modxp1i mulridi + 5cn addridi 3t2e6 dec0h mullidi 1p1e2 2p2e4 decadd 5p4e9 decsucc decaddc2 3cn 9p1e10 decmul1 mul01i oveq12i 0p1e1 3eqtri decma2c 8cn addcomli 1p2e3 8p4e12 decmac 8p6e14 8t8e64 mod2xnegi 1259lem1 7p2e9 3p2e5 decaddi 7p5e12 5p1e6 decaddc 7p3e10 3p1e4 6p1e7 4t2e8 8p2e10 5t3e15 5p2e7 5t4e20 decrmac @@ -203913,9 +203913,9 @@ consisting of identical symbols by at least one position (and not by as ( c8 c6 cdc c9 c1 c2 c3 c4 3nn0 deccl 8nn0 9nn0 1nn0 2nn0 cc0 cmul caddc co cexp cmin cn wcel cn0 2nn 4nn0 nnexpcl mp2an nnm1nn0 ax-mp 6nn0 c5 9nn 5nn0 decnncl eqeltri 1259lem2 6p1e7 eqid decsuc decsucc modsubi 0nn0 cgcd - c7 cz wceq nn0zi gcdcom cdvds wbr wn 3nn 8nn dec0h ax-1cn mulid1i addid2i + c7 cz wceq nn0zi gcdcom cdvds wbr wn 3nn 8nn dec0h ax-1cn mulridi addlidi oveq12i 1p1e2 eqtri 3cn oveq1i 8p3e11 addcomli decmac 8lt10 declti ndvdsi - 8cn 1nn cprime wb 13prm coprm mpbi 2cn mulid2i addid1i 2p1e3 3p1e4 3eqtri + 8cn 1nn cprime wb 13prm coprm mpbi 2cn mullidi addridi 2p1e3 3p1e4 3eqtri decma2c gcdi 3t2e6 mulcomli 6p2e8 4cn 4t2e8 8p1e9 4p3e7 oveq2i 8t4e32 7cn 7nn0 7p2e9 decaddi 9cn 9t4e36 6p4e10 decaddci2 9t2e18 8p8e16 2t0e0 nn0cni decaddci 8p4e12 6cn 9p6e15 eqtr4i gcdmodi ) CDEZFEZGHIJEZUATZGUBTZAYPUCUD @@ -203965,11 +203965,11 @@ consisting of identical symbols by at least one position (and not by as 2503lem1 $p |- ( ( 2 ^ ; 1 8 ) mod N ) = ( ; ; ; 1 8 3 2 mod N ) $= ( c2 c1 cc0 c4 c5 c3 2nn0 5nn0 deccl 0nn0 4nn0 1nn0 co c6 eqid cmul caddc cdc c9 c8 cn 3nn decnncl eqeltri 2nn 9nn0 10nn0 nn0zi 8nn0 3nn0 cmo 8p1e9 - 2exp8 dec0h 2t2e4 ax-1cn addid2i oveq12i 4p1e5 eqtri 5t2e10 decsuc decmac + 2exp8 dec0h 2t2e4 ax-1cn addlidi oveq12i 4p1e5 eqtri 5t2e10 decsuc decmac cexp 6t2e12 decmul1c numexpp1 oveq1i 9cn 2cn 9t2e18 mulcomli 1p1e2 8p3e11 - 6nn0 decaddci 3p1e4 decadd addid1i 2p2e4 decaddi addcomli dec0u 5p1e6 6cn - nn0cni 4t2e8 8p4e12 5cn 4cn 5t4e20 decma2c mul01i 3eqtri 3cn mulid2i 00id - mul02i 4t3e12 decrmac c7 7nn0 6p1e7 oveq2i 5t5e25 7p5e12 mulid1i decrmanc + 6nn0 decaddci 3p1e4 decadd addridi 2p2e4 decaddi addcomli dec0u 5p1e6 6cn + nn0cni 4t2e8 8p4e12 5cn 4cn 5t4e20 decma2c mul01i 3eqtri 3cn mullidi 00id + mul02i 4t3e12 decrmac c7 7nn0 6p1e7 oveq2i 5t5e25 7p5e12 mulridi decrmanc 7cn decmul1 decmul2c eqtr4i mod2xi ) CUADETZFTZDUBTZGDTZCTZXRHTZCTZAACGTZ ETZHTUCBYDHYCECGIJKZLKZUDUEUFUGUHXQXPFUIMKZUJXSCGDJNKZIKZYACXRHDUBNUKKZUL KZIKCUAVFOXTAUMCXTUBUAIUKUNYCPXSCCDCUBVFOIYEVQUOINCGEDCGDDYCDIJLNYCQZDNUP @@ -204007,12 +204007,12 @@ consisting of identical symbols by at least one position (and not by as 2503lem2 $p |- ( ( 2 ^ ( N - 1 ) ) mod N ) = ( 1 mod N ) $= ( c2 c1 cdc c5 cc0 co c3 2nn0 0nn0 1nn0 c4 c8 4nn0 c6 c9 c7 cmul caddc cn cmin 5nn0 deccl 3nn decnncl eqeltri 2nn 0z 8nn0 3nn0 6nn0 9nn0 nn0zi 7nn0 - 4nn 2503lem1 8p1e9 eqid decsuc nn0cni addid1i mulid2i 1p0e1 oveq12i 2p1e3 - 2cn eqtri 5cn oveq1i 5p1e6 dec0h 3eqtri decma2c ax-1cn mul01i 6cn addid2i + 4nn 2503lem1 8p1e9 eqid decsuc nn0cni addridi mullidi 1p0e1 oveq12i 2p1e3 + 2cn eqtri 5cn oveq1i 5p1e6 dec0h 3eqtri decma2c ax-1cn mul01i 6cn addlidi 1z 3cn 3p1e4 8t2e16 decmul1c 3t2e6 decmul1 2t2e4 eqtr4i modxp1i 2t1e2 9cn 9t2e18 mulcomli decmul2c 1p1e2 6p3e9 addcomli decadd 6p1e7 9p2e11 decaddc 7p1e8 oveq2i 5t2e10 decaddi 8p6e14 decmac 0p1e1 5t5e25 5t3e15 5p4e9 3t3e9 - 8cn 4cn 8t5e40 9p3e12 8t3e24 7cn 7p4e11 decaddci 1t1e1 00id 1p2e3 mulid1i + 8cn 4cn 8t5e40 9p3e12 8t3e24 7cn 7p4e11 decaddci 1t1e1 00id 1p2e3 mulridi mod2xi mul02i 7t2e14 4t2e8 8p4e12 4p1e5 9p4e13 7p5e12 9p8e17 9t9e81 4p3e7 decsucc 9t4e36 decrmac 9p6e15 7t7e49 decaddci2 9p1e10 decma 6t6e36 6p6e12 decaddc2 2p2e4 5t4e20 4t3e12 5p2e7 4p2e6 7p6e13 4t4e16 6t2e12 6p2e8 7p2e9 @@ -204155,11 +204155,11 @@ consisting of identical symbols by at least one position (and not by as ( c1 c8 cdc c3 c2 co 1nn0 c5 cc0 2nn0 c6 c7 7nn0 c4 c9 cmul caddc eqtri cexp cmin cn wcel cn0 2nn 8nn0 deccl nnexpcl mp2an nnm1nn0 3nn0 5nn0 0nn0 ax-mp 3nn decnncl eqeltri 2503lem1 1p1e2 eqid modsubi 6nn0 4nn0 9nn0 cgcd - decsuc cz wceq nn0zi gcdcom cdvds wbr 9nn 1nn mulid2i oveq12i 2p2e4 8p1e9 + decsuc cz wceq nn0zi gcdcom cdvds wbr 9nn 1nn mullidi oveq12i 2p2e4 8p1e9 wn 2cn 9t2e18 decmac 1lt9 declt ndvdsi cprime wb 19prm coprm mpbi 4cn 9cn 4p2e6 oveq1i 9p9e18 decma2c gcdi 4p1e5 6p5e11 9p8e17 addcomli 6p1e7 dec0h - 6cn 8cn 1t1e1 00id ax-1cn addid1i 7cn 7p1e8 3eqtri 8p7e15 8t2e16 mulcomli - 2t1e2 6p2e8 decaddi 5cn 5t2e10 addid2i 7p5e12 oveq2i 6t2e12 7t2e14 9p4e13 + 6cn 8cn 1t1e1 00id ax-1cn addridi 7cn 7p1e8 3eqtri 8p7e15 8t2e16 mulcomli + 2t1e2 6p2e8 decaddi 5cn 5t2e10 addlidi 7p5e12 oveq2i 6t2e12 7t2e14 9p4e13 decaddci 2t2e4 7p4e11 nn0cni 3cn 7p3e10 1p2e3 eqtr4i gcdmodi ) CDEZFEZCEZ CGYQUAHZCUBHZAYTUCUDZUUAUEUDGUCUDYQUEUDUUBUFCDIUGUHZGYQUIUJZYTUKUOYRCYQFU UCULUHZIUHZAGJEZKEZFEZUCBUUHFUUGKGJLUMUHUNUHUPUQURZYTCYRGEYSAUUJUUDIUUFAB @@ -204194,8 +204194,8 @@ consisting of identical symbols by at least one position (and not by as 2503prm $p |- N e. Prime $= ( c2 c1 c8 cdc c3 c9 co 1nn0 cc0 caddc cmul 2nn0 deccl 0nn0 c6 3nn0 6nn0 c7 cmin 139prm 8nn decnncl 5nn0 2p1e3 eqid decsuc eqtr4i oveq1i 8nn0 9nn0 - c5 7nn0 6cn ax-1cn 6p1e7 addcomli dec0h eqtri mulid1i addid2i oveq12i 8cn - 1p1e2 8p7e15 decmac 3cn mulid2i 3p3e6 c4 4nn0 8t3e24 4cn 6p4e10 decaddci2 + c5 7nn0 6cn ax-1cn 6p1e7 addcomli dec0h eqtri mulridi addlidi oveq12i 8cn + 1p1e2 8p7e15 decmac 3cn mullidi 3p3e6 c4 4nn0 8t3e24 4cn 6p4e10 decaddci2 decma2c 9cn 9p7e16 9t8e72 mulcomli decmul1c decmul2c nn0cni pncan3oi wcel wceq cn0 eqeltri npcan mp2an eqcomi 1nn 2nn cexp numexp1 oveq2i clt 8lt10 cc 1lt10 declti decltc breqtrri 2503lem2 2503lem3 pockthi ) CDEFZDGFZHFZD @@ -204226,7 +204226,7 @@ consisting of identical symbols by at least one position (and not by as ( c2 c1 cc0 c9 c4 c6 4nn0 0nn0 deccl 9nn0 6nn0 1nn0 2nn0 c5 co cmul caddc cdc cn 1nn decnncl eqeltri 10nn0 nn0zi c8 c7 5nn0 8nn0 7nn0 c3 0z 3nn0 2z 2nn 1z cexp cmo 2exp6 oveq1i 6cn 2cn 6t2e12 mulcomli eqid 9cn dec0h eqtri - addid1i 00id mulid2i oveq12i ax-1cn mul01i 3eqtri decma2c addid2i mulid1i + addridi 00id mullidi oveq12i ax-1cn mul01i 3eqtri decma2c addlidi mulridi 4cn 5cn 5p1e6 addcomli 2p2e4 oveq2i 6t6e36 3p1e4 6p4e10 decaddci2 decaddi 6t4e24 decmac decsuc 4t4e16 decmul1c decmul2c eqtr4i mod2xi nn0cni dec10p 5p4e9 4p1e5 2t2e4 4t2e8 8p1e9 3cn 3p2e5 9t9e81 9p1e10 9t5e45 7cn decaddci @@ -204315,11 +204315,11 @@ consisting of identical symbols by at least one position (and not by as 4001lem2 $p |- ( ( 2 ^ ; ; 8 0 0 ) mod N ) = ( ; ; ; 2 3 1 1 mod N ) $= ( c2 c4 cc0 cdc c9 c8 c1 c3 4nn0 0nn0 deccl 1nn0 2nn0 eqid cmul co caddc c6 cn 1nn decnncl eqeltri 2nn 9nn0 nn0zi 4001lem1 nn0cni 2cn 2t2e4 mul02i - 3nn0 decmul1 mulcomli 6nn0 4p2e6 decaddi 00id decadd addid1i ax-1cn dec0h - eqtri cn0 addid2i 4cn 4t2e8 oveq12i 2p1e3 3cn 4t3e12 decsuc decmac mul01i - decrmanc oveq1i 6cn 3eqtri decma2c mulid1i 3p1e4 8nn0 1p2e3 9t9e81 9t2e18 + 3nn0 decmul1 mulcomli 6nn0 4p2e6 decaddi 00id decadd addridi ax-1cn dec0h + eqtri cn0 addlidi 4cn 4t2e8 oveq12i 2p1e3 3cn 4t3e12 decsuc decmac mul01i + decrmanc oveq1i 6cn 3eqtri decma2c mulridi 3p1e4 8nn0 1p2e3 9t9e81 9t2e18 1p1e2 8p8e16 decaddci decmul2c eqtr4i mod2xi c7 7nn0 7p1e8 4p3e7 addcomli - 9cn 9p1e10 decaddc2 4t4e16 6p3e9 9t4e36 decmul1c 8cn c5 5cn 5p4e9 mulid2i + 9cn 9p1e10 decaddc2 4t4e16 6p3e9 9t4e36 decmul1c 8cn c5 5cn 5p4e9 mullidi 5nn0 4p1e5 0cn 4p4e8 ) CDEFZEFZDGFZEFZHEFZEFZIDFZEFZIFZCJFZIFZIFZAAXOIFUA BXOIXNEDEKLMZLMZUBUCUDZUEYGXQXPEDGKUFMZLMZUGYAIXTEIDNKMZLMZNMZYDIYCICJOUM MZNMZNMCCEFZEFZYPJFZXOGEFZCFZYBAYHUEYPECEOLMZLMZYRYPJUUAUMMZUGYSCGEUFLMZO @@ -204363,9 +204363,9 @@ consisting of identical symbols by at least one position (and not by as 4001lem3 $p |- ( ( 2 ^ ( N - 1 ) ) mod N ) = ( 1 mod N ) $= ( c2 cc0 c1 co c4 0nn0 deccl 2nn0 1nn0 c8 c5 5nn0 eqid cmul caddc decmul1 cdc c9 cmin cn 4nn0 1nn decnncl eqeltri 2nn 0z 10nn0 8nn0 nn0zi 3nn0 9nn0 - c3 4001lem2 4001lem1 8p2e10 decadd dec0h nn0cni addid2i 5cn addid1i eqtri - 00id cn0 5t4e20 4cn 4t2e8 mulcomli mulid2i oveq12i decrmanc mul01i oveq1i - 2cn 3eqtri decma2c mulid1i ax-1cn 1p1e2 c6 6nn0 4p1e5 9t2e18 decaddci2 c7 + c3 4001lem2 4001lem1 8p2e10 decadd dec0h nn0cni addlidi 5cn addridi eqtri + 00id cn0 5t4e20 4cn 4t2e8 mulcomli mullidi oveq12i decrmanc mul01i oveq1i + 2cn 3eqtri decma2c mulridi ax-1cn 1p1e2 c6 6nn0 4p1e5 9t2e18 decaddci2 c7 9cn 7nn0 7p1e8 3cn 9t3e27 decsuc decrmac 9p5e14 9p6e15 decmac 2t2e4 3t2e6 decmul2c eqtr4i modxai dec0u mul02i nncni mod2xi mvrraddi ) CCDSZDSZDSZDA EUAFZEEAAGDSZDSZESZUBBXMEXLDGDUCHIZHIZUDUEUFZUGXIDXHDCDJHIZHIZHIZUHKKCEDS @@ -204399,7 +204399,7 @@ consisting of identical symbols by at least one position (and not by as decnncl eqeltri 4001lem2 0p1e1 eqid decsuc modsubi 6nn0 9nn0 5nn0 cgcd cz wceq nn0zi gcdcom cdvds wbr wn 3nn cmul 4cn 4t3e12 mulcomli 2p2e4 decaddi 3cn 2lt3 ndvdsi cprime wb 3prm coprm mpbi dec0h 2t1e2 oveq12i 2p1e3 4t2e8 - 2cn oveq1i 8p3e11 decma2c gcdi mulid2i addid1i 3p1e4 1t1e1 4p1e5 addcomli + 2cn oveq1i 8p3e11 decma2c gcdi mullidi addridi 3p1e4 1t1e1 4p1e5 addcomli ax-1cn 3eqtri 8p4e12 5cn 5t2e10 4p2e6 5p1e6 1p1e2 6cn 6t2e12 3p2e5 nn0cni 00id 6p5e11 6p3e9 5p2e7 oveq2i 7cn 7p2e9 9cn 9t2e18 decaddci 6p1e7 7p6e13 7nn0 9p2e11 9p1e10 7p3e10 mul01i eqtr4i gcdmodi ) CDEZFEZGEZFCUAGEZGEZUBH @@ -204441,7 +204441,7 @@ consisting of identical symbols by at least one position (and not by as AV, 16-Sep-2021.) $) 4001prm $p |- N e. Prime $= ( c2 c3 cdc c5 c8 cc0 c1 co c4 cmul 0nn0 deccl caddc 5nn0 1nn0 3nn0 2nn0 - c6 cmin 5prm 8nn decnncl2 4nn0 nn0cni ax-1cn addid2i eqid decsuc mvrraddi + c6 cmin 5prm 8nn decnncl2 4nn0 nn0cni ax-1cn addlidi eqid decsuc mvrraddi eqtr4i 8nn0 8t5e40 5cn mul02i decmul1 cc wcel wceq cn0 npcan mp2an eqcomi eqeltri 2nn decnncl 3nn cexp 2p1e3 sqvali 5t5e25 eqtri 2cn 5t2e10 decaddi mulcomli decmul1c numexpp1 6nn0 c7 7nn0 7p1e8 addcomli 3t1e3 oveq1i 3p1e4 @@ -206282,7 +206282,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= $( Utility theorem: index-independent form of ~ df-mulr . (Contributed by Mario Carneiro, 8-Jun-2013.) $) - mulrid $p |- .r = Slot ( .r ` ndx ) $= + mulridx $p |- .r = Slot ( .r ` ndx ) $= ( cmulr c3 df-mulr 3nn ndxid ) ABCDE $. $( The slot for the base set is not the slot for the ring (multiplication) @@ -206334,7 +206334,7 @@ base set is not the slot for the ring (multiplication) operation in an Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) $) rngmulr $p |- ( .x. e. V -> .x. = ( .r ` R ) ) $= - ( cmulr c1 cop rngstr mulrid cnx cfv csn cbs cplusg ctp snsstp3 sseqtrri + ( cmulr c1 cop rngstr mulridx cnx cfv csn cbs cplusg ctp snsstp3 sseqtrri c3 strfv ) DCGEHTIABCDFJKLGMDIZNLOMAIZLPMBIZUBQCUCUDUBRFSUA $. $} @@ -206376,8 +206376,8 @@ base set is not the slot for the ring (multiplication) operation in an $( ` .r ` is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.) $) ressmulr $p |- ( A e. V -> .x. = ( .r ` S ) ) $= - ( cmulr mulrid cnx cbs cfv basendxnmulrndx necomi resseqnbas ) ADCHEBFG - IJKLJHLMNO $. + ( cmulr mulridx cnx cbs cfv basendxnmulrndx necomi resseqnbas ) ADCHEBF + GIJKLJHLMNO $. $} ${ @@ -206417,9 +206417,9 @@ base set is not the slot for the ring (multiplication) operation in an $( The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) $) srngmulr $p |- ( .x. e. X -> .x. = ( .r ` R ) ) $= - ( cmulr c1 c4 cop srngstr mulrid cnx cfv csn cbs cplusg ctp snsstp3 ssun1 - cstv cun sseqtrri sstri strfv ) DCHFIJKABCDEGLMNHODKZPNQOAKZNROBKZUGSZCUH - UIUGTUJUJNUBOEKPZUCCUJUKUAGUDUEUF $. + ( cmulr c1 c4 cop srngstr mulridx cnx cfv csn cbs cplusg ctp snsstp3 cstv + cun ssun1 sseqtrri sstri strfv ) DCHFIJKABCDEGLMNHODKZPNQOAKZNROBKZUGSZCU + HUIUGTUJUJNUAOEKPZUBCUJUKUCGUDUEUF $. $( The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) $) @@ -206615,9 +206615,9 @@ base set is not the slot for the ring (multiplication) operation in an (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) $) ipsmulr $p |- ( .X. e. V -> .X. = ( .r ` A ) ) $= - ( cmulr c1 c8 cop ipsstr mulrid cnx cfv csn cbs ctp cplusg csca cvsca cip - snsstp3 cun ssun1 sseqtrri sstri strfv ) FAJHKLMABCDEFGINOPJQFMZRPSQBMZPU - AQCMZUKTZAULUMUKUEUNUNPUBQDMPUCQEMPUDQGMTZUFAUNUOUGIUHUIUJ $. + ( cmulr c1 c8 cop ipsstr mulridx cnx cfv csn cbs ctp cplusg snsstp3 cvsca + csca cip cun ssun1 sseqtrri sstri strfv ) FAJHKLMABCDEFGINOPJQFMZRPSQBMZP + UAQCMZUKTZAULUMUKUBUNUNPUDQDMPUCQEMPUEQGMTZUFAUNUOUGIUHUIUJ $. $( The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, @@ -207117,9 +207117,9 @@ be used in an extensible structure (slots must be positive integers). $( The multiplication operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) $) odrngmulr $p |- ( .x. e. V -> .x. = ( .r ` W ) ) $= - ( cmulr c1 c2 cdc cop odrngstr mulrid cnx cfv csn ctp cplusg snsstp3 cple - cbs cts cds cun ssun1 sseqtrri sstri strfv ) DHJGKKLMNABCDEFHIOPQJRDNZSQU - DRANZQUARCNZULTZHUMUNULUBUOUOQUERENQUCRFNQUFRBNTZUGHUOUPUHIUIUJUK $. + ( cmulr c1 c2 cdc cop odrngstr mulridx cnx cfv csn ctp cbs cplusg snsstp3 + cts cple cds cun ssun1 sseqtrri sstri strfv ) DHJGKKLMNABCDEFHIOPQJRDNZSQ + UARANZQUBRCNZULTZHUMUNULUCUOUOQUDRENQUERFNQUFRBNTZUGHUOUPUHIUIUJUK $. $( The open sets of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) $) @@ -207849,27 +207849,27 @@ as a product (if ` G ` is a multiplication), a sum (if ` G ` is an ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) ) ) ) $= ( cfv cvv va vc vd ve cds cmpt crn cc0 csn cun cxr clt csup cmpo cplusg cv co cxp c2nd chom cixp c1st cop cco cmulr cbs cvsca cip cgsu cpr cple - wss wbr wral copab ctopn ccom cpt eqid prdsbas prdsplusg prdsval mulrid - wa eqidd cuni cpw cmap wcel ovssunirn strfvss fvssunirn rnss uniss mp2b - wf cnx sstri ovex elpw mpbir a1i fmpttd rnexg uniexg pwexd cdm eqeltrrd - 3syl dmexd elmapd mpbird ralrimivw fmpo sylib xpex fex2 mp3an23 syl ctp - fvexi csca cts snsstp3 ssun1 prdsbaslem ) AGCHICCBJBUPZHUPZSZYGIUPZSZYG - ESZUESUQUFUGUHUIUJUKULUMUNZDUOSZFUAUBCCURZCUCUDUAUPZUSSZUBUPZHICCBJYIYK - YLUTSUQVAUNZUQYPYSSBJYGUCUPSYGUDUPSYGYPVBSSYGYQSVCYGYRSYLVDSUQUQUFUNUNZ - HICCBJYIYKYLVESZUQZUFZUNZHIFVFSZCBJYHYKYLVGSUQUFUNZUUDDVEYSHICCFBJYIYKY - LVHSUQUFVIUQUNZYHYJVJCVLYIYKYLVKSVMBJVNWDHIVOZVPEVQVRSZTABCYMDYNEFYTUUF - UUDUDHIYSUUGJUUEUUHUUIKLUAUBUCMUUEVSQABCDEFJKLMNOPQVTABCDYNEFHIJKLMNOPQ - YNVSWAAUUDWEAUUFWEAUUGWEAUUIWEAUUHWEAYMWEAYSWEAYTWENOWBRWCAYOEUGZWFZUGZ - WFZUGZWFZWGZJWHUQZUUDWPZUUDTWIZAUUCUUQWIZICVNZHCVNUURAUVAHCAUUTICAUUTJU - UPUUCWPABJUUBUUPUUBUUPWIZAYGJWIWDUVBUUBUUOVLUUBUUAUGZWFZUUOUUAYIYKWJUUA - UUMVLUVCUUNVLUVDUUOVLUUAYLUGZWFZUUMYLVEWQVESZWCWKYLUUKVLUVEUULVLUVFUUMV - LEYGWLYLUUKWMUVEUULWNWOWRUUAUUMWMUVCUUNWNWOWRUUBUUOYIYKUUAWSWTXAXBXCAUU - PJUUCTTAUUOTAUUMTWIZUUNTWIUUOTWIAUUKTWIZUULTWIUVHAELWIUUJTWIUVIOELXDUUJ - TXEXIUUKTXDUULTXEXIUUMTXDUUNTXEXIXFAEXGJTQAELOXJXHXKXLXMXMHICCUUCUUQUUD - UUDVSXNXOUURYOTWIUUQTWIUUSCCCDVFPYAZUVJXPUUPJWHWSYOUUQUUDTTXQXRXSUVGUUD - VCZUIZWQVFSCVCZWQUOSYNVCZUVKXTZWQYBSFVCWQVGSUUFVCWQVHSUUGVCXTZUJZUVQWQY - CSUUIVCWQVKSUUHVCWQUESYMVCXTWQUTSYSVCWQVDSYTVCVJUJZUJUVLUVOUVQUVMUVNUVK - YDUVOUVPYEWRUVQUVRYEWRYF $. + wss wbr wral wa copab ctopn ccom eqid prdsbas prdsplusg prdsval mulridx + cpt eqidd cuni cpw cmap wcel ovssunirn cnx strfvss fvssunirn rnss uniss + mp2b sstri ovex elpw mpbir a1i fmpttd rnexg uniexg 3syl pwexd cdm dmexd + eqeltrrd elmapd mpbird ralrimivw fmpo sylib fvexi xpex fex2 mp3an23 syl + wf ctp csca cts snsstp3 ssun1 prdsbaslem ) AGCHICCBJBUPZHUPZSZYGIUPZSZY + GESZUESUQUFUGUHUIUJUKULUMUNZDUOSZFUAUBCCURZCUCUDUAUPZUSSZUBUPZHICCBJYIY + KYLUTSUQVAUNZUQYPYSSBJYGUCUPSYGUDUPSYGYPVBSSYGYQSVCYGYRSYLVDSUQUQUFUNUN + ZHICCBJYIYKYLVESZUQZUFZUNZHIFVFSZCBJYHYKYLVGSUQUFUNZUUDDVEYSHICCFBJYIYK + YLVHSUQUFVIUQUNZYHYJVJCVLYIYKYLVKSVMBJVNVOHIVPZVQEVRWDSZTABCYMDYNEFYTUU + FUUDUDHIYSUUGJUUEUUHUUIKLUAUBUCMUUEVSQABCDEFJKLMNOPQVTABCDYNEFHIJKLMNOP + QYNVSWAAUUDWEAUUFWEAUUGWEAUUIWEAUUHWEAYMWEAYSWEAYTWENOWBRWCAYOEUGZWFZUG + ZWFZUGZWFZWGZJWHUQZUUDXTZUUDTWIZAUUCUUQWIZICVNZHCVNUURAUVAHCAUUTICAUUTJ + UUPUUCXTABJUUBUUPUUBUUPWIZAYGJWIVOUVBUUBUUOVLUUBUUAUGZWFZUUOUUAYIYKWJUU + AUUMVLUVCUUNVLUVDUUOVLUUAYLUGZWFZUUMYLVEWKVESZWCWLYLUUKVLUVEUULVLUVFUUM + VLEYGWMYLUUKWNUVEUULWOWPWQUUAUUMWNUVCUUNWOWPWQUUBUUOYIYKUUAWRWSWTXAXBAU + UPJUUCTTAUUOTAUUMTWIZUUNTWIUUOTWIAUUKTWIZUULTWIUVHAELWIUUJTWIUVIOELXCUU + JTXDXEUUKTXCUULTXDXEUUMTXCUUNTXDXEXFAEXGJTQAELOXHXIXJXKXLXLHICCUUCUUQUU + DUUDVSXMXNUURYOTWIUUQTWIUUSCCCDVFPXOZUVJXPUUPJWHWRYOUUQUUDTTXQXRXSUVGUU + DVCZUIZWKVFSCVCZWKUOSYNVCZUVKYAZWKYBSFVCWKVGSUUFVCWKVHSUUGVCYAZUJZUVQWK + YCSUUIVCWKVKSUUHVCWKUESYMVCYAWKUTSYSVCWKVDSYTVCVJUJZUJUVLUVOUVQUVMUVNUV + KYDUVOUVPYEWQUVQUVRYEWQYF $. $} ${ @@ -208863,17 +208863,17 @@ topology is based on the order and not the extended metric (which would { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } ) $= ( cfv cop eqid vx vy vg vh vi vn cv co csn ciun cnx cbs cplusg ctp csca cmulr cvsca cmpo cip cun cts ctopn cqtop cple ccom cds cvv c1 imasplusg - ccnv c2 eqidd imasds imasval imasvalstr mulrid snsstp3 ssun1 sstri wcel - wral fvex eqeltrdi snex rgenw iunexg sylancl ralrimivw syl2anc strfv3 - cdc ) ADKHJHKUGZGRJUGZGRZSZWLWMEUHGRSZUIZUJZUJZUKULRBSZUKUMRFUMRZSZUKUP - RWSSZUNZUKUORCUORZSUKUQRJHKUAXEULRZWNUIWLWMCUQRZUHGRURUJZSUKUSRKHJHWOWL - WMCUSRZUHSUIUJUJZSUNZUTZUKVARCVBRZGVCUHZSUKVDRGCVDRZVEGVJVEZSUKVFRFVFRZ - SUNZUTZFUPVGVHVHVKWKSAUAUBBXQCUMRZXACWSXGEXHFUCUDUEUFCVFRZGXEXIXJXMXFXP - XOXNHIJKLMXTTZPXETXFTXGTXITXMTYATZXOTABXTXACFGHIJKLMNOYBXATVIAWSVLAXHVL - AXJVLAXNVLAUAUBBXQCFUCUDUEUFYAGHILMNOYCXQTVMAXPVLNOVNBXQXAXEXHWSXSXJXPX - NXSTVOVPXCUIZXLXSYDXDXLWTXBXCVQXDXKVRVSXLXRVRVSAHVGVTZWRVGVTZKHWAWSVGVT - AHCULRVGMCULWBWCZAYFKHAYEWQVGVTZJHWAYFYGYHJHWPWDWEJHWQVGVGWFWGWHKHWRVGV - GWFWIQWJ $. + ccnv c2 cdc eqidd imasds imasval imasvalstr mulridx snsstp3 ssun1 sstri + wcel wral fvex eqeltrdi rgenw iunexg sylancl ralrimivw syl2anc strfv3 + snex ) ADKHJHKUGZGRJUGZGRZSZWLWMEUHGRSZUIZUJZUJZUKULRBSZUKUMRFUMRZSZUKU + PRWSSZUNZUKUORCUORZSUKUQRJHKUAXEULRZWNUIWLWMCUQRZUHGRURUJZSUKUSRKHJHWOW + LWMCUSRZUHSUIUJUJZSUNZUTZUKVARCVBRZGVCUHZSUKVDRGCVDRZVEGVJVEZSUKVFRFVFR + ZSUNZUTZFUPVGVHVHVKVLSAUAUBBXQCUMRZXACWSXGEXHFUCUDUEUFCVFRZGXEXIXJXMXFX + PXOXNHIJKLMXTTZPXETXFTXGTXITXMTYATZXOTABXTXACFGHIJKLMNOYBXATVIAWSVMAXHV + MAXJVMAXNVMAUAUBBXQCFUCUDUEUFYAGHILMNOYCXQTVNAXPVMNOVOBXQXAXEXHWSXSXJXP + XNXSTVPVQXCUIZXLXSYDXDXLWTXBXCVRXDXKVSVTXLXRVSVTAHVGWAZWRVGWAZKHWBWSVGW + AAHCULRVGMCULWCWDZAYFKHAYEWQVGWAZJHWBYFYGYHJHWPWKWEJHWQVGVGWFWGWHKHWRVG + VGWFWIQWJ $. $} ${ @@ -230263,7 +230263,7 @@ proposition to be be proved (the first four hypotheses tell its values ad2ant2l syl2anc nncnd 1cnd addsubassd ax-1cn npcan sylancl 3eltr4d nn0uz cc fveq2d eleqtrdi cfz cfzo ccatcl 3ad2ant2 wrdf ccatlen oveq2d cz fzoval zaddcld eqtrd ffvelcdmda seqsplit simpl2l simpl2r eleq2d biimpar ccatval1 - feq2d mpbid syl3anc seqfveq addid2d eqtr4d seqeq1d addcomd oveq1d addsubd + feq2d mpbid syl3anc seqfveq addlidd eqtr4d seqeq1d addcomd oveq1d addsubd fveq12d ccatval3 eqcomd seqshft2 oveq12d nnaddcld gsumval2 simp2l 3eqtr4d simp2r ) CUCHZDAUDZHZEYBHZIZDUFUGZEUFUGZIZURZDUHJZEUHJZKLZMNLZBDEUILZOPZJ ZYJMNLZBDOPJZYKMNLZBEOPJZBLZCYNUJLCDUJLZCEUJLZBLYIYPYQYOJZYMBYNYQMKLZPZJZ @@ -234028,8 +234028,8 @@ since the target of the homomorphism (operator ` O ` in our model) need ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) ) $= ( wcel wa caddc co wceq cc0 adantr simpr oveq1d eqtrd cmnd cn0 cn mndsgrp w3a csgrp ad2antrr simplr simpr3 mulgnndir syl13anc c0g cfv simpll simpr1 - simplr3 mulgnn0cld eqid mndrid syl2anc syl oveq2d nn0cnd addid1d 3eqtr4rd - mulg0 adantlr wo simpr2 elnn0 sylib mpjaodan simplr2 mndlid addid2d ) DUA + simplr3 mulgnn0cld eqid mndrid syl2anc syl oveq2d nn0cnd addridd 3eqtr4rd + mulg0 adantlr wo simpr2 elnn0 sylib mpjaodan simplr2 mndlid addlidd ) DUA KZEUBKZFUBKZGAKZUEZLZEUCKZEFMNZGCNZEGCNZFGCNZBNZOZEPOZWAWBLZFUCKZWHFPOZWJ WKLDUFKZWBWKVSWHWAWMWBWKVPWMVTDUDQUGWAWBWKUHWJWKRWAVSWBWKVPVQVRVSUIUGABCD EFGHIJUJUKWAWLWHWBWAWLLZWEDULUMZBNZWEWGWDWNVPWEAKWPWEOVPVTWLUNZWNACDEGHIW @@ -234141,7 +234141,7 @@ since the target of the homomorphism (operator ` O ` in our model) need mulgnnass $p |- ( ( G e. Smgrp /\ ( M e. NN /\ N e. NN /\ X e. B ) ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) ) $= ( vn wcel cn cmul co wceq wi c1 oveq1 oveq1d eqeq12d imbi2d vm csgrp nncn - w3a cv caddc mulid2d 3ad2ant1 sgrpmgm mulgnncl syl3an1 3coml mulg1 eqtr4d + w3a cv caddc mullidd 3ad2ant1 sgrpmgm mulgnncl syl3an1 3coml mulg1 eqtr4d cmgm syl wa cplusg cfv cc adantr simpr1 nncnd adddirp1d nnmulcl 3ad2antr1 simpr3 simpr2 eqid mulgnndir syl13anc eqtrd mulgnnp1 sylan2 syl5ibr nnind ex a2d 3expd com4r 3imp2 ) CUBJZDKJZEKJZFAJZDELMZFBMZDEFBMZBMZNZWCWDWEWBW @@ -239834,7 +239834,7 @@ operation is a permutation group (group consisting of permutations), see nn0addcl nn0red syl3anbrc psgnunilem5 oveq2i oveq2d a1i spllen oveq1i 2cn elfz2nn0 cc eqtrid 3eqtrd syl12anc eqid wss eleqtrrd syl3anc wf ffvelcdmd gsumws2 symgov syl2anc simpr gsumspl 3eqtr3d fveqeq2 oveq2 eqeq1d anbi12d - cs2 rspcev adantrr sselda eqtrd addid1d clt 2nn mpbir3an eleqtrri splfv2a + cs2 rspcev adantrr sselda eqtrd addridd clt 2nn mpbir3an eleqtrri splfv2a ex elfzo0 difeq1d dmeqd wb fveq2 eleq2d notbid ad2antrr fveq1 df-2 nn0cnd fzofzp1 addassd eqtr4id 3eltr4d hash0 df-neg eqtr4i pncan2 negeqd oveq12d 1cnd cz elfzel2 zcnd negsub cop csubstr splid cbs symggrp grpmndd symgtrf @@ -240913,7 +240913,7 @@ operation is a permutation group (group consisting of permutations), see odinv $p |- ( ( G e. Grp /\ A e. X ) -> ( O ` ( I ` A ) ) = ( O ` A ) ) $= ( wcel cfv c1 cgcd co cmul cz wceq neg1z cn0 odcl syl cgrp wa cneg odmulg cmg eqid mp3an3 adantl nn0zd gcdcom sylancr 1z gcdneg sylancl gcd1 3eqtrd - mulgm1 fveq2d oveq12d grpinvcl nn0cnd mulid2d 3eqtrrd ) BUAIZAEIZUBZADJZK + mulgm1 fveq2d oveq12d grpinvcl nn0cnd mullidd 3eqtrrd ) BUAIZAEIZUBZADJZK UCZVGLMZVHABUEJZMZDJZNMZKACJZDJZNMVOVDVEVHOIZVGVMPQAVJBVHDEHFVJUFZUDUGVFV IKVLVONVFVIVGVHLMZVGKLMZKVFVPVGOIZVIVRPQVFVGVEVGRIVDABDEHFSUHUIZVHVGUJUKV FVTKOIVRVSPWAULVGKUMUNVFVTVSKPWAVGUOTUPVFVKVNDEVJBCAHVQGUQURUSVFVOVFVOVFV @@ -241543,10 +241543,10 @@ P pGrp ( G |`s S ) ) $= nnrp nn0red pcdvdsb lenltd nnuz fzsubel syl22anc 1m1e0 dvdsval2 divne0d 1zzd oveq1i pcmul npncand ltm1d nnm1nn0 znn0sub 0nn0 nn0addcl pc1 nn0re bcn0 a1d ltp1d simprr lttrd imim1d oveq1 nn0cn addassd nn0addge2 simprl - expr nn0addcld peano2zd znq rpdivcl syl2an rpne0d pcqmul addid1d eqtr2d + expr nn0addcld peano2zd znq rpdivcl syl2an rpne0d pcqmul addridd eqtr2d comraddd zq neneqd sylbid 3imtr3d mt4d ltletrd pcadd2 pm2.61dane eqtr4d dvdssubr eqeltrd 00id eqtr2di eqeq12d syl5ibr animpimp2impd nn0ind pcid - subeq0bd mpcom zsubcld zcnd addid2d 3eqtrd jca ) ADUCUDZUEQBUXERSZBGUCU + subeq0bd mpcom zsubcld zcnd addlidd 3eqtrd jca ) ADUCUDZUEQBUXERSZBGUCU DZRSZFUFSZUGAUXGBFUHSZUISZUXEUEAUXKHUJUCUDUXJUGHGUKULZUCUDZUXEAGUMQZUXJ UNQZUXKUXMUGKAUXJABFABUOQZBUEQLBUPUQMURZUSZHGUXJUTVADUXLUCPVBVCZAUXJTUX GVDSQZUXKUEQAUXTUXJUXGVEVFZAUXJUXGVGVFZUYANAUXOUXGUEQZUYBUYAVHUXRAUYCUX @@ -242104,7 +242104,7 @@ P pGrp ( G |`s S ) ) $= ax-1cn csn c1o cen wbr elin cec wi eqid sseq1 velpw bitr4di breq1 imbi12d simpr erref vex elec sylibr syl5com sylow2alem1 ensn1 eqbrtrdi ex ectocld ssel syld impr sylan2b en1b fveq2d cvv vuniex hashsng ax-mp sumeq2dv wral - eqtrdi ssrab3 mulid1d cmpt rabexd cdm eqeltrd oveq1d 3eqtr4rd crn cxp cpr + eqtrdi ssrab3 mulridd cmpt rabexd cdm eqeltrd oveq1d 3eqtr4rd crn cxp cpr wrel wrex relopabiv relssdmrn xpexd ssexg sylancr sselid ecelqsg syl2an2r erdm errn eqeltrrd snelpwi adantl elind elin2d eqsstrrd eqeq2d syl5ibrcom adantrl unieq unisnv eqtr2di impbid1 f1o2d hasheqf1od eqtr3d diffi eldifi @@ -243966,7 +243966,7 @@ U C_ ( T .(+) U ) ) $= crn syl2anc adantr eleqtrrd chash cs2 csplice cc0 cfz cuz cn0 lencl nn0uz cotp eleqtrdi caddc ccatlen nn0zd uzaddcl eqeltrd elfzuzb sylanbrc simprr uzidd efgtval syl3anc ffvelcdmi ad2antll s2cld ccatrid eqcomd eqidd hash0 - syl oveq1d oveq2i nn0cnd addid1d eqtr2id splval2 s1cld revccat revs1 s1co + syl oveq1d oveq2i nn0cnd addridd eqtr2id splval2 s1cld revccat revs1 s1co oveq1i ccatco 3eqtrd oveq2d ccatass df-s2 eqtr4di 3eqtr2rd wfn cmpo efgtf ffnd fnovrn eqeltrrd efgi2 ersym ertr mpand expcom a2d wrdind mpcom ) EIU FUGZUHZPEKPZEJEUIQZUJZRSZTFULZKUUQEKUUQUMQUUQLUUQUKUNUOUURUAUPZJUVCUIQZUJ @@ -244091,7 +244091,7 @@ U C_ ( T .(+) U ) ) $= -> ( S ` ( A ++ <" B "> ) ) = B ) $= ( cword wcel cs1 cconcat co cdm cfv wceq wa chash cmin efgsval caddc s1cl c1 ccatlen sylan2 s1len oveq2i eqtrdi oveq1d cc lencl nn0cnd ax-1cn pncan - cc0 sylancl addid2d eqtr4d adantr eqtrd fveq2d cfzo adantl cn 1nn eqeltri + cc0 sylancl addlidd eqtr4d adantr eqtrd fveq2d cfzo adantl cn 1nn eqeltri simpl lbfzo0 mpbir a1i ccatval3 syl3anc s1fv 3eqtrd sylan9eqr 3impa ) GRU EZUFZHRUFZGHUGZUHUIZKUJUFZWQKUKZHULWRWNWOUMZWSWQUNUKZUSUOUIZWQUKZHABCDEFI JKLMNOWQPQRSTUAUBUCUDUPWTXCVKGUNUKZUQUIZWQUKZVKWPUKZHWTXBXEWQWTXBXDUSUQUI @@ -244179,7 +244179,7 @@ U C_ ( T .(+) U ) ) $= len0nnbi mpbid lbfzo0 sylibr ccatval1 syl3anc simp2bi eqeltrd cun simp3bi wb w3a fzo0ss1 sseli syl3an3 cuz wss cle wbr elfzoel2 peano2zm zred lem1d eluz2 syl3anbrc fzoss2 elfzo1elm1fzo0 sseldd rneqd eleq12d 3expa ralbidva - cz fveq2d mpbird caddc cn0 lencl nn0cnd addid2d s1len 1nn ccatval3 eqtr3d + cz fveq2d mpbird caddc cn0 lencl nn0cnd addlidd s1len 1nn ccatval3 eqtr3d eqeltri a1i simpr s1fv adantl fzo0end efgsval eqtr4d 3eltr4d fvex fvoveq1 fveq2 ralsn ralunb ccatlen oveq2i eqtrdi oveq2d eleqtrdi fzosplitsn eqtrd nnuz raleqdv ) OJUGZUHZGOJUIZKUIZUJZUHZUKZOGULZUMUNZRUOZUPUQZURZUHZUSUVNU @@ -244391,7 +244391,7 @@ an extension of the previous (inserting an element and its inverse at fviss efgredlemf sselid ccatlen 2nn0 uzaddcl elfzuz3 lencl eleqtrdi nn0uz eluzfz2 oveq12d nn0cnd zaddcl addsubassd pnpcand zsubcl npcan elfzelzd eleqtrrd eluzsub wrd0 efgmf s2cld zred nn0addge1 syl3anbrc - eqeltrd eluz2 splval ccatrid 3eqtr4rd eqcomd oveq2i addid1d eqtr2id + eqeltrd eluz2 splval ccatrid 3eqtr4rd eqcomd oveq2i addridd eqtr2id hash0 splval2 pncan2 eqtrd s2len eqtr4di eqtr3d pfxid wfn cmpo ffnd efgtf fnovrn efgredlemg 0le2 2re subge02 sub32d nncan eqtr2d eqtrid subsub4d jca ) AUBHVMZJOVMZPVMZVNZVOUCIVMZUXQVOAMUEUXPVPZUXNUXQAUXS @@ -244954,7 +244954,7 @@ an extension of the previous (inserting an element and its inverse at bitr4di iserd ovelrn ad2antlr simprl simprr fovcdmd csubstr pfxcl efgmf ffvelcdmi s2cld swrdcl splval syl13anc 3eqtr4rd elfznn0 elfzuz3 eluzadd cop cz nn0zd fveq2d 3eltr3d elfzuzb sylanbrc wrd0 ccatrid eluzfz2 pfxid - ccatpfx 3eqtr2rd pfxlen eqtr2d oveq2i nn0addcld addid1d eqtr2id splval2 + ccatpfx 3eqtr2rd pfxlen eqtr2d oveq2i nn0addcld addridd eqtr2id splval2 hash0 fnovrn efgi2 elec breq2 bitrid syl5ibrcom rexlimdvva sylbid ssrdv 3eqtr4d ralrimiva fvexi mpisyl ereq1 eceq2 sseq2d ralbidv anbi12d elabg erex mpbir2and intss1 eqsstrid ) GUAUQZHUAUQZURZJUAUIVGZUSZUJVGZLUTZVAZ @@ -246437,7 +246437,7 @@ an extension of the previous (inserting an element and its inverse at gcdcld nn0cnd sqvald gcddvds syl2anc simpld dvdsval2 simprd mul4d oveq12d divcan1d 3eqtr2d dvdsmul2 cmg c0g simpl1 simpl2 simpl3 eqid mulgdi oddvds syl13anc dvdsmul1 mulgcl grprid 3eqtr3rd mpbird wi dvdsgcd mp2and breqtrd - mulgcd eqbrtrd dvdsmulcr syl112anc oveq1d grplid c1 1cnd mulid2d 3eqtr2rd + mulgcd eqbrtrd dvdsmulcr syl112anc oveq1d grplid c1 1cnd mullidd 3eqtr2rd mulgcdr mulcan2ad coprmdvds2 syl31anc zsqcl dvdsmulc eqbrtrrd pm2.61dane mpd ) DUAJZAFJZBFJZUBZAEUFZBEUFZKLZABCLZEUFZYDYEUCLZUDUGLZKLZMNYIOYCYIOPZ UEZYFOYKMYMYFQJZYFOMNYCYNYLYCYDYEYCYDYAXTYDUHJYBADEFHGURUIUJZYCYEYBXTYEUH @@ -246480,7 +246480,7 @@ an extension of the previous (inserting an element and its inverse at ( ( O ` A ) gcd ( O ` B ) ) = 1 ) -> ( O ` ( A .+ B ) ) = ( ( O ` A ) x. ( O ` B ) ) ) $= ( wcel cfv co c1 cn0 cmul cdvds wbr syl odcl c2 cabl w3a cgcd wceq simpl1 - wa cgrp ablgrp simpl2 simpl3 grpcl syl3anc nn0mulcld simpr oveq2d mulid1d + wa cgrp ablgrp simpl2 simpl3 grpcl syl3anc nn0mulcld simpr oveq2d mulridd nn0cnd eqtrd odadd1 adantr eqbrtrrd cexp odadd2 oveq1d sq1 eqtrdi breqtrd dvdseq syl22anc ) DUAJZAFJZBFJZUBZAEKZBEKZUCLZMUDZUFZABCLZEKZNJZVNVOOLZNJ VTWBPQWBVTPQVTWBUDVRVSFJZWAVRDUGJZVKVLWCVRVJWDVJVKVLVQUEDUHRVJVKVLVQUIZVJ @@ -246810,7 +246810,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements general use; use ~ cnaddabl instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.) $) cnaddablx $p |- G e. Abel $= - ( vx vy vz cc caddc cv cneg cc0 cnex addex addcl addass addid2 negcl wcel + ( vx vy vz cc caddc cv cneg cc0 cnex addex addcl addass addlid negcl wcel 0cn co addcom wceq mpdan negid eqtr3d isgrpix grpbasex grpplusgx isabli ) CDFGACDEFGACHZIZJKLBUIDHZMUIUKEHNRUIOUIPZUIFQZUIUJGSZUJUIGSZJUMUJFQUNUOUA ULUIUJTUBUIUCUDUEFGAKLBUFFGAKLBUGUIUKTUH $. @@ -246830,7 +246830,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements (New usage is discouraged.) $) cnaddabl $p |- G e. Abel $= ( vx vy vz cc caddc cv cneg cc0 cvv wcel cbs cfv wceq cnex grpbase addcom - ax-mp co cplusg addex grpplusg addcl addass 0cn addid2 negcl mpdan eqtr3d + ax-mp co cplusg addex grpplusg addcl addass 0cn addlid negcl mpdan eqtr3d negid isgrpi isabli ) CDFGACDEFGACHZIZJFKLFAMNOPFGAKBQSZGKLGAUANOUBFGAKBU CSZUNDHZUDUNUREHUEUFUNUGUNUHZUNFLZUNUOGTZUOUNGTZJUTUOFLVAVBOUSUNUORUIUNUK UJULUPUQUNURRUM $. @@ -246840,7 +246840,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements AV, 26-Aug-2021.) (New usage is discouraged.) $) cnaddid $p |- ( 0g ` G ) = 0 $= ( vx cc0 c0g cfv cc wcel wceq 0cn caddc cvv cbs cnex grpbase ax-mp cplusg - eqid co adantl addex grpplusg id cv addid2 addid1 ismgmid2 eqcomi ) DAEFZ + eqid co adantl addex grpplusg id cv addlid addrid ismgmid2 eqcomi ) DAEFZ DGHZDUIIJUJCGKDAUIGLHGAMFINGKALBOPUIRKLHKAQFIUAGKALBUBPUJUCCUDZGHZDUKKSUK IUJUKUETULUKDKSUKIUJUKUFTUGPUH $. @@ -246864,7 +246864,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.) $) zaddablx $p |- G e. Abel $= ( vx vy vz cz caddc cv cneg cc0 zex addex zaddcl wcel cc co addcom syl2an - wceq zcn addass syl3an 0z addid2d znegcl negidd eqtr3d grpbasex grpplusgx + wceq zcn addass syl3an 0z addlidd znegcl negidd eqtr3d grpbasex grpplusgx mpdan isgrpix isabli ) CDFGACDEFGACHZIZJKLBUMDHZMUMFNZUMONZUOFNZUOONZEHZF NUTONUMUOGPZUTGPUMUOUTGPGPSUMTZUOTZUTTUMUOUTUAUBUCUPUMVBUDUMUEZUPUMUNGPZU NUMGPZJUPUNFNZVEVFSZVDUPUQUNONVHVGVBUNTUMUNQRUJUPUMVBUFUGUKFGAKLBUHFGAKLB @@ -246896,12 +246896,12 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements ( va vb cfv co c0 cop cs2 cv crn ciun cdif c2o cxp cword wcel c1o prid1 cpr 0ex df2o3 eleqtrri opelxpi sylancl s2cld cid wceq con0 xpexg wrdexg cvv 2on fvi 3syl eqtrid eleqtrrd wrex wa wne wn 1n0 cc0 chash caddc 2cn - c2 addid2i s2len eqtr4i efgtlen adantll ex 0cnd cn0 simpr efgrcl simprd + c2 addlidi s2len eqtr4i efgtlen adantll ex 0cnd cn0 simpr efgrcl simprd adantl eleqtrd lencl syl nn0cnd 2cnd addcan2d wi cfz csplice rneqd ovex eleq2d cconcat ccatidid hash0 oveq2i eqtrd c1 fveq1d opex s2fv1 3eqtr3g a1i ax-mp s2fv0 elv sylbid wb cs1 cec vrgpval syl2anc s1cld sylibd cotp cmpo wf efgtf simpld eqid elrnmpo simprr ffvelcdmi oveq1i eqtr2i simprl - wrd0 efgmf eleqtrdi elfz1eq eqtr4di cc eqeltrdi addid1d eqtr2id splval2 + wrd0 efgmf eleqtrdi elfz1eq eqtr4di cc eqeltrdi addridd eqtr2id splval2 0cn ccatlid oveq1d ccatrid eqeq2d ad3antrrr 1on fvex efgmval df-ov dif0 fveq2d opeq2i 3eqtr2rd opthg simplbda syl21anc biimtrid expimpd hasheq0 rexlimdvva eleq1 fveq2 anbi12d sylbi eqcoms syl5ibrcom expdimp necon3ad @@ -252166,7 +252166,7 @@ factorization into prime power factors (even if the exponents are wrdf fss c0 cz lencl nn0zd fzosn ineq2d fzodisj eqtr3di cs1 cconcat fveq2i s1cld ccatlen s1len oveq2i nn0uz eleqtrdi fzosplitsn cats1un cuz reseq1d wn fzonel fsnunres sylancr ssun2 snss eleqtrrid fnressn - dprdsn fveq1i nn0cnd addid2d fveq2d eqtr4id eqeltri lbfzo0 ccatval3 + dprdsn fveq1i nn0cnd addlidd fveq2d eqtr4id eqeltri lbfzo0 ccatval3 1nn a1i s1fv mp1i ablcntzd ineq12d incom subg0 eqtr4di dmdprdsplit2 opeq2d 3eqtr4d clsm dprdsplit oveq12d lsmcom sseqtrrd subglsm breq2 cabl subgss eqeq1d anbi12d rspcev syl12anc ) AHDVCZVDZKHVEVFZVGZKHV @@ -252218,7 +252218,7 @@ factorization into prime power factors (even if the exponents are cbs acsmre subg0cl snssd mrcssidd snssg mpbird eqnetrd od1 3syl elsni fveqeq2d syl5ibrcom necon3ad mpd ssnelpssd php3 snfi hashsdom sylancr wn eqbrtrrid cr cc0 1red cn c0 ne0i hashnncl nngt0d cabl wi adantr - ltmul1 syl112anc recnd mulid2d subgabl ablcntzd lsmhash fveq2d eqtr3d + ltmul1 syl112anc recnd mullidd subgabl ablcntzd lsmhash fveq2d eqtr3d ccntz 3brtr3d ltned fveq2 necon3i df-pss sylanbrc psseq1 eqeq2 anbi2d rexbidv imbi12d simpld rspcdva breq2 oveq2 anbi12d cbvrexvw sylib cs1 eqeq1d cconcat cpgp wral simprl simprrl simprrr pgpfaclem1 rexlimddv @@ -255459,11 +255459,11 @@ is also applicable for semirings (without using the commutativity of the $( Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) $) opprmulfval $p |- .xb = tpos .x. $= - ( cmulr cfv ctpos cvv wcel wceq mulrid c0 coppr fvprc eqtrid cnx cop csts - co opprval fveq2i fvexi tposex setsid mpan2 eqtr4id wn str0 eqtr2i fveq2d - tpos0 tposeqd 3eqtr4a pm2.61i eqtri ) CEJKZDLZIBMNZVAVBOVCVABUAJKZVBUBUCU - DZJKZVBEVEJABDEFGHUEUFVCVBMNVBVFODDBJGUGUHMVBJMBPUIUJUKVCULZQJKZQLZVAVBVI - QVHUPJVDPUMUNVGEQJVGEBRKQHBRSTUOVGDQVGDBJKQGBJSTUQURUSUT $. + ( cmulr cfv ctpos cvv wcel wceq mulridx c0 coppr fvprc eqtrid cnx csts co + cop opprval fveq2i fvexi tposex setsid mpan2 eqtr4id wn tpos0 str0 eqtr2i + fveq2d tposeqd 3eqtr4a pm2.61i eqtri ) CEJKZDLZIBMNZVAVBOVCVABUAJKZVBUDUB + UCZJKZVBEVEJABDEFGHUEUFVCVBMNVBVFODDBJGUGUHMVBJMBPUIUJUKVCULZQJKZQLZVAVBV + IQVHUMJVDPUNUOVGEQJVGEBRKQHBRSTUPVGDQVGDBJKQGBJSTUQURUSUT $. $( Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed @@ -259197,7 +259197,7 @@ Absolute value (abstract algebra) abvrcl cgrp ringgrp syl grpinvcl sylan ring1eq0 syl3anc imp wne c2 cexp simpr cmulr ringidcl abvcl mpdan recnd sqvald abvmul mpd3an23 ringmneg2 cr ringnegl grpinvinv 3eqtrd 3eqtr2d abv1z eqtrd sq1 eqtr4di cc0 cle wb - wbr abvge0 0le1 sq11 mpanr12 biimpa syldan adantlr oveq1d simpl mulid2d + wbr abvge0 0le1 sq11 mpanr12 biimpa syldan adantlr oveq1d simpl mullidd 1re eqtr3d pm2.61dane ) DAJZFBJZKZFELZDLZFDLZMCUALZCUBLZWOWSWTMZKWPFDWO XAWPFMZWOCUCJZWPBJZWNXAXBUDWMXCWNACDGUEZNZWMCUFJZWNXDWMXCXGXECUGUHZBCEF HIUIUJWMWNUQZBCWSWPFWTHWSOZWTOZUKULUMPWOWSWTUNZKZQWRRSZWQWRXMWSELZDLZWR @@ -260124,7 +260124,7 @@ Absolute value (abstract algebra) 0vlid.v $e |- V = ( Base ` W ) $. 0vlid.a $e |- .+ = ( +g ` W ) $. 0vlid.z $e |- .0. = ( 0g ` W ) $. - $( Left identity law for the zero vector. ( ~ hvaddid2 analog.) + $( Left identity law for the zero vector. ( ~ hvaddlid analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) $) lmod0vlid $p |- ( ( W e. LMod /\ X e. V ) -> ( .0. .+ X ) = X ) $= @@ -265446,8 +265446,8 @@ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) ) $= (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) $) sramulr $p |- ( ph -> ( .r ` W ) = ( .r ` A ) ) $= - ( cmulr mulrid scandxnmulrndx vscandxnmulrndx ipndxnmulrndx sralem ) ABCG - DEFHIJKL $. + ( cmulr mulridx scandxnmulrndx vscandxnmulrndx ipndxnmulrndx sralem ) ABC + GDEFHIJKL $. $( Obsolete proof of ~ sramulr as of 29-Oct-2024. Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) @@ -266971,11 +266971,11 @@ such that every prime ideal contains a prime element (this is a (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) $) cnfldmul $p |- x. = ( .r ` CCfld ) $= - ( cmul cvv wcel ccnfld cmulr cfv wceq mulex cdc cop cnfldstr mulrid cnx csn - c1 c3 ctp cun ssun1 sstri cbs cc cplusg snsstp3 cstv ccj cts cabs cmin ccom - caddc cmopn cple cle cds cunif cmetu df-cnfld sseqtrri strfv ax-mp ) ABCADE - FGHADEBOOPIJKLMEFAJZNMUAFUBJZMUCFUKJZVBQZDVCVDVBUDVEVEMUEFUFJNZRZDVEVFSVGVG - MUGFUHUIUJZULFJMUMFUNJMUOFVHJQMUPFVHUQFJNRZRDVGVISURUSTTUTVA $. + ( cmul cvv wcel ccnfld cmulr cfv wceq mulex c1 cdc cop cnfldstr mulridx cnx + c3 csn ctp cun ssun1 sstri cbs cplusg caddc snsstp3 cstv ccj cabs cmin ccom + cc cts cmopn cple cle cds cunif cmetu df-cnfld sseqtrri strfv ax-mp ) ABCAD + EFGHADEBIIOJKLMNEFAKZPNUAFUJKZNUBFUCKZVBQZDVCVDVBUDVEVENUEFUFKPZRZDVEVFSVGV + GNUKFUGUHUIZULFKNUMFUNKNUOFVHKQNUPFVHUQFKPRZRDVGVISURUSTTUTVA $. $( The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) @@ -267200,8 +267200,8 @@ such that every prime ideal contains a prime element (this is a cncrng $p |- CCfld e. CRing $= ( vx vy vz ccnfld wcel wtru cc caddc cmul c1 cfv cnfldbas a1i cnfldadd cv wceq cc0 co 3adant1 adantl ccrg cbs cplusg cmulr cnfldmul cgrp cneg addcl - addass 0cn addid2 negcl addcom mpancom negid eqtrd isgrpi mulcl w3a adddi - mulass adddir 1cnd mulid2 mulid1 mulcom iscrngd mptru ) DUAEFABCGHDIJGDUB + addass 0cn addlid negcl addcom mpancom negid eqtrd isgrpi mulcl w3a adddi + mulass adddir 1cnd mullid mulrid mulcom iscrngd mptru ) DUAEFABCGHDIJGDUB KPFLMHDUCKPFNMIDUDKPFUEMDUFEFABCGHDAOZUGZQLNVIBOZUHVIVKCOZUIUJVIUKVIULZVI GEZVJVIHRZVIVJHRZQVJGEVNVOVPPVMVJVIUMUNVIUOUPUQMVNVKGEZVIVKIRZGEFVIVKURSV NVQVLGEUSZVRVLIRVIVKVLIRZIRPFVIVKVLVATVSVIVKVLHRIRVRVIVLIRZHRPFVIVKVLUTTV @@ -267219,7 +267219,7 @@ such that every prime ideal contains a prime element (this is a xrsmcmn $p |- ( mulGrp ` RR*s ) e. CMnd $= ( vx vy vz cxrs cmgp cfv ccmn wcel wtru cxr cxmu cbs wceq eqid c1 3adant1 a1i cv co adantl xrsbas mgpbas cplusg xrsmul mgpplusg xmulcl xmulass rexr - w3a cr 1re mp1i xmulid2 xmulid1 ismndd xmulcom iscmnd mptru ) DEFZGHIABJK + w3a cr 1re mp1i xmullid xmulrid ismndd xmulcom iscmnd mptru ) DEFZGHIABJK USJUSLFMIJDUSUSNZUAUBQZKUSUCFMIDKUSUTUDUEQZIABCJKUSOVAVBARZJHZBRZJHZVCVEK SZJHIVCVEUFPVDVFCRZJHUIVGVHKSVCVEVHKSKSMIVCVEVHUGTOUJHOJHIUKOUHULVDOVCKSV CMIVCUMTVDVCOKSVCMIVCUNTUOVDVFVGVEVCKSMIVCVEUPPUQUR $. @@ -267234,7 +267234,7 @@ such that every prime ideal contains a prime element (this is a $( One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) $) cnfld1 $p |- 1 = ( 1r ` CCfld ) $= - ( vx ccnfld cur cfv c1 cc wcel cv cmul wceq wral ax-1cn mulid2 mulid1 jca + ( vx ccnfld cur cfv c1 cc wcel cv cmul wceq wral ax-1cn mullid mulrid jca co wa rgen pm3.2i crg cnring cnfldbas cnfldmul eqid isringid ax-mp eqcomi wb mpbi ) BCDZEEFGZEAHZIPULJZULEIPULJZQZAFKZQZUJEJZUKUPLUOAFULFGUMUNULMUL NORSBTGUQURUHUAAFBIUJEUBUCUJUDUEUFUIUG $. @@ -267355,7 +267355,7 @@ such that every prime ideal contains a prime element (this is a xrsnsgrp $p |- RR*s e/ Smgrp $= ( c1 cxr wcel cmnf cpnf w3a cxad wne cxrs csgrp wnel 1xr mnfxr wceq mp2an co cc0 ax-mp eqtri mnfaddpnf pnfxr 3pm3.2i xaddcom cr 1re xaddmnf2 oveq1i - renepnf 0ne1 eqnetri oveq2i xaddid1 neeqtrri xrsbas xrsadd isnsgrp mp2 ) + renepnf 0ne1 eqnetri oveq2i xaddrid neeqtrri xrsbas xrsadd isnsgrp mp2 ) ABCZDBCZEBCZFADGPZEGPZADEGPZGPZHIJKURUSUTLMUAUBVBAVDVBQAVBVCQVADEGVADAGPZ DURUSVAVENLMADUCOURAEHZVEDNLAUDCVFUEAUHRAUFOSUGTSUIUJVDAQGPZAVCQAGTUKURVG ANLAULRSUMBIADGEUNUOUPUQ $. @@ -267379,7 +267379,7 @@ such that every prime ideal contains a prime element (this is a ( vx vy vz wcel wtru cxr cmnf cxad cc0 cfv wceq cxrs cv co wne wa eldifsn mp1i cmnd csn cdif wss cbs difss xrsbas ressbas2 cvv cplusg difexi xrsadd xrex ressplusg xaddcl ad2ant2r xaddnemnf sylanbrc syl2anb 3adant1 xaddass - w3a syl3anb adantl cr 0re rexr renemnf eldifi xaddid2 syl xaddid1d ismndd + w3a syl3anb adantl cr 0re rexr renemnf eldifi xaddlid syl xaddridd ismndd mptru ) AUAFGCDEHIUBZUCZJAKVPHUDVPAUELMGHVOUFVPHANBUGUHTVPUIFJAUJLMGHVOUM UKVPJNAUIBULUNTCOZVPFZDOZVPFZVQVSJPZVPFZGVRVQHFZVQIQZRZVSHFZVSIQZRZWBVTVQ HISZVSHISZWEWHRWAHFZWAIQWBWCWFWKWDWGVQVSUOUPVQVSUQWAHISURUSUTVRVTEOZVPFZV @@ -267392,8 +267392,8 @@ such that every prime ideal contains a prime element (this is a xrs10 $p |- 0 = ( 0g ` R ) $= ( vx cc0 c0g cfv wceq wtru cxr cmnf csn cdif cxad wss cbs difss ax-mp cvv cxrs wcel xrsbas ressbas2 eqid cplusg xrex difexi xrsadd ressplusg cr 0re - wne rexr renemnf eldifsn sylanbrc mp1i cv wa co eldifi adantl xaddid2 syl - xaddid1d ismgmid2 mptru ) DAEFZGHCIJKZLZMDAVGVIINVIAOFGIVHPVIIASBUAUBQVGU + wne rexr renemnf eldifsn sylanbrc mp1i cv wa co eldifi adantl xaddlid syl + xaddridd ismgmid2 mptru ) DAEFZGHCIJKZLZMDAVGVIINVIAOFGIVHPVIIASBUAUBQVGU CVIRTMAUDFGIVHUEUFVIMSARBUGUHQDUITZDVITZHUJVJDITDJUKVKDULDUMDIJUNUOUPHCUQ ZVITZURZVLITZDVLMUSVLGVMVOHVLIVHUTVAZVLVBVCVNVLVPVDVEVF $. @@ -267602,7 +267602,7 @@ such that every prime ideal contains a prime element (this is a (Contributed by Mario Carneiro, 15-Oct-2015.) $) zsssubrg $p |- ( R e. ( SubRing ` CCfld ) -> ZZ C_ R ) $= ( vx ccnfld csubrg cfv wcel cz cv wa c1 cmg co cmul wceq ax-1cn cnfldmulg - cc simpr sylancl adantr zcn adantl mulid1d eqtrd csubg subrgsubg subrg1cl + cc simpr sylancl adantr zcn adantl mulridd eqtrd csubg subrgsubg subrg1cl cnfld1 eqid subgmulgcl syl3anc eqeltrrd ex ssrdv ) ACDEFZBGAUOBHZGFZUPAFU OUQIZUPJCKEZLZUPAURUTUPJMLZUPURUQJQFUTVANUOUQRZOUPJPSURUPUQUPQFUOUPUAUBUC UDURACUEEFZUQJAFZUTAFUOVCUQACUFTVBUOVDUQACJUHUGTAUSCUPJUSUIUJUKULUMUN $. @@ -267771,7 +267771,7 @@ such that every prime ideal contains a prime element (this is a regsumfsum $p |- ( ph -> ( ( CCfld |`s RR ) gsum ( k e. A |-> B ) ) = sum_ k e. A B ) $= ( vx ccnfld cgsu co cr cc caddc cvv cc0 wcel a1i cv wa wceq cress csu cfn - cmpt cnfldbas cnfldadd eqid cnfldex ax-resscn fmpttd 0red addid2d addid1d + cmpt cnfldbas cnfldadd eqid cnfldex ax-resscn fmpttd 0red addlidd addridd wss simpr jca gsumress recnd gsumfsum eqtr3d ) AHDBCUDZIJHKUAJZVAIJBCDUBA GBLMKVAHVBNUCOUEUFVBUGHNPAUHQEKLUNAUIQADBCKFUJAUKAGRZLPZSZOVCMJVCTVCOMJVC TVEVCAVDUOZULVEVCVFUMUPUQABCDEADRBPSCFURUSUT $. @@ -268082,7 +268082,7 @@ According to Wikipedia ("Integer", 25-May-2019, ( vx vz czring ccyg wcel wtru cz cmg c1 zringbas eqid ccnfld csubg csubrg cfv cgrp df-zring cv co wceq zsubrg subrgsubg subggrp mp1i 1zzd wrex cmul ax-mp cc ax-1cn cnfldmulg mpan2 1z subgmulg mp3an13 zcn 3eqtr3rd rspceeqv - mulid1d oveq1 mpdan adantl iscygd mptru ) CDEFAGCHOZBCIJVEKZGLMOEZCPEFGLN + mulridd oveq1 mpdan adantl iscygd mptru ) CDEFAGCHOZBCIJVEKZGLMOEZCPEFGLN OEVGUAGLUBUHZGLCQUCUDFUEARZGEZVIBRZIVESZTBGUFZFVJVIVIIVESZTVMVJVIILHOZSZV IIUGSZVNVIVJIUIEVPVQTUJVIIUKULVGVJIGEVPVNTVHUMGVEVOLCVIIVOKQVFUNUOVJVIVIU PUSUQBVIGVLVNVIVKVIIVEUTURVAVBVCVD $. @@ -268117,7 +268117,7 @@ According to Wikipedia ("Integer", 25-May-2019, wo wi eluz2b3 sylibr cui cdiv nnz ad2antrl simprr wb nnne0 dvdsval2 mpbid syl3anc zcnd cc nncn divcan2d simplr eqeltrd zringbas zringmulr zringunit eqid irredmul baib cn0 nnnn0 nn0re nn0ge0 absidd eqeq1d bitrd nnre simprl - nndivred cle clt nnred nngt0 divge0 syl22anc 1cnd divmuld mulid1d orbi12d + nndivred cle clt nnred nngt0 divge0 syl22anc 1cnd divmuld mulridd orbi12d cr 3bitrd expr ralrimiva isprm2 sylanbrc wn prmz 1nprm prmnn id syl5ibcom 3syl eleq1 adantld biimtrid mtoi dvdsmul1 ad2antlr simpr breqtrd absdvdsb simplrl syl2anc breq1 eqeq1 imbi12d simprbi nnne0d simplrr mul02d 3netr4d @@ -268251,7 +268251,7 @@ According to Wikipedia ("Integer", 25-May-2019, mulgrhm2 $p |- ( R e. Ring -> ( ZZring RingHom R ) = { F } ) $= ( vf wcel czring co cv wa wceq cz cmpt cfv zringbas c1 crg crh csn cbs wf eqid rhmf adantl feqmptd cmg rhmghm ad2antlr simpr ghmmulg syl3anc ccnfld - cghm 1zzd cmul ax-1cn cnfldmulg mpan2 adantr zringmulg eqtr4d zcn mulid1d + cghm 1zzd cmul ax-1cn cnfldmulg mpan2 adantr zringmulg eqtr4d zcn mulridd cc 3eqtr3d fveq2d zring1 rhm1 oveq2d mpteq2dva eqtrd eqtr4di velsn sylibr 1z ex ssrdv mulgrhm snssd eqssd ) AUAJZKAUBLZEUCZWEIWFWGWEIMZWFJZWHWGJZWE WINZWHEOWJWKWHDPDMZCBLZQZEWKWHDPWLWHRZQWNWKDPAUDRZWHWIPWPWHUEWEPWPKAWHSWP @@ -268486,8 +268486,8 @@ According to Wikipedia ("Integer", 25-May-2019, $( Ring operation of a ` ZZ ` -module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.) $) zlmmulr $p |- .x. = ( .r ` W ) $= - ( cmulr cfv mulrid cnx csca scandxnmulrndx necomi cvsca vscandxnmulrndx - zlmlem eqtri ) ABFGCFGEFBCDHIJGIFGZKLIMGQNLOP $. + ( cmulr mulridx csca scandxnmulrndx necomi cvsca vscandxnmulrndx zlmlem + cfv cnx eqtri ) ABFNCFNEFBCDGOHNOFNZIJOKNQLJMP $. $( Obsolete version of ~ zlmbas as of 3-Nov-2024. Ring operation of a ` ZZ ` -module (if present). (Contributed by Mario Carneiro, @@ -268782,8 +268782,8 @@ According to Wikipedia ("Integer", 25-May-2019, ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.) $) znmul $p |- ( N e. NN0 -> ( .r ` U ) = ( .r ` Y ) ) $= - ( cmulr mulrid cnx cple cfv plendxnmulrndx necomi znbaslem ) ABHCDEFGIJKL - JHLMNO $. + ( cmulr mulridx cnx cple cfv plendxnmulrndx necomi znbaslem ) ABHCDEFGIJK + LJHLMNO $. $( Obsolete version of ~ znadd as of 3-Nov-2024. The multiplicative structure of ` Z/nZ ` is the same as the quotient ring it is based on. @@ -269242,7 +269242,7 @@ According to Wikipedia ("Integer", 25-May-2019, gcdn0cl nncnd nnne0d divcan2d gcddvds simpld nnzd simprd simpll nndivdvds mpbid dvdsmulc syl3anc mpd eqbrtrrd cmulr c0g wf ffvelcdmd rrgeq0i czring fof crh crg ccrg zncrng crngring zrhrhm zringbas zringmulr rhmmul zmulcld - eqeq1d zndvds0 bitr3d 3imtr3d divcan1d mulid1d dvdscmulr syl112anc gcdcld + eqeq1d zndvds0 bitr3d 3imtr3d divcan1d mulridd dvdscmulr syl112anc gcdcld 3brtr4d 1zzd dvds1 znunit mpbird eleq1 imbi12d syl5ibrcom rexlimdva com23 mpdd ssrdv wss unitrrg eqssd ) CUEHZBAYIUCBAYIUCUFZBHZYJUDUFZDUAIZIZJZUDK UBZYJAHZYIYKYPYIKDUGIZYMUHZYJYRHYPYKYICUSHZYSCUIZYRYMCDEYRUJZYMUJZUKZLBYR @@ -269467,8 +269467,8 @@ According to Wikipedia ("Integer", 25-May-2019, (Contributed by Stefan O'Rear, 28-Aug-2015.) $) cnmsgnsubg $p |- { 1 , -u 1 } e. ( SubGrp ` M ) $= ( vx c1 wcel wceq cc ax-1cn eqeltrdi neg1cn jaoi syl cc0 wne co wa oveq12 - cmul eqeltri cdiv vy cneg cpr cv wo elpri ax-1ne0 eqnetrd neg1ne0 mulid1i - id a1i prid1 negex prid2 mulid2i neg1mulneg1e1 ccase syl2an oveq2 1div1e1 + cmul eqeltri cdiv vy cneg cpr cv wo elpri ax-1ne0 eqnetrd neg1ne0 mulridi + id a1i prid1 negex prid2 mullidi neg1mulneg1e1 ccase syl2an oveq2 1div1e1 1ex divneg2 mp3an negeqi eqtr3i cnmsubglem ) CUADDUBZUCZABCUDZVIEZVJDFZVJ VHFZUEZVJGEZVJDVHUFZVLVOVMVLVJDGVLUKZHIVMVJVHGVMUKZJIKLVKVNVJMNZVPVLVSVMV LVJDMVQDMNZVLUGULUHVMVJVHMVRVHMNVMUIULUHKLVKVNUAUDZDFZWAVHFZUEVJWAROZVIEZ @@ -270106,7 +270106,7 @@ According to Wikipedia ("Integer", 25-May-2019, ( cr wf cc0 cfsupp wcel crefld cgsu co ccnfld cc cnfldbas ax-resscn caddc ccmn wa wceq wbr w3a csupp cres cv cfv cmpt csu cnfld0 crg cnring ringcmn mp1i simp3 wss simp1 fss sylancl ssidd gsumres cnfldadd df-refld a1i 0red - simp2 simpr addid2d addid1d jca gsumress eqtr2d suppssdm feqresmpt oveq2d + simp2 simpr addlidd addridd jca gsumress eqtr2d suppssdm feqresmpt oveq2d fssdm fsuppimpd simpl1 sselda ffvelcdmd sselid gsumfsum 3eqtrd ) CEBFZBGH UAZCDIZUBZJBKLZMBBGUCLZUDZKLZMAWHAUEZBUFZUGZKLWHWLAUHWFWJMBKLWGWFCNBMDWHG OUIMUJIMRIWFUKMULUMZWCWDWEUNZWFWCENUOZCNBFWCWDWEUPZPCENBUQURWFWHUSWCWDWEV @@ -275901,17 +275901,17 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, ( ( f ` x ) .x. ( g ` ( k oF - x ) ) ) ) ) ) ) $= ( cfv c0 cvv wcel wa cv cle cofr wbr crab cmin cof co cmpt cgsu cmpo wceq cmulr cnx cbs cop cplusg ctp csca cvsca csn cxp cts ctopn cpt eqid psrbas - cun simpl psrplusg eqidd simpr psrval fveq2d fvexi mpoex psrvalstr mulrid - c1 c9 snsstp3 ssun1 sstri strfv ax-mp 3eqtr4g wn str0 cmps reldmpsr ovprc - eqcomi eqtrid wo base0 olcd 0mpo0 syl 3eqtr4a pm2.61i ) MUAUBZEUAUBZUCZGI - JCCLDEABUDLUDZUEUFUGBDUHAUDZIUDZSXGXHUIUJUKJUDSHUKULUMUKULZUNZUOXFFUPSZUQ - URSCUSZUQUTSFUTSZUSZUQUPSZXKUSZVAZUQVBSEUSUQVCSAIEURSZCDXHVDVEXIHUJUKUNZU - SUQVFSDEVGSZVDVEVHSZUSVAZVKZUPSZGXKXFFYDUPXFABCDEUTSZXNEFXTHXKIJKLMYBXSYA - UAUANXSVIZYFVIZPYAVIRXFCDEFKMXSUANYGROXDXEVLZVJCYFXNEFMNOYHXNVIVMXKVIXTVI - XFYBVNYIXDXEVOVPVQQXKUAUBXKYEUOIJCCXJCFUROVRZYJVSXKYDUPUAWBWCUSCXNEXTXKYB - VTWAXQVDXRYDXMXOXQWDXRYCWEWFWGWHWIXFWJZTUPSZTGXKTYLUPXPWAWKWOYKGXLYLQYKFT - UPYKFMEWLUKTNMEWLWMWNWPZVQWPYKCTUOZYNWQXKTUOYKYNYNYKFURSTURSCTYKFTURYMVQO - WRWIWSIJCCXJWTXAXBXC $. + simpl psrplusg eqidd simpr psrval fveq2d fvexi mpoex c1 psrvalstr mulridx + cun c9 snsstp3 ssun1 sstri strfv ax-mp 3eqtr4g str0 eqcomi reldmpsr ovprc + wn cmps eqtrid wo base0 olcd 0mpo0 syl 3eqtr4a pm2.61i ) MUAUBZEUAUBZUCZG + IJCCLDEABUDLUDZUEUFUGBDUHAUDZIUDZSXGXHUIUJUKJUDSHUKULUMUKULZUNZUOXFFUPSZU + QURSCUSZUQUTSFUTSZUSZUQUPSZXKUSZVAZUQVBSEUSUQVCSAIEURSZCDXHVDVEXIHUJUKUNZ + USUQVFSDEVGSZVDVEVHSZUSVAZWBZUPSZGXKXFFYDUPXFABCDEUTSZXNEFXTHXKIJKLMYBXSY + AUAUANXSVIZYFVIZPYAVIRXFCDEFKMXSUANYGROXDXEVKZVJCYFXNEFMNOYHXNVIVLXKVIXTV + IXFYBVMYIXDXEVNVOVPQXKUAUBXKYEUOIJCCXJCFUROVQZYJVRXKYDUPUAVSWCUSCXNEXTXKY + BVTWAXQVDXRYDXMXOXQWDXRYCWEWFWGWHWIXFWNZTUPSZTGXKTYLUPXPWAWJWKYKGXLYLQYKF + TUPYKFMEWOUKTNMEWOWLWMWPZVPWPYKCTUOZYNWQXKTUOYKYNYNYKFURSTURSCTYKFTURYMVP + OWRWIWSIJCCXJWTXAXBXC $. psrmulfval.i $e |- ( ph -> F e. B ) $. psrmulfval.r $e |- ( ph -> G e. B ) $. @@ -277866,7 +277866,7 @@ series in the subring which are also polynomials (in the parent ring). fssdm cun noel mtbiri fconstmpt eqtr4di mpt0 ringidval gsum0 mpl1 ssun1 a1d sstr2 ax-mp imim1i cmulr oveq1 cof simprll ssfid eqeltrrd mpbir2and cbs eldifn ssun2 simprr vex snss ffvelcdmd snifpsrbag mplmonmul mplcoe3 - sylibr offval2 addid2d elsni simprlr fveq2d sselid addid1d velsn sylnib + sylibr offval2 addlidd elsni simprlr fveq2d sselid addridd velsn sylnib ad2antrr eqneltrd elun orcom bitri biorf 3eqtr3rd ccntz mgpbas mgpplusg bitr4id cmnd mplring ringmgp cplusg cbvral2vw sselda adantlr gsumzunsnd sylib syldan syl5ibr expr syl5 expcom findcard2s mpcom mpd ssidd eldifi @@ -278258,8 +278258,8 @@ series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.) $) opsrmulr $p |- ( ph -> ( .r ` S ) = ( .r ` O ) ) $= - ( cmulr mulrid cnx cple cfv plendxnmulrndx necomi opsrbaslem ) ABCDJEFGHI - KLMNLJNOPQ $. + ( cmulr mulridx cnx cple cfv plendxnmulrndx necomi opsrbaslem ) ABCDJEFGH + IKLMNLJNOPQ $. $( Obsolete version of ~ opsrmulr as of 1-Nov-2024. The multiplication operation of the ordered power series structure. (Contributed by Mario @@ -280303,7 +280303,7 @@ Additional definitions for (multivariate) polynomials c1 cur cascl csca crg mplsca eqid clmod mpllmod syl2anc mplring ascl1 cbs eqtrd ringidcl syl mhpsclcl eqeltrrd mhpmpl mgpbas ringidval mulg0 nn0cnd mul01d 3eltr4d wa cmulr ad2antrr simplr nn0mulcld simpr mhpmulcl mgpplusg - cmnd ringmgp mulgnn0p1 syl3anc 1cnd adddid mulid1d oveq2d nn0indd mpdan + cmnd ringmgp mulgnn0p1 syl3anc 1cnd adddid mulridd oveq2d nn0indd mpdan cc ) AIUCUDIKEUEZHIUFUEZFUGZUDZSAUAUHZKEUEZHXDUFUEZFUGZUDUIKEUEZHUIUFUEZF UGZUDUBUHZKEUEZHXKUFUEZFUGZUDZXKUPUJUEZKEUEZHXPUFUEZFUGZUDXCUAUBIXDUIUKZX EXHXGXJXDUIKEULXTXFXIFXDUIHUFUMUNUOXDXKUKZXEXLXGXNXDXKKEULYAXFXMFXDXKHUFU @@ -284789,10 +284789,10 @@ represented as an element of (the base set of) ` ( ( 1 ... n ) Mat R ) ` . $( Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.) $) matmulr $p |- ( ( N e. Fin /\ R e. V ) -> .x. = ( .r ` A ) ) $= - ( cfn wcel wa cxp cfrlm co cnx cmulr cfv cop csts cvv wceq ovex cotp mp1i - cmmul ovexi pm3.2i mulrid setsid eqid matval fveq2d eqtr4d ) DHIBEIJZCBDD - KZLMZNOPCQRMZOPZAOPUOSIZCSIZJCUQTUMURUSBUNLUACBDDDUBUDGUEUFSCOSUOUGUHUCUM - AUPOABCUODEFUOUIGUJUKUL $. + ( cfn wcel wa cxp cfrlm co cnx cmulr cfv cop csts cvv wceq pm3.2i mulridx + ovex cotp cmmul ovexi setsid mp1i eqid matval fveq2d eqtr4d ) DHIBEIJZCBD + DKZLMZNOPCQRMZOPZAOPUOSIZCSIZJCUQTUMURUSBUNLUCCBDDDUDUEGUFUASCOSUOUBUGUHU + MAUPOABCUODEFUOUIGUJUKUL $. $} ${ @@ -288229,7 +288229,7 @@ bijection from the singleton containing the empty set (empty matrix) eldifi cofmpt ringabl difssd gsummptfidminv cpmtr crn c2o cen prssd enpr2 wbr pmtrrn pmtrodpm evpmodpmf1o gsummptfif1o eleq1w anbi2d eleq1d imbi12d cabl cneg symggrp sselid grpcl cress cghm psgnghm2 prex cnfldmul mgpplusg - symgtrf ressplusg ghmlin psgnpmtr neg1cn mulid1i eqtrdi psgnodpmr chvarvv + symgtrf ressplusg ghmlin psgnpmtr neg1cn mulridi eqtrdi psgnodpmr chvarvv fveq1 fmptco cbvmptv symgov fvco3 wral rspcdva prcom fveq2i fveq1i necomd psgnevpm a1dd w3a cid wo neanior elpri orcomd con3i sylbi 3adant1 pmtrmvd cdm neleqtrrd wb pmtrf ffnd fnelnfp necon2bbid mpbird 3exp fveq2d grprinv @@ -321590,7 +321590,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed psmetsym $p |- ( ( D e. ( PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) ) $= ( wcel w3a psmetcl 3com23 cxad cle wbr psmettri2 syl13anc cc0 wceq psmet0 - co oveq2d xaddid1d eqtrd cpsmet cfv cxr simp1 simp3 simp2 3adant2 breqtrd + co oveq2d xaddridd eqtrd cpsmet cfv cxr simp1 simp3 simp2 3adant2 breqtrd 3adant3 xrletrid ) CDUAUBEZADEZBDEZFZABCQZBACQZABCDGZUKUMULUPUCEBACDGZHUN UOUPBBCQZIQZUPJUNUKUMULUMUOUTJKUKULUMUDZUKULUMUEZUKULUMUFZVBABBCDLMUNUTUP NIQZUPUNUSNUPIUKUMUSNOULBCDPUGRUKUMULVDUPOUKUMULFUPURSHTUHUNUPUOAACQZIQZU @@ -321993,7 +321993,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric xmetsym $p |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) ) $= ( wcel w3a co xmetcl 3com23 cxad cle wbr xmettri2 syl13anc cc0 wceq xmet0 - oveq2d xaddid1d eqtrd cxmet cfv simp1 simp3 simp2 3adant2 breqtrd 3adant3 + oveq2d xaddridd eqtrd cxmet cfv simp1 simp3 simp2 3adant2 breqtrd 3adant3 cxr xrletrid ) CDUAUBEZADEZBDEZFZABCGZBACGZABCDHZUKUMULUPUIEBACDHZIUNUOUP BBCGZJGZUPKUNUKUMULUMUOUTKLUKULUMUCZUKULUMUDZUKULUMUEZVBABBCDMNUNUTUPOJGZ UPUNUSOUPJUKUMUSOPULBCDQUFRUKUMULVDUPPUKUMULFUPURSITUGUNUPUOAACGZJGZUOKUN @@ -322442,7 +322442,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric wi imbi12d imbi2d cuz resmptd eqtrd breqtrrd adantr crn ciun imasdsval2 cds wfo f1ofo infeq1i eqtr4di cdif xrsex fzfid difss wfn cxmet ffn 3syl xmetf ge0nemnf 3expb sylan ralrimivva ffnov ssrab3 sselda elmapi fco cr - xmetge0 0re rexr renemnf xaddid2 xaddid1 jca gsumress cbs ressbas2 ccmn + xmetge0 0re rexr renemnf xaddlid xaddrid jca gsumress cbs ressbas2 ccmn xrs10 xrs1cmn c0ex fdmfifsupp gsumcl eqeltrd eldifad fmpttd frnd sylibr ralrimiva iunss eqsstrid infxrcl opex f1osn opelxpd fss sylancr elfvexd snssd xpexd snex elmapg sylancl op1stg op2ndg snfi fconst6g xpsn coeq2i @@ -324549,8 +324549,8 @@ S C_ ( P ( ball ` D ) T ) ) $= vex eqid eqeq1d wb eqeq1 bibi1d biidd wn simp3 gt0ne0d neneqd ad2antrr wi 0xr xrltle mpd adantr syl5ibrcom con3dimp 2falsed ifbothda bitrd ad2antrl xrmin1 ifcld ad2antll xrletr syl3anc mpand xrmin2 xrlemin sylibrd adantrr - jctird breq12d xaddcld xaddid2 syl a1i simprbi xleadd1a syl31anc eqbrtrrd - simpr xrletrd oveq2 breq2d ifboth xaddid1d breq2 xleadd2a oveq1 ge0xaddcl + jctird breq12d xaddcld xaddlid syl a1i simprbi xleadd1a syl31anc eqbrtrrd + simpr xrletrd oveq2 breq2d ifboth xaddridd breq2 xleadd2a oveq1 ge0xaddcl ovex oveq12d 3brtr4d comet eqeltrrd ) CFUBUCZIZEJIZKEUDLZUFZHKUGUHMZHNZEO LZUUFEUIZUJZCUKZDYTUUDUUJABFFANZBNZCMZEOLZUUMEUIZULDUUDABHFFUUEUUMUUHUUOC UUIUUDUUAUUKFIZUULFIZPUUMUUEIZUUAUUBUUCUMZUUAUUPUUQUURUUAUUPUUQUFUUMJIKUU @@ -325998,7 +325998,7 @@ the half element (corresponding to half the distance) is also in this rspccva grpsubeq0 3bitrd adantr adantrr ffvelcdmd simprll syl3anc simprlr simprr readdcld grpnnncan2 fveq2d ralrimivva fvoveq1 oveq1d breq12d oveq2 syl13anc oveq2d rspc2va syl21anc eqbrtrrd wi eleq1w anbi2d imbi12d simprl - grpidcl cminusg grpinvval2 grpinvsub eqtr3d pm5.501 bitrdi bitr3d addid2d + grpidcl cminusg grpinvval2 grpinvsub eqtr3d pm5.501 bitrdi bitr3d addlidd eqid bicom recnd 3brtr3d chvarvv adantrlr anbi1d adantrll le2addd oveq12d eqtrd letrd 3brtr4d expr ralrimiv jca cvv cbs fvexi ismet ax-mp sylanbrc ) AGGUDZUEDFUFZUGZUAUHZUBUHZYQPZUIQZYSYTQZUNZUUAUCUHZYSYQPZUUEYTYQPZUJPZR @@ -326949,8 +326949,8 @@ definition of norm (which itself uses the metric). (Contributed by (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.) $) tngmulr $p |- ( N e. V -> .x. = ( .r ` T ) ) $= - ( wcel cmulr cfv mulrid cnx tsetndxnmulrndx necomi dsndxnmulrndx tnglem - cts cds eqtrid ) DEHBCIJAIJGAICDEFKLQJLIJZMNLRJTONPS $. + ( cmulr cfv mulridx cnx cts tsetndxnmulrndx necomi dsndxnmulrndx tnglem + wcel cds eqtrid ) DEQBCHIAHIGAHCDEFJKLIKHIZMNKRITONPS $. $( Obsolete proof of ~ tngmulr as of 31-Oct-2024. The ring multiplication of a structure augmented with a norm. (Contributed by @@ -327789,7 +327789,7 @@ distance function as the normed group (restricted to the base set). ringsubdi ringsubdir ringlidm ringass syl13anc unitlinv oveq2d ringridm 3eqtrd oveq12d rpred rpne0d divmuld mpbird ngpdsr ngpds 3eqtr4rd simprr eqeltrd remulcld 1re min2 lemul1d mpbid eqbrtrid 2halvesd cmin resubcld - caddc nm2dif breqtrrd min1 1red mulid2d lelttrd eqbrtrd breqtrd ltletrd + caddc nm2dif breqtrrd min1 1red mullidd lelttrd eqbrtrd breqtrd ltletrd ltsubadd2d ltadd1d ltmul2dd lttrd ltdivmuld ralrimiva breq2 rspceaimv expr ) AHUCUDZDCUEZFUFZHUGUHZDJUIZUUMJUIZFUFZEUGUHZUJZCIUKUUNBUEZUGUHZU USUJCIUKBUCULAHUMDKUIZEUNUFZUOUHZUMUVDUPZUVCVLUQUFZUNUFZUCUBAUVFUVGAUMU @@ -328297,7 +328297,7 @@ Normed space homomorphisms (bounded linear operators) cbs simpl3 eqid ghmf syl simprl ffvelcdmd nmcl syl2anc rexrd nmocl adantr wf crp nmrpcl 3expb 3ad2antl1 rpxrd xmulcld rpreccld rpge0d nmoix adantrr jca xlemul1a syl31anc cmul rpred rexmul recnd rpcnd rpne0d divrecd eqtr4d - wceq xmulass syl3anc recidd eqtrd oveq2d xmulid1 3eqtrd 3brtr3d ) AUAOZBU + wceq xmulass syl3anc recidd eqtrd oveq2d xmulrid 3eqtrd 3brtr3d ) AUAOZBU AOZCABUBPOZUCZHGOZHIUDZUEZUEZHCQZEQZRHDQZUFPZSPZCFQZXESPZXFSPZXDXEUFPZXHU GXBXDTOXITOXFTOZUHXFUGUIZUEXDXIUGUIZXGXJUGUIXBXDXBWPXCBULQZOXDUJOZWOWPWQX AUKXBGXOHCXBWQGXOCVDWOWPWQXAUMABCGXOKXOUNZUOUPWRWSWTUQURXCBEXOXQMUSUTZVAX @@ -328510,7 +328510,7 @@ Normed space homomorphisms (bounded linear operators) ( N ` ( _I |` V ) ) = 1 ) $= ( vx wcel wa cfv c1 wceq cle wbr co adantr 1red cr syl3anc cngp wpss cres csn cid cnm eqid simpl cgrp cghm ngpgrp syl cc0 0le1 a1i cv wne cmul nmcl - idghm ad2ant2r leidd fvresi ad2antrl fveq2d recnd mulid2d 3brtr4d nmolb2d + idghm ad2ant2r leidd fvresi ad2antrl fveq2d recnd mullidd 3brtr4d nmolb2d wn wex pssnel adantl velsn biimpri necon3bi eqtr4d cnghm cxr nmocl nmoge0 xrrege0 syl22anc wb isnghm2 mpbird simprl nmoi syl2an2r eqbrtrd crp 3expb nmrpcl adantlr lemul1d sylanr2 exlimddv 1xr xrletri3 sylancl mpbir2and ) @@ -329133,9 +329133,9 @@ Normed space homomorphisms (bounded linear operators) caddc cbl cfv cle simpr3 elicc01 sylib simp1d recnd simpr1 eleqtrdi cxmet cnxmet simpll simplr elbl mp3an2i mpbid simpld mulcld 1re resubcl sylancr wb simpr2 addcld wceq eqid cnmetdval syl2anc subdid oveq12d ax-1cn pncan3 - addsub4d sylancl oveq1d adddird mulid2 3eqtr3d 3eqtr2d fveq2d eqtr4d cpnf + addsub4d sylancl oveq1d adddird mullid 3eqtr3d 3eqtr2d fveq2d eqtr4d cpnf ad2antrr subcld abscld simpr abstrid oveq1 mul02d sylan9eqr abs00bd oveq2 - readdcld eqtrdi mulid2d addid2d simprd eqbrtrd adantlr wne absmuld simp2d + readdcld eqtrdi mullidd addlidd simprd eqbrtrd adantlr wne absmuld simp2d 1m0e1 absidd eqtrd eqbrtrrd 0red leltned biimpar syl112anc a1i abssubge0d ltmul2 simp3d subge0 mpbird jca wi ltle sylan mpd lemul2a remulcl ltleadd syl31anc syl22anc mp2and breqtrd pm2.61dane lelttrd ltpnfd breqtrrd cmnf @@ -329354,9 +329354,9 @@ Normed space homomorphisms (bounded linear operators) syl2anc xnegneg xaddcom 3eqtrd xneg0 eqeq12d bitr3d ad2antrr biidd bibi1d eqeq1 ifboth eqtr3d subeq0d pm2.61dane anidms xrleid iftrued xnegid oveq1 syl5ibrcom impbid 3bitr2d w3a simplrr leidd oveq1d simpll1 oveq12 vtoclga - ex eqtrd addid2d simplrl simpll2 eqeq2 wo oveq12d 3brtr4d cabs xrsdsreval + ex eqtrd addlidd simplrl simpll2 eqeq2 wo oveq12d 3brtr4d cabs xrsdsreval syl3anc eqtr2d 3brtr3d oveq2 eqtr4d clt xrleloe neneqd biorf orcom bitrdi - eqeltrrd xrltnle con2bid iffalsed 3eqtr4d addid1d simpll3 simprl abs3difd + eqeltrrd xrltnle con2bid iffalsed 3eqtr4d addridd simpll3 simprl abs3difd simprr abssubd pm2.61da2ne 3adant1 isxmet2d mptru ) AEUBUCFGCDUAAUDEEUDFG UEUFEEUGEAUHZGCUOZDUOZHIZUVHUVGUIZJKZUVGUVHUIZJKZUJZEFZDEUKCEUKUVFUVOCDEE UVGEFZUVHEFZLZUVIUVKUVMEUVQUVQUVJEFZUVKEFZUVPUVQUPUVGULZUVHUVJUMUNZUVQUVP @@ -329981,7 +329981,7 @@ Normed space homomorphisms (bounded linear operators) subm0 eqeltri a1i simprbi adantl wf simplbi fssres syl2an c0ex fdmfifsupp elfpw gsumcl sselid fmpttd frnd cc 0ss 0fin mpbir2an reseq2 eqtrdi oveq2d 0cn res0 gsum0 elrnmpt1s sseldd elin2d diffi ssdifssd fssresd ssfid sstrd - syl xleadd2a syl31anc xaddid1d cplusg xrsadd ressplusg disjdif cun undif2 + syl xleadd2a syl31anc xaddridd cplusg xrsadd ressplusg disjdif cun undif2 ovex ssequn1 sylib eqtr2id gsumsplit resabs1d difss resabs1 mp1i oveq12d eqtr2d 3brtr3d ) AFEDUCZMNZOUDNZUUEFECDUEZUCZMNZUDNZUUEFECUCZMNZUFAOPQUUI PQZUUEPQOUUIUFUGZUUFUUJUFUGAUABUHZRUIZFEUAUJZUCZMNZUKZULZPOAUUPPUUTAUAUUP @@ -330411,7 +330411,7 @@ Normed space homomorphisms (bounded linear operators) ( A e. S /\ B e. X ) ) -> ( F ` B ) <_ ( A D B ) ) $= ( cxmet cfv wcel wa co cxad cle adantrr cc0 wceq syl3anc wss simprr simpr wbr sselda jca metdstri syldan simpll xmetsym metds0 3expa oveq12d xmetcl - cxr xaddid1d eqtrd breqtrd ) EHJKLZFHUAZMZCFLZDHLZMZMZDGKZDCENZCGKZONZCDE + cxr xaddridd eqtrd breqtrd ) EHJKLZFHUAZMZCFLZDHLZMZMZDGKZDCENZCGKZONZCDE NZPVAVDVCCHLZMVFVIPUDVEVCVKVAVBVCUBZVAVBVKVCVAFHCUSUTUCUEQZUFABDCEFGHIUGU HVEVIVJRONVJVEVGVJVHROVEUSVCVKVGVJSUSUTVDUIZVLVMDCEHUJTVAVBVHRSZVCUSUTVBV OABCEFGHIUKULQUMVEVJVEUSVKVCVJUOLVNVMVLCDEHUNTUPUQUR $. @@ -330561,7 +330561,7 @@ Normed space homomorphisms (bounded linear operators) ( wcel c1 co wa cfv cle wbr cif cxr 1xr cc0 cpnf cicc cxmet wss wf adantr cuni ccld eqid cldss syl mopnuni sseqtrrd metdsf syl2anc simprr ffvelcdmd wceq sseldd eliccxr sylancr simprl xmetcl syl3anc xrmin2 metdstri xmetsym - ifcl cxad syl22anc metds0 oveq12d xaddid1d eqtrd breqtrd xrletrd ) ADGRZE + ifcl cxad syl22anc metds0 oveq12d xaddridd eqtrd breqtrd xrletrd ) ADGRZE HRZUAZUAZSEIUBZUCUDZSWIUEZWIDEFTZWHSUFRZWIUFRZWKUFRUGWHWIUHUIUJTZRWNWHKWO EIWHFKUKUBRZGKULZKWOIUMAWPWGNUNZWHGJUOZKWHGJUPUBZRZGWSULAXAWGOUNGJWSWSUQZ URUSWHWPKWSVFWRFJKMUTUSZVAZBCFGIKLVBVCWHHKEWHHWSKWHHWTRZHWSULAXEWGPUNHJWS @@ -331828,9 +331828,9 @@ Normed space homomorphisms (bounded linear operators) wb cico cpnf wf1o ccnv cv cdiv cmpt wa cxr w3a 0re elico2 mp2an simp1bi 1xr crp simp3bi difrp sylancl mpbid rerpdivcld simp2bi divge0d sylanbrc 1re elrege0 simplbi readdcl sylancr a1i simprbi ltp1d cc recnd breqtrrd - ax-1cn addcom lelttrd divge0 syl22anc mulid1d ltdivmul syl112anc mpbird + ax-1cn addcom lelttrd divge0 syl22anc mulridd ltdivmul syl112anc mpbird elrpd syl3anbrc adantr mulcld subadd2d 1cnd subdird oveq1d eqtrd eqeq1d - mulid2d adddid 3bitr4rd 3bitr4g rpne0d divmul3d rpcnd divmul2d f1ocnv2d + mullidd adddid 3bitr4rd 3bitr4g rpne0d divmul3d rpcnd divmul2d f1ocnv2d 3bitr4d mptru ) EFUAGZEUBUAGZCUCCUDBXGBUEZFXHHGZUFGZUGIUHJABXFXGAUEZFXK KGZUFGZXJCDXKXFLZXMXGLZJXNXMMLEXMNOXOXNXKXLXNXKMLZEXKNOZXKFPOZEMLZFUILZ XNXPXQXRUJTUKUOEFXKULUMZUNZXNXRXLUPLZXNXPXQXRYAUQXNXPFMLZXRYCTYBVEXKFUR @@ -331857,7 +331857,7 @@ Normed space homomorphisms (bounded linear operators) vw cpnf wiso crest chmeo wf1o cfv wral ccnv caddc cmpt icopnfcnv simpli cv wa cxr w3a 0re 1xr elico2 simp1bi ssriv sseli adantr crp simp3bi 1re difrp sylancl mpbid rpregt0d lt2mul2div syl22anc remulcld ltsub1d recnd - adantl subdid mulid1d oveq1d eqtrd mulcomd oveq12d breq12d bitr4d oveq2 + adantl subdid mulridd oveq1d eqtrd mulcomd oveq12d breq12d bitr4d oveq2 1cnd id ovex fvmpt breqan12d 3bitr4d rgen2 df-isom mpbir2an cxp cin cvv cordt ctsr letsr elexi inex1 wss icossxr leiso mpbi isores1 isores2 cdm cps tsrps ax-mp ledm psssdm eqcomi ordthmeo mp3an iccssico2 ordtrestixx @@ -332119,7 +332119,7 @@ extended reals extends the topology of the reals (by ~ xrtgioo ), this cc0 cmin clt wbr wss iccss2 adantr iccssre sselda adantrr adantrl simpr3 wi lincmb01cmp ex 3expa an32s sylan sseldd oveq2 unitssre sseli recnd ad2antll imp ax-1cn npcan mpan subcl ancri adddir 3eqtr3d syl2anc 3adantr1 sylan9eqr - mulid2 simplr2 eqeltrd ancom2s iirev sselid syl2anr adantll adantlr addcomd + mullid simplr2 eqeltrd ancom2s iirev sselid syl2anr adantll adantlr addcomd recn mulcl 3adantl3 nncan eqcomd syl eqtrd sylan2 w3o lttri4d mpjao3dan ) A FGBFGHZCABIJZGZDWRGZEUAKIJZGZLZKEUBJZCMJZEDMJZNJZWRGZWQXCHZCDUCUDZXHCDOZDCU CUDZXIXJHCDIJZWRXGXIXMWRUEZXJWQWSWTXNXBWSWTHZXNWQABCDUFPQUGXICFGZDFGZHZXBHX @@ -332706,7 +332706,7 @@ topological space to the reals is bounded (above). (Boundedness below vr cii cuni wa cabs ccom cxp cres cbl wral crp cfz cn df-ii cmet unitssre cnmet ax-resscn sstri metres2 mp2an a1i ccmp iicmp simpl simpr lebnum cfl caddc cn0 rpreccl adantl rpred rpge0d flge0nn0 syl2anc nn0p1nn syl elfznn - wi nnrpd adantr rpdivcld cmul elfzle2 nnred recnd mulid1d breqtrrd nngt0d + wi nnrpd adantr rpdivcld cmul elfzle2 nnred recnd mulridd breqtrrd nngt0d wb ledivmul syl112anc mpbird elicc01 syl3anbrc oveq1 sseq1d rexbidv rspcv 1red cioo cin cxr simplr resubcld readdcld nnm1nn0 nn0red nndivred nnne0d rexrd divsubdird ax-1cn nncan sylancl oveq1d eqtr3d rprecred flltp1 rpgt0 @@ -333265,10 +333265,10 @@ topological space to the reals is bounded (above). (Boundedness below mp1i cmpt cnmpt1st cnmpt21f mulcn cnmpt22f addcn cc crn cnfldtopon wral wb cxp wf wa iiuni cnf syl ffvelcdmda adantrr simprl simprr w3a 0re 1re cr wi mp2an syl3anc ralrimivva fmpo sylib frnd unitssre ax-resscn sstri - icccvx cnrest2 eleqtrrdi eqeltrid sselid mulid2d adantl oveq12d addid1d + icccvx cnrest2 eleqtrrdi eqeltrid sselid mullidd adantl oveq12d addridd mpbid sseli mul02d eqtrd fveq2d simpr 0elunit oveq2d 1m0e1 eqtrdi simpl wceq weq fvex ovmpoa sylancl fvco3 sylan 3eqtr4d 1elunit adantr sylancr - ax-1cn mul01d mulid1d 1m1e0 addid2d subcl 00id npcan isphtpy2d ) ADEUCZ + ax-1cn mul01d mulridd 1m1e0 addlidd subcl 00id npcan isphtpy2d ) ADEUCZ DFGUAAEMMUDNZOZDMGUDNZOUUGUUJOIHEDMMGUEUFHAFBCPQUGNZUUKQCUHZUINZBUHZERZ SNZUULUUNSNZTNZDRZUJMMUKNZGUDNLABCUURDMMMGUUKUUKMUUKULROAUMUNZUVAABCUUK UUKUURUJZUUTVAUORZUUKUPNZUDNZUUTMUDNZAUVBUUTUVCUDNZOZUVBUVEOZABCUUPUUQT @@ -333767,7 +333767,7 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` ) oveq12i mulcomli adantlr simplll ad2antlr letric orcanai readdcl leadd1dd wo simplr eqbrtrrid simp3bi 2lt4 2pos ltrecii le2addd pco0 adddid eqtr3id breqtrdi pnpcan2d divcan1i subidi 1mhlfehlf iccshftli subdid recidi elii2 - remulcl sseli div0i icccntri addid2i iccshftri 3syl 1cnd divdird peano2cn + remulcl sseli div0i icccntri addlidi iccshftri 3syl 1cnd divdird peano2cn divcan2d eqtr3d mvrraddd mpteq2dva wral iitopon 0elunit cioo crn ctg 0red 1red addcomi breq1d recgt0ii 3eqtr4a 0xr rexri lbicc2 sstri retop restabs ovex eqcomi iihalf1cn cnmpt1res ccnfld ctopn unitssre cnfldtopon cnmpt12f @@ -333884,7 +333884,7 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` ) halfre 1re halflt1 ltleii elicc01 2cn oveq1d eqtr4d wss retopon iccssre wa 1m0e1 mp2an resttopon cnmpt2nd cnmpt21 cnmpt1st oveq12 cnmpt22 simpr weq 0elunit simpl breq1d oveq12d ifbieq12d ovmpoa adantl unitssre sseli - recnd mulid2d eqtrd anassrs wn cc ax-1cn subcl sylancr mul02d sylan9eqr + recnd mullidd eqtrd anassrs wn cc ax-1cn subcl sylancr mul02d sylan9eqr syl mpan mpbir3an simprl recidi 0re iihalf1cn iimulcn iihalf2cn cnmpopc 2ne0 cnmpt21f iftrue elii1 pcoval1 iihalf1 sylan2br iffalse elii2 nncan pcoval2 iihalf2 eqtr2d sylan2 pm2.61dan mulcl ifeq12d ifid csn fvconst2 @@ -335829,9 +335829,9 @@ vector spaces (see ~ zclmncvs ), because the ring ZZ is not a division ring cnlmod $p |- W e. LMod $= ( vx vy vz cc0 cc wcel caddc cv cfv wceq eqcomi a1i co adantl cmul ccnfld id c1 cgrp clmod 0cn cneg cbs cnlmodlem1 cplusg cnlmodlem2 3adant1 addass - addcl w3a addid2 negcl addcomd negid eqtrd isgrpd csca cnlmodlem3 cnlmod4 + addcl w3a addlid negcl addcomd negid eqtrd isgrpd csca cnlmodlem3 cnlmod4 cvsca cnfldbas cnfldadd cmulr cnfldmul cur cnfld1 crg cnring mulcl adddir - wa adddi mulass mulid2 islmodd mp2b ) FGHZAUAHZAUBHUCVSCDEGIACJZUDZFGAUEK + wa adddi mulass mullid islmodd mp2b ) FGHZAUAHZAUBHUCVSCDEGIACJZUDZFGAUEK ZLZVSWCGABUFMZNIAUGKZLZVSWFIABUHMZNWAGHZDJZGHZWAWJIOZGHVSWAWJUKUIWIWKEJZG HULZWLWMIOWAWJWMIOZIOLVSWAWJWMUJPVSSWIFWAIOWALVSWAUMPWIWBGHVSWAUNZPVSWIVM WBWAIOZWAWBIOZFWIWQWRLVSWIWBWAWPWISUOPWIWRFLVSWAUPPUQURVTCDEGIIQQTRGAWDVT @@ -336065,7 +336065,7 @@ vector spaces which are also normed vector spaces (that is, normed groups ( cnvc ccvs wcel wa c1 co cfv cabs cmul eqid syl adantr cin cneg csca cbs wceq simpl elin id cvsclm clmneg1 simplbiim simpr ncvsprp syl3anc absnegi cclm ax-1cn abs1 eqtri oveq1i cngp cr cnlm nvcnlm nlmngp sylbi nmcl sylan - recnd mulid2d eqtrid eqtrd ) EIJUAKZADKZLZMUBZABNCOZVPPOZACOZQNZVSVOVMVPE + recnd mullidd eqtrid eqtrd ) EIJUAKZADKZLZMUBZABNCOZVPPOZACOZQNZVSVOVMVPE UCOZUDOZKZVNVQVTUEVMVNUFVMWCVNVMEIKZEJKZWCEIJUGZWEEUPKWCWEEWEUHUIWAWBEWAR ZWBRZUJSUKTVMVNULVPABWAWBCDEFGHWGWHUMUNVOVTMVSQNVSVRMVSQVRMPOMMUQUOURUSUT VOVSVOVSVMEVAKZVNVSVBKVMWDWELWIWFWDWIWEWDEVCKWIEVDEVESTVFAECDFGVGVHVIVJVK @@ -336096,7 +336096,7 @@ vector spaces which are also normed vector spaces (that is, normed groups ( wcel ci c1 co cfv cnvc ccvs cin wa cmul cneg cngp cr elin nvcnlm nlmngp w3a cnlm syl adantr sylbi 3ad2ant1 cgrp clmod nvclmod lmodgrp simp2l cclm id cvsclm simplbiim simp3 simp2r clmvscl syl3anc grpcl nmcl syl2anc recnd - mulid2d cabs ax-icn absnegi absi eqtri oveq1i wceq simp1 wi cminusg sylan + mullidd cabs ax-icn absnegi absi eqtri oveq1i wceq simp1 wi cminusg sylan clmneg clmfgrp eqid grpinvcl eqeltrd imp 3adant2 ncvsprp clmvsdi syl13anc ex mulneg1i negeqi negneg1e1 clmvsass simpr anim12i 3adant3 clmvs1 oveq2d ixi 3eqtr3a cabl clmabl ablcom 3eqtrd fveq2d eqtr3d eqtr3id ) IUAUBUCPZAH @@ -336815,7 +336815,7 @@ of the norm of the sum of two orthogonal vectors (i.e., whose inner ( ( N ` ( A .+ B ) ) ^ 2 ) = ( ( ( N ` A ) ^ 2 ) + ( ( N ` B ) ^ 2 ) ) ) $= ( co cc0 wceq caddc wcel wa cfv c2 cexp cph2di adantr simpr wb cphorthcom ccph syl3anc biimpa oveq12d 00id eqtrdi oveq2d cphipcl addcld 3eqtrd cngp - cc addid1d cgrp cphngp ngpgrp 3syl grpcld nmsq syl2anc 3eqtr4d ) ABCEPZQR + cc addridd cgrp cphngp ngpgrp 3syl grpcld nmsq syl2anc 3eqtr4d ) ABCEPZQR ZUAZBCDPZVNEPZBBEPZCCEPZSPZVNFUBUCUDPZBFUBUCUDPZCFUBUCUDPZSPZVMVOVRVKCBEP ZSPZSPZVRQSPZVRAVOWERVLABCBCDEGHJIKMNONOUEUFVMWDQVRSVMWDQQSPQVMVKQWCQSAVL UGAVLWCQRZAHUJTZBGTZCGTZVLWGUHMNOBCEGHJIUIUKULUMUNUOUPAWFVRRVLAVRAVPVQAWH @@ -337321,7 +337321,7 @@ of the norm of the sum of two orthogonal vectors (i.e., whose inner recnd 3adant3 3adant2 addcld cphipcl 3com23 pnncand clmod cphlmod simpr simplr lmodvscl syl3anc oveq2d cneg cphassi ax-icn negicn mulcld adddid cphassir a1i c1 mulassd mulneg2i negeqi negneg1e1 3eqtri oveq1i eqtr3di - ixi mulid2d oveq1d addneg1mul 3eqtrd subcld ppncand 2timesd eqcomd 2cnd + ixi mullidd oveq1d addneg1mul 3eqtrd subcld ppncand 2timesd eqcomd 2cnd add4d adddird 2p2e4 3eqtr2d 4cn cc0 wne 4ne0 divcan3d 3eqtrrd ) JUAUBZU CGUBZUDZAKUBZBKUBZUEZABCTZIUFUGUHTZABHTZIUFUGUHTZUITZUCAUCBDTZCTZIUFUGU HTZAYSHTZIUFUGUHTZUITZUJTZUKTZULUMTABFTZBAFTZUKTZUUIUKTZUUGUUHUITZUUKUK @@ -337396,7 +337396,7 @@ of the norm of the sum of two orthogonal vectors (i.e., whose inner cngp adantr 3ad2ant1 simp2 clmod cphlmod 3anim1i lmodvscl 3adant2 syl3anc 3expa grpcl nmsq syl2anc cr reipcl recnd eqeltrd mulcld cclm cphclm simp3 clmneg1 addneg1mul clmvsubval eqcomd syl3an1 fveq2d oveq1d oveq2d cminusg - eqtrd simp1r clmvsneg grpsubval eqtr4d oveq12d anim1i clmvs1 mulid2d nnuz + eqtrd simp1r clmvsneg grpsubval eqtr4d oveq12d anim1i clmvs1 mullidd nnuz cn c3 df-4 oveq2 eqtrdi nnnn0 expcld adantl cmodscexp sylan df-3 cn0 df-2 i4 i3 i2 cz 1z exp1 ax-mp fsum1 sylancr 1nn jctil fsump1i simprd grpsubcl eqidd subcld addcomd subadd23d subdid 3eqtr2d negsubd mulneg1d 3eqtr4rd @@ -341428,7 +341428,7 @@ need not be complete (for instance if the given set is infinite cres clss wss eqid lssss syl sselda ovresd eqtrid cngp wceq ccph cphngp c2 ngpds syl3anc eqtrd oveq1d cr clt cinf c0 wne wrex minveclem1 simp1d w3a simp2d simp3d breq1 ralbidv rspcev syl2anc infrecl eqeltrid resqcld - 0red recnd addid1d breq12d clmod cphlmod lmodvsubcl nmcl nmge0 syl31anc + 0red recnd addridd breq12d clmod cphlmod lmodvsubcl nmcl nmge0 syl31anc infregelb mpbird breqtrrdi le2sqd breq2i bitrid 3bitr2d cmpt crn raleqi wb cvv fvex rgenw breq2 ralrnmptw ax-mp bitri bitrdi ) ABUFZMUGZUHZDYFE UIZVNUJUIZGVNUJUIZUKULUIZUMUNZDYFJUIZKUOZUEUFZUMUNZUEFUPZYODCUFJUIZKUOZ @@ -342960,7 +342960,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically ( cr wss covol cfv cc0 cxr ovolcl syl 3ad2ant1 cle wbr clt syl2anc caddc co wcel adantr cmnf wceq w3a cun simp1 simp2 unssd wn wb xrltnle mnfxr ovolge0 wa a1i ge0gtmnf simpr xrre2 syl32anc simpl3 eqeltrdi ovolun syl22anc oveq2d - 0re recnd addid1d eqtrd breqtrd ex sylbird pm2.18d ovolss sylancr xrletrid + 0re recnd addridd eqtrd breqtrd ex sylbird pm2.18d ovolss sylancr xrletrid ssun1 ) ACDZBCDZBEFZGUAZUBZABUCZEFZAEFZVSVTCDZWAHRZVSABCVOVPVRUDZVOVPVRUEZU FZVTIJZVOVPWBHRZVRAIZKZVSWAWBLMZVSWLUGZWBWANMZWLVSWIWDWNWMUHWKWHWBWAUIOVSWN WLVSWNULZWAWBVQPQZWBLWOVOWBCRZVPVQCRWAWPLMVSVOWNWESZWOTHRZWIWDTWBNMZWNWQWSW @@ -343580,7 +343580,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically sstrdi resubcld rexrd cuni c1st c2nd wrex wral 1nn a1i op1stg wb elicc2 w3a biimpa simp2d eqbrtrd simp3d op2ndg breqtrrd fveq2 opex fvmpt ax-mp iftrue eqtrdi fveq2d breq1d anbi12d syl12anc ralrimiva ovolficc ovollb2 - breq2d rspcev mpbird addid1 adantl nnuz eleqtri simpr eleqtrdi rge0ssre + breq2d rspcev mpbird addrid adantl nnuz eleqtri simpr eleqtrdi rge0ssre cc cuz ffvelcdm sselid c2 fveq2i ovolfsval eqtrd c0ex oveq12d eqtrid 1z recnd cfz ad2antrr elfzuz df-2 eleqtrrdi eluz2nn eqeq1 ifex wne eluz2b3 ifbid simprbi neneqd iffalsed op2nd op1st 0m0e0 seqid2 seq1i eqtr3d wfn @@ -344103,7 +344103,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically nulmbl $p |- ( ( A C_ RR /\ ( vol* ` A ) = 0 ) -> A e. dom vol ) $= ( vx cr wss covol cfv cc0 wceq wa cv wcel cin cdif caddc co cle wi mp3an1 wbr cpw wral cdm simpl elpwi inss2 ovolssnul adantr oveq1d difss ovolsscl - cvol adantl recnd addid2d eqtrd simprl ovolss sylancr eqbrtrd expr sylan2 + cvol adantl recnd addlidd eqtrd simprl ovolss sylancr eqbrtrd expr sylan2 ralrimiva ismbl2 sylanbrc ) ACDZAEFGHZIZVEBJZEFZCKZVHALZEFZVHAMZEFZNOZVIP SZQZBCTZUAAUKUBKVEVFUCVGVQBVRVHVRKVGVHCDZVQVHCUDVGVSVJVPVGVSVJIZIZVOVNVIP WAVOGVNNOVNWAVLGVNNVGVLGHZVTVKADVEVFWBVHAUEVKAUFRUGUHWAVNWAVNVTVNCKZVGVMV @@ -344508,7 +344508,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically mnfxr 0xr xrltletr mp3an12 mpani sylc wb xrrebnd cdif co simpl sseq1d cin simpll ineq1d fnfvelrn elssuni adantll sseqin2 eqtrd eqtr4d eqtrdi adantr adantrr c0 imbi12d 3ad2ant1 sylbird mpbird 3eqtr4g eqtr4di fveq1d oveq12d - difeq1 difid ovol0 recnd addid1d fveq2 breq12d pm5.74d pm5.74da w3a wdisj + difeq1 difid ovol0 recnd addridd fveq2 breq12d pm5.74d pm5.74da w3a wdisj expr simp2 simp3 voliunlem1 3exp1 vtoclg mpcom mpd mpand wn nltpnft pnfge rexr 3syl ex breq2 imbi2d syl5ibrcom pm2.61d ralrimiv breq1 ralrn syl2anc supxrleub xrletrid ) AFUAZUBZUCOZUWBPOZCUAZUDUEUFZAUWBUCUGZQUWCUWDUHABDEF @@ -345000,7 +345000,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically prssi ovolfi sylancr nulmbl syl2anc csn df-pr ineq2i indi eqtri simp1 ltnrd eliooord simpld nsyl disjsn sylibr simp2 simprd uneq12d un0 eqtrdi ioossicc eqtrid iccssre ovolicc resubcld eqeltrd ovolsscl mp3an2i eqeltrid eqtrd 0re - wn eqeltrdi volun syl32anc cxr rexr id prunioo syl3an fveq2d oveq2d addid1d + wn eqeltrdi volun syl32anc cxr rexr id prunioo syl3an fveq2d oveq2d addridd recnd 3eqtr3d eqtr3id ) ACDZBCDZABUAEZUBZABUCFZGHZXGIHZBAUDFZXGIUEZDZXIXHSA BUFZXGJUGZXFABUHFZIHZXOGHZXIXJXCXDXPXQSZXEXCXDUIXOXKDXRABUJXOJKLXFXGABUKZMZ IHZXIXSIHZNFZXPXIXFXLXSXKDZXGXSOZPSXICDYBCDYAYCSXLXFXMULXFXSCQZXSGHZRSZYDXC @@ -348101,7 +348101,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by wex crn cdif ccnv cima csu i1f1 adantr itg1val syl cpr i1f1lem simpli frn co wf ssdif difprsnss sstri mblss sselda eleq1w ifbid 1ex c0ex ifex fvmpt iftrue adantl wfn ffn fnfvelrn sylancr eqeltrrd ax-1ne0 eldifsn sylanblrc - snssd eqssd sumeq1d cc 1re simpri ad2antrr oveq2d simplr recnd mulid2d id + snssd eqssd sumeq1d cc 1re simpri ad2antrr oveq2d simplr recnd mullidd id eqeltrd sneq imaeq2d oveq12d sumsn ex exlimdv biimtrid pm2.61dne ) BGUBZH ZBGIZJHZUAZCKIZYGLZBMBMLZYKUCYIYLJNOZUDZKIZMGIZYJYGMUEIZNYPYOUFMYEHYPYQLU GMUHPUIUJYLCYNKYLAJAUKZBHZQNULZUMAJNUMCYNYLAJYTNYLYSQNYLYSYRMHYRUNBMYRUPU @@ -349564,7 +349564,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( B x. ( vol ` A ) ) ) $= ( cvol wcel cfv cr cc0 co cif cmpt citg1 cmul cle wbr wceq c1 cvv a1i 0re cdm cpnf cico w3a cv citg2 c0p cofr csn cxp cof reex simpl3 1re fconstmpt - ifcli eqidd offval2 ovif2 simp3 elrege0 sylib simpld recnd mulid1d mul01d + ifcli eqidd offval2 ovif2 simp3 elrege0 sylib simpld recnd mulridd mul01d wa ifeq12d eqtrid mpteq2dv eqid i1f1 3adant3 i1fmulc eqeltrrd wral simprd eqtrd 0le0 ifboth sylancl ralrimivw cc wss ax-resscn wfn adantr ralrimiva breq2 ifcl fnmpt syl 0pledm ofrfval2 bitrd mpbird itg2itg1 syl2anc fveq2d @@ -349827,7 +349827,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cof wceq cle cpnf cico crp elrege0 sylib simpld anim1i sylibr itg2mulclem vz elrp cdiv c1 ge0mulcl adantl fconst6g syl reex a1i inidm cicc icossicc off wss sylancl remulcld itg2lecl syl3anc rpreccld cmpt feqmptd ax-resscn - cv fss rge0ssre sstri ffvelcdmda mulid2d mpteq2dva eqtr4d ofc12 fconstmpt + cv fss rge0ssre sstri ffvelcdmda mullidd mpteq2dva eqtr4d ofc12 fconstmpt 1red eqtrdi recnd rpne0d recid2d mpteq2dv eqtrd offval2 rpcnd w3a caofass mulass 3eqtr2d fveq2d divrec2d 3brtr4d lemuldiv2d mpbird cxr itg2cl rexrd wb xrletri3 syl2anc mpbir2and 0re simplr oveq1d mul02 eqtr3d itg20 mul02d @@ -349874,13 +349874,13 @@ or are almost disjoint (the interiors are disjoint). (Contributed by wss inss1 mblss sstrid covol wceq cof cvv reex a1i fvex c0ex ifex offval2 eqidd i1fadd eqeltrrd cdif wf i1ff eldifi ffvelcdm syl2an leidd iftrue wn adantl eldifn elin sylnib imnan sylibr imp iffalse oveq12d cc recnd eqtrd - addid1d breqtrrd ad2antrr bitrdi ffnd adantlr ifclda fmptd r19.21bi eldif + addridd breqtrrd ad2antrr bitrdi ffnd adantlr ifclda fmptd r19.21bi eldif 0e0iccpnf nfcv cid fveq2d breqtrd syldan pm2.61dan eleq1w ifbieq1d sselda sseqtrrid nfv nfan ofrfval2 iftrued 3brtr4d 0le0 ralrimi wb mpbird itg2ub a1d syl3anc wo cun eleq2d elun notbid ioran biimpar simprr wfn cpnf inidm cicc ofrfval mpbid sylan2 nfmpt1 nffv nfeq1 fveqeq2 fvmpt2i 0cn fvi ax-mp nfcxfr eqtrdi sylan9eq sylbi vtoclgaf sylbir sylan anassrs oveq1d sylancl - ifcl addid2d fveq2 ovex fvmpt itg1lea itg1add eqtr3d ssun1 feqmptd biimpd + ifcl addlidd fveq2 ovex fvmpt itg1lea itg1add eqtr3d ssun1 feqmptd biimpd 0re nfbr impr ssun2 le2addd letrd expr ralrimiva cxr rexrd itg2leub ) AIU CUDGUCUDZHUCUDZUEUFZUGUHZUAUIZIUGUJZUHZUWTUKUDZUWRUGUHZUQZUAUKULZUMZAUXEU AUXFAUWTUXFUNZUXBUXDAUXHUXBURZURZUXCBTBUIZCUNZUXKUWTUDZUOUPZUSZUKUDZBTUXK @@ -349946,8 +349946,8 @@ or are almost disjoint (the interiors are disjoint). (Contributed by wfn wb sseqtrrid sselda syldan simpr dffn5 sylib ofrfval2 r19.21bi adantl mpbid breqtrd recnd syl2anc breqtrrd xrletrd fveq2 expr resubcld itg2leub ralrimiva mpbird cun ssun2 eldifn elin sylnib imnan imp iffalsed leadd2dd - sylibr addid1d simprlr ad2antrr iftrued fvmpt2 3eqtr4d leadd1dd ad3antrrr - ssun1 iftrue iffalse addid2d eleq2d wo elun biorf bitr4id bitrd pm2.61dan + sylibr addridd simprlr ad2antrr iftrued fvmpt2 3eqtr4d leadd1dd ad3antrrr + ssun1 iftrue iffalse addlidd eleq2d wo elun biorf bitr4id bitrd pm2.61dan ifbid eqtrd eqbrtrd ralrimi nffv breq12d cbvralw itg2uba eqbrtrrd adantrr ex leaddsub2d anassrs lesubd leaddsub xrletrid ) AIUDUEZGUDUEZHUDUEZUFUGZ ATUHUIUJUGZIUKZUWFULUMABTBUNZFUMZEUHUOZUWJIAUWLTUMZUQZUWMEUHUWJAUWMEUWJUM @@ -350039,11 +350039,11 @@ or are almost disjoint (the interiors are disjoint). (Contributed by bitr4i mpbird mpteq2dv sylibr syl2an2r fveq2d 1zzd cvol ccnv cima simpr cdif cin readdcl cico w3a 1xr elioo2 mp2an simp1d renegcld off elpreima inidm mpbirand elioomnf biantrurd bitr4id cmin ofc1 cc mulneg1d negsubd - ax-mp ofval ltsubaddd addid2d notbid eldif baib 3bitr4d rabbi2dva rembl + ax-mp ofval ltsubaddd addlidd notbid eldif baib 3bitr4d rabbi2dva rembl lenltd i1fmbf mbfadd syl2anc cmmbl inmbl sylancr eqeltrrd fmptd fvoveq1 mbfima cofr rspccva syl2an ofrfval wi letr mpan2d ss2rabdv 3sstr4d ciun cuni an32s dmmptd eleqtrrid dm0rn0 necon3bii breq1 ralrn bitrdi rexbidv - 1nn ne0d suprcld simp3d 1red simprr ltmul1 syl112anc mulid2d rneqd ltso + 1nn ne0d suprcld simp3d 1red simprr ltmul1 syl112anc mullidd rneqd ltso supeq1d supex ltletrd suprlub syl31anc rexbiia simplr ffvelcdmd elrege0 breq2 rexrn simpld adantlrr ltle reximdva anassrs simplrr simp2d lemul2 mpd ne0ii mul01d simprd r19.2z ltlecasei rabid2 iunrab eqtr4di iuneq2dv @@ -350197,7 +350197,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by fnfvelrn syl2anc breqtrrdi xrletrd ltaddrpd lelttrd gt0ne0d c2 redivcld div23d cif remulcld halfre ifcl max2 sylancr wb wn rexrd xrltnle mpbird lemul1 syl112anc mpbid crab cmbf adantlr cofr wrex cioo halfgt0 ltletrd - max1 mulid1d breqtrrd 1red ltdivmul breq1 halflt1 ifboth w3a 1xr elioo2 + max1 mulridd breqtrrd 1red ltdivmul breq1 halflt1 ifboth w3a 1xr elioo2 syl3anbrc oveq2d breq12d cbvrabv mpteq2i itg2monolem1 eqbrtrd ledivmul2 mp2an letrd mulgt0d lemul2 alrple ) ADUBUCZEUDRZUUSEUAUEZUFULZUDRZUAUGU HZAUVCUAUGAUVAUGSZUMZUVCEUUSUIULZEUVBUIULUDRZUVFUVGUVBUJULZEUDRZUVHUVFU @@ -351187,7 +351187,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by oveq2 i3 itgvallem oveq12d cc ax-icn a1i expcl sylan cz nn0z eqidd iblitg recnd sylan2 mulcld c1 df-2 i2 1e0p1 exp1 ax-mp cibl cmbf iblmbf mbfmptcl syl div1d fveq2d ibllem mpteq2dv eqtr4id oveq2d w3a iblcnlem mpbid simp2d - 0z simpld mulid2d eqtr3d eqeltrd exp0 fsum1 sylancr eqtrd jctil cim imval + 0z simpld mullidd eqtr3d eqeltrd exp0 fsum1 sylancr eqtrd jctil cim imval eqtr2id fsump1i renegd ax-1cn negnegi oveq2i negcld eqtrid negcli neg1ne0 0nn0 wne div2neg mp3an23 simprd mulm1d simp3d mulcl addcld negsubd 3eqtrd addsubd imnegd eqcomi ine0 negne0i 3eqtr3d mulneg12 subcld addassd adddid @@ -351296,7 +351296,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( cr wcel cc0 cfv cle wbr wa cif cmpt citg2 cneg cmin co eqid citg cv cre ci cmul caddc itgcnlem rered ibllem mpteq2dv fveq2d negeqd oveq12d reim0d itgvallem3 neg0 eqtrdi 0m0e0 oveq2d it0e0 cmbf cibl iblrelem mpbid simp2d - cim w3a simp3d resubcld recnd addid1d 3eqtrd ) ABCDUABGBUBCHZIDUCJZKLMVNI + cim w3a simp3d resubcld recnd addridd 3eqtrd ) ABCDUABGBUBCHZIDUCJZKLMVNI NZOZPJZBGVMIVNQZKLMVRINZOZPJZRSZUDBGVMIDVFJZKLMWCINOPJZBGVMIWCQZKLMWEINOP JZRSZUESZUFSBGVMIDKLMDINZOZPJZBGVMIDQZKLMWLINZOZPJZRSZIUFSWPABCDVQWAWDWFG VQTWATWDTWFTEFUGAWBWPWHIUFAVQWKWAWORAVPWJPABGVOWIABCVNDAVMMZDEUHZUIUJUKAV @@ -352117,7 +352117,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by wn cvv reex recld recnd elrege0 0e0icopnf imcld eqidd offval2 wceq iftrue oveq12d eqtr4d iffalse 3eqtr4a pm2.61i mpteq2i eqtr2di fveq2d iblcn mpbid 00id eqid simpld iblabslem simprd itg2add eqtrd eqeltrd cofr addge0d wral - readdcld ci cmul ax-icn mulcl sylancr abstrid replimd absmul absi mulid2d + readdcld ci cmul ax-icn mulcl sylancr abstrid replimd absmul absi mullidd c1 oveq1i eqtrid eqtr2d oveq2d 3brtr4d adantl pm2.61d1 ralrimivw ofrfval2 ex 0le0 mpbird itg2le itg2lecl iblpos mpbir2and ) ABCDHIZJZUAKUUBLKBMBUBZ CKZUUANUCZJZOIZMKZAHBCDJZUEZUUBLABCDPMHPMHUDAUFUGABCDEAUUIUAKZUUILKZGUUIU @@ -352418,8 +352418,8 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) ) $= ( cioo co caddc wceq cr wcel cc0 c0 clt wbr citg wn cle wo cicc wb elicc2 w3a syl2anc mpbid simp2d simp1d leloed ord cc cxr wss rexrd iooss1 sselda - cv syldan itgcl addid2d eqcomd oveq1 itgeq1 syl iooid eqtrdi itg0 eqeq12d - oveq1d syl5ibrcom simp3d iooss2 addid1d oveq2 eqtr3id oveq2d syl5ibcom wa + cv syldan itgcl addlidd eqcomd oveq1 itgeq1 syl iooid eqtrdi itg0 eqeq12d + oveq1d syl5ibrcom simp3d iooss2 addridd oveq2 eqtr3id oveq2d syl5ibcom wa csn cun cin covol cfv indir jca adantr leidd ioodisj syl21anc incom ltnrd syld eliooord simpld nsyl disjsn sylibr eqtrid uneq12d un0 fveq2d ioojoin ovol0 3jca sylan adantlr cmpt cibl ssun1 ioossre snssd unssd cdif difeq1i @@ -352509,7 +352509,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by E. x e. RR A. y e. dom F ( abs ` ( F ` y ) ) <_ x ) -> F e. L^1 ) $= ( vz cmbf wcel cdm cvol cfv cr cv cabs cle wbr wral c1 cmul cibl 3ad2ant1 cc wrex w3a csn cxp cof co mbfdm wf mbff ffnd wfn 1cnd fnconstg syl eqidd - wceq 1ex fvconst2 adantl ffvelcdmda mulid1d offveq simp2 iblconst syl3anc + wceq 1ex fvconst2 adantl ffvelcdmda mulridd offveq simp2 iblconst syl3anc wa bddmulibl syld3an2 eqeltrrd ) CEFZCGZHIJFZBKCILIAKMNBVKOAJUAZUBZCVKPUC UDZQUEUFZCRVNDVKDKZCIZPQCVOCHGZVJVLVKVSFZVMCUGSZVNVKTCVJVLVKTCUHVMCUISZUJ ZVNPTFZVOVKUKVNULZVKPTUMUNWCVNVQVKFZVFZVRUOWFVQVOIPUPVNVKPVQUQURUSWGVRVNV @@ -352867,8 +352867,8 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( S_ [ A -> B ] D _d x + S_ [ B -> C ] D _d x ) ) $= ( caddc co wa cc0 cdit wceq cr wcel cle wbr cicc w3a elicc2 syl2anc mpbid wb simp1d adantr ad2antrr biid ditgsplitlem adantlr oveq1d ditgcl addassd - ditgswap oveq2d negidd eqtrd addid1d 3eqtrd eqtr2d adantllr lecasei ancom - cneg addid2d 3eqtr3d eqtr3d ) ABCEFUAZBCDFUAZBDEFUAZQRZUBZCDACUCUDZHCUEUF + ditgswap oveq2d negidd eqtrd addridd 3eqtrd eqtr2d adantllr lecasei ancom + cneg addlidd 3eqtr3d eqtr3d ) ABCEFUAZBCDFUAZBDEFUAZQRZUBZCDACUCUDZHCUEUF ZCIUEUFZACHIUGRZUDZWAWBWCUHZLAHUCUDZIUCUDZWEWFULJKHICUIUJUKUMZADUCUDZHDUE UFZDIUEUFZADWDUDZWJWKWLUHZMAWGWHWMWNULJKHIDUIUJUKUMZACDUEUFZSZVTCEAWAWPWI UNAEUCUDZWPAWRHEUEUFZEIUEUFZAEWDUDZWRWSWTUHZNAWGWHXAXBULJKHIEUIUJUKUMZUNW @@ -354121,7 +354121,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by difss ax-mp oveq1i sseqtri subidd subcn ccncf cncfmptid syl3anc cncfmpt2f ccn cncfmptc oveq1 cnmptlimc eqeltrrd sselid cop mulcn dvcl 0cn toponunii opelxpi cncnpi limccnp2 mul01d simpr wne eldifsni adantl subne0d divcan1d - mpteq2dva oveq1d 3eltr3d fmpttd limcdif eqtrdi eleqtrrd addid2d npcand wb + mpteq2dva oveq1d 3eltr3d fmpttd limcdif eqtrdi eleqtrrd addlidd npcand wb eqidd addcn feqmptd eqtr4d cnplimc mpbir2and ex exlimdv eldmg ibi impel ) CJKZAJDUFZACKZUBZBUAUGZCDUCLZUDZUAUEZDBEFUHLMNZBUUTUIZNZUURUVAUVCUAUURUVA UVCUURUVAUJZUVCUUPBDMZDBOLZNZUUOUUPUUQUVAUKZUVFPUVGQLIAIUGZDMZUVGRLZUVGQL @@ -354261,7 +354261,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( cc wcel co wa cn0 cfv caddc wceq wi cc0 fveq2 fveq2d eqeq12d imbi2d wss oveq2 vn vk cr cpr cpm cdvn cv c1 recnprss ad2antrr cdm cvv ssidd wf cnex wb elpm2g mpan simplbda a1i simpl syl22anc adantr dvnff ffvelcdmda sseldd - pmss12g dvn0 syl2anc nn0cn adantl addid1d eqtr4d simpr dvnp1 syl3anc 1cnd + pmss12g dvn0 syl2anc nn0cn adantl addridd eqtr4d simpr dvnp1 syl3anc 1cnd cdv addassd simpllr nn0addcl adantll eqtr3d syl5ibr expcom a2d com12 impr nn0ind ) AUCEUDZFZBEAUEGZFZHZCIFZDIFZDACABUFGZJZUFGZJZCDKGZWQJZLZWPWNWOHZ XCXDUAUGZWSJZCXEKGZWQJZLZMXDNWSJZCNKGZWQJZLZMXDUBUGZWSJZCXNKGZWQJZLZMXDXN @@ -354665,7 +354665,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by wf cdm cr cpr recnprss dvbss sseldd cres dvconst dmeqd c0ex fconst fdmi wss eqtrdi sseqtrrd dvres3 xpssres oveq2d reseq1d eqtrd 3eqtr3d fconst2 syl22anc sylibr fdmd eleqtrrd dvmul fveq1d fvconst2 ffvelcdmd fvconst2g - oveq1d mul02d syl2anc dvfg mulcomd oveq12d mulcld addid2d 3eqtrd ) ACDD + oveq1d mul02d syl2anc dvfg mulcomd oveq12d mulcld addlidd 3eqtrd ) ACDD BUAZLZEMUBNONPCDWIONZPZCEPZMNZCDEONZPZCWIPZMNZUCNQBWOMNZUCNWRACDWIEDFAB RUDZDRWIUGIDBRUESADUFHJGACDWJUHAFDCJAWNUHZFCAFDEADUIRUJUDZDRUTZGDUKSZHJ ULKUMZUMZADQUAZWJAWJDXFLZTDXFWJUGADRWHLZDUNZONZRXHONZDUNZWJXGAXARRXHUGZ @@ -354690,7 +354690,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ctopon ntrss2 syl2anc dvbssntr eqssd reseq12d 3eqtr4d feq1d mpbiri dvmulf cuni fdmd cin cv sseqin2 sylib mpteq1d cvv ssexd fvconst2g sylan wa eqidd ffnd offval inidm dvfg feq2d mpbid 0cnd ovexd oveq1d mul02 adantl caofid2 - fvexd adantr feqmptd offval2 ffvelcdmda mulcld addid2d mulcomd mpteq2dva + fvexd adantr feqmptd offval2 ffvelcdmda mulcld addlidd mulcomd mpteq2dva ) ACEBUAZUBZDKUCZLZMLCYOMLZDYPLZCDMLZYOYPLZUDUCLZCCYNUBZDYPLZMLUUCYTYPLZA CYODEFAEYNNYOABNUEZEYNYOOHEBNUGPZABNHUFZUHGAEQUAZYRAEUUIYROEUUIEUUIUBZOEQ UIUJAEUUIYRUUJACUUCEUTZMLZCUUCMLZEUKULRZCUMLZUNRRZUTZYRUUJACNUOZCNUUCOCCU @@ -354939,7 +354939,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( cc cexp co cmpt cdv c1 cmin cmul wceq caddc mpteq2dv oveq2d id wcel a1i oveq1 cvv wf vn vk cv oveq2 oveq12d eqeq12d cid cres exp1 mpteq2ia eqtr4i mptresid oveq2i csn cxp cc0 1m1e0 exp0 eqtrid 1t1e1 eqtrdi fconstmpt dvid - cn cof nncn adantr ax-1cn pncan sylancl cn0 nnnn0 syl2anr adddird mulid2d + cn cof nncn adantr ax-1cn pncan sylancl cn0 nnnn0 syl2anr adddird mullidd wa expcl 3eqtrd mpteq2dva cnex mulcld nnm1nn0 simpr eqidd offval2 mulassd expm1t ancoms eqtr4d eqtrd eqtri oveq1d eqcomd sylan9eq cr cpr cnelprrecn fmpttd wf1o f1oi f1of mp1i cdm dmeqd fdmd fconst feq1i mpbir dvmulf expp1 @@ -355182,7 +355182,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by crest dvmptc cdm dmeqd wral ralrimiva dmmptg syl eqtrd dvbsss eqsstrrdi wceq eqid cnt ctop cuni wss ctopon cnfldtopon cr cpr recnprss resttopon sylancr topontop toponuni sseqtrd ntrss2 fmpttd dvbssntr eqsstrrd eqssd - syl2anc dvmptres2 dvmptmul mul02d oveq1d dvmptcl mulcld addid2d mulcomd + syl2anc dvmptres2 dvmptmul mul02d oveq1d dvmptcl mulcld addlidd mulcomd 3eqtrd mpteq2dva ) AFBHECNOPUAOBHQCNOZDENOZUBOZPBHEDNOZPABEQCDFRGHIAERS ZBUCZHSZMUDZAWRUEZUFABEQFUGUHTZFUIOZXARFHHIAWPWQFSZMUDAXCUEUFABEFIMUJAH FBHCPZUAOZUKZFAXFBHDPZUKZHAXEXGLULADGSZBHUMXHHUTAXIBHKUNBHDGUOUPUQZFXDU @@ -355368,7 +355368,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( cdiv co cmul cc0 cmpt cdv c1 c2 cexp cneg caddc cmin cv wcel wa eldifad cc csn wne cdif eldifsn sylib simprd divrecd mpteq2dva oveq2d reccld 1cnd mulcld sqcld neneqd sqeq0 syl mtbird neqned divcld negcld dvrecg dvmptmul - wceq dvmptcl negsubd div12d mulid2d sqvald divcan5rd eqtr2d 3eqtrd oveq1d + wceq dvmptcl negsubd div12d mullidd sqvald divcan5rd eqtr2d 3eqtrd oveq1d wb negeqd mulneg1d div23d eqcomd oveq12d divsubdird 3eqtr4d ) AGBICEQRZUA ZUBRGBICUCEQRZSRZUAZUBRBIDWPSRZUCFSRZEUDUERZQRZUFZCSRZUGRZUABIDESRZFCSRZU HRXAQRZUAAWOWRGUBABIWNWQABUIIUJUKZCEKXIEUMTUNZNULZXIEUMUJZETUOZXIEUMXJUPU @@ -355610,11 +355610,11 @@ or are almost disjoint (the interiors are disjoint). (Contributed by c3 abscl ad2antrr subcl sylancl remulcld resqcld cn 3re 4nn nndivre mp2an cexp remulcl cfa cuz cn0 csu divcan2d divsubdird dividd oveq2d eqtrd 0cnd 1e0p1 eqtr2di efsep oveq2i a1i oveq1i 1re breqtrd wtru wf mptru mpbir2an - subsub4d addcl sylancr 2nn0 eftlcl df-2 1nn0 efval2 nn0uz sumeq1i addid2d + subsub4d addcl sylancr 2nn0 eftlcl df-2 1nn0 efval2 nn0uz sumeq1i addlidd 0nn0 eqtr2d eft0val eqtr4di exp1 fac1 eqtrdi div1 mvrladdd 3eqtr3d eqtr3d absmuld 2nn simpr ltled eftlub eqbrtrrd df-3 fac2 2t2e4 oveq12i breqtrrdi - eqtr2i sqge0d 3lt4 4cn mulid1i breqtrri 4pos pm3.2i ltdivmul mp3an ltleii - mpbir lemul2ad recnd sqcld mulid1d letrd sqvald absgt0 mpbid elrpd mpbird + eqtr2i sqge0d 3lt4 4cn mulridi breqtrri 4pos pm3.2i ltdivmul mp3an ltleii + mpbir lemul2ad recnd sqcld mulridd letrd sqvald absgt0 mpbid elrpd mpbird 4re lemul2d ad2ant2l lelttrd eqbrtrd sylbid adantld sylan2b brimralrspcev ex ralrimiva syl2anc rgen eldifi eldifsni fmpti difssd ellimc3 cnfldtopon toponrestid ef0 mpteq2ia ssidd eff eldv ) CDEUEUFFGZCEUGUHHZUIHHZIZDUAECU @@ -355671,7 +355671,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by 1cnd cop c0ex snid opelxpi mpan2 dvconst eleqtrrd df-br sylibr ccom oveq1 cofmpt eqid ovex fvmpt subid eqtrd dveflem eqbrtrdi 1ex cr cpr cnelprrecn simpr dvmptid simpl id dvmptc dvmptsub 1m0e1 mpteq2i eqtr4i eqtrdi dvcobr - breqtrdi breqdi dvmulbr ffvelcdmd mul02d fvconst2 mulid2d oveq12d addid2d + breqtrdi breqdi dvmulbr ffvelcdmd mul02d fvconst2 mullidd oveq12d addlidd 1t1e1 fvex breqtrd breldm ssriv eqssi feq2i mpbi wfun ax-mp funbrfv mpsyl vex ffun mpteq2ia eqtr4d mptru ) CDEFZDGHYQACAUAZDIZJZDHYQACYRYQIZJYTHACC YQCCYQKZHYQUBZCYQKZUUBDUCZUUCCCYQUUCCCDUDACUUCYRCLZYRYSYQUEZYRUUCLUUFCCYS @@ -355699,7 +355699,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( CC _D cos ) = ( x e. CC |-> -u ( sin ` x ) ) ) $= ( vy cc cdv co ccos wceq cneg cmpt wtru ci cmul ce cdiv caddc c2 wcel a1i ax-icn negicn csin cv cfv wa cpr cnelprrecn simpr mulcld efcl syl cc0 wne - cr ine0 divcld mulcl sylancr negcld addcld adantl c1 1cnd dvmptid mulid1i + cr ine0 divcld mulcl sylancr negcld addcld adantl c1 1cnd dvmptid mulridi dvmptcmul mpteq2i eqtrdi wf eff feqmptd oveq2d dvef eqtrid eqtr3d dvmptco fveq2 dvmptdivc divcan4d eqtrd sylancl dvmptneg divneg2d negne0i dvmptadd mpteq2dva 2cnd 2ne0 cmin df-sin subcld divdiv1d mulcomi oveq2i divsubdird @@ -356113,7 +356113,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by a1i recnd dvmptid tgioo2 cnt iccntr dvmptres2 eqtr3id dmeqd dmmpti eqtrdi eqid 1ex cmvth cxr cle wbr rexrd ubicc2 syl3anc fvresi syl lbicc2 oveq12d ltled adantr oveq1d fveq1d eqidd sylan9eq oveq2d cncff ffvelcdmd resubcld - fvmpt3i wf mulid1d eqtrd eqeq12d dvf feq2d ffvelcdmda cc0 wne clt posdifd + fvmpt3i wf mulridd eqtrd eqeq12d dvf feq2d ffvelcdmda cc0 wne clt posdifd mpbii mpbid gt0ne0d divmuld bitr4d eqcom 3bitr4g rexbidva ) ADELZCELZMNZB UHZOUACDUBNZUCZUDNZLZPNZDYJLZCYJLZMNZYHOEUDNZLZPNZQZBCDUENZUFYRYGDCMNZUGN ZQZBUUAUFABCDEYJFGHIAYJKYIKUHZUIZYIOURNZKYIUJZAYIOUKZORUKUUFUUGSACOSZDOSZ @@ -356264,7 +356264,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ccom iirev adantl oveq12d cxr ad2antrr blcvx syl23anc wceq fvres mpteq2ia feqmptd eqtrdi fveq2 eqid a1i cncfmptc mp3an23 mulcncf cncfmpt2f syl22anc wf syl cnt sylancr reseq2d eqtrd dmeqd sylib eqtrid cioo sylancl cvv cneg - sylan2 mulcld recnd dvmptres dvmptcmul mulid1d mpteq2dv negex 1cnd oveq2d + sylan2 mulcld recnd dvmptres dvmptcmul mulridd mpteq2dv negex 1cnd oveq2d dvfcn wral abscld fveq2d 2fveq3 breq1d ralrimiva weq nfcv nffv oveq2 fvex fvmpt ax-mp mul01d 1m0e1 eqeltrd eqidd fssresd fmptco fmpttd ccnfld ctopn wb ctx ccn addcn ssid cncfmptid mp2an subcn ax-1cn cncfcdm syl2anc mpbird @@ -356276,7 +356276,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by dvmptadd feq2d mpbii feq1d mpbid 3eqtr3d dvmptco ovex rgenw mp1i remulcld dmmptg fveq1d sylan9eq ffvelcdm absmuld absge0d cbvralvw rspcdva lemul1ad fvmpt2 mpan2 eqbrtrd nfv nfmpt1 nfov cbvralw r19.21bi dvlip mpanr12 1m1e0 - nfbr addid1d addid2d fveq2i abs1 eqtri oveq2i 3brtr3d ) AIDUCZJDUCZUDZUDZ + nfbr addridd addlidd fveq2i abs1 eqtri oveq2i 3brtr3d ) AIDUCZJDUCZUDZUDZ UESUFUEUGUHZISUIZUJUHZJUEVUNUKUHZUJUHZULUHZFUMZUNZUMZUFVUTUMZUKUHZUOUMZGI JUKUHZUOUMZUJUHZUEUFUKUHZUOUMZUJUHZIFUMZJFUMZUKUHZUOUMVVGUPVULUEVUMUCZUFV UMUCZVVDVVJUPUQURUSVULUAUFUEVUTVVGUEUFVULUTZVULVAVULFDVBZSVUMVURUNZWAVUTV @@ -356776,7 +356776,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( vx vy cmnf cc0 co clt cv wcel wa wbr cr mpbid cioo ccnv gtso cicc cfv caddc cmin cmul cdiv dvgt0lem1 eliooord syl simprd wb wf ccncf ad2antrr cncff simplrr ffvelcdmd simplrl resubcld 0red wss iccssre syl2anc simpr - sseldd posdifd ltdivmul syl112anc mul01d breqtrd ltsubaddd addid2d fvex + sseldd posdifd ltdivmul syl112anc mul01d breqtrd ltsubaddd addlidd fvex recnd brcnv sylibr dvgt0lem2 ) AIJBCKLUAMZDNUBZEFGHUCAIOZBCUDMZPZJOZWDP ZQZQZWCWFNRZQZWFDUEZWCDUEZNRWMWLWBRWKWLLWMUFMZWMNWKWLWMUGMZLNRWLWNNRWKW OWFWCUGMZLUHMZLNWKWOWPUIMZLNRZWOWQNRZWKKWRNRZWSWKWRWAPXAWSQABCWADWCWFEF @@ -356884,7 +356884,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cdm dvfre eleqtrrd ffvelcdmd iccssre syl2anc sseldd adantr a1i remulcld sselda resubcld fmptd iccssioo2 fssresd cc wb ax-resscn fss cmpt oveq2i cpr reelprrecn recnd feq2d mpbid feqmptd oveq2d eqtr3d crn ccnfld ctopn - ctg remulcl sylan c1 1cnd dvmptid dvmptcmul mulid1d mpteq2dv eqtrd eqid + ctg remulcl sylan c1 1cnd dvmptid dvmptcmul mulridd mpteq2dv eqtrd eqid cvv ovex mpbird simpld fvres wi ad2antrr cc0 fveq2 oveq1d simprl simprr weq sylib exp32 wo clt eliooord cxr w3a elioo2 mpbir3and eqeltrd simprd c0 rexrd xrltled raleqdv fveq2d letri3d mpbir2and tgioo2 iooretop dmeqd @@ -357109,7 +357109,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by eqtr2i txtopon resmpt sseqtri cncfmptc eqidd cnmptlimc feqresmpt oveq1d toponrestid sseqtrid ccn cop ccnp 0cn opelxpi toponunii cncnpi limccnp2 subcn eqeltrrd divcn opelxpd resttopon fmpttd ellimc2 notrab ccld sylan - abssub rabbidva blcld cldopn eqeltrrid sylbird mt3d c1 mulid2d ltmul1dd + abssub rabbidva blcld cldopn eqeltrrid sylbird mt3d c1 mullidd ltmul1dd 1red 1lt2 eqbrtrrd lelttrd ) ALJUPZLKUPZUQURZFUSURZUTUPZIVAIVBURZAVYSAV YRFAVYPVYQAVYPADEVCURZVDLJQADGVCURZWUBLAEVEVFZGEVGVIWUCWUBVJZOUHDGEVHVK ZUIVLZVMZWBZAVYQAWUBVDLKRWUGVMZWBZAVNKVOZVFZVPVYQVNVQZUCAWUMVYQVNAVYQWU @@ -357739,7 +357739,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( ( T x. ( F ` A ) ) + ( ( 1 - T ) x. ( F ` B ) ) ) ) $= ( cr co cmin clt wbr wcel vx vy cv cicc cres cdv cdiv wceq cioo wrex cmul cfv caddc cc0 elioore syl remulcld resubcl sylancr readdcld eqeltrid 1cnd - c1 1re recnd subdird mulid2d oveq1d eqtrd wb eliooord simprd posdif mpbid + c1 1re recnd subdird mullidd oveq1d eqtrd wb eliooord simprd posdif mpbid wa sylancl ltmul2 syl112anc eqbrtrrd ltsubadd2d breqtrrdi wss ccncf leidd cle simpld ltsub2dd eqbrtrd ltaddsub2d ltled syl22anc rescncf sylc cdm cc iccss wf ax-resscn iccssre syl2anc dvres iccntr reseq2d dmeqd dmres rexrd @@ -357833,7 +357833,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by fzval2 sselda cmpt wf ccncf cncff eqid sylibr nfcsb1v nfel1 weq csbeq1a fmpt eleq1d rspc syldan ralrimiva fzofzp1 csbeq1 rspccva syl2an elfzofz mpan9 resubcld cmul cc elfzoelz adantl zred recnd ax-1cn pncan2 sylancl - oveq2d peano2re subdid mulid1d 3eqtr3d ccnfld ctopn wss adantr elfzole1 + oveq2d peano2re subdid mulridd 3eqtr3d ccnfld ctopn wss adantr elfzole1 mulcn wbr elfzle2 iccss syl22anc iccssre sstrd ax-resscn sstrdi syl3anc cioo cdv cxr rexrd 1cnd ioossre dvmptres eqtrd cbvmpt sseli cdm adantlr nfcv oveq2 cvv eqeq2 biimpa csbied cncfmptc cncfmptid cncfmpt2ss iooss1 @@ -357950,7 +357950,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ralrimiva fsumsub eqeq2 biimpa csbied oveq2d eqtrd fveq2d abscld fsumrecl cvv vex cmul wbr cxr adantl rexrd cr syl3anc wss adantr ax-resscn sylancl zred cdv cdm cioo sstrd mulcld adantlr sylan2 ovex 1cnd iooretop dvmptres - sstrid mulid1d eqtrdi nfcv nfov nffv oveq2 oveq12d fvmptf recnd telfsumo2 + sstrid mulridd eqtrdi nfcv nfov nffv oveq2 oveq12d fvmptf recnd telfsumo2 fsumcl fsumabs elfzoelz peano2re lep1d ubicc2 cres elfzole1 elfzle2 iccss lbicc2 syl22anc resmptd ccnfld ctopn ctx ccn iccssre sstrdi ssid cncfmptc cncfmptid mulcncf cncfmpt2f rescncf sylc eqeltrrd crn ctg r19.21bi tgioo2 @@ -358082,7 +358082,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by wb flbi baibd syl21anc biimpar oveq2d oveq1d sumeq1d oveq12d adantr simpr peano2zd eqeltrd flid eqtrd recnd subcld 1cnd cc wral wss a1i dvmptrecl ralrimiva nfcsb1v nfel1 csbeq1a eleq1d rspc sylc subsub4d - cv subdird mulid2d 3eqtr3d cuz zred peano2rem 1red lesubaddd mpbird + cv subdird mullidd 3eqtr3d cuz zred peano2rem 1red lesubaddd mpbird peano2zm flge syl2anc mpbid eluz2 syl3anbrc elfzuz eleqtrrdi syl2an rspccva cvv sylancl 3eqtr4d nfcv nfov eqvisset csbied eqcomd fsumm1 eqeq2 ax-1cn pncan csbeq1d fzfid fsumcl mulcld nppcan3d wo peano2re @@ -358131,7 +358131,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by a1i fmpttd nfcv nfcsb1v csbeq1a cbvmpt fmpt sylibr rspcdva remulcld ralrimiva weq recnd subdid cc ccncf sstrdi cncfmptc syl3anc remulcl ax-resscn cncfmpt2ss cdv c1 sstrid 1cnd dvmptres dvmptcmul mpteq2dv - sselda mulid1d eqtrd cdm mpbird r19.21bi adantr w3a letrd wi anbi2d + sselda mulridd eqtrd cdm mpbird r19.21bi adantr w3a letrd wi anbi2d eleq1 breq2 breq1 3anbi23d eqtr3id imbi12d breq2d oveq2 cneg oveq1d vex subdird cc0 3brtr4d oveq2d cfl cfz csu ioossre eqsstri eleqtrdi sselid elioopnf simpld reflcl dvmptrecl fzfid cuz eleqtrrdi rspccva @@ -358146,7 +358146,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by 3eqtrd subge0d lemul2ad lesub1dd leadd1dd dvfsumlem1 xrletrd fllep1 flle leidd peano2rem subcld addcomd subsub2d 3eqtr4d vtoclg1f vtocl simp2l subled renegcld lesubadd2d suble0d le0neg1d mulneg1d 3brtr3d - 1red lenegd mulid2d sub32d addsubd jca ) AOKURZNKURZUSUTVVCBNDVAZVB + 1red lenegd mullidd sub32d addsubd jca ) AOKURZNKURZUSUTVVCBNDVAZVB VCZVVBBODVAZVBVCZUSUTAONUUAURZVBVCZVVFVDVCZBOCVAZVBVCZLVVHUUBVCZEJU UCZVEVCZNVVHVBVCZVVDVDVCZBNCVAZVBVCZVVNVEVCZVVBVVCUSAVVLVVSVVNAVVJV VKAVVIVVFAOVVHAGVFOGHVGVHVCZVFQHVGUUDZUUEZULUUGZANVFVIZVVHVFVIAVWEH @@ -358336,7 +358336,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by fveq2 wbr caddc cmul cpnf nfv csbeq1 rspcv sselid syl remulcld readdcld reflcl cc0 fracge0 w3a rexrd 3jca wa simpr1 nfbr nfim eleq1 breq2 breq1 wi 3anbi123d anbi2d imbi12d vtoclg1f mpcom mpdan mulge0d addge02d mpbid - breq2d id oveq1d 3brtr3d recnd addcomd c1 letrd fracle1 mulid2d breqtrd + breq2d id oveq1d 3brtr3d recnd addcomd c1 letrd fracle1 mullidd breqtrd lemul1ad rspccva fveq2d wss ioossre eqsstri a1i dvmptrecl cbvralw sylib cioo xrletrd lesub1dd cmpt eqid dvfsumlem3 simprd subsub3d 1red subge0d mpbird subge02d eqbrtrrd simpld leadd1dd absdifled mpbir2and eqbrtrd @@ -358574,7 +358574,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cuz rspccva fveq2d abssubd wss cioo ioossre eqsstri dvmptrecl csbeq1 expr cbvralw lesub1dd cmpt cc cpr reelprrecn dvmptneg adantrr 3adant3 3ad2ant1 simp2r eqid dvfsumlem3 simprd id csbnegg adantl 3brtr4d 1red nfbr fracle1 - csbied nfim lemul1ad mulid2d leadd1dd lesubadd2d simpld addcomd absdifled + csbied nfim lemul1ad mullidd leadd1dd lesubadd2d simpld addcomd absdifled mpbir2and eqbrtrd ) APLUTZOLUTZVAVBZVCUTZMOVDUTZVEVBZEJVFZBOCVGZVAVBZMPVD UTZVEVBZEJVFZBPCVGZVAVBZVAVBVCUTZKVHAUWHUWRUWMVAVBZVCUTUWSAUWGUWTVCAUWEUW RUWFUWMVAAPGVIZUWRVJVIUWEUWRVKUNAUWPUWQAUWOEJAMUWNVLZADVJVIZBQVOZJVMZQVIZ @@ -360859,7 +360859,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by c1 ralbidv imbi2d ply1ring syl2anc oveq2d sylan fveq2d nn0cnd ralrimiva weq cco1 cle ad2antlr nn0addcld cc eqid simplr ffvelcdmd ringcl adantrr wf coe1f syl13anc adantlrr rspcdva ad3antrrr rexlimdva ringrz grpsubid1 - ring0cl cgrp ringgrp eqtr2d addid1d breq12d biimpa ex cv1 cmgp nn0addcl + ring0cl cgrp ringgrp eqtr2d addridd breq12d biimpa ex cv1 cmgp nn0addcl cmg cvsca simprl csn cun cz wb deg1cl peano2nn0 nn0zd degltlem1 syl2an2 impr nn0cn peano2cn 1cnd addsubassd ax-1cn pncan eqtrd breqtrd ply1tmcl leidd deg1tmle deg1mulle2 coe1tmmul2fv addcomd oveq1d ringlidm 3eqtr3rd @@ -362256,7 +362256,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by plypow $p |- ( ( S C_ CC /\ 1 e. S /\ N e. NN0 ) -> ( z e. CC |-> ( z ^ N ) ) e. ( Poly ` S ) ) $= ( cc wss c1 wcel cn0 w3a cv cexp co cmul cmpt cply cfv wa id simp3 expcl - syl2anr mulid2d mpteq2dva eqid ply1term eqeltrrd ) BDEZFBGZCHGZIZADFAJZCK + syl2anr mullidd mpteq2dva eqid ply1term eqeltrrd ) BDEZFBGZCHGZIZADFAJZCK LZMLZNZADULNBOPUJADUMULUJUKDGZQULUOUOUIULDGUJUORUGUHUISUKCTUAUBUCAFBUNCUN UDUEUF $. @@ -362265,7 +362265,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by plyconst $p |- ( ( S C_ CC /\ A e. S ) -> ( CC X. { A } ) e. ( Poly ` S ) ) $= ( vz cc wss wcel wa cv cc0 cexp co cmul cmpt csn cxp cply cfv wceq exp0 - c1 adantl oveq2d ssel2 mulid1d eqtrd mpteq2dva fconstmpt eqtr4di cn0 0nn0 + c1 adantl oveq2d ssel2 mulridd eqtrd mpteq2dva fconstmpt eqtr4di cn0 0nn0 adantr eqid ply1term mp3an3 eqeltrrd ) BDEZABFZGZCDACHZIJKZLKZMZDANOZBPQZ URVBCDAMVCURCDVAAURUSDFZGZVAATLKAVFUTTALVEUTTRURUSSUAUBVFAURADFVEBDAUCUKU DUEUFCDAUGUHUPUQIUIFVBVDFUJCABVBIVBULUMUNUO $. @@ -362325,7 +362325,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by nnge1 simplr leaddsub2d negsubdi2d lenegcon2d neg1z eluz leexp2ad expn1 zltp1le lemul2ad divrecd 3brtr4d mulge0d climsqz2 an32s adantlr absmuld rpge0d absidd 3eqtr4rd climabs0 wn ltned velsn necon3bbii mul02d ifeq1d - iffalsed eqtrdi mulid1d nn0ssre anim1i eldifsn difun2 suprubd breqtrrdi + iffalsed eqtrdi mulridd nn0ssre anim1i eldifsn difun2 suprubd breqtrrdi ifid ad4ant14 simpllr letri3d subidd exp0d iftrued pm2.61dane mpteq2dva fconstmpt eqtr4di ifcl eqimss2i ltlecasei snex xpex anasss fveq1d 0pval eqbrtrd sumeq2sdv sumex expne0d div0d fsumdivc expsubd divassd sumeq2dv @@ -362610,7 +362610,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by a1i wa sumex offval2 weq fveq2 oveq2 oveq12d oveq2d cn0 elfznn0 wf adantr ffvelcdmda expcl adantll mulcld sylan2 anim12dan mulcl syl fsum0diag2 cuz wceq nn0cnd ad2antrr adantl addsubd fznn0sub nn0uz eleqtrdi nn0zd eluzadd - wss cz syl2anc eqeltrd addid2d fveq2d eleqtrd fzss2 adantlr cdif csn cima + wss cz syl2anc eqeltrd addlidd fveq2d eleqtrd fzss2 adantlr cdif csn cima c1 wn eldifn cun wo eldifi peano2nn0 uzsplit eqtrid ax-1cn sylancl uneq1d pncan eqtrd ad3antrrr elun sylib ord mpd wi wfun cdm ffund sseqtrrid fdmd ssun2 sseqtrrd funfvima2 elsni oveq1d simplr syl2an mul02d fzfid sumeq2dv @@ -362834,7 +362834,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by co wf wcel cvv cv wa subcl adantl cnex nn0ex elmap sylib a1i off sylibr inidm wss 0cn snssi ax-mp ssequn2 mpbi oveq1i c1 cuz cfv wne cle wbr wi cima wral wn cr wb nn0red syl2an ffnd eqidd adantrr adantr nn0cnd nn0uz - clt cz eleqtrdi nn0zd eluzadd syl2anc addid2d fveq2d eleqtrd eluzle syl + clt cz eleqtrdi nn0zd eluzadd syl2anc addlidd fveq2d eleqtrd eluzle syl lelttrd ltnled mpbid plyco0 r19.21bi necon1bd mpd c0p cmpt cfz cmul csu eqtrd adantlr oveq1d ffvelcdmda sylan2 mulcld simpr eqid fvmpt2 sylancl sumex fzss2 sselda syldan eldifn eldifi elfz5 sylibrd mul02d fsumss @@ -363169,7 +363169,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cv csu wf plyf syl ffvelcdmda feqmptd coeid oveq1 oveq2d sumeq2sdv fmptco eqid cn0 dgrcl wi c1 caddc oveq2 sumeq1d mpteq2dv eleq1d imbi2d csn wa cz cxp 0z exp0d cun wss plybss 0cnd snssd unssd coef ffvelcdm sylancl sseldd - 0nn0 adantr mulid1d eqtrd eqeltrd fveq2 oveq12d fsum1 mpteq2dva fconstmpt + 0nn0 adantr mulridd eqtrd eqeltrd fveq2 oveq12d fsum1 mpteq2dva fconstmpt sylancr eqtr4di plyconst syl2anc plyun0 eleqtrdi simprr peano2nn0 nn0p1nn cof syl2an cn exp1d eqtr4d adantlr plymul expr cvv cnex a1i ovexd offval2 eqidd nnnn0 ad2antlr expp1d sylibd expcom a2d nnind impcom adantrr plyadd @@ -363285,7 +363285,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( vz vk cc wcel csn cxp cdgr cfv cc0 wceq cn0 cv co cmpt cexp cmul oveq2d simpl c1 cle wbr wss cply ssid plyconst mpan 0nn0 a1i cfz fconstmpt wa cz csu 0z exp0 sylan9eqr eqeltrd oveq2 fsum1 sylancr eqtrd mpteq2dva eqtr4id - mulid1 dgrle wb dgrcl nn0le0eq0 3syl mpbid ) ADEZDAFGZHIZJUAUBZVNJKZVLBAD + mulrid dgrle wb dgrcl nn0le0eq0 3syl mpbid ) ADEZDAFGZHIZJUAUBZVNJKZVLBAD CVMJDDUCVLVMDUDIEZDUEADUFUGZJLEVLUHUIVLCMZJJUJNZESVLVMBDAOBDVTABMZVSPNZQN ZCUNZOBDAUKVLBDWDAVLWADEZULZWDAWAJPNZQNZAWFJUMEWHDEWDWHKUOWFWHADWEVLWHATQ NAWEWGTAQWAUPRAVEUQZVLWESURWCWHCJVSJKWBWGAQVSJWAPUSRUTVAWIVBVCVDVFVLVQVNL @@ -363297,7 +363297,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( ( deg ` F ) = 0 <-> F = ( CC X. { ( F ` 0 ) } ) ) ) $= ( vz vk cfv wcel cdgr cc0 wceq cc csn cxp wa cfz co cexp cmul cn0 eqtrd c1 cply ccoe cmpt cv eqid coeid adantr simplr oveq2d sumeq1d cz 0z adantl - csu exp0 coef3 0nn0 ffvelcdm sylancl ad2antrr mulid1d eqeltrd fveq2 oveq2 + csu exp0 coef3 0nn0 ffvelcdm sylancl ad2antrr mulridd eqeltrd fveq2 oveq2 wf oveq12d fsum1 sylancr mpteq2dva fconstmpt eqtr4di fveq1d fvex fvconst2 0cn ax-mp eqtrdi xpeq2d eqtr4d ex plyf 0dgr syl fveqeq2 syl5ibrcom impbid sneqd ) BAUAEFZBGEZHIZBJHBEZKZLZIZWHWJWNWHWJMZBJHBUBEZEZKZLZWMWOBCJWQUCZW @@ -363332,7 +363332,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( vk cfv wcel cc0 cfz co cexp cmul csu cc wceq cn0 syl wa c1 ffvelcdm 0cn cply cdgr cv eqid coeid2 mpan2 cuz wss dgrcl nn0uz eleqtrdi fzss2 elfz1eq fveq2 oveq2 0exp0e1 eqtrdi oveq12d wf coef3 0nn0 sylancl sylan9eqr adantr - mulid1d eqeltrd cdif cn eldifn wn wo eldifi elfznn0 elnn0 sylib ord id cz + mulridd eqeltrd cdif cn eldifn wn wo eldifi elfznn0 elnn0 sylib ord id cz 0z elfz3 ax-mp eqeltrdi syl6 mt3d adantl 0expd oveq2d syl2an mul01d eqtrd fzfid fsumss fsum1 sylancr 3eqtr2d ) CBUBFGZHCFZHCUCFZIJZEUDZAFZHXAKJZLJZ EMZHHIJZXDEMZHAFZWQHNGWRXEOUAABECWSHDWSUEUFUGWQXFWTXDEWQWSHUHFZGXFWTUIWQW @@ -363629,7 +363629,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by coeidp $p |- ( A e. NN0 -> ( ( coeff ` Xp ) ` A ) = if ( A = 1 , 1 , 0 ) ) $= ( vz c1 cc wcel cn0 cidp ccoe cfv wceq cc0 cif ax-1cn 1nn0 cid cres cv co - cmpt cmul cexp mptresid df-idp exp1 oveq2d mulid2 eqtrd mpteq2ia coe1term + cmpt cmul cexp mptresid df-idp exp1 oveq2d mullid eqtrd mpteq2ia coe1term 3eqtr4i mp3an12 ) CDECFEAFEAGHIIACJCKLJMNBCGACODPBDBQZSGBDCULCUARZTRZSBDU BUCBDUNULULDEZUNCULTRULUOUMULCTULUDUEULUFUGUHUJUIUK $. @@ -363638,7 +363638,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by dgrid $p |- ( deg ` Xp ) = 1 $= ( vz c1 cc wcel cc0 wne cn0 cidp cdgr cfv wceq ax-1cn ax-1ne0 1nn0 cid cv cres cmpt co cmul cexp mptresid df-idp exp1 oveq2d eqtrd mpteq2ia 3eqtr4i - mulid2 dgr1term mp3an ) BCDBEFBGDHIJBKLMNABHBOCQACAPZRHACBULBUASZTSZRACUB + mullid dgr1term mp3an ) BCDBEFBGDHIJBKLMNABHBOCQACAPZRHACBULBUASZTSZRACUB UCACUNULULCDZUNBULTSULUOUMULBTULUDUEULUIUFUGUHUJUK $. dgreq0.1 $e |- N = ( deg ` F ) $. @@ -363707,7 +363707,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by plyaddcl 3adant3 dgrcl syl nn0red eqeltrid 3ad2ant2 3ad2ant1 ifcld dgradd cply cr leidd simp3 ltled breq1 ifboth syl2anc letrd coeadd fveq1d cvv wf ccoe coef3 nn0ex a1i inidm wn ltnled mpbid wi simp1 dgrub 3expia necon1bd - ffnd mpd adantr wa eqidd ofval mpdan ffvelcdmd addid2d simp2 0red nn0ge0d + ffnd mpd adantr wa eqidd ofval mpdan ffvelcdmd addlidd simp2 0red nn0ge0d 3eqtrd lelttrd gt0ne0d c0p dgreq0 fveq2 dgr0 3eqtr4g syl6bir necon3d sylc eqcomi eqnetrd syl3anc letri3d mpbir2and ) BAUOHZIZCXIIZDEUAJZUBZBCKUCZLZ MHZENXPEOJEXPOJZXMXPDEOJZEDUDZEXMXPXMXOPUOHIZXPQIXJXKXTXLABCUEUFZPXOUGUHU @@ -363755,7 +363755,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( cc wcel cc0 wne cply cfv csn cxp cmul cdgr wceq c0p fveq2d dgr0 caddc 0cn co w3a cof oveq2 fveq2 eqtrdi eqeq12d wa wss ssid simpl1 plyconst fvconst2g sylancr sylancl simpl2 ne0p plyssc simpl3 sselid simpr eqid dgrmul syl22anc - eqnetrd 0dgr syl oveq1d cn0 dgrcl nn0cnd addid2d 3eqtrd cvv a1i simp1 ofc12 + eqnetrd 0dgr syl oveq1d cn0 dgrcl nn0cnd addlidd 3eqtrd cvv a1i simp1 ofc12 cnex mul01d sneqd xpeq2d eqtrd df-0p oveq2i 3eqtr4g pm2.61ne ) ADEZAFGZCBHI ZEZUAZDAJKZCLUBZTZMIZCMIZNWKOWLTZMIZFNCOCONZWNWQWOFWRWMWPMCOWKWLUCPWRWOOMIZ FCOMUDQUEUFWJCOGZUGZWNWKMIZWORTZFWORTWOXAWKDHIZEZWKOGZCXDEWTWNXCNXADDUHWFXE @@ -363794,7 +363794,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( deg ` ( x e. CC |-> ( ( G ` x ) ^ M ) ) ) = ( M x. N ) ) $= ( wcel cc cfv cexp co cdgr cmul wceq adantr c0p vy vd vz vw cn cv cmpt wi c1 caddc oveq2 mpteq2dv fveq2d oveq1 eqeq12d imbi2d wa cply wf ffvelcdmda - plyf syl exp1d mpteq2dva feqmptd eqtr4d eqtr4di nncnd mulid2d cof adantlr + plyf syl exp1d mpteq2dva feqmptd eqtr4d eqtr4di nncnd mullidd cof adantlr cn0 nnnn0 adantl expp1d cvv cnex a1i ovexd eqidd offval2 wne fmptco ssidd ccom wss 1cnd plypow syl3anc plyssc sselid addcl mulcl plyco eqeltrrd cc0 simpr nnmulcld nnne0d eqnetrd fveq2 eqtrdi necon3i eqtrid dgrmul syl22anc @@ -364276,7 +364276,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( c0p cfv cc vz cply wcel cmul cof co cmin wceq cdgr clt wbr wrex wss wo plybss ply0 3syl cc0 cv cvv cnex a1i wf wfn plyf ffn inidm offn wa eqidd 0pval adantl ofval ffvelcdmda mul01d eqtrd subid1d offveq caddc - syl eqeq1d fveq2d dgrcl nn0red recnd addid2d eqcomd breq12d ltsubaddd + syl eqeq1d fveq2d dgrcl nn0red recnd addlidd eqcomd breq12d ltsubaddd 0red bitr4d orbi12d mpbird oveq2 oveq2d eqtrid breq1d rspcev syl2anc cn0 ) AREUBSZUCZFGRUDUEZUFZUGUEZUFZRUHZXFUISZGUISZUJUKZUNZDRUHZDUISZX IUJUKZUNZHXAULAFXAUCZETUMXBMEFUOEUPUQZAXKFRUHZFUISZXIUGUFURUJUKZUNQAX @@ -364845,7 +364845,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by 1red mpbir2and necon3ad disjsn fsumsplit negnegd eqeq2d anbi12d sumeq1d sumsn cnveq imbi12d wral rspcdva mp2and simp2d dgrmul syl22anc coemulhi ssid plyid mp2an plyconst coesub 1nn0 mp2b nn0ex inidm iftruei clt 0lt1 - 0re 1re ltnlei mpbi mtbiri sylan 1m0e1 ffvelcdmd mulid2d negcld nnm1nn0 + 0re 1re ltnlei mpbi mtbiri sylan 1m0e1 ffvelcdmd mullidd negcld nnm1nn0 0dgr dgreq0 divcld negdid mulcld divdird coemul 1e0p1 sumeq1i cuz nn0uz 0nn0 eleqtri elfznn0 ffvelcdm pncan nnuz eleqtrdi fzss2 fznn0sub syldan sylancl sylan2br oveq2d fsump1 eldifn eldifi cz elfzuz 1z elfz5 sylibrd @@ -364955,7 +364955,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by coef3 0nn0 ffvelcdm sylancl 1nn0 c0p wne ax-1ne0 a1i eqnetrrd dgr0 eqtrdi simpr necon3i syl wb dgreq0 necon3bid mpbid eqnetrd divcld negcld syl2anc sumsn adantrr cfn wss cen wbr cle fta1 syldan simpld cfz cexp cmul coeid2 - id oveq2d nn0uz 1e0p1 oveq2 ffvelcdmda expcl sylan cz exp0d mulid1d eqtrd + id oveq2d nn0uz 1e0p1 oveq2 ffvelcdmda expcl sylan cz exp0d mulridd eqtrd mulcld 0z sylancr eqeltrd fsum1 jctil exp1d mulneg2d 3eqtrd negidd simprd divcan2d fsump1i 3eqtr2d wfn plyf fniniseg mpbir2and snssd hashsng simprr ffnd eqtr4d snfi hashen fisseneq syl3anc 1m1e0 eqtr3id 3eqtr3d ex cbvsumv @@ -365283,7 +365283,7 @@ of all kernels (preimages of ` { 0 } ` ) of all polynomials in c0ex ifex ax-mp nn0ge0d iftrued nn0cnd subid1d eqtrd eqtrid 3eqtrd simprd wb dgreq0 necon3bid mpbid eqnetrd sylancr sylanbrc oveq1 oveq2d sumeq2sdv ne0p sumex caddc coef3 elfznn0 ffvelcdm syl2an expcl mulcld expcld simplr - ad2antrr nn0zd expne0d divcld fveq2 oveq12d oveq1d addid2d expsubd dividd + ad2antrr nn0zd expne0d divcld fveq2 oveq12d oveq1d addlidd expsubd dividd fsumrev2 divassd divdiv32d exprecd syl2anc 3eqtr4d sumeq2dv coeid2 simprr eqtr3d fzfid fsumdivc div0d 3eqtr3d 3eqtr2d fveq1 eqeq1d rspcev rexlimddv ) AUCEZAFGZHZAUAIZJZFKZUDALMZUCEZUANUEJZOUFUGZUUOUUTUAUVDUHZUUPUUOAPEZUVE @@ -365829,7 +365829,7 @@ of all kernels (preimages of ` { 0 } ` ) of all polynomials in wa cc0 cr clt cfa cneg crp eluznn negeqd ovex fvmpt syl 2rp nnnn0d faccld cv cz nnzd znegcld rpexpcl sylancr eqeltrd rpred rpgt0d wi breq12d imbi2d caddc cdiv cmin nnnn0 leidd nncn subidd cc halfcn exp0 ax-mp eqtrdi rpcnd - weq mulid1d eqtrd breqtrrd uzid oveq1 3brtr4d rpge0d simpl halfre halfgt0 + weq mulridd eqtrd breqtrrd uzid oveq1 3brtr4d rpge0d simpl halfre halfgt0 nnz 3syl elrpii eluzelz zsubcl syl2anr rpmulcld adantr adantl zmulcld a1i jca31 simpr 1le2 nncnd zcnd mulneg1d nnmulcld nnge1d wb nnred leneg mpbid 2re 1re eqbrtrd neg1z eluz sylancl mpbird leexp2a mp3an12i expn1 breqtrdi @@ -366240,7 +366240,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the ovex id jaoi elicc1 sylancr mpbid simp2d elnn0z sylanbrc ffvelcdmd faccld dvnf nnne0d divcld 0cnd expcld mulcld sylan2 oveq2d fveq1d eqtrdi oveq12d nncnd oveq1d wfun fmpttd csn cdif cn eldifi eldifsni adantl elnnne0 0expd - wne mul01d zex inex2 suppss2 gsumpt fac0 oveq2 0exp0e1 eqid fvmpt mulid1d + wne mul01d zex inex2 suppss2 gsumpt fac0 oveq2 0exp0e1 eqid fvmpt mulridd div1d 3eqtrd ccmn ringcmn ctps cnfldtps cfn csupp cfsupp mptexg c0ex snfi funmpt suppssfifsupp syl32anc tsmsid eqeltrrd mpteq2dv eleqtrrd mpbir2and subidd eltayl taylf ffun funbrfv2b 3syl ) ACCGUAZEUBZCEUCZOCEUAUWHUDUEZAU @@ -366361,7 +366361,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the ccnfld csubrg subrgss elplyd cnfld1 subrg1cl plyid plyconst csubg caddc c1 syl2anc subrgsubg cnfldadd subgcl 3expb sylan cnfldmul subrgmcl cneg cminusg ax-1cn cnfldneg ax-mp subginvcl eqeltrrid plysub eqeltrd fveq2d - eqid plyco dgrco ccnv cima w3a simp2d dgrcl nn0cnd mulid1d eqtrd 3eqtrd + eqid plyco dgrco ccnv cima w3a simp2d dgrcl nn0cnd mulridd eqtrd 3eqtrd plyremlem elfznn0 adantl dvn2bss ffvelcdmd faccld nnne0d divcld eqbrtrd dvnf nncnd dgrle jca ) AFDUEUFZSFUGUFZIUHUIAFUATUJIUKULZCGUMZEHUNULZUFZ UFZUVCUOUFZUPULZUAUMZUVCUQULZURULZGUSZUTZVATCVBZVCZVDVEULZVFZUUTAFUBTUV @@ -366422,7 +366422,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the 0cnd 3expa 1cnd ad2antrr oveq1 oveq2d mpteq2dva eqtrd 1nn0 dvnadd syl2anc fveq1d fveq2d 3eqtr3d fveq2 oveq12d oveq1d sylan2 cuz taylpfval c0ex ovex 3ad2ant2 neqned elnnne0 nnm1nn0 ifclda ifex dvmptid dvmptc dvmptsub 1m0e1 - wne mpteq2i eqtrdi dvexp2 ifeq2d dvmptco mulid1d dvmptcmul dvmptfsum 1zzd + wne mpteq2i eqtrdi dvexp2 ifeq2d dvmptco mulridd dvmptcmul dvmptfsum 1zzd dvfg dvbss dvn1 addcomd dmeqd eleqtrrd taylplem2 oveq2 fsumshft ifnefalse elfznn simpll fz1ssfz0 sseli simplr mulassd facp1 npcand div23d divcan5rd pncan3d 3eqtr3rd 3eqtr2d sumeq2dv 0p1e1 oveq1i sumeq1i eqtr4di an32s cdif @@ -366498,7 +366498,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the wa dvnbss fdmd sseqtrd sstrd fzofzp1 fznn0sub elfzofz dvnadd 1cnd addassd nppcan2d eqtrd fveq2d pncan3d subcld add12d eqtr3d 3eqtr4d dmeqd eleqtrrd cdm dvtaylp oveq1d eqcomd oveqd 3eqtr3rd syl5ibr expcom a2d fzind2 subidd - mpcom addid1d ) AFNGFOPZCDEUBPZPZUCPZQZGFFUDPZOPZCDFDEUCPZQZUBPZPZGCUUAPF + mpcom addridd ) AFNGFOPZCDEUBPZPZUCPZQZGFFUDPZOPZCDFDEUCPZQZUBPZPZGCUUAPF RFUEPZSZAYPUUBTZAFRUFQZSZUUDAFUJUUFKUGUHRFUKULAUAUMZYOQZGFUUHUDPZOPZCDUUH YSQZUBPZPZTZUNARYOQZGFRUDPZOPZCDRYSQZUBPZPZTZUNZAUIUMZYOQZGFUVDUDPZOPZCDU VDYSQZUBPZPZTZUNAUVDUOOPZYOQZGFUVLUDPZOPZCDUVLYSQZUBPZPZTZUNAUUEUNUAUIFRF @@ -366593,7 +366593,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the cfzo dmeqd taylpf dvntaylp feq1d mpbird syldan dvn0 eqtr4di ctopon subcld eleqtrrdi funbrfvb nnm1nn0 dvnbss fzo0end elfzofz 3syl dvn2bss nncnd 1cnd fssdmd npcand recnprss 3eqtr3rd 1nn0 dvn1 pncan3d eqtr4id 0nn0 cnfldtopon - sstrd addid2d toponmax mp1i df-ss ssid cmap mapsspm sylibr sselid 3eqtr3d + sstrd addlidd toponmax mp1i df-ss ssid cmap mapsspm sylibr sselid 3eqtr3d cin elmap dvmptres3 ctop cuni resttopon topontop toponuni ntrss2 dvbssntr sseqtrd dvmptres2 dvmptsub breqd fmpttd eldv simprd eldifi subid1d eqtr2d oveqan12rd sylan2 ssdifssd sseldd adantr mpteq2dva expr expcom a2d fzind2 @@ -366702,7 +366702,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the elfzofz wb isopn3 mpbird difss dvnf fvexd recn cnfldtopon toponmax mp1i cin df-ss sylib ssid cmap mapsspm sylibr dmmpti sseqtrrid addcld mulcld elmap prid2 elfznn ovexd dvmptid 0cnd dvmptc 1m0e1 mpteq2i dvexp pncand - mpteq2dv dvmptco mulid1d dvntaylp0 subidd ctx subcn dvcn syl31anc plycn + mpteq2dv dvmptco mulridd dvntaylp0 subidd ctx subcn dvcn syl31anc plycn ccn sstrdi cncfmptid cncfmpt1f cncfmpt2f wrex ssdifssd eldifsni subne0d 0expd wne nnzd necomd neneqd nrexdv df-ima eleq2i elrnmpti bitri nnne0d sylnibr mulne0d imaeq1d eleq2d bitrdi mtbird eldifi divcld nnrecred cxp @@ -367062,7 +367062,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the cuz cv cc cmap wa wf adantr cz simpr eleqtrdi eluzsub eleqtrrdi ffvelcdmd syl3anc fmpt3d cmpt eluzelz syl zcnd subnegd wceq eluzadd syl2anc fvoveq1 fveq2d fveq1d fvex fvmpt pncand eqtrd 3eqtrd mpteq2dva cc0 addassd negidd - oveq2d addid1d eqtr4di mpteq1d feqmptd 3eqtr4rd impbid ) ADEBUARZUBFEWLUB + oveq2d addridd eqtr4di mpteq1d feqmptd 3eqtr4rd impbid ) ADEBUARZUBFEWLUB ABCDEFGHIJKLMNOPUCABQFEDGUDZHGUESZWNWMUESZUJRZILWPUFAHGMNUGAGNUHZACICUKZG UISZDRZULBUMSZFPAWRITZUNZJXAWSDAJXADUOXBOUPXCWSHUJRZJXCHUQTZGUQTZWRWNUJRZ TWSXDTAXEXBMUPAXFXBNUPXCWRIXGAXBURLUSGHWRUTVCKVAVBVDAQJQUKZWMUISZFRZVEQJX @@ -367932,11 +367932,11 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the caddc cli cdm cn0 nn0uz 0zd crp 1rp a1i wceq pserval2 sylan cvv fvexd cc psergf ffvelcdmda serf0 climi0 wa cdiv cmpt simprl cr nn0re adantl adantr abscld wne 0red absge0d lelttrd gt0ne0d redivcld remulcld eqid - reexpcl fmptd wf recnd cle divge0 syl22anc absidd mulid1d breqtrrd wb + reexpcl fmptd wf recnd cle divge0 syl22anc absidd mulridd breqtrrd wb ltdivmul syl112anc mpbird eqbrtrd geomulcvg syl2anc eluznn0 ffvelcdmd 1red ad2antrr reexpcld nn0red nn0ge0d expcld mulcld expge0d weq fveq2 simprr oveq12d fveq2d breq1d rspccva wi 1re ltle sylancl mpd lemul1ad - absmuld mul32d absexpd oveq2d 3eqtr3d mulid2d 3brtr3d cz 3brtr4d ovex + absmuld mul32d absexpd oveq2d 3eqtr3d mullidd 3brtr3d cz 3brtr4d ovex oveq2 eluzelz expgt0 syl3anc lemuldiv expdivd lemul2ad mulge0d fvmpt2 mpbid id fvmpt syl cvgcmpce rexlimddv ) AUAUDZCQZIUUOUERZSRZUFQZUGUHU IZUAUBUDZUJQZUKZUNGULUMUOUPZTUBUQAUURUGUBUAIFQZULUQURAUSZUGUTTAVAVBAI @@ -367986,7 +367986,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the 1nn0 cr id 2fveq3 oveq12d eqid ovex fvmpt adantl nn0re cc psergf abscld ffvelcdmda remulcld eqeltrd wf fvco3 sylan recnd cbvmptv radcnvlem1 cuz 1red cle cn elnnuz sylbir sylan2 wbr absge0d eluzle lemul1ad absidm syl - nnnn0 fveq2d mulid2d 3eqtr4d oveq2d eqtrd 3brtr4d cvgcmpce ) ANUAUBOUBU + nnnn0 fveq2d mullidd 3eqtr4d oveq2d eqtrd 3brtr4d cvgcmpce ) ANUAUBOUBU CZWNFEPZPQPZRUDZUEZQWOUFZUGNOUHNOSAUKUIAUAUCZOSZUJZWTWRPZWTWTWOPZQPZRUD ZULXAXCXFTAUBWTWQXFOWRWNWTTZWNWTWPXERXGUMWNWTQWOUNUOZWRUPWTXERUQURUSZXB WTXEXAWTULSZAWTUTUSZXBXDAOVAWTWOABCDEFHIJVBZVDZVCZVEZVFXBWTWSPZXEVAAOVA @@ -368123,12 +368123,12 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the fvmpt syl2an oveq2d eqtrd 3eqtrd nn0red wf ffvelcdm expcl mulcld remulcld abscld eqeltrd weq oveq1 oveq2 nn0cnd sylan cseq cli cdm wbr ccom cbvmptv radcnvlt1 simpld climdm sylib cz wb neg1z isershft mp2an seqex breldm cuz - 0z fvex cle neg1cn addid2i 0le1 le0neg2 ax-mp mpbi eqbrtri eqeltri eluz1i + 0z fvex cle neg1cn addlidi 0le1 le0neg2 ax-mp mpbi eqbrtri eqeltri eluz1i 1re mpbir2an eluzelcn nn0re psergf ffvelcdmda fmpttd breq2d sylan2 adantr recnd crp simpr 3brtr4d ad2antrr eluzp1p1 oveq1i 1pneg1e0 addcomli fveq2i eqtri eqtr4i eleqtrrdi iserex mpbid 1red wn neqne absrpcl rprecred ifclda wne cn elnnuz nnnn0 sylbir nn0ge0d absge0d mulge0d sylibr 0expd sylan9eqr - mul01d abs00bd mulid2d df-ne mulassd absmuld absidd oveq1d eqled rpreccld + mul01d abs00bd mullidd df-ne mulassd absmuld absidd oveq1d eqled rpreccld rpcnd mul12d absdivd divassd cmin pncan nn0zd expm1d eqtr3d eqtr4d rpne0d sylancl divrec2d 3eqtr3rd breqtrrd sylanl2 sylan2br ifbothda cvgcmpce ) A HPUCZQQHUDUEZUFRZUGZUAUBUHUBUIZUXAHFUEZUEZUDUEZSRZUJZQUKZUMRZGPQUHULQUHTA @@ -368618,7 +368618,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the ( ( z e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( z ^ n ) ) ) ) ` r ) ) e. dom ~~> } , RR* , < ) ) $= ( c1 cfv caddc cv cc cn0 cexp co cmul cmpt cc0 cseq wcel cabs cli cr crab - cdm cxr clt csup abs1 eqid feqmptd wa ffvelcdmda mulid1d mpteq2dva eqtr4d + cdm cxr clt csup abs1 eqid feqmptd wa ffvelcdmda mulridd mpteq2dva eqtr4d cle 1cnd wceq ax-1cn oveq1 cz nn0z 1exp sylan9eq oveq2d nn0ex mptex fvmpt syl ax-mp eqtr4di seqeq3d eqeltrrd radcnvle eqbrtrrid ) AHHUAIJEKBLDMDKZC IZBKZVQNOZPOZQZQZIRSUBUEZTEUCUDUFUGUHZUQUIABCWEDWCHEWCUJZFWEUJAURAJCRSJHW @@ -368642,7 +368642,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the eqtrdi abs1 oveq2d breq12d elrab2 sylanbrc cun crab w3a wo wne necon3bbii wn velsn wi clt simprll 0cn cnmetdval sylancl subid1d fveq2d eqtrd abscld eqid 1red caddc 1re resubcl ax-1cn sylancr simpll remulcld oveq2i abs2dif - subcl eqbrtrrid abssub breqtrd simprlr letrd mpbid adantr addsubd mulid1d + subcl eqbrtrrid abssub breqtrd simprlr letrd mpbid adantr addsubd mulridd lesubaddd subdid oveq1d eqtr4d breqtrrd peano2re syl leaddsub2d adddirp1d mpbird 3brtr4d 0red simplr lelttrd lemul2 syl112anc lensymd simprr necomd wb ltp1d subeq0 necon3bid absgt0 eqbrtrd eqtr3id breq1d syl5ibcom leneltd @@ -368678,7 +368678,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the seq 0 ( + , ( n e. NN0 |-> ( ( A ` n ) x. ( X ^ n ) ) ) ) e. dom ~~> ) $= ( wcel c1 cc0 cfv co caddc cn0 cc vr csn cabs cmin ccom wo cexp cmul cmpt cbl cv cseq cli cdm wa cun cdif wss abelthlem2 simprd ssundif sylibr elun - sselda sylib feqmptd ffvelcdmda mulid1d mpteq2dva eqtr4d oveq1d wceq nn0z + sselda sylib feqmptd ffvelcdmda mulridd mpteq2dva eqtr4d oveq1d wceq nn0z elsni cz 1exp sylan9eq oveq2d eqcomd seqeq3d adantr eqeltrrd cxmet cnxmet syl cxr 0cn 1xr blssm mp3an simpr sselid oveq1 mpteq2dv nn0ex mptex fvmpt eqid cr crab clt csup wf abscld rexrd 1re rexr mp1i cpnf iccssxr radcnvcl @@ -368764,8 +368764,8 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the shftval sylancr eqidd eleqtrdi fsumser eqtr4d 3eqtr4d cz ax-mp eqtrdi nncn 0z c0 eqtrd simpr oveq1d subdird expm1t mulcomd mul12d isermulc2 nnnn0 sylan2br seqfeq wb isershft mp2an sylib eqbrtrrd clim2ser2 seq1 - risefall0lem sum0 sylancl mul02d eqtrid isumcl addid1d breqtrd fsumm1 - 1z simpl eqtr3d pncan2d eqtr2d climsub 1cnd mulid2d breqtrrd isumclim + risefall0lem sum0 sylancl mul02d eqtrid isumcl addridd breqtrd fsumm1 + 1z simpl eqtr3d pncan2d eqtr2d climsub 1cnd mullidd breqtrrd isumclim sersub ) AIGUDZRFUEZDUDZIUVPUFSZUGSZFUHZUIIUJSRUVPUODTUKZUDZUVRUGSZFU HZUGSZAIEULUVOUVTUMAIEUIUNZQUPZBIRUVQBUEZUVPUFSZUGSZFUHUVTEGUWHIUMZRU WJUVSFUWKUWIUVRUVQUGUWHIUVPUFUQURUSORUVSFUTVAVBAUVSUWEFUARUAUEZDUDZIU @@ -368844,11 +368844,11 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the fveq2d eqbrtrd clt fsumrecl absmuld absge0d 1red adantr sylancl eqtrd cr 0cn wb 1re exple1 syl31anc lemul2ad recnd letrd lelttrd ltled cdiv syl112anc mpbird absidm seqex lemul1ad wne rerpdivcld isumclim2 rpcnd - breldm eqtr4d breqtrd rpne0d mulid2d mul12d eluznn0 fveq2 oveq12d cdm + breldm eqtr4d breqtrd rpne0d mullidd mul12d eluznn0 fveq2 oveq12d cdm ccom cbl cdif abelthlem2 sseldd abelthlem5 mpdan adantl iserex isumcl abelthlem6 isumsplit adddid abstrid fsumabs reexpcl cnmetdval subid1d wss readdcld cxmet cxr cnxmet 1xr elbl3 mpanl12 eqbrtrrd wi mpd simpr - ltle absexp eqtr2d mulid1d 3brtr3d fsumle ltp1d 0red fsumge0 ltmuldiv + ltle absexp eqtr2d mulridd 3brtr3d fsumle ltp1d 0red fsumge0 ltmuldiv fvex cz uzid geolim2 expge0d breq1d rspccva 3brtr4d cvgcmpce isumrecl wral eldifsni necomd subeq0 necon3bid absrpcld iserabs reexpcld difrp crp isermulc2 isumle isumclim rpdivcld div12d resubcl lemul2d mulcomd @@ -368931,7 +368931,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the wrex cli climi0 cfz csu cr fzfid cc wf ffvelcdmda serf elfznn0 ffvelcdm syl2an abscld fsumrecl ad2antrr absge0d fsumge0 rpdivcld wceq cneg cmul cle csn cdif ccom cbl wss abelthlem2 simpld cexp oveq1 cz nn0z 1exp syl - sylan9eq oveq2d sumeq2dv sumex fvmpt mulid1d eqcomd eqtrd oveq1d df-neg + sylan9eq oveq2d sumeq2dv sumex fvmpt mulridd eqcomd eqtrd oveq1d df-neg eqeltrd isumclim eqtr4di fveq2d absnegd adantlr fveq2 sylan9eqr abs00bd abelthlem4 ad2ant2r simpllr rpgt0d eqbrtrd wne ad3antrrr simprll simprr ad5ant15 cdm eldifsn sylanbrc simplrl 2fveq3 simplrr weq cbvralvw sylib @@ -368977,7 +368977,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the ifbieq2d eqid ovex fvex ifex fvmpt syl wne nnne0 neneqd iffalsed sylan2br eqtrd seqfeq isumclim2 clim2ser 0z seq1 oveq2i breqtrdi eqbrtrd clim2ser2 ax-mp breqtrd wss oveq1 nn0z 1exp sylan9eq oveq12d sumeq2dv sumex adantlr - eqtrid mulid1d oveq2d oveq1d cbvsumv breqtrrd isumclim oveq2 expcl mulcld + eqtrid mulridd oveq2d oveq1d cbvsumv breqtrrd isumclim oveq2 expcl mulcld sylan iftrue sylancl npncan2 syl2anc seqex c0ex breldm abelthlem8 wb cdif csn cbl abelthlem2 simpld ad2antrr eqtr4d eqtrdi subidd oveqan12rd ssrab3 ccom eqeltrd sselda 3eqtr4d abelthlem3 sumeq2sdv exp0d abelthlem4 npncand @@ -369116,7 +369116,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the ( vz c1 cv cmin co cfv cmul cle wbr cc cr wcel cabs crab cn0 cexp csu cc0 cmpt cicc ccncf wss unitssre ax-resscn sstri a1i wa 1re w3a simpr elicc01 sylib simp1d resubcl sylancr leidd simp3d abssubge0d simp2d absidd oveq2d - cres 1red recnd mulid2d eqtrd 3brtr4d ssrabdv resmptd eqtr4di 0le1 abelth + cres 1red recnd mullidd eqtrd 3brtr4d ssrabdv resmptd eqtr4di 0le1 abelth eqid rescncf sylc eqeltrrd ) ABJIKZLMZUANZJJWEUANZLMZOMZPQZIRUBZUCDKZCNBK WMUDMOMDUEZUGZUFJUHMZVJZEWPRUIMZAWQBWPWNUGEABWLWPWNAWKIRWPWPRUJAWPSRUKULU MUNAWEWPTZUOZWFWFWGWJPWTWFWTJSTWESTZWFSTUPWTXAUFWEPQZWEJPQZWTWSXAXBXCUQAW @@ -369421,7 +369421,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the 2t2e4 breq2i bitr2i cxr w3a 0xr elioc2 mp2an mpbir3an sin02gt0 breq2 mto wo mpbii caddc sq1 resincl gt0ne0ii neii 2ne0 recni 2cn divcan2i fveq2i eqtr3i cc sin2t sinpi sincl coscl mulcli mul0ori mtpor oveq1i eqtri sincossq sqcli - sq0 oveq2i addid1i 3eqtr2ri ax-1cn sqeqori ori mt3 ) ABUACZDEZFGZXJUBEZHGZX + sq0 oveq2i addridi 3eqtr2ri ax-1cn sqeqori ori mt3 ) ABUACZDEZFGZXJUBEZHGZX LXKFUCZGZXPHXOIJZFHIJZXQHFIJXRUDUEHFUFUMUGKFLMXRXQUHUMFUIKUJXPHXKIJZXQXJHBU KCMZXSXTXJLMZHXJIJZXJBNJZAOULZABOPUNUOUPAQNJZYCAQOUQBAIJAQIJURUSUTYCABBRCZN JZYEALMBLMZYHHBIJZVAYCYGUHOPYHYIPUOVBABBVCVDYFQANVEVFVGSHVHMYHXTYAYBYCVIUHV @@ -369472,7 +369472,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the Carneiro, 9-May-2014.) $) efhalfpi $p |- ( exp ` ( _i x. ( _pi / 2 ) ) ) = _i $= ( ci cpi c2 cdiv co cmul ce cfv ccos csin caddc cc wcel wceq picn halfcl c1 - cc0 ax-icn eqtri efival coshalfpi sinhalfpi oveq2i mulid1i oveq12i addid2i + cc0 ax-icn eqtri efival coshalfpi sinhalfpi oveq2i mulridi oveq12i addlidi mp2b ) ABCDEZFEGHZUIIHZAUIJHZFEZKEZABLMUILMUJUNNOBPUIUAUHUNRAKEAUKRUMAKUBUM AQFEAULQAFUCUDASUETUFASUGTT $. @@ -369489,7 +369489,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.) $) efipi $p |- ( exp ` ( _i x. _pi ) ) = -u 1 $= ( ci cpi cmul co ce cfv ccos csin caddc c1 cneg wcel wceq picn efival ax-mp - cc cc0 cospi eqtri sinpi oveq2i it0e0 oveq12i neg1cn addid1i ) ABCDEFZBGFZA + cc cc0 cospi eqtri sinpi oveq2i it0e0 oveq12i neg1cn addridi ) ABCDEFZBGFZA BHFZCDZIDZJKZBQLUGUKMNBOPUKULRIDULUHULUJRISUJARCDRUIRACUAUBUCTUDULUEUFTT $. $( Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised @@ -369534,7 +369534,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the efper $p |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( A + ( ( _i x. ( 2 x. _pi ) ) x. K ) ) ) = ( exp ` A ) ) $= ( cc wcel cz wa ci c2 cpi cmul co caddc ce cfv wceq ax-icn 2cn picn mulcli - c1 mulcl sylancr efadd sylan2 ef2kpi oveq2d efcl mulid1d sylan9eqr eqtrd + c1 mulcl sylancr efadd sylan2 ef2kpi oveq2d efcl mulridd sylan9eqr eqtrd zcn ) ACDZBEDZFAGHIJKZJKZBJKZLKMNZAMNZUPMNZJKZURUMULUPCDZUQUTOUMUOCDBCDVAGU NPHIQRSSBUKUOBUAUBAUPUCUDUMULUTURTJKURUMUSTURJBUEUFULURAUGUHUIUJ $. @@ -369582,7 +369582,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the 10-May-2014.) $) sin2kpi $p |- ( K e. ZZ -> ( sin ` ( K x. ( 2 x. _pi ) ) ) = 0 ) $= ( cz wcel cc0 c2 cpi cmul co caddc csin cfv cc zcn 2cn mulcli mulcl sylancl - picn addid2d fveq2d wceq 0cn sinper mpan sin0 eqtrdi eqtr3d ) ABCZDAEFGHZGH + picn addlidd fveq2d wceq 0cn sinper mpan sin0 eqtrdi eqtr3d ) ABCZDAEFGHZGH ZIHZJKZUJJKDUHUKUJJUHUJUHALCUILCUJLCAMEFNROAUIPQSTUHULDJKZDDLCUHULUMUAUBDAU CUDUEUFUG $. @@ -369591,7 +369591,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the 10-May-2014.) $) cos2kpi $p |- ( K e. ZZ -> ( cos ` ( K x. ( 2 x. _pi ) ) ) = 1 ) $= ( cz wcel cc0 c2 cpi cmul co caddc ccos cfv c1 cc zcn 2cn picn mulcli mulcl - sylancl addid2d fveq2d wceq 0cn cosper mpan cos0 eqtrdi eqtr3d ) ABCZDAEFGH + sylancl addlidd fveq2d wceq 0cn cosper mpan cos0 eqtrdi eqtr3d ) ABCZDAEFGH ZGHZIHZJKZUKJKLUIULUKJUIUKUIAMCUJMCUKMCANEFOPQAUJRSTUAUIUMDJKZLDMCUIUMUNUBU CDAUDUEUFUGUH $. @@ -369600,7 +369600,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the sin2pim $p |- ( A e. CC -> ( sin ` ( ( 2 x. _pi ) - A ) ) = -u ( sin ` A ) ) $= ( cc wcel cneg csin cfv c2 cpi cmul co cmin c1 caddc cz wceq sinper sylancl - negcl 1z eqtr3d 2cn picn mulcli mulid2i oveq2i wa negsubdi negsubdi2 eqtrid + negcl 1z eqtr3d 2cn picn mulcli mullidi oveq2i wa negsubdi negsubdi2 eqtrid mpan2 fveq2d sinneg ) ABCZADZEFZGHIJZAKJZEFZAEFDUMUNLUPIJZMJZEFZUOURUMUNBCL NCVAUOOARSUNLPQUMUTUQEUMUTUNUPMJZUQUSUPUNMUPGHUAUBUCZUDUEUMUPBCZVBUQOVCUMVD UFAUPKJDVBUQAUPUGAUPUHTUJUIUKTAULT $. @@ -369609,7 +369609,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the Chapman, 15-Mar-2008.) $) cos2pim $p |- ( A e. CC -> ( cos ` ( ( 2 x. _pi ) - A ) ) = ( cos ` A ) ) $= ( cc wcel cneg ccos cfv c2 cpi cmul co cmin c1 caddc cz wceq cosper sylancl - negcl 1z eqtr3d 2cn picn mulcli mulid2i oveq2i wa negsubdi negsubdi2 eqtrid + negcl 1z eqtr3d 2cn picn mulcli mullidi oveq2i wa negsubdi negsubdi2 eqtrid mpan2 fveq2d cosneg ) ABCZADZEFZGHIJZAKJZEFZAEFUMUNLUPIJZMJZEFZUOURUMUNBCLN CVAUOOARSUNLPQUMUTUQEUMUTUNUPMJZUQUSUPUNMUPGHUAUBUCZUDUEUMUPBCZVBUQOVCUMVDU FAUPKJDVBUQAUPUGAUPUHTUJUIUKTAULT $. @@ -369629,7 +369629,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the cosmpi $p |- ( A e. CC -> ( cos ` ( A - _pi ) ) = -u ( cos ` A ) ) $= ( cc wcel cpi cmin ccos cfv cmul csin caddc cneg wceq picn cossub mpan2 cc0 co oveq2i eqtrd eqtrid c1 cospi coscl neg1cn mulcom mulm1 syl sinpi oveq12d - sincl mul01d negcld addid1d ) ABCZADEQFGZAFGZDFGZHQZAIGZDIGZHQZJQZUPKZUNDBC + sincl mul01d negcld addridd ) ABCZADEQFGZAFGZDFGZHQZAIGZDIGZHQZJQZUPKZUNDBC UOVBLMADNOUNVBVCPJQVCUNURVCVAPJUNURUPUAKZHQZVCUQVDUPHUBRUNUPBCZVEVCLAUCZVFV EVDUPHQZVCVFVDBCVEVHLUDUPVDUEOUPUFSUGTUNVAUSPHQPUTPUSHUHRUNUSAUJUKTUIUNVCUN UPVGULUMSS $. @@ -369638,7 +369638,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the sinppi $p |- ( A e. CC -> ( sin ` ( A + _pi ) ) = -u ( sin ` A ) ) $= ( cc wcel cpi caddc co csin cfv ccos cmul cneg wceq sinadd mpan2 cc0 oveq2i picn c1 eqtrd eqtrid cospi sincl neg1cn mulcom mulm1 syl sinpi coscl mul01d - oveq12d negcld addid1d ) ABCZADEFGHZAGHZDIHZJFZAIHZDGHZJFZEFZUOKZUMDBCUNVAL + oveq12d negcld addridd ) ABCZADEFGHZAGHZDIHZJFZAIHZDGHZJFZEFZUOKZUMDBCUNVAL QADMNUMVAVBOEFVBUMUQVBUTOEUMUQUORKZJFZVBUPVCUOJUAPUMUOBCZVDVBLAUBZVEVDVCUOJ FZVBVEVCBCVDVGLUCUOVCUDNUOUESUFTUMUTUROJFOUSOURJUGPUMURAUHUITUJUMVBUMUOVFUK ULSS $. @@ -369668,8 +369668,8 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the sinhalfpip $p |- ( A e. CC -> ( sin ` ( ( _pi / 2 ) + A ) ) = ( cos ` A ) ) $= ( cc wcel cpi c2 cdiv co caddc csin cfv ccos cmul cc0 halfpire recni sinadd - wceq c1 oveq1i eqtrid mpan sinhalfpi coscl mulid2d coshalfpi mul02d oveq12d - sincl addid1d 3eqtrd ) ABCZDEFGZAHGIJZULIJZAKJZLGZULKJZAIJZLGZHGZUOMHGUOULB + wceq c1 oveq1i eqtrid mpan sinhalfpi coscl mullidd coshalfpi mul02d oveq12d + sincl addridd 3eqtrd ) ABCZDEFGZAHGIJZULIJZAKJZLGZULKJZAIJZLGZHGZUOMHGUOULB CUKUMUTQULNOULAPUAUKUPUOUSMHUKUPRUOLGUOUNRUOLUBSUKUOAUCZUDTUKUSMURLGMUQMURL UESUKURAUHUFTUGUKUOVAUIUJ $. @@ -369678,7 +369678,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the sinhalfpim $p |- ( A e. CC -> ( sin ` ( ( _pi / 2 ) - A ) ) = ( cos ` A ) ) $= ( cc wcel cpi c2 cdiv co cmin csin cfv ccos cmul wceq halfpire recni sinsub - cc0 c1 oveq1i eqtrid sinhalfpi coscl mulid2d coshalfpi sincl mul02d oveq12d + cc0 c1 oveq1i eqtrid sinhalfpi coscl mullidd coshalfpi sincl mul02d oveq12d mpan subid1d 3eqtrd ) ABCZDEFGZAHGIJZULIJZAKJZLGZULKJZAIJZLGZHGZUOQHGUOULBC UKUMUTMULNOULAPUHUKUPUOUSQHUKUPRUOLGUOUNRUOLUASUKUOAUBZUCTUKUSQURLGQUQQURLU DSUKURAUEUFTUGUKUOVAUIUJ $. @@ -369688,7 +369688,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the coshalfpip $p |- ( A e. CC -> ( cos ` ( ( _pi / 2 ) + A ) ) = -u ( sin ` A ) ) $= ( cc wcel cpi c2 cdiv co ccos cfv cmul csin cmin cc0 caddc coshalfpi oveq1i - cneg eqtrid c1 wceq coscl mul02d sinhalfpi sincl mulid2d oveq12d recni mpan + cneg eqtrid c1 wceq coscl mul02d sinhalfpi sincl mullidd oveq12d recni mpan halfpire cosadd df-neg a1i 3eqtr4d ) ABCZDEFGZHIZAHIZJGZUOKIZAKIZJGZLGZMUTL GZUOANGHIZUTQZUNURMVAUTLUNURMUQJGMUPMUQJOPUNUQAUAUBRUNVASUTJGUTUSSUTJUCPUNU TAUDUERUFUOBCUNVDVBTUOUIUGUOAUJUHVEVCTUNUTUKULUM $. @@ -369698,8 +369698,8 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the coshalfpim $p |- ( A e. CC -> ( cos ` ( ( _pi / 2 ) - A ) ) = ( sin ` A ) ) $= ( cc wcel cpi c2 cdiv cmin ccos cfv cmul csin caddc cc0 wceq halfpire recni - co oveq1i eqtrid c1 cossub coshalfpi coscl mul02d sinhalfpi mulid2d oveq12d - mpan sincl addid2d 3eqtrd ) ABCZDEFQZAGQHIZUMHIZAHIZJQZUMKIZAKIZJQZLQZMUSLQ + co oveq1i eqtrid c1 cossub coshalfpi coscl mul02d sinhalfpi mullidd oveq12d + mpan sincl addlidd 3eqtrd ) ABCZDEFQZAGQHIZUMHIZAHIZJQZUMKIZAKIZJQZLQZMUSLQ USUMBCULUNVANUMOPUMAUAUHULUQMUTUSLULUQMUPJQMUOMUPJUBRULUPAUCUDSULUTTUSJQUSU RTUSJUERULUSAUIZUFSUGULUSVBUJUK $. @@ -369716,7 +369716,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is ( ( ( sin ` A ) x. ( sin ` B ) ) + ( ( sin ` C ) x. ( sin ` D ) ) ) = ( ( sin ` ( B + C ) ) x. ( sin ` ( A + C ) ) ) ) $= ( cc wcel wa caddc co cpi wceq cmin ccos cfv c2 cdiv csin cneg syl 3adant3 - cmul addcl 3ad2ant2 coscld negnegd addid2 oveq1d 0cnd adantr pnpcan2d simp3 + cmul addcl 3ad2ant2 coscld negnegd addlid oveq1d 0cnd adantr pnpcan2d simp3 w3a oveq2d 3eqtr3rd df-neg eqtr4di fveq2d cosmpi cosneg negeqd eqtr3d subcl cc0 3eqtr3d adantl subnegd wne 3ad2ant1 subcld addcld 2cnne0 divdir syl3anc eqtrd a1i nppcan3d sinmul oveq12d simplr simpll simprl 3jca addass 3ad2ant3 @@ -369796,7 +369796,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is ( cpi c3 c2 co cmul wcel cr clt wbr csin cfv ccos wa wb pire halfpire caddc cc0 cxr cdiv cioo w3a 3re remulcli rexr elioo2 mp2an cmin cneg pidiv2halves syl2an breq1i ltaddsub mp3an12 bitr3id ltsubadd mp3an23 c1 oveq1i 2cn recni - df-3 ax-1cn adddiri divcan2i mulid2i oveq12i 3eqtrri breq2i bitr2di anbi12d + df-3 ax-1cn adddiri divcan2i mullidi oveq12i 3eqtrri breq2i bitr2di anbi12d 2ne0 resubcl mpan2 sincosq2sgn ancom 3imtr3i syl3an1 3expib sylbid resincld lt0neg2d anbi2d sylibd wceq recn pncan3 sylancr fveq2d recnd sinhalfpip syl cc eqtr3d breq1d coshalfpip sylibrd 3impib sylbi ) ABCBDUAEZFEZUBEGZAHGZBAI @@ -369817,7 +369817,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is ( ( sin ` A ) < 0 /\ 0 < ( cos ` A ) ) ) $= ( c3 cpi c2 co cmul wcel cr clt wbr csin cfv cc0 wa halfpire caddc c1 c4 cc ccos cdiv cioo w3a cxr wb 3re remulcli rexri 2re pire elioo2 cmin cneg df-3 - mp2an oveq1i 2cn ax-1cn recni adddiri 2ne0 divcan2i mulid2i oveq12i 3eqtrri + mp2an oveq1i 2cn ax-1cn recni adddiri 2ne0 divcan2i mullidi oveq12i 3eqtrri breq1i ltaddsub mp3an12 bitr3id ltsubadd mp3an23 df-4 oveq2i wne 4cn 2cnne0 div12 mp3an 4d2e2 mulcomi eqtri breq2i bitr2di anbi12d resubcl mpan2 sylbir wceq sincosq3sgn syl3an1 3expib sylbid resincld lt0neg1d anbi1d sylibd recn @@ -369892,11 +369892,11 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is ccos ctan syl 3nn nndivre sylancl resubcl 2re remulcl resincld 2cn wne 2ne0 divcan2d fveq2d wceq cos2t eqtr3d cioc wa eliooord simpld 2pos divgt0d pire cle simprd pigt2lt4 simpri 2t2e4 breqtrri pm3.2i ltdivmul mp3an mpbir lttrd - mp1i c4 mulid2i breqtrrdi ltdivmul2 syl112anc mpbird ltled cxr elioc2 mp2an + mp1i c4 mullidi breqtrrdi ltdivmul2 syl112anc mpbird ltled cxr elioc2 mp2an w3a 0xr syl3anbrc cos01bnd cos01gt0 wi 0re mpd lt2sqd mpbid ltmul2 ltsub1dd ltle eqbrtrd 3re 4re readdcl 3lt4 gt0ne0d sqgt0d 3pos ltmul1 mpbii ltsub2dd - ax-1cn addcl subcld mulid2d oveq2d oveq1d subdid oveq1i eqtrid eqtrd 3eqtrd - oveq12d mulassd mulid1d eqtr4d df-3 breqtrrd cn0 sq1 addsubd 3eqtr4d adddid + ax-1cn addcl subcld mullidd oveq2d oveq1d subdid oveq1i eqtrid eqtrd 3eqtrd + oveq12d mulassd mulridd eqtr4d df-3 breqtrrd cn0 sq1 addsubd 3eqtr4d adddid binom2sub 2timesi addassd eqtr3di assraddsubd mvrladdd subcl subdird mul12d sqvald subadd4d addcomi joinlmuladdmuld 3brtr4d mulgt0d oveq2i 2nn0 mulcomd eqtri expp1 3cn 3ne0 divassd eqtr2d 3eqtr3d sin01bnd reexpcl resubcld sin2t @@ -370016,7 +370016,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is negative. (Contributed by NM, 17-Aug-2008.) $) sinq34lt0t $p |- ( A e. ( _pi (,) ( 2 x. _pi ) ) -> ( sin ` A ) < 0 ) $= ( cpi c2 cmul co cioo wcel csin cfv cc0 clt cneg cmin cr wb caddc picn pire - wbr syl elioore addid2i eqcomi 2timesi oveq12i eleq2i wa 0re iooshf mpanr12 + wbr syl elioore addlidi eqcomi 2timesi oveq12i eleq2i wa 0re iooshf mpanr12 mpan2 bitr4id ibi sinq12gt0 cc wceq sinmpi breqtrd resincld lt0neg1d mpbird recnd ) ABCBDEZFEZGZAHIZJKSJVFLZKSVEJABMEZHIZVGKVEVHJBFEGZJVIKSVEVJVEANGZVE VJOABVCUAZVKVEAJBPEZBBPEZFEZGZVJVDVOABVMVCVNFVMBBQUBUCBQUDUEUFVKBNGZVJVPOZR @@ -370058,7 +370058,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is ( sin ` ( A x. ( _pi / 2 ) ) ) = ( cos ` ( B x. ( _pi / 2 ) ) ) ) $= ( cc wcel caddc co c1 wceq w3a cpi c2 cdiv cmul cmin ccos cfv mulcl 3adant3 mpan2 eqtr3d csin picn 2cn 2ne0 divcli coshalfpim syl 3ad2ant1 adddir oveq1 - mp3an3 mulid2i eqtrdi 3ad2ant3 wb subadd mp3an3an mpbird fveq2d ) ACDZBCDZA + mp3an3 mullidi eqtrdi 3ad2ant3 wb subadd mp3an3an mpbird fveq2d ) ACDZBCDZA BEFZGHZIZJKLFZAVEMFZNFZOPZVFUAPZBVEMFZOPUTVAVHVIHZVCUTVFCDZVKUTVECDZVLJKUBU CUDUEZAVEQSZVFUFUGUHVDVGVJOVDVGVJHZVFVJEFZVEHZVDVBVEMFZVQVEUTVAVSVQHZVCUTVA VMVTVNABVEUIUKRVCUTVSVEHVAVCVSGVEMFVEVBGVEMUJVEVNULUMUNTUTVAVPVRUOZVCVMUTVL @@ -370070,7 +370070,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is ( cos ` ( _pi / 4 ) ) = ( 1 / ( sqrt ` 2 ) ) ) $= ( cpi c4 cdiv co csin cfv c1 c2 csqrt wceq cmul cc wcel ax-mp pire 2ne0 cc0 recni wbr clt ccos caddc halfcn ax-1cn 2halves sincosq1eq oveq2i divmuldivi - mp3an 2cn mulid2i 2t2e4 oveq12i eqtri fveq2i recidi oveq1i redivcli mulassi + mp3an 2cn mullidi 2t2e4 oveq12i eqtri fveq2i recidi oveq1i redivcli mulassi 2re 3eqtr3i mulcli sin2t sinhalfpi 4re 4ne0 resincl remulcli 0le2 sqrtmulii cr msqge0i sqrt1 3eqtr3ri wne wa cle sqrtcli sqrt2re sqrt00 mp2an necon3bii wb mpbir pm3.2i divmul2 0re cioc 4pos divgt0ii 1re pigt2lt4 simpri ltdiv1ii @@ -370106,7 +370106,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is cexp csin ccos csqrt pire 6re gt0ne0ii redivcli recni sincl recoscl mulassi 6pos 2ne0 sin2t eqtr4i caddc 3cn divcli reccli oveq1i dividi ax-1cn divdiri 3ne0 df-3 3eqtr3ri sincosq1eq mp3an picn divmuldivi 3t2e6 oveq2i 6cn 3eqtri - divassi fveq2i eqtr3i mulid2i oveq12i eqtri wne wa wb mulcli pipos divgt0ii + divassi fveq2i eqtr3i mullidi oveq12i eqtri wne wa wb mulcli pipos divgt0ii cioo 2lt6 2re 2pos pm3.2i ltdiv2 mpbi w3a halfpire rexr elioo2 syl2an mp2an 0re cxr mpbir3an sincosq1sgn simpri mulcan2 mvllmuli 3re 3pos sqrtpclii cle sqdivi ltleii sqsqrti sq2 sqrtge0i divge0i sqrtsqi cmin 4cn divsubdir 4m1e3 @@ -370169,11 +370169,11 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is (Proof modification is discouraged.) $) pige3ALT $p |- 3 <_ _pi $= ( vx c1 c3 cmul co cpi cle ci ce cfv cabs cc0 wtru wcel ax-mp cr a1i wceq - cc ax-icn vy 3cn mulid2i wbr cdiv cneg cicc cv cmpt cmin tru cxr 0xr pirp + cc ax-icn vy 3cn mullidi wbr cdiv cneg cicc cv cmpt cmin tru cxr 0xr pirp crp 3rp rpdivcl mp2an rpxr rpge0 lbicc2 mp3an ubicc2 pm3.2i 0re pire 3ne0 wa 3re redivcli ccncf efcn cres wss iccssre ax-resscn sstri resmpt wf cdv mp1i cdm ssidd simpr mulcl sylancr fmpttd cpr cnelprrecn ax-1cn dvmptcmul - dvmptid mulid1i mpteq2i eqtrdi dmeqd elexi eqid dmmpti dvcn rescncf mpsyl + dvmptid mulridi mpteq2i eqtrdi dmeqd elexi eqid dmmpti dvcn rescncf mpsyl syl31anc eqeltrrd cncfmpt1f cioo crn ctg ccnfld ctopn reelprrecn recn syl efcl sylan2 sylancl ctopon cnfldtopon toponmax cin df-ss sylib adantl eff dvef feqmptd oveq2d 3eqtr3a fveq2 dvmptco 1re oveq2 fveq2d fvmpt3i absefi @@ -370222,7 +370222,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is ( cc wcel wa c2 co cpi cmul caddc csin cfv cabs wceq 2cn picn mp3an23 eqtrd c1 syl cz cdiv zcn halfcl mulass cc0 wne oveq1d eqtr3d adantl oveq2d fveq2d 2ne0 divcan1 eqcomd adantr sinper adantlr cneg cmin peano2cn mulcli sylancl - mulcl mp3an2 sylan2 ax-1cn adddir mulid2i oveq2i eqtr2di eqtr2d mpan2 pncan + mulcl mp3an2 sylan2 ax-1cn adddir mullidi oveq2i eqtr2di eqtr2d mpan2 pncan subadd23 subcl sylan sinmpi ad2antrr sincl absnegd wo zeo mpjaodan ) ACDZBU ADZEZBFUBGZUADZABHIGZJGZKLZMLZAKLZMLZNBSJGZFUBGZUADZWGWIEZWLWNMWSWLAWHFHIGZ IGZJGZKLZWNWGWLXCNWIWGXCWLWGXBWKKWGXAWJAJWFXAWJNZWEWFBCDZXDBUCZXEWHFIGZHIGZ @@ -370239,7 +370239,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is $( The sine of an integer multiple of ` _pi ` is 0. (Contributed by NM, 11-Aug-2008.) $) sinkpi $p |- ( K e. ZZ -> ( sin ` ( K x. _pi ) ) = 0 ) $= - ( cz wcel cc0 cpi cmul co caddc csin cfv cc zcn picn sylancl addid2d fveq2d + ( cz wcel cc0 cpi cmul co caddc csin cfv cc zcn picn sylancl addlidd fveq2d mulcl 0cn addcl cabs sylancr sincld wceq abssinper mpan fveq2i eqtri eqtrdi sin0 abs0 abs00d eqtr3d ) ABCZDAEFGZHGZIJZUNIJDUMUOUNIUMUNUMAKCEKCUNKCZALMA EQNZOPUMUPUMUOUMDKCZUQUOKCRURDUNSUAUBUMUPTJZDIJZTJZDUSUMUTVBUCRDAUDUEVBDTJD @@ -370374,7 +370374,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is wa subcl eqtr3d wne cz wn adantr pipos gt0ne0ii divcan1d zre adantl remulcl cmul sylancl eqeltrrd resubcl rered simplr caddc elioore eliooord simprd wb 0zd posdif mpbid divgt0d negcli negsubi pidiv2halves subaddrii eqtri simpld - eqbrtrid ltsub23d mulid1i breqtrrdi 1red ltdivmul syl112anc mpbird breqtrdi + eqbrtrid ltsub23d mulridi breqtrrdi 1red ltdivmul syl112anc mpbird breqtrdi 1e0p1 btwnnz syl3anc pm2.01da sineq0 necon3abid eqnetrd ) ABCZAUADZEUBUCQZU DZXDUEQZCZULZAFDZXDAGQZUFDZHXHXDXJGQZFDZXIXKXHXLAFXHXDBCZXBXLAIZXDJUGZXBXGU HZXDAUIKZUJXHXJBCZXMXKIXHXNXBXSXPXQXDAUMKZXJUKLUNXHXKHUOXJEUCQZUPCZUQZXHYBX @@ -370790,7 +370790,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is cle 0re absid oveq2i eqtrdi adantr pm3.2i cn0 divcan5 a1i subdid fveq2d mp2an cz syl2anc simpr oveq2 fvex fvmpt 3eqtr3d 3eqtrd efeq1 mpbid wrex eqeltrrd oveq1 oveq1d ralrimiva cneg cicc csin caddc eqtrd cexp sylancl - eleq1d oveq12d sqcld mulid1i breqtrrdi abscld 1re ltdivmul eqbrtrd ine0 + eleq1d oveq12d sqcld mulridi breqtrrdi abscld 1re ltdivmul eqbrtrd ine0 mpbird efsub efne0 diveq1bd nn0abscl nn0lt10b abs00d subeq0d ralrimivva diveq0 dff13 csqrt rexbidv neghalfpire halfpire iccssre efif1olem3 cres ex cim resinf1o f1oeq1 mpbir f1ocnv f1of mp2b ffvelcdmi remulcl rspcdva @@ -370798,7 +370798,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is sylib simpld sqrtcld recld cosq14ge0 sqrtrege0d sincossq sqsqrtd absexp 2z 2nn0 simprd absvalsq2d 3eqtr2d fveq1i fvresd eqtrid f1ocnvfv2 eqtr3d sincld coscld pncan2d pncand sq11d oveq2d efival replimd 3eqtr4d bitr2d - mulid2d eqeq12d imbitrid adantl eqeq2d 3imtr4d reximdva dffo3 df-f1o + mullidd eqeq12d imbitrid adantl eqeq2d 3imtr4d reximdva dffo3 df-f1o mpd ) AGFIUAZGFIUBZGFIUCAGFIUDZBUEZIPZCUEZIPZQZBCUFZUGZCGUHBGUHUYGAEGRE UEZUISZUJPZFIAUYQGTUKUYQULTZUYSFTAGULUYQLUNUYTUYSUMUOUPUQURZFUYTUYSUSTZ UYSUMPUPQZUYSVUATZUYTUYRUSTZVUBUYTRUSTZUYQUSTVUEUTUYQVARUYQVBVCUYRVDVEU @@ -370879,7 +370879,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is cmul cfv ccnv csn cima ax-icn recn mulcl sylancr efcl syl absefi absf ffn wa wb fniniseg mp2b sylanbrc eleqtrrdi fmpti wss frn cpi cres df-ima cmpt c2 reseq1i clt wbr cle cxr w3a 0xr 2re pire remulcli elioc2 mp2an simp1bi - ssriv resmpt rneqi wf1o 0re eqid caddc recni addid2i oveq2i eqcomi efif1o + ssriv resmpt rneqi wf1o 0re eqid caddc recni addlidi oveq2i eqcomi efif1o f1ofo forn imassrn eqsstrri eqssi df-fo mpbir2an ) FBCGCFHZCIZBJFBCUAZXDA FBKAUBZUDLZUCUEZCDXGFMZXINUFOUGUHZBXJXIPMZXINUEOJZXIXKMZXJXHPMZXLXJKPMXGP MXOUIXGUJKXGUKULXHUMUNXGUOPFNUANPHXNXLXMURUSUPPFNUQPOXINUTVAVBEVCVDZFBCUQ @@ -371217,7 +371217,7 @@ imaginary part lies in the interval (-pi, pi]. See logrnaddcl $p |- ( ( A e. ran log /\ B e. RR ) -> ( A + B ) e. ran log ) $= ( clog crn cr caddc co cc cpi cim cfv clt wbr cle syl2an ellogrn adantr cc0 wcel wceq wa cneg logrncn recn addcl simp2bi imadd reim0 adantl imcld recnd - oveq2d addid1d 3eqtrd breqtrrd simp3bi eqbrtrd syl3anbrc ) ACDZSZBESZUAZABF + oveq2d addridd 3eqtrd breqtrrd simp3bi eqbrtrd syl3anbrc ) ACDZSZBESZUAZABF GZHSZIUBZVCJKZLMVFINMVCUSSUTAHSZBHSZVDVAAUCZBUDZABUEOVBVEAJKZVFLUTVEVKLMZVA UTVGVLVKINMZAPZUFQVBVFVKBJKZFGZVKRFGVKUTVGVHVFVPTVAVIVJABUGOVBVORVKFVAVORTU TBUHUIULVBVKVBVKVBAUTVGVAVIQUJUKUMUNZUOVBVFVKINVQUTVMVAUTVGVLVMVNUPQUQVCPUR @@ -371324,7 +371324,7 @@ imaginary part lies in the interval (-pi, pi]. See 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.) $) logm1 $p |- ( log ` -u 1 ) = ( _i x. _pi ) $= ( c1 cneg clog cfv ci cpi cmul co caddc cc0 crp wcel wceq logneg ax-mp log1 - 1rp oveq1i ax-icn picn mulcli addid2i 3eqtri ) ABCDZACDZEFGHZIHZJUFIHUFAKLU + 1rp oveq1i ax-icn picn mulcli addlidi 3eqtri ) ABCDZACDZEFGHZIHZJUFIHUFAKLU DUGMQANOUEJUFIPRUFEFSTUAUBUC $. $( If a number has imaginary part equal to ` _pi ` , then it is on the @@ -371467,7 +371467,7 @@ imaginary part lies in the interval (-pi, pi]. See 3eqtrd subcl efeq1 syl ax-icn 2cn picn mulcli a1i ine0 2ne0 pire gt0ne0ii mpbid pipos mulne0i divcan2d pncan3 mpancom oveq2 rspceeqv 3ad2ant1 fveq2 eqtr2d oveq1d eqeq2d rexbidv syl5ibcom wa logcl 3adant1 zcn mulcl sylancr - adantl efadd ef2kpi oveqan12d simpl2 mulid1d fveqeq2 syl5ibrcom rexlimdva + adantl efadd ef2kpi oveqan12d simpl2 mulridd fveqeq2 syl5ibrcom rexlimdva syl2an2r impbid ) ADEZBDEZBFGZUAZAHIZBJZABKIZLMNOPZOPZCUBZOPZQPZJZCRUCZXC AXDKIZXJQPZJZCRUCZXEXMWTXAXQXBWTAXNUDPZXHSPZREZAXNXHXSOPZQPZJXQWTXRHIZTJZ XTWTYCXDXNHIZSPZXDXDSPTWTXNDEZYCYFJWTXDAUEZAUFZUGZAXNUHUIWTYEXDXDSWTXDDEX @@ -371626,7 +371626,7 @@ imaginary part lies in the interval (-pi, pi]. See crp divcld immul2d divcan2d fveq2d eqtr3d 3brtr4d a1i ltmul2d mpbird efiarg breqtrrd resinval resincld lt0neg2d mpbid caddc cicc readdcl sylancl df-neg wn cmin logimcl simpld wi renegcli ltle sylancr eqbrtrrid lesubaddd leadd1d - mpd biimpa picn addid2i breqtrdi elicc2i syl3anbrc sinq12ge0 sinppi breqtrd + mpd biimpa picn addlidi breqtrdi elicc2i syl3anbrc sinq12ge0 sinppi breqtrd ex con3d renegcld 3imtr4d simprd rpre negneg eleq1d imbitrid lognegb reim0b ltnle 3imtr3d necon3d necomd leneltd cxr w3a 0xr rexri elioo2 mp2an ) AUABZ CADEZFGZUMZAUBEZDEZHBZCYPFGZYPIFGZYPCIUCRBZYNYOYKYMACUDZYOUABYNYLCUDZUUAYKY @@ -371737,7 +371737,7 @@ imaginary part lies in the interval (-pi, pi]. See ( cc wcel cc0 wne clog cfv cabs caddc co cpi abscld recnd readdcld a1i cmul cr ci fveq2d c1 cim logcl crp absrpcl relogcl syl imcld pire cre cle ax-icn recld mulcld abstrid replimd relog eqcomd absmuld absi oveq1i eqtrid eqtr2d - wa mulid2d oveq12d 3brtr4d abslogimle leadd2dd letrd ) ABCADEVCZAFGZHGZAHGZ + wa mullidd oveq12d 3brtr4d abslogimle leadd2dd letrd ) ABCADEVCZAFGZHGZAHGZ FGZHGZVKUAGZHGZIJZVOKIJVJVKAUBZLVJVOVQVJVNVJVNVJVMUCCVNQCAUDVMUEUFMLZVJVPVJ VPVJVKVSUGMZLZNVJVOKVTKQCVJUHOZNVJVKUIGZRVPPJZIJZHGWDHGZWEHGZIJVLVRUJVJWDWE VJWDVJVKVSULMVJRVPRBCVJUKOZWAUMUNVJVKWFHVJVKVSUOSVJVOWGVQWHIVJVNWDHVJWDVNAU @@ -371809,8 +371809,8 @@ imaginary part lies in the interval (-pi, pi]. See ( cr wcel ceu cle wbr clt wa cfv co cmul cmin c1 crp cc0 syl 1re recnd wb w3a clog cdiv simpl2 simpl3 simpr ere simpl1 lelttr mp3an2i mp2and epos 0re wi lttr mp3an12i mpani mpd elrpd ltletr rpdivcld relogcl rerpdivcld resubcl - sylancl remulcld ce caddc wceq reeflog ax-1cn pncan3 sylancr eqtr4d mulid2d - cc eqbrtrd 1red ltmuldiv syl112anc mpbid efgt1p eflt syl2anc mpbird mulid1d + sylancl remulcld ce caddc wceq reeflog ax-1cn pncan3 sylancr eqtr4d mullidd + cc eqbrtrd 1red ltmuldiv syl112anc mpbid efgt1p eflt syl2anc mpbird mulridd difrp df-e breqtrrd eqbrtrrid efle posdif eqbrtrrd ltletrd relogdiv subdird lemul2 1cnd rpne0d div32d oveq12d eqtrd 3brtr3d ltsub1d ltdivmuld ) ACDZBCD ZEAFGZUAZABHGZIZBUBJZBUCKAUBJZAUCKZHGXLBXNLKZHGZXKXPXLXMMKZXOXMMKZHGXKBAUCK @@ -372121,7 +372121,7 @@ imaginary part lies in the interval (-pi, pi]. See syl3anbrc eqcomd fveq2d eqtr3d ctan abscld ccos c2 cioo 0red rerpdivcld cre 1re absrpcld resubcl sylancr recld caddc rpred 1rp rpaddcl remulcld cmul eqeltrid cif wa rpre adantl adantr ifclda ltmin mpbid ltp1d ax-1cn - addcom sylancl breqtrd ltled mulid1d ledivmuld lemul2d eqbrtrid ltletrd + addcom sylancl breqtrd ltled mulridd ledivmuld lemul2d eqbrtrid ltletrd wn a1i ltdivmuld posdif divsubdird dividd oveq1d eqtrdi absdivd gt0ne0d oveq2d rpne0d lelttrd releabsd resub re1 oveq1i 3brtr3d subled argregt0 cosq14gt0 retancld tanabsge tanarg absidd 3eqtrd absimle oveq2i subid1d @@ -372273,7 +372273,7 @@ imaginary part lies in the interval (-pi, pi]. See logdmnrp lognegb syl2anc necon3bbid mpbid necomd leneltd cxr w3a renegcli rexri elioo2 syl3anbrc wfn imf elpreima mp2b sylanbrc mprgbir ce eliooord ffn simpl adantl imcl adantr ltle sylancl mpd ellogrn logef syl efcl picn - recnd pipos gt0ne0ii cdiv caddc cmul mulid1i breqtrrdi ltdivmul syl112anc + recnd pipos gt0ne0ii cdiv caddc cmul mulridi breqtrrdi ltdivmul syl112anc 1re mpbird 1e0p1 breqtrdi cz csin ccos ci recoscld resincld crimd sylancr ad2antrr redivcld eqeltrrd 0z efeul oveq1d ax-icn mulcl addcld recl efne0 cre divcan3d eqtrd simpr reefcld reim0d eqtr3d sineq0 zleltp1 cmin df-neg @@ -372387,7 +372387,7 @@ imaginary part lies in the interval (-pi, pi]. See crn ctg recn 1red dvmptid rpssre eqid tgioo2 cpnf ioorp iooretop eqeltrri wss dvmptres relogcl peano2rem rpreccl rpcnd cres wf1o relogf1o f1of mp1i syl wf feqmptd fvres mpteq2ia eqtrdi oveq2d dvrelog eqtr3di 0cnd dvmptsub - dvmptc subid1d mpteq2dva dvmptmul mulid2d wne rpne0 recid2d oveq12d npcan + dvmptc subid1d mpteq2dva dvmptmul mullidd wne rpne0 recid2d oveq12d npcan ax-1cn sylancl mptru ) BACAUAZWSDEZFUBGZHGIJGZACWTIZUCKXBACFXAHGZFWSUDGZW SHGZLGZIXCKAWSFXAXEBMMCBBMUENKUFOZKWSCNZUGZWSXIWSBNZKWSUHPQZXJRZKAWSFBUIU LUMEZUJUKEZBBCXHXKWSMNKWSUNPKXKUGZUOKABXHUPCBVDKUQOZXOXOURZUSZXRCXNNKSUTU @@ -372410,7 +372410,7 @@ imaginary part lies in the interval (-pi, pi]. See vy cn0 wa cv cfz clog cfa csu cfzo caddc cmin rpcn adantl rpdivcl adantlr fzfid relogcld elfznn0 reexpcl syl2an faccld nndivred recnd fsummulc2 cuz simplr nn0uz eleqtrdi mulcld oveq2 fveq2 fac0 eqtrdi oveq12d oveq2d exp0d - fsum1p oveq1d 1div1e1 mulid1d eqtrd 1zzd cz nn0z ad2antlr fz1ssfz0 sylan2 + fsum1p oveq1d 1div1e1 mulridd eqtrd 1zzd cz nn0z ad2antlr fz1ssfz0 sylan2 sseli fsumshftm 0p1e1 oveq1i sumeq1i 1m1e0 fzoval eqtr3id sumeq1d 3eqtr4d syl 3eqtrd mpteq2dva cpr reelprrecn 1cnd cioo crn ctg ccnfld recn dvmptid ctopn wss rpssre eqid tgioo2 cpnf iooretop eqeltrri dvmptres cfn elfzonn0 @@ -372420,7 +372420,7 @@ imaginary part lies in the interval (-pi, pi]. See dvmptc cres wf1o wf relogf1o f1of feqmptd fvres mpteq2ia dvrelog dvmptsub eqtr3di df-neg mpteq2i eqtr4di ovexd nn0p1nn dvexp dvmptdivc nn0cnd pncan mp1i facp1 peano2cn mulcomd divcan5d dvmptco rpcnd mulneg2d rpne0 divrecd - dvmptmul mulid2d rerpdivcld mulneg1d divcan1d negsubd dvmptfsum telfsumo2 + dvmptmul mullidd rerpdivcld mulneg1d divcan1d negsubd dvmptfsum telfsumo2 eqtr4d dvmptadd pncan3 sylancr ) BEFZDUCFZUDZGAEAUEZHDUFIZBUXAJIZUGKZCUEZ LIZUXEUHKZJIZCUIMIZNZOIGAEUXAHDUJIZUXAUXDUAUEZPUKIZLIZUXMUHKZJIZMIZUAUIZU KIZNZOIAEPUXDDLIZDUHKZJIZPULIZUKIZNAEUYCNUWTUXJUXTGOUWTAEUXIUXSUWTUXAEFZU @@ -372631,9 +372631,9 @@ imaginary part lies in the interval (-pi, pi]. See logf1o fssres mp2an feqmptd dvlog2 eqtr3di dvmptco dvmptneg reccld mulcom fvres mulm1d negnegd dmeqd dmmptg sumex cbvsumv pserdv2 ssriv nnnn0 nnne0 mprg neneqd iffalsed nncn recidd nnm1nn0 nnuz 1e0p1 fveq2i eqtri isumshft - mulid2d 1zzd pncan2 sumeq2i geoisum 3eqtrd crp 1rp blcntr 1m0e1 log1 neg0 + mullidd 1zzd pncan2 sumeq2i geoisum 3eqtrd crp 1rp blcntr 1m0e1 log1 neg0 eqeq1d simpll eqeltrdi simplr expcld mul02d 0expd mul01d ifbothda eqimssi - cfn orci sumz dv11cn negex sumeq2sdv 3eqtr3d breqtrrd seqex elnnuz addid2 + cfn orci sumz dv11cn negex sumeq2sdv 3eqtr3d breqtrrd seqex elnnuz addlid sylbi 1eluzge0 1nn0 ffvelcdm elfz1eq 1m1e0 oveq2i eleq2s iftrue sylan9eqr 0nn0 eqtrid seqid divrec2d id 3eqtr4d sylan2br seqfeq climeq ) AFGZAHIZJU EUFZKZUGBLBUHZMNZMJVYJOTZUIZAVYJPTZQTZRZMUJZJAUKTZULIZUMZUNUFUGBURVYNVYJO @@ -372917,7 +372917,7 @@ imaginary part lies in the interval (-pi, pi]. See (Contributed by Mario Carneiro, 2-Aug-2014.) $) ecxp $p |- ( A e. CC -> ( _e ^c A ) = ( exp ` A ) ) $= ( cc wcel ceu ccxp co clog cfv cmul ce cc0 wne wceq ere recni cxpef mp3an12 - ene0 c1 loge oveq2i mulid1 eqtrid fveq2d eqtrd ) ABCZDAEFZADGHZIFZJHZAJHDBC + ene0 c1 loge oveq2i mulrid eqtrid fveq2d eqtrd ) ABCZDAEFZADGHZIFZJHZAJHDBC DKLUFUGUJMDNORDAPQUFUIAJUFUIASIFAUHSAITUAAUBUCUDUE $. $( Closure of the complex power function. (Contributed by Mario Carneiro, @@ -373103,7 +373103,7 @@ imaginary part lies in the interval (-pi, pi]. See ad2antrr eqtrd vx vk cn0 cv caddc imbi2d mul01 adantl cxpcl exp0d 3eqtr4d cxp0 oveq1 ax-mp 1t1e1 eqtr4i simplr simpr oveq1d cn nn0p1nn nncnd mul02d 0cn oveq12d nn0cn eqtrdi 3eqtr4a simpll mulcld syl2anc mul01d 0cxp nnne0d - wne mulne0d 3eqtr4rd pm2.61dane 1cnd adddid mulid1d syl211anc expp1 sylan + wne mulne0d 3eqtr4rd pm2.61dane 1cnd adddid mulridd syl211anc expp1 sylan cxpadd syl5ibr expcom a2d nn0ind com12 3impia ) ADEZBDEZCUCEZABCFGZHGZABH GZCIGZJZWNWLWMKZWSWTABUAUDZFGZHGZWQXAIGZJZLWTABMFGZHGZWQMIGZJZLWTABUBUDZF GZHGZWQXJIGZJZLWTABXJNUEGZFGZHGZWQXOIGZJZLWTWSLUAUBCXAMJZXEXIWTXTXCXGXDXH @@ -373655,7 +373655,7 @@ the complex square root function (the branch cut is in the same place for ce c1 cv cmin cvv cpr reelprrecn relogcl rpreccl recn wa mulcl syl sylan2 efcl ovexd cres wf1o wf relogf1o f1of mp1i feqmptd fvres mpteq2ia dvrelog eqtrdi eqtr3di ccnfld ctopn eqid ctopon cnfldtopon toponmax wss ax-resscn - cin wceq df-ss sylib cnelprrecn simpl simpr 1cnd dvmptid dvmptcmul mulid1 + cin wceq df-ss sylib cnelprrecn simpl simpr 1cnd dvmptid dvmptcmul mulrid id mpteq2dv eqtrd eff eqcomd 3eqtr4a fveq2 dvmptco dvmptres3 oveq2 fveq2d dvef oveq1d rpcn cc0 rpne0 cxpefd mpteq2dva cxpsubd cxpcld divrecd 3eqtrd wne cxp1d rpcnd mul12d mulassd eqtr4d 3eqtr4d ) BDEZFAGBAUAZHIZJKZSIZLZMK @@ -373680,7 +373680,7 @@ the complex square root function (the branch cut is in the same place for ( x e. CC |-> ( ( log ` A ) x. ( A ^c x ) ) ) ) $= ( vy wcel cc cv cfv cmul co ce cmpt cdv cr adantr recnd mulcomd mpteq2dva a1i c1 oveq2d crp clog ccxp cpr cnelprrecn wa simpr relogcl mulcld adantl - efcl 1cnd dvmptid dvmptcmul mulid1d 3eqtrd dvef wf feqmptd eqcomd 3eqtr4a + efcl 1cnd dvmptid dvmptcmul mulridd 3eqtrd dvef wf feqmptd eqcomd 3eqtr4a eff fveq2 dvmptco rpcn cc0 wne rpne0 cxpefd cxpcld oveq1d eqtrd 3eqtr4d ) BUADZEAEAFZBUBGZHIZJGZKZLIAEVRVPHIZKEAEBVOUCIZKZLIAEVPWAHIZKVNACVQVPCFZJG ZWEEEVRVRMEEEEMEUDDVNUERZWFVNVOEDZUFZVOVPVNWGUGZWHVPVNVPMDWGBUHZNZOZUIWKW @@ -373717,7 +373717,7 @@ the complex square root function (the branch cut is in the same place for c1 cv cmin cvv cr cpr cnelprrecn a1i cmnf cioc cdif difss eqsstri logdmn0 cc0 sseli logcld reccld wa mulcl efcl syl ovexd cres ccncf wf logcn cncff mp1i feqmptd fvres mpteq2ia eqtrdi dvlog eqtr3di simpl simpr 1cnd dvmptid - dvmptcmul mulid1 mpteq2dv eqtrd dvef eff 3eqtr3a fveq2 dvmptco wceq oveq2 + dvmptcmul mulrid mpteq2dv eqtrd dvef eff 3eqtr3a fveq2 dvmptco wceq oveq2 id fveq2d wne cxpefd mpteq2dva cxpsubd cxp1d cxpcld divrecd 3eqtrd mul12d mulassd eqtr4d 3eqtr4d ) BFGZFACBAUAZHIZJKZLIZMZNKACXHBJKZTXEOKZJKZMFACXE BPKZMZNKACBXEBTUBKPKZJKZMXDAEXFXKBEUAZJKZLIZXSBJKFFXHXJFUCCFFUDFUEGXDUFUG @@ -373985,7 +373985,7 @@ the complex square root function (the branch cut is in the same place for cxpge0 difrp sylancl biimpa cxple2d mpbid recnd 1cxpd breqtrd simpr 1m1e0 wb oveq2d cc cxp0d eqtrd 1le1 eqbrtrdi wo leloe mpjaodan lemul1a syl31anc eqtrdi caddc ax-1cn npcan anim1i elrp sylibr rpne0d cxpaddd cxp1d 3eqtr3d - cxpcld mulid2d 3brtr3d cxpge0d breq1 syl5ibcom imp 0re ) AIBUAJZBBCKLZMJZ + cxpcld mullidd 3brtr3d cxpge0d breq1 syl5ibcom imp 0re ) AIBUAJZBBCKLZMJZ IBNZAWSOZBPCUBLZKLZWTUDLZPWTUDLZBWTMXCXEQRZPQRZWTQRZIWTMJZOZXEPMJZXFXGMJA XHWSABXDDEAXICQRZXDQRUEACGUFZPCUCUGZUHSXCUIAXLWSABQRZIBMJZXNXLDEXOXQXRXNU JXJXKBCUKBCUQULUMSXCCPUAJZXMCPNZXCXSOZXEPXDKLZPMYABPMJZXEYBMJAYCWSXSFTYAB @@ -374010,8 +374010,8 @@ the complex square root function (the branch cut is in the same place for cxpaddle $p |- ( ph -> ( ( A + B ) ^c C ) <_ ( ( A ^c C ) + ( B ^c C ) ) ) $= ( cc0 co wbr ccxp cle c1 cr wcel adantr recnd caddc wceq wa cmul readdcld - clt addge0d rpred recxpcld mulid2d cdiv crp anim1i elrp sylibr rerpdivcld - simpr syl22anc addge01d mpbid mulid1d breqtrrd wb 1red ledivmul syl112anc + clt addge0d rpred recxpcld mullidd cdiv crp anim1i elrp sylibr rerpdivcld + simpr syl22anc addge01d mpbid mulridd breqtrrd wb 1red ledivmul syl112anc divge0 mpbird cxpaddlelem le2addd rpne0d divdird dividd eqtr3d cc divcxpd addge02d oveq12d rpcxpcld eqtr4d 3brtr3d lemuldivd eqbrtrrd 0cxpd cxpge0d eqbrtrd oveq1 breq1d syl5ibcom imp wo 0re leloe sylancr mpjaodan ) AKBCUA @@ -374155,7 +374155,7 @@ the complex square root function (the branch cut is in the same place for syl22anc subcld root1id eqtr3d diveq1d 3eqtr4d eflog eqeq12d zmodfz eqcom divcan2d 3eqtr3rd bitrid rspcev ex sylbid syl5 sylbird mpd eqeq2d rexbidv syl5ibcom pm2.61dane simp3 nnrecre elfznn0 syl2an mulexpd cxproot mulcomd - 3ad2ant2 elfzelz mulid1d syl5ibrcom impbid ) AEFZDUBFZBEFZUCZADGHZBIZABJD + 3ad2ant2 elfzelz mulridd syl5ibrcom impbid ) AEFZDUBFZBEFZUCZADGHZBIZABJD KHZLHZJUDZMDKHZLHZCUEZGHZNHZIZCODJUFHZUGHZUHZUWPUWRUXJUIAOUWPAOIZUWRUXJUW PUXKUWRUJZUJZOUXIFZAUWTUXCOGHZNHZIUXJUXMUXHOUKPZFUXNUXMUXHUPUXQUXMUWNUXHU PFUWMUWNUWOUXLULZDUMQUQUNOUXHUOQUXMOUXONHOUXPAUXMUXOUXMUXCEFZOUPFUXOEFUWP @@ -374218,7 +374218,7 @@ the complex square root function (the branch cut is in the same place for mpteq2i eqtrdi tgioo2 cpnf ioorp iooretop eqeltrri dvmptres wf1o relogf1o cdv fveq2 rpcnd eqtrd clt cexp rpred oveq1d mpbid cxr 0xr mp3an1 sylan f1of feqmptd fvres mpteq2ia oveq2d dvrelog eqtr3di oveq2 dvmptco ioossicc - mp1i mulid1d sseli sylan2 eliooord simpld elrpd elrege0 sylanbrc resabs1d + mp1i mulridd sseli sylan2 eliooord simpld elrpd elrege0 sylanbrc resabs1d cico sqrtf feqresmpt resqrtcn rescncf mpisyl sqrtcld 2rp rpsqrtcl rpmulcl rpcn dvsqrt 1re resubcl sqge0d sqsqrtd binom2sub1 addsubd 3eqtr4d breqtrd syl subge0d lerecd syldan rexr lbicc2 ubicc2 fv0p1e1 log1 fvoveq1 ge0p1rp @@ -374523,7 +374523,7 @@ be obtained in Metamath (without the need for a new definition) by the curry -> ( B logb ( B ^ M ) ) = M ) $= ( crp wcel c1 wne cz w3a cexp co clogb cmul cc cc0 cdif wceq wa rpcn adantr cpr rpne0 3jca eldifpr sylibr relogbzexp stoic4a 3adant3 logbid1 syl oveq2d - simpr zcn 3ad2ant3 mulid1d 3eqtrd ) ACDZAEFZBGDZHZAABIJKJZBAAKJZLJZBELJBUPU + simpr zcn 3ad2ant3 mulridd 3eqtrd ) ACDZAEFZBGDZHZAABIJKJZBAAKJZLJZBELJBUPU QAMNETODZURUTVBPUPUQQZAMDZANFZUQHZVCVDVEVFUQUPVEUQARSUPVFUQAUASUPUQUKUBZAMN EUCUDAABUEUFUSVAEBLUSVGVAEPUPUQVGURVHUGAUHUIUJUSBURUPBMDUQBULUMUNUO $. @@ -375015,7 +375015,7 @@ be obtained in Metamath (without the need for a new definition) by the curry cfv cr simp1 subcl sylancr simp3 necomd subeq0 necon3bid mpbird recne0d reccld logcld sylancl simp2 divcld divne0d imcld logcl 3adant3 logimcld wb addcld simpld lt2addd negpicn 2timesi imaddd 3brtr4d logimcl adddiri - cle df-3 mulid2i oveq2i 3eqtri fveq2i cre eqeltrid imval syl ang180lem1 + cle df-3 mullidi oveq2i 3eqtri fveq2i cre eqeltrid imval syl ang180lem1 eqtrid cz simprd eqtrd eqbrtrid syl112anc mpbid breqtrrdi eqtr3d adantr le2addd crp recnd resubcl subge0 eqeltrrd relogcld rered pipos mulgt0ii breqtrd ltmuldiv gt0ne0ii redivcld ltaddsubd subid1i eqnetri negsub 1rp @@ -375084,7 +375084,7 @@ be obtained in Metamath (without the need for a new definition) by the curry 0re cfv simp3 necomd subeq0 necon3bid reccld logcld simp2 divcld addcld recne0d divne0d logcl 3adant3 eqeltrid ax-icn ine0 2cn picn mulcli 2ne0 pire pipos gt0ne0ii mulne0i halfcn mpbid divmuld divreci eqtr3i eqtr3di - divcan3i eqcomd mulneg1i mulcomi negeqi divcan1i oveq1i mulassi mulid2i + divcan3i eqcomd mulneg1i mulcomi negeqi divcan1i oveq1i mulassi mullidi olcd eqtri negsubdii 1mhlfehlf simpr eqtrdi oveq1d npcan eqtrd divcan1d 3eqtr3i eqtr2d orcd df-2 negdi2 mp2an eqbrtrrid neg1z neg1rr leloe elpr mpjaodan ovexi sylibr ) CJKZCLUAZCMUAZUBZDNOPQZUCZRZDUUJRZUDZDUUKUUJUEK @@ -375124,7 +375124,7 @@ be obtained in Metamath (without the need for a new definition) by the curry ( wcel cc0 wne c1 cmin co caddc cdiv clog cfv cim cpi cneg angvald wceq w3a cpr 1cnd simp1 subcld simp3 necomd subne0d ax-1ne0 a1i oveq12d divcld simp2 recne0d logcld divne0d imaddd eqtr4d div1d fveq2d eqtrd addcld cmul - cc ci c2 eqid ang180lem3 wo fveq2 ax-icn mulcli imnegi addid2i fveq2i 0re + cc ci c2 eqid ang180lem3 wo fveq2 ax-icn mulcli imnegi addlidi fveq2i 0re picn pire crimi eqtr3i negeqi eqtri eqtrdi orim12i ovex elpr fvex 3imtr4i syl eqeltrd ) CVDFZCGHZCIHZUAZICJKZIDKZCCIJKZDKZLKZICDKZLKZIWOMKZNOZWQCMK ZNOZLKZCNOZLKZPOZQRZQUBZWNXAXFPOZXGPOZLKXIWNWSXLWTXMLWNWSXCPOZXEPOZLKXLWN @@ -375144,7 +375144,7 @@ be obtained in Metamath (without the need for a new definition) by the curry ( A F B ) ) e. { -u _pi , _pi } ) $= ( cc wcel cc0 wne cmin co caddc cmul oveq12d wceq angcan syl222anc eqtr3d c1 w3a cdiv cpi cneg cpr simp1l 1cnd simp2l simp1r divcld subdid divcan2d - wa mulid1d eqtrd subcld necomd divne1d subne0d ax-1ne0 a1i simp2r divne0d + wa mulridd eqtrd subcld necomd divne1d subne0d ax-1ne0 a1i simp2r divne0d simp3 ang180lem4 syl3anc eqeltrd ) CGHZCIJZUMZDGHZDIJZUMZCDJZUAZCDKLZCELZ DDCKLZELZMLZCDELZMLTDCUBLZKLZTELZWBWBTKLZELZMLZTWBELZMLZUCUDUCUEZVOVTWGWA WHMVOVQWDVSWFMVOCWCNLZCTNLZELZVQWDVOWKVPWLCEVOWKWLCWBNLZKLVPVOCTWBVHVIVMV @@ -375313,7 +375313,7 @@ line segment AB (the hypotenuse), and ` O ` is the signed right angle cjnegd eqtr2 sylan2 ax-1ne0 neneq mp1i pm2.65da adantl df-ne cexp oveq1 sq1 absvalsq eqtr3d 3adant3 oveq1d cjcld simp3 divcan3d eqtrd syl3an3br mpd3an3 eqcomd negeqd cjsub 1red cjred oveq2d 3eqtrd 3ad2ant2 simpl divnegd syl2anc - simpr divcld reccld cjne0d eqnetrrd divcan5d divcan2d subdid mulid1d recidd + simpr divcld reccld cjne0d eqnetrrd divcan5d divcan2d subdid mulridd recidd c2 oveq12d 3eqtr2d subcl negnegd negsubdi2 reim0bd eqeltrd recne0d rereccld resubcld negrebd absord eqeq1 orim12d sylc reim0d logcld imcld recnd fveq2d relogcld ord mpd rpaddcld rpdivcld eqtr4d negne0d divne0d logcj isosctrlem1 @@ -375364,7 +375364,7 @@ line segment AB (the hypotenuse), and ` O ` is the signed right angle simp1l simp21 simp1r subcld simp23 angneg syl22anc negsubdi2d oveq2d 1cnd subne0d ax-1ne0 a1i divcld necomd divne1d angvald div1d fveq2d wn absdivd simp3 eqcomd oveq1d abscld recnd absne0d dividd 3eqtrd neneqd isosctrlem2 - syl3anc negcld simp22 divne0d negne0d eqtr4d mulid1d subdid oveq12d eqtrd + syl3anc negcld simp22 divne0d negne0d eqtr4d mulridd subdid oveq12d eqtrd divcan2d angcan syl222anc eqtr3d mulneg2d negeqd 3eqtr4d 3eqtr3d ) CGHZDG HZUAZCIJZDIJZCDJZUBZCUCKZDUCKZLZUBZCMZCDNOZMZEOZCXEEOZXDDCNOZEOXEDMZEOZXC WMWPXEGHXEIJXGXHLWMWNWSXBUDZWOWPWQWRXBUEZXCCDXLWMWNWSXBUFZUGXCCDXLXNWOWPW @@ -375453,8 +375453,8 @@ line segment AB (the hypotenuse), and ` O ` is the signed right angle affineequiv $p |- ( ph -> ( B = ( ( D x. A ) + ( ( 1 - D ) x. C ) ) <-> ( C - B ) = ( D x. ( C - A ) ) ) ) $= ( cmul co c1 cmin caddc wceq cc0 mulcld subsubd subcld addcomd 1cnd eqtrd - eqtr2d subdird mulid2d oveq1d oveq2d pncan3d subdid oveq12d eqtr3d eqeq2d - addsubassd 3eqtr4d addid1d eqeq1d 0cnd addcand 3bitr2d eqcom bitrdi bitrd + eqtr2d subdird mullidd oveq1d oveq2d pncan3d subdid oveq12d eqtr3d eqeq2d + addsubassd 3eqtr4d addridd eqeq1d 0cnd addcand 3bitr2d eqcom bitrdi bitrd subeq0ad ) ACEBJKZLEMKDJKZNKZOZDCMKZEDBMKZJKZMKZPOZVHVJOAVGPVKOZVLAVGCCVK NKZOCPNKZVNOVMAVFVNCAVDDEDJKZMKZNKZDVPVDMKZMKZVFVNAVTVQVDNKVRADVPVDHAEDIH QZAEBIFQZRAVQVDADVPHWASWBTUCAVEVQVDNAVELDJKZVPMKVQALEDAUAIHUDAWCDVPMADHUE @@ -375467,7 +375467,7 @@ line segment AB (the hypotenuse), and ` O ` is the signed right angle affineequiv2 $p |- ( ph -> ( B = ( ( D x. A ) + ( ( 1 - D ) x. C ) ) <-> ( B - A ) = ( ( 1 - D ) x. ( C - A ) ) ) ) $= ( cmul co c1 cmin caddc wceq affineequiv subcld mulcld subcanad nnncan1d - 1cnd subdird mulid2d oveq1d eqtr2d eqeq12d 3bitr2d ) ACEBJKLEMKZDJKNKODCM + 1cnd subdird mullidd oveq1d eqtr2d eqeq12d 3bitr2d ) ACEBJKLEMKZDJKNKODCM KZEDBMKZJKZOUJUIMKZUJUKMKZOCBMKZUHUJJKZOABCDEFGHIPAUJUIUKADBHFQZADCHGQAEU JIUPRSAULUNUMUOADBCHFGTAUOLUJJKZUKMKUMALEUJAUAIUPUBAUQUJUKMAUJUPUCUDUEUFU G $. @@ -375674,9 +375674,9 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal ( ( ( abs ` ( Q - M ) ) ^ 2 ) + ( ( abs ` ( P - M ) ) ^ 2 ) ) ) $= ( cmin co cabs cc wcel adantr vx vy cfv c2 cexp caddc wceq wa cdiv addcld - cc0 halfcld eqeltrd subcld abscld recnd sqcld addid1d cmul c1 mulcld 1cnd + cc0 halfcld eqeltrd subcld abscld recnd sqcld addridd cmul c1 mulcld 1cnd simpr subeq0bd abs00bd sq0id oveq2d abssubd fveq2d eqtr3d oveq1d 3eqtr4rd - addid2d wne csn cdif cv clog cim cmpo cpi cneg simprl simprr chordthmlem2 + addlidd wne csn cdif cv clog cim cmpo cpi cneg simprl simprr chordthmlem2 cpr eqid cr pythag syl321anc pm2.61da2ne ) ADEOPZQUCZUDUEPZEFOPZQUCZUDUEP ZDFOPZQUCZUDUEPZUFPZUGZDFEFADFUGZUHZWQUKUFPWQXAWNXDWQAWQRSXCAWPAWPAWOAEFJ AFBCUFPZUDUIPZRLAXEABCHIUJULUMZUNUOUPUQTURXDWTUKWQUFXDWSXDWRXDDFADRSZXCAD @@ -375759,7 +375759,7 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal ( ( abs ` ( P - Q ) ) ^ 2 ) ) ) $= ( caddc co c2 cmin cabs cfv cmul c1 cdiv cexp addcld halfcld subcld recnd abscld sqcld cc cc0 cicc unitssre sselid mulcld 1cnd eqeltrd pnpcand 0red - cr eqidd mul02d subid1d oveq1d mulid2d eqtrd oveq12d addid2d chordthmlem3 + cr eqidd mul02d subid1d oveq1d mullidd eqtrd oveq12d addlidd chordthmlem3 eqtr2d chordthmlem4 3eqtr4rd ) AEBCMNZOUANZPNZQRZOUBNZCVMPNZQRZOUBNZMNZVP DVMPNZQRZOUBNZMNZPNVSWCPNCEPNQROUBNZDEPNQROUBNZPNDBPNQRDCPNQRSNAVPVSWCAVO AVOAVNAEVMIAVLABCGHUCUDZUEUGUFUHAVRAVRAVQACVMHWGUEUGUFUHAWBAWBAWAADVMADFB @@ -375857,7 +375857,7 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal sq2 oveq2d 3eqtr3d resincld absresq syl oveq12d sqcld caddc readdcld 4ne0 eqtr2d mulassd ccos addcld coscld sincossq adddid 2timesd ppncand pnncand 4cn eqtr4d 2t2e4 subdid sqvald 3eqtr4d mulcomd eqtrd mul4d syl2anc nncand - subsq subsubd mulid1d addcomd addsubassd lawcos syl32anc addcan2ad binom2 + subsq subsubd mulridd addcomd addsubassd lawcos syl32anc addcan2ad binom2 add32d 3eqtrd oveq2i divcan2d eqtr3d pnpcan2d resubcld subsub2d binom2sub eqtrid addassd pnpcand assraddsubd subsub3d mul12d mulcanad div23d fveq2d 3eqtr2d divcan3d ) AUDUEUFUGZJKUHUGZUHUGZIUIUJZUKUJZUHUGZUEULUGZURUJUUSGG @@ -376028,10 +376028,10 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal ( vn wcel cc c3 cexp co c1 wceq cmul caddc ax-1cn ax-mp cc0 eqtrdi oveq2d c2 cdiv wa cneg ci csqrt cfv cmin ctp w3a wss neg1cn ax-icn sqrtcl mulcli 3cn addcli halfcl subcli 3pm3.2i elexi ovex tpss mpbi sseli pm4.71ri ccxp - eqsstri cv cfz cn wb 3nn cxpeq mp3an23 w3o eltpg eleq2i 3m1e2 2cn addid2i + eqsstri cv cfz cn wb 3nn cxpeq mp3an23 w3o eltpg eleq2i 3m1e2 2cn addlidi wrex eqtr4i oveq2i cz 0z fztp eqtri rexeqi 3ne0 reccli 1cxp oveq1i eqeq2i rexbii oveq2 divcli cxpcl mp2an exp0 1t1e1 eqeq2d id exp1 1cubrlem simpli - mulid2i sqcli simpri rextp 3bitri 3bitr4g bitr4d pm5.32i bitr4i ) ABEZAFE + mullidi sqcli simpri rextp 3bitri 3bitr4g bitr4d pm5.32i bitr4i ) ABEZAFE ZXNUAXOAGHIJKZUAXNXOBFABJJUBZUCGUDUEZLIZMIZSTIZXQXSUFIZSTIZUGZFCJFEZYAFEZ YCFEZUHYDFUIYEYFYGNXTFEYFXQXSUJUCXRUKGFEXRFEUNGULOUMZUOXTUPOYBFEYGXQXSUJY HUQYBUPOURJYAYCFJFNUSXTSTUTYBSTUTVAVBVFVCVDXOXPXNXOXPAJJGTIZVEIZXQSGTIZVE @@ -376072,7 +376072,7 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal cn0 expcl sqvald oveq1d a1i expne0d divcan4d eqtr2d 3cn wne 3ne0 divcld cz 3z eqeltrd negsubd eqtr4d mulcld negcld add42d mulneg2d negeqd eqtrd negsubdid oveq2d eqtr3d adddid 3eqtrd addassd addcomd negidd divsubdird - addcld addid2d 3eqtr3d 3eqtr4d oveq12d binom3 syl2anc sqcld mulassd syl + addcld addlidd 3eqtr3d 3eqtr4d oveq12d binom3 syl2anc sqcld mulassd syl div12d divcan2d sqneg 3eqtr2d cdvds wbr n2dvds3 oexpneg syl3anc expdivd cn wn 3nn subcld divdird eqeq1d diveq0ad bitrd ) AIUCUDUEZBIUFUEZCUGUEZ UGUEZUHUIEUCUDUEZUJUDUEZCYBUFUEZGUCUDUEZUKUEZUGUEZYBULUEZUHUIYGUHUIAYAY @@ -376117,7 +376117,7 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal wne subsq syl2anc sqcld pncan2d 3eqtrd negeqd addcld mulneg1d cdvds wbr cn wn n2dvds3 oexpneg syl3anc 3eqtr4d div23d divcan4d neg2subd divneg2d 3nn eqcomd simpr eqnetrrd mulne0d 0exp ax-mp eqtrdi necon3i syl negne0d - ddcand wo mulid2d negsubd mpbid 2cn mulcl sylancr c4 sqmul 2t1e2 oveq2i + ddcand wo mullidd negsubd mpbid 2cn mulcl sylancr c4 sqmul 2t1e2 oveq2i neg0 addcomd eqtrid dcubic1lem 1cnd ax-1ne0 sqcli subnegd 2ne0 divcan2d mulcld 4cn mulneg2 oveq1i negeqi eqtr4di adddid 3eqtr4rd divdird eqtr2d sq2 quad2 divcan3d divsubdird negdi2d orbi12d mpjaodan ) AEUEUFUGZFHUHU @@ -376206,10 +376206,10 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal ( cc0 vu c3 cexp co cmul caddc wceq cv c1 cdiv cmin wa cc wrex cdif c2 c4 csn csqrt cfv wn wne adantr wcel wi cz expne0i mp3an3 syl ad2antrr simprl 3z ex oveq2d mul01d eqtrd oveq1d cn 0exp ax-mp eqtrdi simplr 0cnd eqeltrd - 3nn addcld addid2d 3eqtr3rd 2cn 2ne0 div0i sq0id 3cn a1i 3ne0 divcld 4ne0 + 3nn addcld addlidd 3eqtr3rd 2cn 2ne0 div0i sq0id 3cn a1i 3ne0 divcld 4ne0 4cn sqcld mulcl sylancr simprr sqr00d mul01i eqtr4di mulcanad 00id sqeq0d oveq12d necon3ad syld mpd oveq12 sqrtcld halfaddsub syl2anc simpld eqeq1d - 0m0e0 simprd wo adantl eqcom mulcld ax-1ne0 negcld mulid2d eqeq2d orbi12d + 0m0e0 simprd wo adantl eqcom mulcld ax-1ne0 negcld mullidd eqeq2d orbi12d jca cneg bitrdi risset wb halfcld eldifsn baib bitr3id ad2antrl 3nn0 1cnd anbi12d imbitrid eldifi eldifsni subaddd 3bitr4g subeq0ad sqvald divcan1d con3d adddird addcomd 3eqtrrd eqeq12d 3bitrd sqneg mulneg2 subnegd eqtr2d @@ -376297,7 +376297,7 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal 3t3e9 divcan2d mulcomd addassd negsubdi2d subdird 3eqtr2d 9t3e27 divcan5d eqtr3id eqeq1d adantr wbr n2dvds3 2cnd 2ne0 sqneg eqtr2d sqne0 mpbird 9nn wn mulcomi sq3 eqnetri divnegd eqidd negne0d dcubic binom3 syl2anc adddid - sqvali 3eqtr4a divmuldivd expp1 eqtr2id ax-1cn subaddrii mulid2d mulcomli + sqvali 3eqtr4a divmuldivd expp1 eqtr2id ax-1cn subaddrii mullidd mulcomli df-3 2p1e3 npncan3d eqeltrrd 3eqtr3d addsub4d pncan3d oveq1 ax-mp eqnetrd 0exp necon2i eqcom simprl simprr mulne0d subnegd bitrid subadd2d mulneg2d divcan7d div2negd eqeq2d 3bitrrd anassrs sylan2 pm5.32da rexbidva 3bitr3d @@ -376538,7 +376538,7 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal ( cc wcel caddc co c4 cexp c3 c2 c1 oveq2i sylancl 3cn addcld mulassd eqtrd cmul oveq2d oveq12d wa df-4 cn0 wceq addcl expp1 eqtrid binom3 oveq1d simpl c6 3nn0 expcl sqcld simpr mulcld mulcl sylancr adddid eqtr2id a1i df-3 2nn0 - mul32d joinlmuladdmuld sqvald eqtr4d mulcomd adddird addassd 3eqtrd mulid2d + mul32d joinlmuladdmuld sqvald eqtr4d mulcomd adddird addassd 3eqtrd mullidd 4nn0 add4d oveq1i ax-1cn 3p3e6 eqtr3id 3p1e4 addcomli ) ACDZBCDZUAZABEFZGHF ZWDIHFZWDRFZAIHFZIAJHFZBRFZRFZEFZIABJHFZRFZRFZBIHFZEFZEFZWDRFZAGHFZGWHBRFZR FZEFZUKWIWMRFZRFZGAWPRFZRFZBGHFZEFZEFZEFZWCWEWDIKEFZHFZWGGXLWDHUBLWCWDCDIUC @@ -376588,7 +376588,7 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal oveq1d divdird divsubdird div23d sqvald oveq2d sqmul sqvali oveq1i eqtrdi divrec2d mulassd 3eqtrd divcan3d eqtrd divcan4d addsub12d negcld subsub4d 3eqtr4d 2timesd subnegd negdi2d eqtr3d oveq12d eqeq1d bitr3d mp1i ax-1ne0 - wb ax-1cn halfcl divne0i 2cnne0 divmuldiv syl22anc mulid2d 2t2e4 divcan2d + wb ax-1cn halfcl divne0i 2cnne0 divmuldiv syl22anc mullidd 2t2e4 divcan2d nncand eqtr2d quad2 recidi oveq2i div1d eqtrid eqeq2d orbi12d 3bitrd ) AG PUAQZFBRQZPSQZRQZFPSQZGTQZCUBSQZUCQZDSQZRQZUDUEZUFPSQZYETQZDGTQZDPUAQZEPU AQZUCQZPSQZRQZRQZUDUEZGDUGZERQZPYPTQZSQZUEZGUUFEUCQZUUHSQZUEZUHGUUGUEZGUU @@ -376753,7 +376753,7 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal oveq2i 2ne0 divmuldivd eqtr4d oveq12d subdird expp1 eqtrid div23d 3eqtr4d df-4 wceq sqcld 4nn0 subadd23d addcomd 3eqtrd 1nn0 6nn decnncl nncni 2nn0 3cn 5nn0 deccl divcli mulcl sylancr addsubd addcld cneg add12d negsubdi2d - ppncand oveq2d addsub4d 1p2e3 oveq1i 1cnd eqtr3id mulid2d divcan2d 4t4e16 + ppncand oveq2d addsub4d 1p2e3 oveq1i 1cnd eqtr3id mullidd divcan2d 4t4e16 3eqtr3d divdiv1d eqtrdi mulcli mulne0i mul32i 2exp4 8t2e16 eqtr4i oveq12i 4p4e8 expadd mp3an 2exp8 3eqtr3i 3eqtr2d 3eqtr2i addsub12d 2timesd sqdivd pnpcan2d mvrladdd eqtri mulcomd mulassd expaddd eqeltrid eqnetri 3eqtr3rd @@ -376824,7 +376824,7 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal 2nn0 divdiv1d 3eqtr4a 2ne0 eqtr4d expmul 4t2e8 2exp8 oveq12d addcld mulcl 4nn0 sq2 halfcld addassd eqtrd adddid add4d adddird eqtr3d subcld subdird 5nn0 cmin divassd 3eqtr2d deccl simp1d eqeltrd simp2d simp3d binom2 2t2e4 - quart1cl pncan3d eqtrid div23d 3eqtr3d addsub12d subsub2d mulid2d 3eqtr3a + quart1cl pncan3d eqtrid div23d 3eqtr3d addsub12d subsub2d mullidd 3eqtr3a 3m1e2 1cnd addcomd ppncand npcand quart1lem 3eqtrrd ) AJUAUBTZFJUCUBTZUDT ZUETZGJUDTZHUETZUETIUAUBTZBIUFUBTZUDTZUETZUFUGUHTZBUCUBTZUDTZIUCUBTZUDTZB UFUBTZUGUHTZUCUHTZIUDTZBUAUBTZUCUIUJZUQUJZUHTZUETZUETZUVFUETZUETZUVIUETUV @@ -376905,10 +376905,10 @@ Pythagorean theorem ( ~ pythag ) in the case where P and Q are unequal ( ( P ^ 2 ) - ( 4 x. R ) ) ) ) ) + ( ; 2 7 x. -u ( Q ^ 2 ) ) ) ) ) $= ( c2 cmul co c3 c4 cmin cc wcel c8 cexp wceq c9 c7 cdc caddc c1 2cn sqmul cneg sylancr sq2 oveq1i eqtrdi oveq1d 4cn a1i sqcld subdird eqtr4d ax-1cn - 3cn 3p1e4 subaddrii mulid2 eqtrid syl eqtr2d mulcl 1nn0 2nn decnncl nncni + 3cn 3p1e4 subaddrii mullid eqtrid syl eqtr2d mulcl 1nn0 2nn decnncl nncni subsubd subdid 4t3e12 mulcomli mulassd eqtr3id oveq2d 3eqtr4d cn0 mulexpd 3nn0 cu2 8cn expcl sylancl mul12d eqtrd 9cn mulcld df-3 oveq2i 2nn0 expp1 - mulcomd mul4d 4t2e8 oveq12d 9t8e72 negsubdi2d 8p1e9 mulid2d eqtr3d negeqd + mulcomd mul4d 4t2e8 oveq12d 9t8e72 negsubdi2d 8p1e9 mullidd eqtr3d negeqd 7nn0 3eqtrd 7nn mulneg2 negcld addsubd addcld negsubd jca ) AELBMNZLUANZO BLUANZPDMNZQNZMNZQNZUBFLXPOUANZMNZUCXPXTMNZMNZQNZLUDUEZCLUANZUJMNZUFNZUBA XRUGLUEZDMNZUFNZXQOXRMNZYMQNZQNZEYBAYNXQYOQNZYMUFNYQAXRYRYMUFAYRPOQNZXRMN @@ -377018,7 +377018,7 @@ ridiculously long (see ~ quartfull ) if all the substitutions are wcel a1i wne 4ne0 divcld subnegd eqtrd quart1 eqeq1d simp1d simp2d negcld quart1cl eqeltrd subcld quartlem3 csqrt sqrtcld 2cnd 2ne0 divcan2d oveq1d cfv sqsqrtd eqtr2d quartlem4 sqcld halfcld addcld simp3d cv c1 wa wrex cz - 1cnd 1exp mp1i mulid2d oveq12d negeqd eqtr4d oveq1 eqeq2d rspcev syl12anc + 1cnd 1exp mp1i mullidd oveq12d negeqd eqtr4d oveq1 eqeq2d rspcev syl12anc 3z anbi12d 2cn mulcl sylancr cn quartlem2 3nn cxproot sylancl cn0 subaddd ccxp addassd eqtr3d bitr4d eqcom bitrdi addsubassd orbi12d 3nn0 expcl cdc c9 c7 quartlem1 simpld simprd mcubic mpbird dquart negsubd 3bitrd ) ARUSU @@ -377352,7 +377352,7 @@ already know is total (except at ` 0 ` ). There are branch points at ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) ) $= ( cc wcel ci casin cmul co ce c1 c2 cexp cmin csqrt caddc clog cneg asinval cfv ax-icn a1i oveq2d negicn mulcl mpan ax-1cn subcl sylancr sqrtcld addcld - sqcl asinlem logcld mulassd mulneg2i negeqi negneg1e1 3eqtri oveq1i mulid2d + sqcl asinlem logcld mulassd mulneg2i negeqi negneg1e1 3eqtri oveq1i mullidd ixi eqtrid 3eqtr2d fveq2d cc0 wne wceq eflog syl2anc eqtrd ) ABCZDAERZFGZHR DAFGZIAJKGZLGZMRZNGZORZHRZVQVJVLVRHVJVLDDPZVRFGZFGDVTFGZVRFGZVRVJVKWADFAQUA VJDVTVRDBCZVJSTVTBCVJUBTVJVQVJVMVPWDVJVMBCSDAUCUDVJVOVJIBCVNBCVOBCUEAUJIVNU @@ -377394,7 +377394,7 @@ already know is total (except at ` 0 ` ). There are branch points at fveq2d c1 cpi c2 cdiv cioo wa simpl sylancr recld reefcld simpr neghalfpirx cr w3a wb halfpire rexri elioo2 mp2an sylib simp1d recoscld efgt0 cosq14gt0 cxr adantl mulgt0d cim csin caddc efeul imcld recnd resincld addcld remul2d - crred mulneg1i ixi negeqi negneg1e1 3eqtri oveq1i negicn a1i mulassd mulid2 + crred mulneg1i ixi negeqi negneg1e1 3eqtri oveq1i negicn a1i mulassd mullid imre adantr 3eqtr3a eqtrd oveq2d 3eqtrd breqtrrd ) ABCZADEZUAUBUCFZGZWPUDFC ZUEZHIAJFZDEZKEZWOLEZJFZWTKEZDEZMWSXBXCWSXAWSWTWSIBCZWNWTBCZNWNWRUFZIAOUGZU HZUIZWSWOWSWOULCZWQWOMPZWOWPMPZWSWRXMXNXOUMZWNWRUJWQVDCWPVDCWRXPUNUKWPUOUPW @@ -377427,7 +377427,7 @@ already know is total (except at ` 0 ` ). There are branch points at eleqtrdi eqeltrd mulneg12 fveq2d breqtrrd addgt0d readdd mulgt0d cosval wne recni 2ne0 divrec2d remul2 mvrraddd eqsqrt2d crn cim cle elioore adantl crp pire pirp rphalflt ax-mp ltnegi mpbi eliooord simpld lttrd imre ixi mulassd - mulneg1i negeqi 3eqtri mulid2 3eqtr3a ltled eqbrtrd ellogrn syl3anbrc logef + mulneg1i negeqi 3eqtri mullid 3eqtr3a ltled eqbrtrd ellogrn syl3anbrc logef simprd ) ABCZADEZFGUAHZIZUUPUBHZCZUCZAUDEZUEEZJIZJUVAKHZRUVAGUFHZLHZUGEZMHZ UHEZKHZUVCJAKHZKHZAUUTUVABCZUVBUVJNUUNUVMUUSAUIOZUVAUJUKUUTUVIUVKUVCKUUTUVK VDEZUHEZUVIUVKUUTUVOUVHUHUUTUVDUVOUVDLHZMHUVOUVHUUTUVDUVOUUTJBCZUVMUVDBCZUL @@ -377484,11 +377484,11 @@ already know is total (except at ` 0 ` ). There are branch points at 2-Apr-2015.) $) asin1 $p |- ( arcsin ` 1 ) = ( _pi / 2 ) $= ( c1 cfv ci cneg cmul cmin csqrt caddc clog cpi wcel ax-mp cc0 ax-icn eqtri - co fveq2i clt wbr halfpire c2 cexp cdiv wceq ax-1cn asinval addid1i mulid1i + co fveq2i clt wbr halfpire c2 cexp cdiv wceq ax-1cn asinval addridi mulridi casin cc sq1 oveq2i 1m1e0 sqrt0 oveq12i efhalfpi 3eqtr4i crn cim cle mulcli ce recni pipos cr wb pire lt0neg2 mpbi crp pirp rphalfcl rpgt0 renegcli 0re - lttri mp2an addid2i crimi eqtr3i breqtrri rphalflt eqbrtri ellogrn mpbir3an - ltleii logef mulneg1i negeqi negneg1e1 3eqtri oveq1i negicn mulassi mulid2i + lttri mp2an addlidi crimi eqtr3i breqtrri rphalflt eqbrtri ellogrn mpbir3an + ltleii logef mulneg1i negeqi negneg1e1 3eqtri oveq1i negicn mulassi mullidi ixi ) AUIBZCDZCAEPZAAUAUBPZFPZGBZHPZIBZEPZWRCJUAUCPZEPZEPZXFAUJKWQXEUDUEAUF LXDXGWREXDXGVBBZIBZXGXCXIICMHPCXCXICNUGWSCXBMHCNUHXBMGBMXAMGXAAAFPMWTAAFUKU LUMOQUNOUOUPUQQXGIURKZXJXGUDXKXGUJKJDZXGUSBZRSXMJUTSCXFNXFTVCZVAZXLXFXMRXLM @@ -377674,7 +377674,7 @@ already know is total (except at ` 0 ` ). There are branch points at 2ne0 cji 2re cjre oveq12i eqtr4i oveq1i cjcld mulneg12 eqtrid cjsub syl2anc cim imsub reim adantr oveq2d eqtr4d df-neg im1 eqtr4di recl negne0d eqnetrd recnd logcj 1re mp1i cjcl mulneg1 eqtrd oveq12d subneg 3eqtrd fveq2d eqtr3d - cr imadd addid2d 3eqtr2d negsub atandmcj simp3bi simp2bi negsubdi2d atanval + cr imadd addlidd 3eqtr2d negsub atandmcj simp3bi simp2bi negsubdi2d atanval cjadd negeqd 3eqtr4d jca ) ABCZAUADZEFZUBZAUCUDZCZAUCDZGDZAGDZUCDZHYIYFAIJZ FZAIFZYKYFYHUEZYIYHYQYFYHUFZAYPYGEAYPHYGYPUADZEAYPUAUGUUAIUADZJEJEIKUHUUBEU LUIUJUKUMUNUOYIYHYRYTAIYGEAIHYGUUBEAIUAUGULUMUNUOAUPUQZYIILURMZNIAOMZPMZQDZ @@ -377740,8 +377740,8 @@ already know is total (except at ` 0 ` ). There are branch points at wne leloe biimpa cneg ax-1cn ax-icn mulcl addcl simp3bi logcld subcl addcld simp2bi pire renegcli a1i imcld readdcld cioo im1 oveq1i df-neg eqtr4i reim imsub negeqd 3eqtr4a lt0neg2d eqbrtrd argimlt0 eliooord simpld simpr oveq2d - imadd recnd addid2d eqtrid 3eqtr2d breqtrrd argimgt0 ltaddpos2d mpbid lttrd - imaddd 0red simprd ltadd2dd addid1d breqtrd ltled ellogrn syl3anbrc reim0bd + imadd recnd addlidd eqtrid 3eqtr2d breqtrrd argimgt0 ltaddpos2d mpbid lttrd + imaddd 0red simprd ltadd2dd addridd breqtrd ltled ellogrn syl3anbrc reim0bd syl eqtr2d addcomd ad2antrr logrncl readdcl 1red 0lt1 addge01 ltletrd elrpd 1re relogcld logrnaddcl eqeltrd resubcl 1m0e1 lesub2d lecasei jaodan syldan eqbrtrrid ) AUAUBBZCAUCDZEFZCYKGFZCYKHZUDZIUEAUFJZKJZUGDZIYPSJZUGDZKJZUGUHZ @@ -377799,13 +377799,13 @@ already know is total (except at ` 0 ` ). There are branch points at oveq2i subneg eqtrid addcom 3eqtrd fveq2d cle resub rei recld recnd subid1d cdiv eqtrd gt0ne0 sylancom eqnetrd fveq2 re0 eqtrdi necon3i syl wi 0re ltle simpr mpd breqtrrd logimul syl3anc eqtr3d halfpire recni mulcli imadd ax-mp - reim rere eqtr3i readd addid1d logneg2 syl2anc adddid negsub negsubdi2 picn + reim rere eqtr3i readd addridd logneg2 syl2anc adddid negsub negsubdi2 picn negeqd oveq1d assraddsubd 3eqtr3d subcli imsub mp2an oveq12i negcli negsubi pire pidiv2halves subaddrii eqtri oveq12d imcld logimcld wb mpbid ltsubaddd ctan imi 3brtr4d tanarg argregt0 mpbird cxr rexri addsub4d subnegi resubcld 3eqtri readdcl renegcli renegcld simpld leneg ltleaddd negsubd breqtrd 0red - simprd eqbrtrrid peano2rem peano2re ltm1d ltp1d lttrd ltdiv1 tanord addid2d - syl112anc ltadd1dd addid2i breqtrdi elioo2 syl3anbrc eqeltrd ) AUAUBBZCADEZ + simprd eqbrtrrid peano2rem peano2re ltm1d ltp1d lttrd ltdiv1 tanord addlidd + syl112anc ltadd1dd addlidi breqtrdi elioo2 syl3anbrc eqeltrd ) AUAUBBZCADEZ FGZUCZHIAUDJZKJZLEZHUVOMJZLEZMJNEZAIMJZLEZNEZAIKJZLEZNEZMJZOKJZOUEZOUFJZUVN UVTUVQNEZUVSNEZMJUWCOUGVRJZKJZUWFUWMUEZKJZMJZUWHUVNUVQUVSUVNUVPUVNHPBZUVOPB ZUVPPBUHUVNIPBZAPBZUWSSUVNUXAUVRCUIZUVPCUIZUVNUVKUXAUXBUXCUJUVKUVMUKAULUMZU @@ -377924,7 +377924,7 @@ already know is total (except at ` 0 ` ). There are branch points at cdiv wceq catan cdm clog atanval oveq2d atancl mulassd halfcl ax-mp mulassi cneg mul12i 2ne0 divcan2i oveq2i eqtri 3eqtri oveq1i ax-1cn atandm2 simp1bi ixi mulcl subcl simp2bi logcld simp3bi subcld mulm1d eqtrid 2mulicn 3eqtr3d - addcl negsubdi2d fveq2d efsub syl2anc eflog oveq12d negsub mulid1d pnpcan2d + addcl negsubdi2d fveq2d efsub syl2anc eflog oveq12d negsub mulridd pnpcan2d 3eqtr3a 3eqtr4d adddid 2timesd oveq1d subdid subneg addcom divcan5d sylancl eqtrd 3eqtrd ine0 atandm negicn subeq0 necon3bid mpbird eqnetrrd divsubdird wb wa dividd ) AUAUBBZCDAUAEZFGFGZHEIDAFGZJGZUCEZIXIKGZUCEZKGZHEZXKHEZXMHEZ @@ -378031,7 +378031,7 @@ already know is total (except at ` 0 ` ). There are branch points at rexri mvllmuld eqeq1d biimpa reim0bd tanhbnd sselid readdcl 0red ltsubadd2d df-neg elrpd crp difrp resubcld relogrn gt0ne0d w3o recl 0re mpjao3dan picn lttri4 divneg mp3an renegcli 2pos ltdivmul breqtrrd remulcl ltmuldiv2 ltled - immul2 eqbrtrd ellogrn negeqd halfcl mp1i divcan1i mul2neg mulid2 3eqtrd ) + immul2 eqbrtrd ellogrn negeqd halfcl mp1i divcan1i mul2neg mullid 3eqtrd ) ABCZAUADZEFGHZIZUWQUBHCZUCZAUDDZUEDZJFGHZKJUXALHZUFHZUGDZKUXDUHHZUGDZUFHZLH ZUXCFJALHZLHZIZLHZAUWTUXAUEUMCZUXBUXJSUWOUWSAUIDZMUNZUXOAUJZAUKULZUXAUONUWT UXIUXMUXCLUWTUXHUXFUFHZIUXIUXMUWTUXHUXFUWTUXGUWTKBCZUXDBCZUXGBCZUPUWTJBCZUX @@ -378278,13 +378278,13 @@ already know is total (except at ` 0 ` ). There are branch points at wa cdm simpr sselid atandm2 sylib simp1d mulcl sylancr subcl simp2d addcl w3a logcld simp3d subcld ovexd cneg atans2 simp2bi negex csn cdif logdmss adantl crn wf1o wf logf1o f1of ax-mp ffvelcdmi logrncn ccnfld ctopn sylan - 3syl 1cnd dvmptc dvmptid dvmptcmul mulid1i mpteq2i eqtrdi dvmptsub df-neg + 3syl 1cnd dvmptc dvmptid dvmptcmul mulridi mpteq2i eqtrdi dvmptsub df-neg 0cnd eqtr4di wss ssrab3 eqid cnfldtopon dvmptres fveq2 oveq2 dvmptco irec oveq2d oveq2i oveq1i eqtrid eqtrd oveq12d subneg 3eqtrd 3eqtr3d mpteq2dva - eqtr3id negicn addcom sylancl 3eqtr3rd oveq1d eqtr3d 2cn eqtri mulid1d wb + eqtr3id negicn addcom sylancl 3eqtr3rd oveq1d eqtr3d 2cn eqtri mulridd wb ine0 subeq0 necon3bid mpbird divcan5d mp1i toponrestid fssres mp2an fvres - atansopn feqmptd mpteq2ia eqtr2di recdiv2d reccld divrecd subdird mulid2i - mul32d mulm1d comraddd simp3bi dvmptadd addid2i divdiv2d divdird divcan3d + atansopn feqmptd mpteq2ia eqtr2di recdiv2d reccld divrecd subdird mullidi + mul32d mulm1d comraddd simp3bi dvmptadd addlidi divdiv2d divdird divcan3d dvlog ixi negsub div23d sqcld atandm4 simprbi syl pnpcand divreci 2timesi divsubdird negsubi 3eqtri mulcomd i2 subsq atandm eqnetrrd halfcl df-atan reseq1i atanf fdmi sseqtri resmpt 2ne0 divcan6 mp4an divcli divassd mptru @@ -378358,7 +378358,7 @@ already know is total (except at ` 0 ` ). There are branch points at ( cc wcel cfv c1 caddc ci cdiv cmul cmin cexp sylancr wceq oveq12d adantl co cn vm vk cabs clt wbr wa cseq c2 clog catan cli cneg cv cmpt nnuz 1zzd ax-icn halfcl mp1i simpl mulcl negcld absnegd absmul absi oveq1i cr abscl - cvv adantr recnd mulid2d eqtrid 3eqtrd simpr eqbrtrd ax-1cn subneg fveq2d + cvv adantr recnd mullidd eqtrid 3eqtrd simpr eqbrtrd ax-1cn subneg fveq2d logtayl syl2anc negeqd breqtrd seqex a1i eqbrtrrd oveq2 id eqid fvmpt cn0 ovex nnnn0 expcl syl2an nncn cc0 wne nnne0 divcld eqeltrd serf ffvelcdmda cuz eleqtrdi cfz elfznn eqtr4d sersub climsub addcl cdm bndatandm atandm2 @@ -378495,7 +378495,7 @@ already know is total (except at ` 0 ` ). There are branch points at wbr cv cexp cmpt cdvds cif 2cn nn0cn mulcl sylancr ax-1cn pncan sylancl oveq1d wne 2ne0 divcan3 mp3an23 syl eqtrd oveq2d mpteq2ia eqtr4i seqeq3 cn ax-mp breq1i wb cr reexpcl mpan 2nn0 nn0mulcl nn0p1nn nndivred recnd - neg1rr eqeltrd adantl oveq1 id oveq12d iserodd cuz cfv cres addid2 0cnd + neg1rr eqeltrd adantl oveq1 id oveq12d iserodd cuz cfv cres addlid 0cnd mptru 1eluzge0 a1i 1nn0 wo wa wn ioran leibpilem1 simprd simpld sylan2b ifclda fmpti ffvelcdmi mp1i cfz simpr 1m1e0 oveq2i eleqtrdi fveq2d 0nn0 elfz1eq iftrue orcs c0ex fvmpt eqtrdi seqid 1zzd eleqtrrdi nnne0 neneqd @@ -378544,7 +378544,7 @@ already know is total (except at ` 0 ` ). There are branch points at peano2re mpbid nnred nngt0d lerec syl22anc oveq2 oveq1d oveq2d eqid fvmpt ovex weq 3brtr4d nnuz 1zzd mp1i mptex eqeltrd nnre syl2anc breqtrrd nngt0 wb cvv rpreccld rpge0d expcl nnne0d oveq12d 3eqtr4d sylib leibpilem2 ccom - catan crp mulid1d eqtr4d 1elunit climcncf oveq1 fmptco wral adantll eqtrd + catan crp mulridd eqtr4d 1elunit climcncf oveq1 fmptco wral adantll eqtrd 1re c0ex ifex fvmpt2 sylancl nfv nffvmpt1 nfcv nfeq fveq2 eqeq12d cbvralw r19.21bi ax-mp wf cdvds wo cneg ioran leibpilem1 simprd reexpcl peano2nn0 simpld cdm nn0re 2re 2pos lemul2 ax-1cn divcnv nn0ex nnrecre nnnn0 sylan2 @@ -378555,7 +378555,7 @@ already know is total (except at ` 0 ` ). There are branch points at nnex nn0cn mul02d ifeq1d ovif eqtr4di ralrimiva nfov mulcld eqeltrrd 0nn0 simpr 0p1e1 seqeq1 cuz elnnuz nnne0 neneqd biorf bicomd ifbid rgen seqfeq mpan2 sylan2br abssubge0d ltsubrp eqbrtrd atantayl2 eqbrtrid clim2ser2 cz - cabs 0z seq1 iftrue eqtri oveq2i atanrecl addid1d eqtrid breqtrd isumclim + cabs 0z seq1 iftrue eqtri oveq2i atanrecl addridd eqtrid breqtrd isumclim orcs mpteq2dva nn0z 1exp sylan9eq mptru ffvelcdmi sumex 3brtr3d cmnf cioc sumeq2dv cdif crab cres ccncf atancn wss unitssre ressatans sselii sselid sstri fss cncff feqmptd fvres mpteq2ia eqtrdi atan1 climuni mpbir ) FBGUB @@ -378663,7 +378663,7 @@ already know is total (except at ` 0 ` ). There are branch points at cr mp2an oveq12i eqtri 1lt3 wb recgt1 mpbi eqbrtri atantayl3 oveq2 oveq1d eqid oveq2d oveq12d ovex fvmpt 2nn0 nn0mulcl peano2nn0 syl expdivd neg1cn expcl sylancr cn 3nn nnexpcl nncnd nnne0d divassd expp1 expmul mp3an12 i2 - mpan oveq1i eqtrdi eqtrd mulassd expaddd neg1sqe1 2timesd 3eqtr3d mulid2i + mpan oveq1i eqtrdi eqtrd mulassd expaddd neg1sqe1 2timesd 3eqtr3d mullidi id nn0cn cz 3eqtr2d divdiv1d 3eqtrd nnmulcld divcld adantl mulcom sylancl c9 nnmulcl c4 ax-1cn 3eqtri 3eqtr2i oveq2i 3eqtr2ri fveq2i wa crp rpdivcl 3rp 2ne0 nn0z eqtr3d nn0p1nn mulcomd eqeltrd sq3 mul32d dmdcand isermulc2 @@ -378892,12 +378892,12 @@ already know is total (except at ` 0 ` ). There are branch points at ( c3 c7 co c5 cmul cc0 c2 c1 caddc c9 c6 cdc 2nn0 5nn0 deccl 1nn0 eqid 3nn0 c4 cexp cfz cv cdiv csu c8 cle cmin 0le0 c0 risefall0lem sumeq1i sum0 eqtri oveq2i wcel cn0 3cn 7nn0 expcl mp2an 5cn 7cn mulcli mul01i 2cn 3brtr4i 0nn0 - nn0cni addid2i addid1i mulid2i oveq1i 0p1e1 nn0mulcli 9nn0 2p1e3 8nn0 1p1e2 + nn0cni addlidi addridi mullidi oveq1i 0p1e1 nn0mulcli 9nn0 2p1e3 8nn0 1p1e2 cc wceq 9cn ax-mp 9t9e81 numexpp1 8cn 9t8e72 mulcomli decmul1 7t5e35 7p3e10 exp1 addcomli ax-1cn 3p1e4 4nn0 7t3e21 4cn 4p1e5 decaddi decmac dec0h 3t2e6 oveq12i 6p1e7 5t2e10 decsuc 9t3e27 7p4e11 decaddci 9t5e45 decmul1c decmul2c decma2c 3eqtr4ri log2ublem2 1m1e0 6nn0 9p5e14 decadd 5p5e10 decaddc2 sqvali - 5p1e6 3t3e9 mulassi 3eqtr2i mul12i 6p3e9 mulid1i eqtr4i 3eqtri 2m1e1 6p6e12 + 5p1e6 3t3e9 mulassi 3eqtr2i mul12i 6p3e9 mulridi eqtr4i 3eqtri 2m1e1 6p6e12 df-3 1p2e3 9t7e63 df-5 2t2e4 3m1e2 5p3e8 mulcomi exp0 df-7 3eqtr4i 00id 6cn 6t2e12 dec10p 8t2e16 breqtri ) BCUADZECFDZFDZGBUBDHBHAUCZFDIJDFDKUUOUADFDUD DZAUEFDHHLMZEMZHMZUFMZFDEBMZGMZEMZLMUGUUSBMZAEUUTHGBUUQTMZLMZGMZALBMZUVDIIH @@ -378941,8 +378941,8 @@ already know is total (except at ` 0 ` ). There are branch points at cicc oveq2i sumeq1i cn0 4nn0 ax-mp nnmulcli nnexpcl mp2an nndivre mpbi wtru cn 9nn wa 3nn sylancr 0nn0 6nn0 6p1e7 decsuc 5nn 7nn nnrei declt 7cn 7t5e35 5cn mulcomli c8 4cn 2cn 4t2e8 oveq1i eqtri 9cn 3brtr4i nngt0i nncni mulcomi - wb 3cn 3eqtr3i wceq mp3an mulcli mulid1i nn0rei pm3.2i 8nn0 ax-1cn addcomli - 9nn0 5p1e6 decaddi decaddci nn0cni addid1i 7t3e21 6cn decmac mul02i addid2i + wb 3cn 3eqtr3i wceq mp3an mulcli mulridi nn0rei pm3.2i 8nn0 ax-1cn addcomli + 9nn0 5p1e6 decaddi decaddci nn0cni addridi 7t3e21 6cn decmac mul02i addlidi 1p2e3 dec0h 7t7e49 4p1e5 5t3e15 5p2e7 decmul1c decmul2c 3t2e6 2t2e4 oveq12i 4p3e7 7t2e14 1p1e2 8cn decadd decma2c decmul1 nn0mulcli mulassi lelttri 0re log2tlbnd eqeltrri 3re 4nn 1nn numnncl elicc2i simp3i crp 2rp relogcl fzfid @@ -378951,7 +378951,7 @@ already know is total (except at ` 0 ` ). There are branch points at df-8 expp1 sq3 log2ublem1 readdcli lemuldiv2 1lt10 6lt7 decltc 9p4e13 5p3e8 cc 3p1e4 6p5e11 9p7e16 00id 3t3e9 decma 3exp3 4p4e8 numexp2x 6p2e8 decrmanc 8p4e12 9t3e27 numexpp1 7p3e10 decaddc2 2p2e4 5t2e10 7p6e13 8t2e16 decaddci2 - 6p4e10 8t5e40 8t3e24 2p1e3 7p4e11 decrmac mulid2i nn0expcli 3brtr3i decnncl + 6p4e10 8t5e40 8t3e24 2p1e3 7p4e11 decrmac mullidi nn0expcli 3brtr3i decnncl ltmul1ii lt2mul2div mp4an ) AUBUCZBCUDDZACAUAUEZEDZFGDZEDZUFUYCUGDZEDZUHDZU AUIZCHAHEDZFGDZEDZUFHUGDZEDZUHDZGDZUJUKZUYQAIJZCJZCKJZIJZUHDZULUKZUYAVUCULU KUYAUYJUMDZUYPUJUKZUYRVUEUNLZBVUEUJUKZVUFVUEBUYPUQDZLVUGVUHVUFUOUYABHFUMDZU @@ -379113,7 +379113,7 @@ already know is total (except at ` 0 ` ). There are branch points at div0d adantr resqcld resubcld rehalfcld nndivre sylancom renegcld rpefcld eqtrd rpge0d eqbrtrd clog csu simplr simpr cz simpll nn0uz eleqtrdi elfz5 cuz nnz mpbird birthdaylem2 elfznn0 adantl nn0red peano2rem nnred elfzle2 - crp cmul ltm1d ltletrd lelttrd mulid1d breqtrrd nngt0d ltdivmul syl112anc + crp cmul ltm1d ltletrd lelttrd mulridd breqtrrd nngt0d ltdivmul syl112anc 1red nndivred mpbid recnd efle 1re difrp sylancl relogcld divge0 syl22anc elfzle1 eflegeo reefcld efgt0 rpregt0d syl21anc reeflogd cc efneg 3brtr4d lerec2 fsumle fsumneg fsumdivc arisum2 eqtr3d breqtrd fsumrecl ltlecasei @@ -379155,7 +379155,7 @@ already know is total (except at ` 0 ` ). There are branch points at birthday $p |- ( ( # ` T ) / ( # ` S ) ) < ( 1 / 2 ) $= ( cdiv co c2 c1 clt wcel c3 cdc 2nn0 cr ax-mp chash cfv cexp cmin cneg ce cle wbr cn0 cn 3nn0 deccl eqeltri c6 6nn0 decnncl birthdaylem3 mp2an clog - 5nn log2ub cmul nn0cni sqvali mulid1i eqcomi oveq12i ax-1cn subdii eqtr4i + 5nn log2ub cmul nn0cni sqvali mulridi eqcomi oveq12i ax-1cn subdii eqtr4i oveq1i subcli 2cn 2ne0 divassi 1nn0 caddc 2p1e3 eqid decsuc mvrraddi wceq c5 11multnc divmuli mpbir eqtri 3p2e5 decaddi decmul2c 3eqtri crp relogcl breqtrri 2rp 5nn0 nn0rei nndivre ltnegi mpbi wb renegcli eflt recni efneg @@ -379557,7 +379557,7 @@ exponential function (see also ~ dfef2 ). (Contributed by Mario crn cnpco eqeltrd cxpcld ax-1ne0 divdiv2d sylan2 resmpt eqtr4di 0e0icopnf mulcl div1d wo cbl cxmet abscl peano2re absge0 ltp1d elrpd rpreccld rpxrd 0red blssm eqsstrid sselda mul0or syldanl adantlr 1cxpd mul02d ef0 iftrue - ifid jaodan mulid1 eqtr4d mulne0b df-ne bitrdi simprr neneqd iffalsed csn + ifid jaodan mulrid eqtr4d mulne0b df-ne bitrdi simprr neneqd iffalsed csn cmnf cioc dvlog2lem logdmss pncan2 remulcld 1red cle absmul absge0d lep1d lemul1ad 0cn eleqtrdi eqbrtrrd ltmuldiv2 syl112anc sselid eldifsni cxpefd rpxr logcld mulcomd simprl dividd divdiv1d eqtr3d div12d sylbird ifbothda @@ -379977,7 +379977,7 @@ exponential function (see also ~ dfef2 ). (Contributed by Mario wbr cmin cv wral ad2antrr simpr1 simpr2 oveq1 sseq1d oveq2 rspc2v syl2anc wi mpd cc ax-1cn unitssre simpr3 sselid recnd nncan sylancr oveq1d sseldd simpr simplr3 iirev syl lincmb01cmp syl31anc eqeltrrd sylancl 1re resubcl - pncan3 adddird mulid2d 3eqtr3d sylan9eqr eqeltrd mulcld addcomd mpjao3dan + pncan3 adddird mullidd 3eqtr3d sylan9eqr eqeltrd mulcld addcomd mpjao3dan lttri4d ) AFDJZGDJZEUAKLMZJZUBZNZFGUCUEZEFOMZKEUFMZGOMZPMZDJFGQZGFUCUEZWN WONZFGLMZDWSXBBUGZCUGZLMZDRZCDUHBDUHZXCDRZAXHWMWOAXGBCDDIUDZUIWNXHXIUQZWO WNWIWJXKAWIWJWLUJZAWIWJWLUKZXGXIFXELMZDRBCFGDDXDFQXFXNDXDFXELULUMXEGQXNXC @@ -380011,10 +380011,10 @@ exponential function (see also ~ dfef2 ). (Contributed by Mario oveq1d fveq2d breq12d wral jca simprr simpll breq1 fvoveq1d fveq2 ralbidv oveq2d imbi2d imbi12d breq2 3expia ralrimiv expcom vtocl2ga syl3c rspcdva wi simprl orcd expr cc unitssre simpr3 sselid recnd ax-1cn pncan3 sylancl - subcl sylancr adddird mulid2d 3eqtr3d ffvelcdmd eqtr4d eqeq12d syl5ibrcom + subcl sylancr adddird mullidd 3eqtr3d ffvelcdmd eqtr4d eqeq12d syl5ibrcom wf olc syl6 1re elioore resubcl eliooord simprd wb posdif simpld ltsubpos mpbid cxr 0xr 1xr elioo2 mp2an syl3anbrc ad2antrl remulcld nncan comraddd - 3brtr3d 3jaod mpd ex addid2d mul02d eqtrd 3eqtr4rd eqtrdi addid1d prunioo + 3brtr3d 3jaod mpd ex addlidd mul02d eqtrd 3eqtr4rd eqtrdi addridd prunioo elpri 1m0e1 1m1e0 jaod syl56 0le1 mp3an eleqtrrdi elun sylib mpjaod cvxcl cun readdcld leloed mpbird ) AHEUAZIEUAZFPQUBRZUAZUCZUDZFHSRZQFUERZISRZTR ZGUFZFHGUFZSRZUVEIGUFZSRZTRZUGUHUVHUVMUIUHZUVHUVMUJZUKZUVCFPQULRZUAZUVPFP @@ -380124,7 +380124,7 @@ exponential function (see also ~ dfef2 ). (Contributed by Mario vex snss jensenlem1 rpred elrege0 simplbi 0red simprbi addge01d ltletrd rpgt0d gt0ne0d sselda adantr rpne0d divsubdird mvrladdd dividd 3eqtr3rd ccmn ringcmn cicc rpge0d divge0 syl22anc 1red ledivmul mpbird syl3anbrc - mulid1d elicc01 3jca cvxcl mpdan fco syl2anc 1re resubcl sylancr expcom + mulridd elicc01 3jca cvxcl mpdan fco syl2anc 1re resubcl sylancr expcom fvoveq1d vtocl3ga syl3anc pm2.43i breqtrd divgt0d lemul2 leadd1dd letrd eqbrtrd fmptco 3brtr4d jca ) AUKJMULUMZUNZGDUOZUPZUQZURZUSUNZLUTUNZHVAU XNKVBZUKJKMVCZUXGUNZUXKURZUSUNZLUTUNZVDVEAUXNILUTUNZUKUXHGURZUSUNZIUTUN @@ -380456,7 +380456,7 @@ group sum in the additive group (i.e. the sum of the elements). This is ( 1 / ( A + 1 ) ) <_ ( ( log ` ( A + 1 ) ) - ( log ` A ) ) ) $= ( crp wcel c1 caddc co cdiv ce cfv clog cmin cle wbr rpre ge0p1rpd rprecred rpge0 clt recnd oveq1d 1red cc0 0le1 a1i divge0d cmul id ltaddrp2d readdcld - mulid1d breqtrrd ltdivmuld mpbird eflegeo rpne0d divsubdird pncand 3eqtr3rd + mulridd breqtrrd ltdivmuld mpbird eflegeo rpne0d divsubdird pncand 3eqtr3rd dividd oveq2d rpne0 recdivd eqtrd breqtrd rpefcld rpdivcld logled relogdivd mpbid relogefd 3brtr3d ) ABCZDADEFZGFZHIZJIZVMAGFZJIZVNVMJIAJIKFLVLVOVQLMVP VRLMVLVODDVNKFZGFZVQLVLVNVLVMVLAANZAQOZPZVLDVMVLUAZWBUBDLMVLUCUDUEVLVNDRMDV @@ -380584,7 +380584,7 @@ group sum in the additive group (i.e. the sum of the elements). This is wf ltaddrp eqeltrrd rpge0d subge0d mpbid fveq2 breq1d leidi simpld wi syl peano2nn letr syl3anc mpand nnind ralrimiva brralrspcev climsup eqbrtrrid letrd climrel releldmi isumclim2 df-em 3brtr4g cmpt nnex eqeltri emcllem4 - cfz mptex eqeltrd pncan3d eqtr2d climadd cc mptru climcl addid1i breqtrdi + cfz mptex eqeltrd pncan3d eqtr2d climadd cc mptru climcl addridi breqtrdi pm3.2i ) DUBUCLZEUBUCLZYTMDUBUDUENUBUCMUBUDKEFDOUFPUGMUHZMUEAOUIZPOKVBZUJ NZOUUEUENZUKQZULNZKUMEUBUCMUUHKAOPUGUUBUUDPRZUUDAQUUHUNMCUUDOCVBZUJNZOUUK UENZUKQZULNUUHPACKUOZUUKUUEUUMUUGULUUJUUDOUJUPZUUNUULUUFUKUUNUUKUUEOUEUUO @@ -380761,7 +380761,7 @@ group sum in the additive group (i.e. the sum of the elements). This is wb oveq2 fsumm1 cc wceq nn0cnd ax-1cn pncan sylancl oveq2d sumeq1d oveq1d eqtrd mvrraddd breqtrd logleb mpdan lesub2dd readdcld relogdivd rerpdivcl letrd id mpancom reefcld wne rpcnne0 divdir syl3anc reflcl cmul flle rpcn - ce mulid1d breqtrrd ledivmul mpbird leadd1dd eqbrtrd efgt1p ltled rpdivcl + ce mulridd breqtrrd ledivmul mpbird leadd1dd eqbrtrd efgt1p ltled rpdivcl rpefcld relogefd eqbrtrrd lesubadd2d harmonicbnd3 0re lesubaddd absdifled logled simp3bi mpbir2and ) ACDZEAUAFZGHZEBULZIHZBUBZAUCFZTHZJTHZUDFZUUDUU EJKHTHZUDFEAIHZLYSUUGUUIUDYSUUDUUEJYSUUDYSUUAUUCBYSEYTUEYSUUBUUADZMUUBUUK @@ -380913,7 +380913,7 @@ group sum in the additive group (i.e. the sum of the elements). This is syl cmin remul renegd rered oveqan12d reim0d oveq2d imcl mul01d sylan9eqr cim oveq12d recld renegcld subid1d 3eqtrd eqtr4d cxpcld eqeltrd caddc cli cseq cdm cn0 cexp cdiv crp 2rp 1re resubcl sylancr rpcxpcl rpcnd clt recl - addid2d breqtrrd wb 0re ltsubadd mp3an13 mpbird 2re cxplt mpanl12 sylancl + addlidd breqtrrd wb 0re ltsubadd mp3an13 mpbird 2re cxplt mpanl12 sylancl 1lt2 mpbid rprege0d absid 2cn cxp0 ax-mp 3brtr4d reeflogd syl2anc remulcl a1i nnrpd efle mulneg1 nncnd nnne0d cxpmuld cxpexp 2nn 3eqtr3d oveq2 nnre eqcomi geolim seqex breldm syl2anr rpred rpge0d lep1d peano2nn 0lt1 lttrd @@ -381172,7 +381172,7 @@ group sum in the additive group (i.e. the sum of the elements). This is $( If ` A ` is not a nonpositive integer, then ` A ` is nonzero. (Contributed by Mario Carneiro, 3-Jul-2017.) $) dmgmn0 $p |- ( ph -> A =/= 0 ) $= - ( cc0 caddc co cc cz cdif eldifad addid1d wcel cn0 0nn0 dmgmaddn0 sylancl + ( cc0 caddc co cc cz cdif eldifad addridd wcel cn0 0nn0 dmgmaddn0 sylancl cn wne eqnetrrd ) ABDEFZBDABABGHQIZCJKABGUAILDMLTDRCNBDOPS $. ${ @@ -381229,7 +381229,7 @@ group sum in the additive group (i.e. the sum of the elements). This is cn unitssre ax-resscn cncfmptc syl3anc sylancr wf syl crp adantr mulcld wa wi simpr 1cnd addcld cre wceq adantl recld clt cneg abscld rehalfcld absge0d absdivd nnrpd absidd oveq2d eqtr2d cn0 wral fveq2 breq1d simpld - eqbrtrrd lemul12ad absmuld mulid1d wne divrecd breqtrd ledivmuld mpbird + eqbrtrrd lemul12ad absmuld mulridd wne divrecd breqtrd ledivmuld mpbird mpbid letrd lelttrd oveq2i eqtrdi breqtrrd eqeltrrd eqtrd cncfmpt2f cdv c2 wb cioo logdmn0 logcld subcld sylan2 mpteq2dv dvmptres2 reccld oveq2 cdm remulcld resubcld fveq2d nfcv cvv oveq12d oveq1d divsubdird recdivd @@ -381241,7 +381241,7 @@ group sum in the additive group (i.e. the sum of the elements). This is 2ne0 ellogdm sylanbrc cofmpt fvresd mpteq2dva fmpttd difss addcn cncfco cncfcdm crn ctg tgioo2 iccntr dvmptntr reelprrecn ioossicc sseli sselda cnt 0re dvmptid dvmptcmul sstrid ctop retop iooretop isopn3i cnelprrecn - cpr mp2an eldifi dvmptc dvmptadd addid1d feqmptd fvres mpteq2ia eqtr2di + cpr mp2an eldifi dvmptc dvmptadd addridd feqmptd fvres mpteq2ia eqtr2di 0cnd dvlog dvmptco dvmptsub dmeqd ovex dmmpti 2re 2timesd ltletrd difrp ltaddrpd syl2anc rprecred nnrecred fveq1d nfv nffvmpt1 nffv nfbr eleq1w nfim anbi2d 2fveq3 imbi12d fvmpt2 sylancl subdid mulcomd div23d 3eqtr4d @@ -381452,7 +381452,7 @@ group sum in the additive group (i.e. the sum of the elements). This is cdif wss lgamgulmlem1 sseldd eldifad simprl peano2nnd rpdivcld relogcld nnrpd recnd mulcld nncnd nnne0d divcld 1cnd addcld logcld subcld abscld dmgmdivn0 readdcld nnred remulcld rpmulcld pire abs2dif2d absmuld rpred - cr a1i mulid2d lep1d eqbrtrd 1red lemuldivd mpbid logge0d absidd oveq2d + cr a1i mullidd lep1d eqbrtrd 1red lemuldivd mpbid logge0d absidd oveq2d eqtrd elrab2 ad2antll cc0 crp wceq nnrecred oveq2 fveq2d eqtr2d absdivd 1rp letrd rpge0d 3brtr3d logled leadd1dd cmpt oveq12d fvoveq1d ad2antrl oveq1 fvmpt ovex simprbi simpld lemul1ad absrpcld abslogle syl2anc cneg @@ -381608,7 +381608,7 @@ group sum in the additive group (i.e. the sum of the elements). This is cdif addcld ad2antrr cmin cneg negcld abs2difd absnegd subnegd lesubadd2d oveq2d fveq2d 3brtr3d mpbid absrpcld cc0 wne abslogle syl2anc df-neg log1 recnd oveq1i eqtr4i wceq 1rp relogdiv sylancr eqtr4id oveq2 breq2d breq1d - cn0 fvoveq1 ralbidv anbi12d elrab2 simprbi adantl rspcdva addid1d breqtrd + cn0 fvoveq1 ralbidv anbi12d elrab2 simprbi adantl rspcdva addridd breqtrd fveq2 0nn0 rpreccld logled eqbrtrd absled leadd1dd letrd simpllr leadd2dd mpbir2and ex ralimdva impr brralrspcev rexlimddv ) ACUIZUBMZUUEUCMZNOZPMZ UAUIZQRZCEUDZUUFPMZIUIQRCEUDISUEZUASANUFHUJUGZCEUUHUHEUKMZRZUULUASUEZAUUF @@ -381861,12 +381861,12 @@ group sum in the additive group (i.e. the sum of the elements). This is 3eqtr4d breqtrd cfz csu elfznn oveq12d oveq2 eleqtrdi recnd mulcld subcld relogcld fsumser fsumcl 3eqtr3d divdird eqtr3d dividd logdiv2 syl3anc cbl ccncf logcn dvlog2lem eluznn ad2antrl imp cnmetdval absdivd rpge0d absidd - simplrr cle eluzle nnleltp1 lttrd mulid1d breqtrrd 1red ltdivmuld eqbrtrd + simplrr cle eluzle nnleltp1 lttrd mulridd breqtrrd 1red ltdivmuld eqbrtrd ex cxmet cxr cnxmet rpxr elbl3 syl22anc fmpttd ssrdv divcnvshft climaddc1 simpr breqtrdi eqbrtrrd ellogdm mpbir2an climcncf csn wf1o logf1o logdmss 0p1e1 f1of cofmpt wss frn cores 3syl log1 eqtrdi 3brtr3d rexlimddv dmgmn0 fvres fvex climsubc2 subid1d rpdivcld fzfid eqeltrrd nncand sub4d pncan2d - id subdird mulid2d subsubd addcomd subsub2d wne nnnn0d dmgmaddn0 nnncan2d + id subdird mullidd subsubd addcomd subsub2d wne nnnn0d dmgmaddn0 nnncan2d add32d addassd 3eqtr3rd relogdivd eqtr2d sumeq2dv fsumsub telfsum climsub div1d eqcomd lgamcl ) AHDIUDZBIHJZUEKZUYHLKZHJZUYJMJUYIUFAUYKUYJUAHCNUYHC URZIHJZUYLOJZLKZUGJZUYHUYLOJZIHJLKZMJZUHZIUDZCNUYJBUYMOJZIHJZLKZMJZUHZUYG @@ -382077,9 +382077,9 @@ group sum in the additive group (i.e. the sum of the elements). This is lgam1 $p |- ( log_G ` 1 ) = 0 $= ( vm caddc c1 cfv cc0 cxp cli wbr clog co wtru cn cdiv cmin cmpt wcel cc cz cdif ax-mp cuz csn cseq clgam wceq cv cmul peano2nn nnrpd rpdivcld relogcld - nnrp recnd mulid2d nncn nnne0 dividd oveq1d divdird reccld addcomd 3eqtr4rd + nnrp recnd mullidd nncn nnne0 dividd oveq1d divdird reccld addcomd 3eqtr4rd 1cnd fveq2d oveq12d subidd eqtrd mpteq2ia fconstmpt nnuz xpeq1i 3eqtr2ri wn - ax-1cn 1nn eldifn mt2 eldif mpbir2an a1i lgamcvg log1 oveq2i lgamcl addid1i + ax-1cn 1nn eldifn mt2 eldif mpbir2an a1i lgamcvg log1 oveq2i lgamcl addridi mptru eqtri breqtri 1z serclim0 climuni mp2an ) BCUADZEUBZFZCUCZCUDDZGHWPEG HZWQEUEWPWQCIDZBJZWQGWPWTGHKCAWOALCAUFZCBJZXAMJZIDZUGJZCXAMJZCBJZIDZNJZOALE OLWNFWOALXIEXALPZXIXDXDNJEXJXEXDXHXDNXJXDXJXDXJXCXJXBXAXJXBXAUHUIXAULUJUKUM @@ -382143,7 +382143,7 @@ group sum in the additive group (i.e. the sum of the elements). This is biantrurd bitrd simpl euclemma wn cn prmnn fzm1ndvds eqid prmdiveq 3bitr3rd sylan moddvds crp cle clt elfznn nnred nnrpd nnnn0d nn0ge0d elfzle2 zltlem1 1zzd prmz syl2anr mpbird modid syl22anc cuz prmuz2 eluz2gt1 syl2anc eqeq12d - 1mod bitr3d cneg znegcld mulid2d oveq2d neg1cn addcom sylancr negsub 3eqtrd + 1mod bitr3d cneg znegcld mullidd oveq2d neg1cn addcom sylancr negsub 3eqtrd cfv nncnd oveq1d neg1rr modcyc peano2rem cn0 nnm1nn0 3eqtr3d subneg orbi12d ltm1d ) AUACZBDADEFZUBFZCZUCZBBAGEFHFAIFZJZABDEFZKLZABDMFZKLZUDZBDJZBYBJZUD YEAYHYJNFZKLZBUEYBUBFZCZABBNFZDEFZKLZUCZYLYGYEYPUUAUUBYEYOYTAKYEYOYJYHNFZBG @@ -382200,7 +382200,7 @@ group sum in the additive group (i.e. the sum of the elements). This is ( c1 co cmo wceq wcel cc cz vz vw cmin csn wss cid cres cgsu wa cv cmpt simpr cexp wral cfz eleq2 raleqbi1dv anbi12d elrab2 sylib simprd simpld adantr reseq2d mptresid oveq2d oveq1d caddc cn syl nncnd sylancl ccnfld - cmul mp1i cn0 nn0cnd id gsumsn syl3anc mulid2d 3eqtr4d cr 1zzd eqtrd ex + cmul mp1i cn0 nn0cnd id gsumsn syl3anc mullidd 3eqtr4d cr 1zzd eqtrd ex a1i wn cpr ax-mp wf1o f1oi f1of sstrdi fss sylancr wbr cvv cfn ssfi 1ex wf fdmfifsupp cin c0 oveq1 rspcdva oveq2i gsumsubmcl ssdifssd sseldd wo cun cfv wb syl2anc eqtrid sylnib ovex eqcom bitri orcom cdvds fzm1ndvds @@ -382623,7 +382623,7 @@ sum notation (which we used for its unordered summing capabilities) into wf syl2an2r adantr expcld fsumcl elfznn0 expcl wceq syl2anc nnred ltp1d cle c0 cr wss nnuz fveq2 neeq1d nnne0d c0p cdgr eqtrdi eqtrid infssuzle wb sylancr eqbrtrid cz eleqtrdi mpbird eqtrd fveq2d cneg oveq12d oveq1i - mulid1d elfznn adantl lelttrd oveq1d mulexpd oveq2d mulassd eqtr4d cdiv + mulridd elfznn adantl lelttrd oveq1d mulexpd oveq2d mulassd eqtr4d cdiv wn ccxp negcld 3eqtrd subdid 3eqtr4d 1re absmuld rpge0d absidd fsumrecl oveq2 eqbrtrd ltmul1dd simp3d cply 0cn sylancl resubcld coef3 peano2nn0 plyf elfzuz eluznn0 readdcld abstrid coeid2 cin fzdisj crab cinf ssrab2 @@ -382631,7 +382631,7 @@ sum notation (which we used for its unordered summing capabilities) into mpd fsumsplit coefv0 eqcomd exp0d 1e0p1 sumeq1i eqtr3id peano2rem ltm1d fsumm1 elfzle2 ltnled mpbid mpan nsyl elrab3 necon2bbid mul02d sumeq2dv cfn fzfi olci sumz ax-mp recid2d divcld nnrecred cxpmul2d cxp1d 3eqtr3d - wo mulneg2d divcan2d negeqd mulneg1d addid2d fsum1p negsubd resubcl cif + wo mulneg2d divcan2d negeqd mulneg1d addlidd fsum1p negsubd resubcl cif 1cnd simp2d min1 exple1 syl31anc subge0 3eqtrrd 3brtr4d fsumabs elfzelz rpexpcl absge0d leexp2rd lemul2ad fsumle fsummulc1 expp1d mulcomd nnssz breqtrrd ne0d infssuzcl eqeltrid sselid rpexpcld peano2re min2 breqtrdi @@ -382735,7 +382735,7 @@ sum notation (which we used for its unordered summing capabilities) into df-idp id subneg syl2anr mpteq2dva eqtrd plyf syl feqmptd fmptco plyssc wf fveq2 sselid ccnv cima w3a plyremlem simp1d addcl adantl mulcl plyco c1 cmul eqeltrrd cn fveq2d simp2d 1nn eqeltrdi nnmulcld eqeltrd fvoveq1 - dgrco fvex fvmpt ax-mp addid2d eqtrid eqnetrd ftalem6 breq12d ffvelcdmd + dgrco fvex fvmpt ax-mp addlidd eqtrid eqnetrd ftalem6 breq12d ffvelcdmd 0cn abscld cr ltnled bitrd biimpd 2fveq3 breq2d notbid rspcev rexlimdva syl6an mpd rexnal sylib ) AGEOZPOZBUCZEOPOZUDUEZUKZBQUFZUUABQUGUKAUAUCZ NQNUCZGUHRZEOZUIZOZPOZUJUUHOZPOZULUEZUAQUFUUCAUAUUHUMOZQUUHUUHUNOZUUNUO @@ -382828,7 +382828,7 @@ sum notation (which we used for its unordered summing capabilities) into nncn biimpa 2pos pm3.2i lemul2 syl3anc mpbid eqbrtrrd nnzd syl2anc mpbird 2re olcd bcval4 oveq1d adantr mul02d 3eqtrd ex sylbird necon1ad ralrimiva plyco0 sumeq2i mpteq2i eqtr4i subidd exp0 ax-mp eqtrdi breqtrrdi eleqtrdi - nnred lep1d nn0uz elfz5 bccl2 mulid1d nnne0d eqnetrd dgreq coeeq 3jca ) E + nnred lep1d nn0uz elfz5 bccl2 mulridd nnne0d eqnetrd dgreq coeeq 3jca ) E UAIZBJUBUCZIBUDUCEKBUGUCDLFMDUEZNOZUFOZPUHZEUVFUIOZUJOZNOZUKZKUVDBAJQEULO ZFMCUEZNOZUFOZUVIEUVOUIOZUJOZNOZAUEUVOUJOZNOZCUMZUKZUVEHUVDAUVTJCEUVDJUNE UOZUVDUVOUVNIZUTUVOUVMUCZUVTJUWFUWGUVTKZUVDUWFUVOLIZUWHUVOEUPZDUVOUVLUVTL @@ -382887,7 +382887,7 @@ sum notation (which we used for its unordered summing capabilities) into 2t1e2 adddid 3eqtr4a peano2zd peano2cn divsubdir divcan3d zsubcld eqeltrd nncn peano2re subge02d ltp1d lelttrd breqtrd zred ltdivmul elfzd divcan2d zleltp1 wfn ffnd fnfvelrn orim12d mpd orcomd ord impr sylan2b reim0d bccl - fsumss 2z znegcl ax-mp rpexpcl rpred addid2d mul12d bccmpl expneg mulneg1 + fsumss 2z znegcl ax-mp rpexpcl rpred addlidd mul12d bccmpl expneg mulneg1 exprecd expmulz 3eqtr2d addsubd subdid eqtr4d expp1 expmuld eqtrdi mulcom syl22anc mulcomd 3eqtr2rd 0re sumeq2dv fsumim oveq1 sumeq2sdv sumex crimd crim ) EUCIZBJUGKUDLZUELIZUFZJFUHLZFUAUIZUJLZMBUKULZUDLZFVWENLZOLZPVWEOLZ @@ -383055,7 +383055,7 @@ sum notation (which we used for its unordered summing capabilities) into exp1 mpan peano2nnd eqeltrid nnnn0d 2z nnz sylancr syl2anc 3eqtrd cle wbr nnred cuz wb nnuz eleqtrdi elfz5 mpbird bccl2 nncnd eqtrd nnne0d c3 bcm1k cr nndivre recnd a1i wne 3eqtr3d peano2zm zmulcl bccl nn0cnd mulcom lep1d - mulm1d negcld subidd exp0 fz1ssfz0 breqtrrdi nnzd mulid1d eqnetrd divnegd + mulm1d negcld subidd exp0 fz1ssfz0 breqtrrdi nnzd mulridd eqnetrd divnegd sselid negeqd negnegd 1cnd pnncand oveq1i 3eqtr4g eqeltrdi nn0sub 2timesd df-2 2nn0 addsubd nn0nnaddcl mpancom 2timesi eqcomi subsub4d 2cnd 3eqtr4a oveq2i subdid subsubd df-3 eqtr4di 3re 2re mulassd 3cn divmuldivd mulcomd @@ -383185,7 +383185,7 @@ sum notation (which we used for its unordered summing capabilities) into cuz ccos 0re ltle mpd le2sqd breqtrrd lemuldiv2d bitr3d peano2cn eqtr2d lemuldivd recoscld sincossq simprd tanval syl2anc sqne0 recdivd 3eqtrrd wb mpbird 3eqtr3d addcom sumeq2dv fsumadd chash fsumconst nnnn0 hashfz1 - cfn mulid1d 3cn adddid pncan3oi 3eqtri subadd23d addassd 3eqtr4a mul32d + cfn mulridd 3cn adddid pncan3oi 3eqtri subadd23d addassd 3eqtr4a mul32d df-3 2cnd 3t2e6 mulcomi eqtr3i mulcl divcan3d ) GUDOZGEUEZGBUEZUFUGVUPG FUEZUFUGVUNPGUHQZUIHRQZUJUKQZUAULZUISQHRQZUMUEZUJUNZUKQZSQZUAUOZVURVVAV VDUKQZUAUOZVUOVUPUFVUNVURVVFVVHUAVUNPGUUAZVUNVVAVUROZUPZVUTVVEVUNVUTUQO @@ -383292,10 +383292,10 @@ sum notation (which we used for its unordered summing capabilities) into 3eqtr4d cc recn wbr recni climconst2 ax-resscn sylancl basellem7 ffnd wfn c3 fnconstg syl offn eqidd ofval climmul breqtrdi eqbrtrid 3cn mul01i 2cn ofc1 eqbrtrrd cle zrei npcand mpteq2dva w3a syl3an caofdi oveq12i eqtr4di - subdi cuz eqimss2i mp3an2i neg1cn eqbrtrrdi mulid1i basellem6 3ex mulid2d + subdi cuz eqimss2i mp3an2i neg1cn eqbrtrrdi mulridi basellem6 3ex mullidd ofnegsub mulneg1 negeqd mulcl negnegd eqtr2d oveq12d negsubd eqtrd ax-1cn 3re df-3 addcomi eqtri oveq1i 1cnd adddird eqtrid pnpcand 3eqtr4rd simprd - basellem8 lesub1dd 3brtr4d simpld subge0d mpbird breqtrrd climadd addid2i + basellem8 lesub1dd 3brtr4d simpld subge0d mpbird breqtrrd climadd addlidi climsqz2 3brtr3g isumclim mptru ) MAUDZUEUFZUGNZAUHUIUEUGNZUJUKNZULOUWOUW QABMBUDZUWNUGNZUMZPMUNOUOZUWMMQZUWMUWTUPZUWOULOBUWMUWSUWOMUWTUWRUWMUWNUGU QUWTURZUWMUWNUGUSUTVAZOUXBVBZUWOUXFUXCUWORUXEOMRUWMUWTOBMUWSRUWTUWRMQZUWS @@ -383873,7 +383873,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 0sgm $p |- ( A e. NN -> ( 0 sigma A ) = ( # ` { p e. NN | p || A } ) ) $= ( vk cn wcel cc0 csgm co cv cdvds wbr crab cexp csu chash c1 cmul cz wceq cfv 0z sgmval2 mpan elrabi nncnd exp0d sumeq2i cfn cc cfz fzfid dvdsssfz1 - ssfid ax-1cn fsumconst sylancl eqtrid cn0 hashcl nn0cnd mulid1d 3eqtrd + ssfid ax-1cn fsumconst sylancl eqtrid cn0 hashcl nn0cnd mulridd 3eqtrd syl ) ADEZFAGHZBIAJKZBDLZCIZFMHZCNZVGOTZPQHZVKFREVDVEVJSUAFACBUBUCVDVJVGP CNZVLVGVIPCVHVGEZVHVNVHVFBVHDUDUEUFUGVDVGUHEZPUIEVMVLSVDPAUJHVGVDPAUKABUL UMZUNVGPCUOUPUQVDVKVDVKVDVOVKUREVPVGUSVCUTVAVB $. @@ -384240,7 +384240,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 22-Sep-2014.) $) cht2 $p |- ( theta ` 2 ) = ( log ` 2 ) $= ( c2 ccht c1 caddc co clog df-2 fveq2i cz wcel cprime wceq 1z 2prm eqeltrri - cfv chtprm mp2an cc0 cht1 eqcomi oveq12i crp cr relogcl ax-mp recni addid2i + cfv chtprm mp2an cc0 cht1 eqcomi oveq12i crp cr relogcl ax-mp recni addlidi 2rp eqtr3i 3eqtri ) ABPCCDEZBPZCBPZULFPZDEZAFPZAULBGHCIJULKJUMUPLMAULKGNOCQ RSUQDEUPUQSUNUQUODUNSTUAAULFGHUBUQUQAUCJUQUDJUIAUEUFUGUHUJUK $. @@ -384661,7 +384661,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) -> ( m e. ( 1 ... ( |_ ` A ) ) /\ n e. ( 1 ... ( |_ ` ( A / m ) ) ) ) ) ) $= ( c1 cfl cfv co wcel wa cle wbr cr wb fznnfl syl mpbid cc0 syl112anc cdiv - cv cfz cn simprr adantr simprl simpld nndivred nnred simprd recnd mulid2d + cv cfz cn simprr adantr simprl simpld nndivred nnred simprd recnd mullidd cmul nnge1d clt 1red 0red nnmulcld nngt0d lemuldiv2 mpbird ltletrd lemul1 eqbrtrrd ledivmul letrd mpbir2and lemuldiv jca ex ) ADUBZFBGHUCIZJZCUBZFB VLUAIZGHUCIJZKZVOVMJZVLFBVOUAIZGHUCIJZKAVRKZVSWAWBVSVOUDJZVOBLMZWBWCVOVPL @@ -384762,7 +384762,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 sum_ k e. { n e. NN | n || m } ( ( mmu ` k ) x. B ) = C ) $= ( cn cmul co csu c1 wcel cc0 wceq cdvds wbr crab cmu cfv csn cif sselda cv wa musum syl oveq1d cfz fzfid dvdsssfz1 ssfid cc cz elrabi mucl zcnd - wss adantl fsummulc1 ovif wb velsn bicomi mulid2 mul02 ifbieq12d eqtrid + wss adantl fsummulc1 ovif wb velsn bicomi mullid mul02 ifbieq12d eqtrid a1i 3eqtr3d sumeq2dv wral cuz cfn wo snssd syldan ralrimiva olcd sumss2 syl21anc eleq1d rspcdva sumsn syl2anc 3eqtr2d ) ABGUIFUIZUAUBZGMUCZEUIZ UDUEZCNOEPZFPBWLQUFZRZCSUGZFPZWRCFPZDABWQWTFAWLBRZUJZWNWPEPZCNOWLQTZQSU @@ -384792,7 +384792,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 cr eqid fzfid wss dvdsssfz1 ssfid mucl ffvelcdm syl2an fsummulc2 sumeq2dv zcnd wf simpr adantrr anasss mulcld fsumdvdsdiag cif csn ssrab2 dvdsdivcl adantll sselid musum oveq1d adantr fsummulc1 ovif nncn nncnd 1cnd divmuld - mulid1d eqeq1d bitrd mulid2d mul02d ifbieq12d eqtrid 3eqtr3d iddvds snssd + mulridd eqeq1d bitrd mullidd mul02d ifbieq12d eqtrid 3eqtr3d iddvds snssd elrabd sselda syldan 0cn ifcl sylancl eldifsni neneqd iffalsed ffvelcdmda cdif fsumss iftrue fveq2 sumsn 3eqtr2d 3eqtrd mpteq2dva eqtr4d ) AGEKEUAZ GUBZUCEKBUAZUUFLUDZBKUEZCUAZUFUBZUUFUUKUHMZHUBZNMZCOZUCAEKPGIUGAEKUUPUUGA @@ -385081,7 +385081,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 expcom ssdomg mp2 inss1 hashdom eqbrtrdi leadd1dd le2addd cun ovex rabbii wo elpr unrab inrab wn rabeq0 1lt5 ltneii necon3ai mprgbir hashun elfzelz eqtr2 crp nnrp 0le1 1lt6 eleqtrrdi df-3 mvrraddi nnne0i redivcli divgt0ii - 4z 2pos 2lt6 mulid1i breqtrri ltdivmul 1e0p1 subid1d 5pos 5lt6 negsubdi2i + 4z 2pos 2lt6 mulridi breqtrri ltdivmul 1e0p1 subid1d 5pos 5lt6 negsubdi2i 5nn nnzi 3p2e5 pncan2 negeqi 3eqtr2i divneg lenegi ltnegi 1pneg1e0 neg1cn 0z neg0 renegcli neg1z 2timesd df-6 addsub4 mulcl divsubdir 3t2e6 mulcomi subneg 3ne0 2cnne0 divcan5 dividi divdir mp3an3 addassd 3brtr4d lesubaddd @@ -385233,7 +385233,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 pceq0 cmpt prmorcht cexp ralrimiva pcmpt efchtcl 3brtr4d csu wss eqeltrrd recnd efle oveq2 mulcom nnge1 leaddsub mpbid elfz2nn0 syl3anbrc bccl2 crp nnrpd relogcld 4pos elrpii relogcl ax-mp wral simpr pccld nn0addge1 prmnn - remulcl ad2antlr simprl prmz dvdsfac id pcelnn nnge1d addid2d assraddsubd + remulcl ad2antlr simprl prmz dvdsfac id pcelnn nnge1d addlidd assraddsubd ad2antll bcval2 mvrladdd pcdiv simprr prmfac1 3expia sylan simpld sylibrd dvdsmultr1 facnn2 oveq12d 00id pccl subid1d expr eqbrtrd ex 0nn0 nn0addcl ifcli breq1d cseq exp1d ifeq1d mpteq2ia eqcomi eqidd simpl pc2dvds dvdsle @@ -385315,7 +385315,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( theta ` N ) < ( ( log ` 2 ) x. ( ( 2 x. N ) - 3 ) ) ) $= ( vk cr wcel c2 clt wbr wceq ccht cmul co c3 cmin c1 caddc adantr sylancr cfv cle syl vx vn cfl clog cuz cc0 crp 2re 1lt2 rplogcl mp2an elrp simpli - wa mpbi recni mulid1i eqtr4i fveq2 eqtr4id chtfl sylan9eqr c4 2t2e4 eqtri + wa mpbi recni mulridi eqtr4i fveq2 eqtr4id chtfl sylan9eqr c4 2t2e4 eqtri cht2 df-4 simplr wb simpl 2pos a1i ltmul2 mpbid eqbrtrrid remulcl resubcl 3re 1red sylancl syl3anc eqbrtrrd chtcl ad2antrr simpr df-3 fveq2i cv cfz oveq2 oveq1d oveq2d breq12d wral raleqdv c6 elrpii fveq2d eqtrdi 3cn cexp @@ -385699,8 +385699,8 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 csu cfn simpl 0red 1red a1i simpr ltletrd elrpd rpge0d resqrtcld ppifi cv 0lt1 crp relogcld fsumrecl remulcld cdiv cfl elin2d prmnn nnrpd nnred cuz cn prmuz2 eluz2gt1 rplogcld rerpdivcld reflcl recnd fsumsub wss 0le0 cexp - lemul2ad sqsqrtd mulid1d eqtr4d sqvald 3brtr4d le2sqd mpbird iccss ssrind - sqrtge0d syl22anc sselda syldan cdif eldifi mulid2d w3a elin1d 0re elicc2 + lemul2ad sqsqrtd mulridd eqtr4d sqvald 3brtr4d le2sqd mpbird iccss ssrind + sqrtge0d syl22anc sselda syldan cdif eldifi mullidd w3a elin1d 0re elicc2 cc wb sylancr simp3d logled eqbrtrd lemuldivd wn eldifn adantl elin rbaib df-3an bitrdi baibd bitrd mtbid ltnled lt2sqd eqbrtrrd nnsqcld syl2anc cz logltb sylancl breqtrd eqtrd flle letrd chash cn0 cfz relogexp ltdivmul2d @@ -385776,7 +385776,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 readdcld wb rpregt0 lemul2 syl3anc mpbid le2subd remulcl syl2an peano2rem flle adantr peano2re nnred fllep1 lesub1dd rpreccld lemul1d nncnd subdird nnne0d recidd eqtr2d chash cfn fsumconst syl2anc cuz elfzuz3 1cnd addsubd - hashfz eqtr4d oveq1d eqtrd 3brtr4d fsumle sylancl hashfz1 mulid1d 3eqtrrd + hashfz eqtr4d oveq1d eqtrd 3brtr4d fsumle sylancl hashfz1 mulridd 3eqtrrd fsummulc2 ax-1cn oveq12d fsumsub eqid uztrn2 biantrurd wss ad2antll sseld cc uzss pm4.71rd bitr3d pm5.32da ancom an4 3bitr4g elfzuzb anbi12i anasss fsumcom2 letrd cz nnz flid sumeq1d nnre nnge1 harmonicubnd fsumadd logfac @@ -385830,7 +385830,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 nnrpd rpre adantr peano2rem remulcld resubcld recnd abscld ax-1cn sylancl cc subcl abs1 oveq2i abs2dif eqbrtrrid cfz csu cmpt oveq2d sumeq1d oveq1d cv id oveq12d eqid ovex fvmpt3i logfac eqtr4d 1rp cz flid ax-mp 0cn fsum1 - 1z log1 mp2an subcli mulid2i nncan mp1i fveq2d cpnf cioo ioorp eqcomi 1re + 1z log1 mp2an subcli mullidi nncan mp1i fveq2d cpnf cioo ioorp eqcomi 1re cn nnuz cxr 1nn0 nn0addge1i 0red adantl nnrp sylan2 cdv advlog w3a simp32 pnfxr logleb 3ad2ant2 mpbid simprr sylancr 1le1 simpr rexrd pnfge dvfsum2 wb simprl eqbrtrrd letrd lesubaddd ) ADEZFAGHZIZAUAJZUBJZKJZAAKJZFLMZUCMZ @@ -385901,7 +385901,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 id faccl nnred adantl elfznn0 faccld nndivred remulcld resubcld rerpdivcl rerpdivcld sylancom 1red nncnd simpl oveq2d oveq1d mpteq2dva breq1d recnd ccxp cxpexp rpcn oveq12d 1rp mulcld subcld wne simprd cabs adantrr abscld - cn 1z a1i cdv nnne0d eqtrd oveq2 fveq2d w3a rpdivcld log1 mulid2d eqbrtrd + cn 1z a1i cdv nnne0d eqtrd oveq2 fveq2d w3a rpdivcld log1 mullidd eqbrtrd rpcnd rpred lemuldivd mpbid wb logleb sylancr eqbrtrrid 3ad2antr1 sumeq1d ad2antrl cvv eqtrdi sumeq2dv ax-1cn cdif div0d ovexd fvmptd ax-mp 3brtr3d syl 3eqtrd divsubdir syl3anc wss rpssre rlimconst breqtrd breqtrdi simpld @@ -385912,10 +385912,10 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 cpr eqid expge0d simprr leidd dvfsumlem4 expcld fsumcl sub32d eqidd divid 1le1 csn sylan9eqr cuz nn0uz eleqtrdi eluzfz1 snssd elsni 0exp0e1 1div1e1 fveq2 fac0 eqeltrdi eldifi eldifsni eldifsn sylanbrc dfn2 eleqtrrdi 0expd - fsumss eqtr4d 0cn sumsn mp2an mulid1d flid div1d eqeltrd subdird divcan1d + fsumss eqtr4d 0cn sumsn mp2an mulridd flid div1d eqeltrd subdird divcan1d fsum1 3eqtr4d absmuld rprege0 absid 1cnd csbied leabsd subid1d rlimsqzlem letrd breqtrrd divass fsumdivc rpcnne0d divdiv32 anasss an32s rlimdiv cfn - fsumrlim wo fzfi olci rlimmul mul01d rlimsub 0m0e0 rlimadd npcand addid1d + fsumrlim wo fzfi olci rlimmul mul01d rlimsub 0m0e0 rlimadd npcand addridd sumz ) CUAEZAFGAUPZUCHZUDIZVUTBUPZJIZUEHZCUFIZBUGZVUTUEHZCUFIZCUHHZKCUDIZ VVHDUPZUFIZVVLUHHZJIZDUGZLIZMIZMIZVUTJIZVVRVUTJIZUIIZNVVJKUIIAFVVGVUTJIZN VVJUJVUSAFVVTVWAVVJKOVUSVUTFEZPZVVSVUTVWEVVGVVRVWEVVBVVFBVWEGVVAUKVWEVVCV @@ -386035,7 +386035,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 nnnn0d 0lt1 eqbrtrrid nnred lttrd elnnz divcan2d eqeltrd eluz2b2 csu cneg nnne0d cc wceq wb ax-1cn subeq0 sylancl necon3bid simpr nndivdvds syl2anc ad2antrr geoser negsubdi2 oveq2d expmuld oveq12d div2negd 3eqtr2d elfznn0 - eqtr3d eqtrd fzfid zexpcl syl2an fsumzcl eqeltrrd mulid2d cle w3a elfzm11 + eqtr3d eqtrd fzfid zexpcl syl2an fsumzcl eqeltrrd mullidd cle w3a elfzm11 2z biimpa simp3d adantr ltsub1dd eqbrtrd ltmuldiv syl112anc nprm pm2.65da ralrimiva isprm3 ) ADEZFAGHZIJHZUAEZUBZAFUCUDZEZBUEZAUFKZUNZBFAIJHZUGHZUH AUAEUUFUUBIALKZUUHUUBUUEUIZUUFUUNFIGHZUUCLKUUFUUPIIUJHZUUCLUUPFUUQFUKULUO @@ -386134,7 +386134,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 syl32anc 0lt1 peano2rem expgt1 posdif nngt0d ltdiv2 div1d lelttrd eluz2b2 syl222anc cpr cfz fzfid dvdsssfz1 ssfid ssrab2 prssd simplrl snssd simp2d crab dvdsmul2 nnne0d divcan2d simplrr incom disjsn2 eqtr3id df-pr divdird - iddvds prfi 1cnd subdird mulid2d pncan3d divassd 3eqtr4d ccxp 2nn nnexpcl + iddvds prfi 1cnd subdird mullidd pncan3d divassd 3eqtr4d ccxp 2nn nnexpcl expp1 mulcom 2cnd mulassd coprm 2z rpexp1i sgmmul syl13anc pncan 1sgm2ppw 2prm eqtr3d 3eqtrd sgmnncl sgmval sselid cxp1d sumeq2dv remulcld ltaddrpd 1nn0 3eqtrrd readdcld condan elpri expr ralrimiva orbi12d imbi12d rspcdva @@ -386593,13 +386593,13 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 SAVADUCVBDUCUNUNVCRULUOUPARTUQUR $. $} - dchrmulid2.t $e |- .x. = ( +g ` G ) $. - dchrmulid2.x $e |- ( ph -> X e. D ) $. + dchrmullid.t $e |- .x. = ( +g ` G ) $. + dchrmullid.x $e |- ( ph -> X e. D ) $. $( Left identity for the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.) $) - dchrmulid2 $p |- ( ph -> ( .1. .x. X ) = X ) $= + dchrmullid $p |- ( ph -> ( .1. .x. X ) = X ) $= ( cc0 co cmul cof wcel cn dchrrcl syl dchr1cl dchrmul cv c1 cif cmpt wceq - cfv wa oveq1 eqeq1d cc dchrf ffvelcdmda adantr mulid2d wn 0cn mul02i cmgp + cfv wa oveq1 eqeq1d cc dchrf ffvelcdmda adantr mullidd wn 0cn mul02i cmgp wne ccnfld cmhm wral dchrelbas2 mpbid simprd r19.21bi necon1bd imp oveq2d 3eqtr4a ifbothda mpteq2dva cvv cbs fvexi a1i ax-1cn ifcli feqmptd offval2 wi 3eqtr4d eqtrd ) AFJDUAFJUBUCUAZJACDHIFJKLMNRABCEFGHIKLMNOPQAJCUDZIUEUD @@ -386651,7 +386651,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( vx vy vk va vb wcel cbs cfv cv co w3a eqid wa cmul cc dchrf dchrmul czn vz vc cn cplusg eqidd cui cdiv cc0 cif cmpt simp2 simp3 dchrmulcl cof cvv c1 fvexd 3adant3r3 simpr3 wceq mulass adantl caofass simpr1 simpr2 oveq1d - wf oveq2d 3eqtr4d dchr1cl simpr dchrmulid2 dchrinvcl simpld simprd isgrpd + wf oveq2d 3eqtr4d dchr1cl simpr dchrmullid dchrinvcl simpld simprd isgrpd id mulcom caofcom isabld ) BUDIZDEAJKZAUEKZAWBWCUFZWBWDUFZWBDEUBWCWDAFBUA KZJKZFLZWGUGKZIZUQWIDLZKUHMUIUJUKZFWHWKUQUIUJUKZWEWFWBWLWCIZELZWCIZNZWCWD ABWLWPWGCWGOZWCOZWDOZWBWOWQULZWBWOWQUMZUNZWBWOWQUBLZWCIZNPZWLWPWDMZXEQUOZ @@ -386759,7 +386759,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 Carneiro, 28-Apr-2016.) $) dchr1 $p |- ( ph -> ( .1. ` A ) = 1 ) $= ( vk vx wcel c1 cc0 wceq eqid cv cif cbs cc cmpt cplusg co dchr1cl eleq1w - cfv ifbid cbvmptv dchrmulid2 cn cabl cgrp wa dchrabl ablgrp isgrpid2 4syl + cfv ifbid cbvmptv dchrmullid cn cabl cgrp wa dchrabl ablgrp isgrpid2 4syl wb mpbi2and simpr adantr eqeltrd iftrued unitss sselid 1cnd fvmptd ) ANBN UAZCPZQRUBZQGUCUJZDUDANVOVNUEZEUCUJZPZVPVPEUFUJZUGVPSZDVPSZAVOVQCVPNEFGHI VQTZVOTZKVPTLUHZAVOVQVSCVPOEFVPGHIWBWCKNOVOVNOUAZCPZQRUBVLWESVMWFQRNOCUIU @@ -386963,7 +386963,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 cn cneg cdiv ccxp 2re znfi unitss ssfi sylancl odcl2 syl3anc ad2antrr wss nndivre recnd cxpcl eqeltrid neg1ne0 cxpne0d neeq1i sylibr zsubcl expaddz syl22anc npcand oveq2d oveq1i root1eq1 syl2an biimpar expclzd - eqtrid oveq1d mulid2d 3eqtr3d ex sylbird mpd biimpd expimpd rexlimdva + eqtrid oveq1d mullidd 3eqtr3d ex sylbird mpd biimpd expimpd rexlimdva syl12anc oveq1 oveq2 anbi12d rspcev expr adantl impbid iota5 mpan2 ) AEKVGZVHZTVIVGZESGVJZVJZTSUCVJZJVKZVLZVHZVHZEUDVJZUWOOVMZUWPJVKZVLZMV MZIUXBVNVKZVLZVHZOVIVOZMVTZITVNVKZUWKUXAUXJVLAUWSBEBVMZUWNVJUXCVLZUXG @@ -387135,14 +387135,14 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( wceq cfv c1 wcel co vk vx vb vc cv csu cphi cc0 cif eqeq2 wa fveq1 cn dchrrcl syl adantr simpr dchr1 sylan9eqr an32s sumeq2dv cmul cfn cc cn0 chash znunithash phicld nnnn0d eqeltrd cvv wb cui hashclb sylibr ax-1cn - fvexi ax-mp fsumconst sylancl oveq1d nncnd mulid1d 3eqtrd eqtrd wn wrex + fvexi ax-mp fsumconst sylancl oveq1d nncnd mulridd 3eqtrd eqtrd wn wrex wral cabl cgrp dchrabl ablgrp grpidcl 4syl dchreq notbid rexnal bitr4di wne df-ne neeq2d cmin cbs wf dchrf unitss sseli ffvelcdm syl2an adantlr eqid fsumcl 0cnd simprl sselid ffvelcdmd simprr subeq0 necon3bid mpbird subcl cmulr oveq2 fveq2d cbvsumv cmgp ccnfld cmhm dchrmhm adantl mgpbas ad2antrr mgpplusg cnfldmul mhmlin syl3anc eqtrid cmpt fveq2 sylbid wf1o ccrg crg zncrng crngring unitgrp unitgrpbas cplusg ressplusg grplactf1o - cress syl2an2r grplactval sylan fsumf1o 3eqtr4rd mulid2d oveq12d subidd + cress syl2an2r grplactval sylan fsumf1o 3eqtr4rd mullidd oveq12d subidd fsummulc2 1cnd subdird mul01d 3eqtr4d mulcanad expr biimtrrid rexlimdva imp ifbothda ) GDPZCIUEZGQZIUFZFUGQZPUVNUHPZUVNUVKUVOUHUIZPAUVOUHUVOUVQ UVNUJUHUVQUVNUJAUVKUKZUVNCRIUFZUVOUVRCUVMRIAUVLCSZUVKUVMRPUVKAUVTUKZUVM @@ -387205,11 +387205,11 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( wceq c1 wcel co vy vz va vb cv cfv csu chash cc0 cif eqeq2 fveq2 cmgp wa ccnfld cmhm dchrmhm simpr sselid eqid ringidval cnfld1 syl sylan9eqr mhm0 an32s sumeq2dv cmul cfn dchrfi ax-1cn fsumconst sylancl cn0 hashcl - cc cn 3syl nn0cnd mulid1d eqtrd adantr wne df-ne dchrpt dchrf ffvelcdmd + cc cn 3syl nn0cnd mulridd eqtrd adantr wne df-ne dchrpt dchrf ffvelcdmd wn cmin fsumcl simprl subcl simprr subeq0 necon3bid mpbird cplusg oveq2 0cnd wb fveq1d cbvsumv cof dchrmul wfn cvv wf ffnd cbs fvexi a1i fnfvof syl22anc eqtrid cmpt fveq1 cgrp wf1o cabl dchrabl grplactf1o grplactval - ablgrp syl2anc sylan fsumf1o fsummulc2 3eqtr4rd mulid2d oveq12d subdird + ablgrp syl2anc sylan fsumf1o fsummulc2 3eqtr4rd mullidd oveq12d subdird subidd mul01d 3eqtr4d mulcanad rexlimddv sylan2br ifbothda ) CFQZECBUEZ UFZBUGZEUHUFZQUUBUIQZUUBYSUUCUIUJZQAUUCUIUUCUUEUUBUKUIUUEUUBUKAYSUNZUUB ERBUGZUUCUUFEUUARBAYTESZYSUUARQYSAUUHUNZUUAFYTUFZRCFYTULUUIYTIUMUFZUOUM @@ -387485,7 +387485,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 oveq12i vx vn cv wbr oveq2 oveq12d breq12d 6nn0 7nn0 4nn0 0nn0 4lt10 6lt7 id cdc decltc cmin cc 2nn0 expmul mp3an sq2 eqcomi 4m1e3 eqtr4i 3cn 3t2e6 3nn0 mulcomli oveq2i 2exp6 eqtri wne cz 4cn 4ne0 4z 3eqtr3ri df-4 bcp1ctr - expm1 ax-mp c5 df-3 df-2 1nn0 1e0p1 nn0mulcli bcn0 oveq1i 1div1e1 mulid2i + expm1 ax-mp c5 df-3 df-2 1nn0 1e0p1 nn0mulcli bcn0 oveq1i 1div1e1 mullidi 2t0e0 2t1e2 3eqtri 2ne0 divcan2i 2t2e4 df-5 5cn 3ne0 divassi 6cn nnmulcli wa 2nn 5nn nncni pm3.2i div12 5t2e10 divmuli mpbir df-7 7cn mulassi cn wi 4nn mpan nnexpcl sylancr nnrpd rpdivcld rpred nnmulcl nnnn0d bccl syl2anc @@ -387589,7 +387589,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 deccl cn cle cv cmul co wa cprime wrex cuz cfv wi elnnuz ax-1 c5 cc0 6nn0 wn cr nn0rei 8nn0 eluzelre 4lt10 6lt8 decltc eluzle ltletrd ltnle sylancr a1i wb mpbid pm2.21d 83prm eqid 4t2e8 3t2e6 decmul1 3lt10 4lt8 3lt6 declt - 6nn 43prm 2t2e4 2lt4 23prm 1nn0 2cn mulid2i 1lt2 13prm 7nn0 7t2e14 declti + 6nn 43prm 2t2e4 2lt4 23prm 1nn0 2cn mullidi 1lt2 13prm 7nn0 7t2e14 declti 7lt10 3lt4 7prm 5nn0 5t2e10 5lt7 5prm 3lt5 5lt6 3prm 2lt3 2prm olci sylbi 1nn 4nn imp ) AUACZADEFZUBGZABUCZHGXOIAUDUEUBGUFBUGUHZXLAJUIUJCXNXPUKZAUL XQJIIABXPXNUMZXQIEKABXRXQKDUNABXRXQUNJUOFZLABXRXQLJEFZJKFZABXRXQYAIDFZIKF @@ -387620,7 +387620,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 wn ltdivmul syl112anc bitrd impr 0p1e1 breqtrrdi 0z flbi mpbir2and eqtrdi eqcomd ce cn0 3brtr4d 3syl df-2 breqtrdi zleltp1 iftrue syl5ibrcom nnge1d biantrurd prmuz2 eluz2gt1 jca elfzelz 1lt2 2t1e2 nnre 0le2 nnge1 syl31anc - lemul2a eqbrtrrid ltletrd rpdivcld 3bitr4rd notbid ltnled mulid1d biimprd + lemul2a eqbrtrrid ltletrd rpdivcld 3bitr4rd notbid ltnled mulridd biimprd elfz bitr4d nngt0d ltaddrp2d 2timesd breqtrrd wi lttr mpand sylibrd 2t0e0 oveq2d 0m0e0 0le0 expr sylbid iffalse mpbidi pm2.61d fsumle pcbcctr chash eqbrtrdi cfn bernneq3 reeflogd reexplog relogcld remulcld efle ledivmul2d @@ -387688,14 +387688,14 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( c2 cmul co c1 cc0 wcel cle wbr clt cr c3 adantr vk cbc cpc cv cexp cdiv cfz cfl cfv cmin csu cn cprime wceq pcbcctr syl2anc cuz wo elnn1uz2 sylib elfznn wa oveq2 prmnn syl nncnd exp1d sylan9eqr oveq2d fveq2d caddc 2t1e2 - mulid2d eqbrtrd wb nnred nngt0d lemuldiv syl112anc mpbid nndivred 1re 2re + mullidd eqbrtrd wb nnred nngt0d lemuldiv syl112anc mpbid nndivred 1re 2re 1red 2pos pm3.2i lemul2 mp3an13 eqbrtrrid nnne0d divassd breqtrrd nnmulcl 2cnd 2nn sylancr 3pos ltdiv23 mp3an2 syl12anc df-3 breqtrdi cz 2z sylancl 3re flbi mpbir2and eqtrd c4 remulcl a1i 4re eqbrtrrd 3lt4 lttrd breqtrrdi 2t2e4 ltmul2 mp3an23 mpbird df-2 eqtrdi oveq12d 2cn subidi crp eluzge2nn0 1z nnrpd nnexpcl syl2an rpdivcld rpge0d ltdivmul ltdivmuld 1e0p1 rpred 0z cn0 mp3an3 remulcld nnltp1le eqbrtrid lemul1 mp3an1 sqvald simpr leexp2ad - nnge1d letrd ltletrd mulid1d ltaddrpd 2timesd 2t0e0 0m0e0 jaodan sumeq2dv + nnge1d letrd ltletrd mulridd ltaddrpd 2timesd 2t0e0 0m0e0 jaodan sumeq2dv sylan2 cfn fzfid wss sumz olcs ) ABICJKZCUBKUCKZLUVFUGKZUVFBUAUDZUEKZUFKZ UHUIZICUVJUFKZUHUIZJKZUJKZUAUKZMACULNZBUMNZUVGUVQUNDEBUACUOUPAUVQUVHMUAUK ZMAUVHUVPMUAUVIUVHNZAUVILUNZUVIIUQUINZURZUVPMUNZUWAUVIULNUWDUVIUVFVAUVIUS @@ -387821,7 +387821,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 mtbiri eqtri 1le1 eqbrtri zcnd exp0d breqtrrid ffvelcdmda ad2antlr lemul1 1z nnz ppiprm cc expp1d eqtrd breq2d bitr4d simpr eleqtrdi seqp1 peano2nn nnuz oveq1 oveq12d ifbieq1d ovex ifex iftrue bposlem1 nnmulcld letr mpand - simpld sylbid iffalse mulid1d 3eqtrd ppinprm biimprd pm2.61dan expcom a2d + simpld sylbid iffalse mulridd 3eqtrd ppinprm biimprd pm2.61dan expcom a2d wn nncnd nnind mpcom cxpexp nn0red nn0ge0d ppiub flle eqbrtrid 3re lediv1 3pos leadd1dd 2t1e2 nnge1d 1re eluz2gt1 cxpled eqbrtrrd ) AEUCCMUDZUEZNFU COZEUFUEZUGOZUYHUYHUHUEZUIUJOZNUKOZULOZAUYGAPPEUYFAPPCUMZPPUYFUMZABUNZUYH @@ -388058,7 +388058,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 rereccli 6cn df-7 3t2e6 oveq12i 3eqtr4ri divdiv32i 9nn0 0nn0 9lt10 decltc 7cn 4lt5 7t7e49 mul4i 5t2e10 dec0u 7pos rpge0 rpre lt2msqi rpgt0 ltdivmul mp3an12 ltdiv1ii divsubdir 5p3e8 pncan3oi dividi 5lt8 ltadd2i df-9 adddii - mulid2i mulid1i nnmulcli divmuldivi df-4 3nn0 4p3e7 5p2e7 addcomi 3eqtr2i + mullidi mulridi nnmulcli divmuldivi df-4 3nn0 4p3e7 5p2e7 addcomi 3eqtr2i 6re expadd 2nn0 sq2 ltsub2 eqbrtrri resubcli lttri ltaddsubi eqbrtri 1lt2 rplogcl ltmul1ii eqcomi pm3.2i ) UAGUBZCUCZUEHUYHIUFUCZJKUYHLGMNZIUDUCZMN ZUGGMNZOIOUHNZMNZPNZQNZRSMNZUYKMNZQNZUYIPNZUEUYGUIHUYHVUATUAGUUAUJUUBZBUY @@ -388146,7 +388146,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 efadd eflt mulass fveq2i 3eqtr2i oveq2i eqtrdi 3rp rpdivcl adddir eqtr3id 4p2e6 4cn 6cn pm3.2i div23 3t2e6 divcan3i divdir 3eqtr3d mvrladdd subdird mp3an3 nnncan2d adddird mulcli nnncan1d addsubassd 3p2e5 pncan2 df-3 1cnd - subdiri eqtrid mulid2d assraddsubd relogdiv subdi divmuldivi 3t3e9 eqtr2i + subdiri eqtrid mullidd assraddsubd relogdiv subdi divmuldivi 3t3e9 eqtr2i oveq12i divcli mul4 mpanr12 divcan6 2cnne0 div12 divcld npncan3d remulcli subcl rpge0d div23d 0le2 rprege0d sqrtmul mul4d 2timesd relogmuld addcomd addcld 3eqtr4rd addcl 3brtr4d ltmul2d ltnsymd pm2.21dd ) AGEUBZUCUDUUAZEU @@ -388501,7 +388501,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ad2antlr syl2an2 mtbird simpr ad2antrr pceq0 mpbird oveq2d cc neg1z ifcli syl2anc 0z cn0 simpl csn wne simplr neqned eldifsn sylanbrc oddprm zexpcl cdif peano2zd zmodcld nn0zd ifclda zcnd eqtrd oveq1d 3eqtrd oveq12d exp1d - peano2zm adantlr exp0d ifeq1da eqtrdi seqid3 wf syl3anc ffvelcdmd mulid2d + peano2zm adantlr exp0d ifeq1da eqtrdi seqid3 wf syl3anc ffvelcdmd mullidd ifid lgsfcl iftrue nncnd sylancl eqtr3d eqeq1 oveq1 id ifbieq2d ralrimiva pcid eleq1d rspcdva ) AFGZDHGZIZADUAJZDKLZAMNJOLOKPZDKUDQZAKUDQZIZOUEZOPZ DUFRZSCOUGZRZSJZPZUVRDMLZMAUBQZKAUHUIJOURUJGZOUVMPZPZADOUKJZMULJZNJZOUMJZ @@ -388659,7 +388659,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 cif cabs cseq wne wceq simpl nnz nnne0 lgsval4 wn nngt0 cr wi nnre ltnsym 0re sylancr mpd intnanrd iffalsed cn0 nnnn0 nn0ge0d absidd fveq2d oveq12d cuz simpr nnuz eleqtrdi wf cv lgsfcl3 elfznn ffvelcdm syl2an zmulcl seqcl - cfz zcnd mulid2d 3eqtrd ) AHIZDJIZKZADUALZDMNOZAMNOZKZPUBZPUCZDUDQZRCPUEZ + cfz zcnd mullidd 3eqtrd ) AHIZDJIZKZADUALZDMNOZAMNOZKZPUBZPUCZDUDQZRCPUEZ QZRLZPDWOQZRLWRWGWEDHIZDMUFZWHWQUGWEWFUHZWFWSWEDUISZWFWTWEDUJSZABCDEUKTWG WMPWPWRRWGWKWLPWGWIWJWGMDNOZWIULZWFXDWEDUMSWGMUNIDUNIZXDXEUOURWFXFWEDUPSZ MDUQUSUTVAVBWGWNDWOWGDXGWGDWFDVCIWEDVDSVEVFVGVHWGWRWGWRWGFGRHCPDWGDJPVIQW @@ -388713,7 +388713,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 lgsneg1 $p |- ( ( A e. NN0 /\ N e. ZZ ) -> ( A /L -u N ) = ( A /L N ) ) $= ( cn0 wcel cz wa cneg clgs co wceq cc0 simpr negeqd 3eqtr4a oveq2d wne c1 neg0 cmul 3ad2ant1 w3a clt wbr cif nn0z lgsneg syl3an1 wn iffalsed oveq1d - nn0nlt0 simp2 lgscl syl2anc zcnd mulid2d 3eqtrd 3expa pm2.61dane ) ACDZBE + nn0nlt0 simp2 lgscl syl2anc zcnd mullidd 3eqtrd 3expa pm2.61dane ) ACDZBE DZFZABGZHIZABHIZJZBKVBBKJZFZVCBAHVHKGKVCBRVHBKVBVGLZMVINOUTVABKPZVFUTVAVJ UAZVDAKUBUCZQGZQUDZVESIZQVESIVEUTAEDZVAVJVDVOJAUEZABUFUGVKVNQVESVKVLVMQUT VAVLUHVJAUKTUIUJVKVEVKVEVKVPVAVEEDUTVAVPVJVQTUTVAVJULABUMUNUOUPUQURUS $. @@ -388758,7 +388758,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 wb w3a wne wn cle simplrr biantrud 0re simpl2 zred ltlen sylancr renegcld simpl1 mul01d mulneg1d breq12d lt0neg1d biimpa syl112anc remulcld 3bitr4d 0red ltmul2 3bitr2rd lenlt bitrd oveq2 neg1mulneg1e1 eqtrdi ax-1cn mulm1i - ifsb eqtr4i eqtr4di iftrue adantl oveq1d eqtr4d iffalse cc neg1cn mulid2i + ifsb eqtr4i eqtr4di iftrue adantl oveq1d eqtr4d iffalse cc neg1cn mullidi ifnot ifcli biimpar simplrl ne0gt0d breq2d eqtrid pm2.61dan simpr oveq12d eqtr2d biantrurd 3eqtr3d intnanrd iffalsed 1t1e1 ) ADEZBDEZCDEZUAZAFUBZBF UBZGZGZCFHIZXGABJKZFHIZGZLMZLNZXGAFHIZGZXKLNZXGBFHIZGZXKLNZJKZOXFXGGZXIXK @@ -388782,7 +388782,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) ) $= ( c1 c8 cmo co wceq c7 c3 c5 wcel cc0 cle wbr clt modid mp4an caddc eqtri cr 8cn 3cn cneg wa crp 1re 8re 8pos elrpii 0le1 1lt8 oveq2i ax-1cn negcli - cmul mulid2i cmin negsubi 8m1e7 addcomli oveq1i cz renegcli 1z modcyc 7re + cmul mullidi cmin negsubi 8m1e7 addcomli oveq1i cz renegcli 1z modcyc 7re mp3an 0re 7pos ltleii 7lt8 3eqtr3i pm3.2i 3re 3pos 3lt8 5cn subaddrii 5re 5p3e8 5pos 5lt8 ) ABCDAEZAUAZBCDZFEZUBGBCDGEZGUAZBCDZHEZUBWAWDARIBUCIZJAK LABMLWAUDBUEUFUGZUHUIABNOWBABUMDZPDZBCDZFBCDZWCFWLFBCWLWBBPDFWKBWBPBSUNZU @@ -388842,9 +388842,9 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 -> ( ( ( A x. B ) mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) ) $= ( c8 cmo co c1 c7 wcel cz wa wceq wo cmul cr a1i simpr cneg oveq1d eleq1d eqtr4di vx cpr wb ovex elpr crp zre ad2antrr 1red simplr 8re elrpii c3 c5 - lgsdir2lem1 simpli modmul1 syl221anc cc zcn ad2antlr mulid2d eqtrd neg1rr + lgsdir2lem1 simpli modmul1 syl221anc cc zcn ad2antlr mullidd eqtrd neg1rr 8pos simpri mulm1d wi znegcl cv oveq1 negeq neg1cn mulcom mpan2 mulm1 syl - imbi12d adantr neg1z eqtr3d mulid2i oveq1i eqtri ex neg1mulneg1e1 orim12d + imbi12d adantr neg1z eqtr3d mullidi oveq1i eqtri ex neg1mulneg1e1 orim12d eqtrdi orcom bitri 3imtr4g vtoclga negnegd sylibd impbid jaodan sylan2b bitrd ) ACDEZFGUBZHAIHZBIHZJZWSFKZWSGKZLABMECDEZWTHZBCDEZWTHZUCZWSFGACDUD UEXCXDXJXEXCXDJZXFXHWTXKXFFBMEZCDEZXHXKANHZFNHZXBCUFHZWSFCDEZKXFXMKXAXNXB @@ -388873,7 +388873,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 3cn wtru cpr cneg c7 wo ovex elpr anbi12i simpll simplr crp elrpii simprl 8pos lgsdir2lem1 simpri simpli simprr orcd ex znegcl mp1i mulneg1i eqtrdi 8re olcd mulneg2i mul2negi ccased biimtrid imp sylibr caddc c9 8cn ax-1cn - df-9 addcomi eqtri 3t3e9 mulid2i oveq2i 3eqtr4i cr 1re 1z modcyc nnmulcli + df-9 addcomi eqtri 3t3e9 mullidi oveq2i 3eqtr4i cr 1re 1z modcyc nnmulcli mp3an 3nn nnzi eqidd mptru mulcli mulm1i 3eqtr3i preq12i eleqtrdi ) ACDZB CDZEZAFGHZIJUAZDZBFGHZXBDZEZEZABKHZFGHZIIKHZFGHZXJUBZFGHZUAZLUCUAXGXIXKMZ XIXMMZUDZXIXNDWTXFXQXFXAIMZXAJMZUDZXDIMZXDJMZUDZEWTXQXCXTXEYCXAIJAFGUEUFX @@ -388898,8 +388898,8 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( ( A x. B ) /L 2 ) = ( ( A /L 2 ) x. ( B /L 2 ) ) ) $= ( cz wcel wa c2 cc0 c8 cmo co c1 cif cmul wceq cc ifcli adantl 3eqtr4a wn iffalse cdvds wbr c7 cpr cneg clgs 0cn ax-1cn neg1cn mul02i iftrue oveq1d - wi 2z dvdsmultr1 mp3an1 imp iftrued mul01i dvdsmultr2 mulid2i lgsdir2lem4 - oveq2d wb adantlr ifbid mulid1i zcn mulcom syl2an ad2antrr eleq1d ancom1s + wi 2z dvdsmultr1 mp3an1 imp iftrued mul01i dvdsmultr2 mullidi lgsdir2lem4 + oveq2d wb adantlr ifbid mulridi zcn mulcom syl2an ad2antrr eleq1d ancom1s bitrd neg1mulneg1e1 oveqan12d c3 c5 cun lgsdir2lem3 ad2ant2r elun orcanai wo sylib ad2ant2l anim12dan lgsdir2lem5 syldan pm2.61ddan cprime euclemma ioran 2prm notbid biimpar sylan2br syl 3eqtr4d lgs2 zmulcl 3eqtr4rd ) ACD @@ -388967,8 +388967,8 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( A =/= 0 /\ B =/= 0 ) ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) ) $= ( vn cz wcel cc0 wa cmul co clgs wceq cexp c1 cif cc wbr adantr oveq12d - cn vk vx w3a wne c2 ax-1cn ifcli mulid2i iftrue adantl oveq1d simpl1 zcnd - ad2antrr simpl2 sqmuld simpr sqcld mulid2d 3eqtrd eqeq1d ifbid 3eqtr4a wn + cn vk vx w3a wne c2 ax-1cn ifcli mullidi iftrue adantl oveq1d simpl1 zcnd + ad2antrr simpl2 sqmuld simpr sqcld mullidd 3eqtrd eqeq1d ifbid 3eqtr4a wn 0cn mul02i iffalse cdvds dvdsmul1 syl2anc wb dvdssq mpbid breq2 syl5ibcom zmulcld cle wi simprl neneqd sqeq0 syl mtbird cn0 zsqcl2 elnn0 sylib mt3d wo ord nnzd 1nn dvdsle sylancl nnge1d jctird cr nnred letri3 sylibrd syld @@ -389030,7 +389030,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 $( Lemma for ~ lgsdi . (Contributed by Mario Carneiro, 4-Feb-2015.) $) lgsdilem2 $p |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) = ( seq 1 ( x. , F ) ` ( abs ` ( M x. N ) ) ) ) $= - ( cfv c1 co wcel cn cz cc0 syl2anc vk vx cmul cc cabs mulid1 adantl cuz + ( cfv c1 co wcel cn cz cc0 syl2anc vk vx cmul cc cabs mulrid adantl cuz wceq wne nnabscl nnuz eleqtrdi cle wbr nnzd zmulcld zcnd mulne0d abscld cv absge0d nnge1d lemulge11d absmuld breqtrrd eluz2 syl3anbrc cfz wa wf lgsfcl3 syl3anc elfznn ffvelcdm syl2an mulcl seqcl cprime clgs cpc cexp @@ -389287,7 +389287,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 -> ( ( ( A ^ 2 ) x. B ) /L N ) = ( B /L N ) ) $= ( cz wcel cc0 wne wa cgcd co c1 wceq w3a cexp cmul clgs simpl syl anim12i c2 zsqcl adantr 3anim123i cc zcn sqne0 biimpar simpr 3adant3 lgsdir syl2anc - wb 3anass biimpri 3adant2 lgssq oveq1d 3adant1 lgscl zcnd mulid2d 3eqtrd ) + wb 3anass biimpri 3adant2 lgssq oveq1d 3adant1 lgscl zcnd mullidd 3eqtrd ) ADEZAFGZHZBDEZBFGZHZCDEZACIJKLZHZMZATNJZBOJCPJZVMCPJZBCPJZOJZKVPOJVPVLVMDEZ VFVIMVMFGZVGHZVNVQLVEVRVHVFVKVIVCVRVDAUAUBVFVGQZVIVJQZUCVEVHVTVKVEVSVHVGVCV SVDVCAUDEVSVDULAUEAUFRUGVFVGUHSUIVMBCUJUKVLVOKVPOVLVEVIVJMZVOKLVEVKWCVHWCVE @@ -389306,7 +389306,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( vx cz wcel cmul co clgs wceq cc0 wa oveq1 oveq1d cc zcnd adantr sylancr c1 0z cn0 w3a cv eqeq2d wral id nn0z lgscl syl2anr mul01d oveq2d 3eqtr4rd simpr cgcd wb lgsne0 gcdcom nn0gcdid0 eqtrd eqeq1d lgs1 adantl syl5ibrcom - wne oveq2 sylbid imp ad2antrr mulid2d pm2.61dane ralrimiva 3ad2ant3 simp2 + wne oveq2 sylbid imp ad2antrr mullidd pm2.61dane ralrimiva 3ad2ant3 simp2 eqtr2d rspcdva syl2anc mulcomd eqtr4d zcn 3ad2ant2 mul02d sylan9eqr simp1 3eqtr4d lgsdir syl3anl3 pm2.61da2ne ) AEFZBEFZCUAFZUBZABGHZCIHZACIHZBCIHZ GHZJZAKBKWKAKJZLZKCIHZWTWOGHZWMWPWSWTWOWTGHZXAWKWTXBJZWRWKWTDUCZCIHZWTGHZ @@ -389333,7 +389333,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( vx wcel cz cmul co clgs wceq cc0 wa oveq2 oveq1d c1 adantr lgscl oveq2d zcnd wne cn0 w3a cv eqeq2d wral c2 cexp sq1 eqeq2i wb cr cle nn0re nn0ge0 wbr 1re 0le1 sq11 mpanr12 syl2anc bitr3id biimpa 1lgs ad2antlr eqtrd nn0z - ad2antrr 0z sylancl mulid2d eqtr2d cc sylan mul01d cif lgs0 syl ifnefalse + ad2antrr 0z sylancl mullidd eqtr2d cc sylan mul01d cif lgs0 syl ifnefalse sylan9eq 3eqtr4rd pm2.61dane ralrimiva simp3 rspcdva mulcomd eqtr4d oveq1 3ad2ant1 mul02d sylan9eqr simpr 3eqtr4d simp2 lgsdi syl3anl1 pm2.61da2ne ) AUAEZBFEZCFEZUBZABCGHZIHZABIHZACIHZGHZJZBKCKWTBKJZLZAKIHZXIXDGHZXBXEXHX @@ -389832,7 +389832,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and iftrued halfcn subsubd prmnn nngt0 syl21anc halfgt0 addgt0d 0red readdcld divgt0 resubcl ancoms ltletr mpand elnnz omoe syl2an2r syl2anc 1red nnge1 lesub2dd lediv1 3imtr4g 3imp21 subcld zre halfge0 rehalfcl subge02d mpbii - mpan2d lesub lenltd eqcomd recnd mulcomd mulsubfacd 2m1e1 mulid2d 3bitr3d + mpan2d lesub lenltd eqcomd recnd mulcomd mulsubfacd 2m1e1 mullidd 3bitr3d letr 3imtr4d com3l com13 eqnbrtrd iffalsed 3adant1 3eqtrrd pm2.61i impbid nncan bitrid eqrdv ) AUGDUAZIEUBJZUGUHZVUCKZVUEBUHZLUCJZCLUDJZMNZVUHCVUHO JZUEZUFZBVUDUIZAVUEVUDKZVUFVUNUJUGBVUDVULVUEDUKHULUMAVUNVUOAVUMVUOBVUDVUJ @@ -389986,7 +389986,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and cn wb 3nn0 4nn divfl0 sylancl mpbid eqtrid c0 adantr fz10 eqtrdi prodeq1d oveq2 prod0 oveq1 0p1e1 oveq1d oveq12d fzfid cc cmin cif breq1d ifbieq12d oveq2d elfzelz zcnd 2cnd mulcld adantl eldifi prmz subcld fvmptd3 eqeltrd - simpr 3syl ifcld adantll fprodcl mulid2d eqtr2d syl a1d gausslemma2dlem0d + simpr 3syl ifcld adantll fprodcl mullidd eqtr2d syl a1d gausslemma2dlem0d ex cin nn0red ltp1d fzdisj cun cle w3a cz eluzelre 4re a1i redivcld flcld cr 4ne0 crp mp3an2i sylbi nnrp ax-mp eluz2 4lt5 5re ltleletr mpani 3impia zre divge1 1zzd flge elnnz1 sylanbrc oddprm prmuz2 fldiv4lem1div2uz2 3jca @@ -390094,7 +390094,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and gausslemma2dlem7 $p |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 ) $= ( cmo co c1 c2 cmul wceq cz wcel cneg cexp gausslemma2dlem6 nnnn0d faccld - cfa cfv gausslemma2dlem0b nncnd mulid2d eqcomd oveq1d eqeq1d cn cgcd 1zzd + cfa cfv gausslemma2dlem0b nncnd mullidd eqcomd oveq1d eqeq1d cn cgcd 1zzd wb cn0 neg1z gausslemma2dlem0h zexpcl sylancr 2z zmulcld nnzd cprime cdif csn eldifi prmnn 3syl gausslemma2dlem0c cncongrcoprm syl32anc bitrd simpr wa cr clt wbr nnred prmgt1 jca syl 1mod adantr eqtr3d ex sylbid mpd ) AEU @@ -390116,7 +390116,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and 1mod crp cn0 neg1z gausslemma2dlem0h zexpcl sylancr 2nn gausslemma2dlem0b a1i nnnn0d nnexpcld nnzd zmulcld zred 1red adantr gausslemma2dlem0a nnrpd cn simpr modmul1 syl3anc ex zcnd nncnd mul32d caddc nn0cnd 2timesd oveq2d - neg1cn expaddd nn0zd m1expeven 3eqtr3d oveq1d mulid2d 3eqtrd eqeq12d cmin + neg1cn expaddd nn0zd m1expeven 3eqtr3d oveq1d mullidd 3eqtrd eqeq12d cmin cc cdiv oveq2i oveq1i eqeq1i 2z lgsvalmod eqeq1d gausslemma2dlem0i sylbid biimtrid syld mpd ) AMUAZGNOZPENOZQOZCROZMSZPCUGOZXPSZABCDEFGHIJKLUBAXTXS MCROZSZYBAMYCXSAYCMACUCTZMCUDUHZUEZYCMSACUFPUIZUJTZCUFTZYGHCUFYHUKYJYEYFY @@ -390179,7 +390179,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and cv wa csn cdif adantr oddprm syl nnzd caddc wne wceq neg1cn a1i neg1ne0 cn cc 2z simpr expmulz syl22anc cn0 eldifad prmz elfzelz adantl sylancr zmulcl zmulcld prmnn zmodfz syl2anc eqeltrid elfznn0 zcnd 2cnd divcan2d - nn0zd 2ne0 oveq2d neg1sqe1 oveq1i 1exp eqtrid 3eqtr3d oveq1d mulid2d cr + nn0zd 2ne0 oveq2d neg1sqe1 oveq1i 1exp eqtrid 3eqtr3d oveq1d mullidd cr eqtrd crp cle clt nnrpd nn0ge0d zred modlt eqbrtrid modid eqeltrd nncnd nn0red renegcld recnd addcomd negsubd 3eqtr2d modcyc nnred ltled mpbird syl3anc resubcld wn cdvds 2nn elfznn elfzle2 wb cuz syl112anc fzm1ndvds @@ -390253,7 +390253,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and adantr 2z elfzelz zmulcl sylancr zmulcld zmodcld eqeltrid nn0zd m1expcl syl cn nn0cnd a1i syl112anc eqidd oveq1i cr zred modabs2 syl2anc eqtrid modmul12d cdvds wbr moddvds syl3anc zcnd subdid mul12d eqtrd cgcd prmrp - breq2d mpbird coprmdvds mpan2d mulassd ax-1cn mulid2d eqeq2d cle elfznn + breq2d mpbird coprmdvds mpan2d mulassd ax-1cn mullidd eqeq2d cle elfznn caddc nnred elfzle2 3syl sylbid oveq1 eqeq1d syl5ibrcom clt 2nn nnmulcl imbi1d nnnn0d nn0ge0d lemuldiv2 zltlem1 modid syl22anc nfcv nffv nfv wn fvmpt mpan2 csn cdif prmnn 2cnd 2ne0 div11 nnrpd crp zsubcld dvdsmultr2 @@ -390411,7 +390411,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and mgpplusg neg1z prmnn zmodcld eqeltrid gsummptfidmadd2 offval2 zringmulr c0ex rhmmul zred nnrpd modval nnne0d div23d pncan3d 2cnd mul12d expaddz crp nn0zd expmulz 1cnd eldifsni necomd neneqd 2z dvdsprm mtbird oexpneg - cuz uzid negeqd syl2an 2nn0 expmuld oveq1i mulassd mulid2d lgseisenlem3 + cuz uzid negeqd syl2an 2nn0 expmuld oveq1i mulassd mullidd lgseisenlem3 neg1sqe1 mpteq2dva 3eqtr3rd ringridm gsumconst hashfz1 mhmmulg submmulg gsumcl chash oddprm cnfldexp csubg subrgsubg subgsubm df-zring gsumsubm gsumfsum fsumzcl zndvds mpbid moddvds mpbird ) AEDUBUCUDUEUFUDZUGUDZDUH @@ -390861,13 +390861,13 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and lgsquad2lem1 $p |- ( ph -> ( ( M /L N ) x. ( N /L M ) ) = ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) ) $= ( c1 co c2 cmul wcel cneg cmin cdiv cexp caddc clgs cc wceq nnzd ax-1cn - zcnd npcan sylancl oveq12d cz peano2zm syl muladdd 1t1e1 oveq2d mulid1d + zcnd npcan sylancl oveq12d cz peano2zm syl muladdd 1t1e1 oveq2d mulridd a1i eqtrd eqtr3d oveq1d mulcld addcl addcld addsubd 3eqtrd 2cnd cc0 wne pncan divdird divassd divcan2d cdvds wbr wn dvdsmul1 syl2anc breqtrd wa 2ne0 wi dvdstr mp3an2i mpan2d mtod 1zzd cprime 2prm nprmdvds1 mp1i omoe 2z syl22anc wb dvdsval2 dvdsmul2 mulassd 3eqtr2d zmulcld zaddcld neg1cn mpbid adddird neg1ne0 expaddz expmulz oveq1i 1exp eqtrid expclzd eqtr4d - neg1sqe1 mulid2d lgscl mul4d nnne0d lgsdir syl32anc lgsdi 3eqtr2rd ) AP + neg1sqe1 mullidd lgscl mul4d nnne0d lgsdir syl32anc lgsdi 3eqtr2rd ) AP UAZDPUBQZRUCQZEPUBQZRUCQZSQZUDQZYFBPUBQZRUCQZYJSQZCPUBQZRUCQZYJSQZUEQZU DQZBEUFQZEBUFQZSQZCEUFQZECUFQZSQZSQZDEUFQZEDUFQZSQZAYLYFRYNYQSQZYJSQZSQ ZYNYQUEQZYJSQZUEQZUDQZPYFUUOUDQZSQZYTAYKUUPYFUDAYKRUUKSQZUUNUEQZYJSQUUT @@ -391005,7 +391005,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and ( cn wcel c2 cdvds wbr wn wa co c1 wceq cmul cexp cz syl syl2anc wb cc0 wne cgcd clgs cneg cmin cdiv cfv cr simplrl nnz ad3antrrr lgscl absresq gcdcomd cabs zred simpr lgsabs1 mpbird oveq1d sq1 eqtrdi zcnd sqvald 3eqtr3d oveq2d - eqtrd mulassd eqtr4d mulid1d simplll simpllr simplrr lgsquad2 cc neg1cn a1i + eqtrd mulassd eqtr4d mulridd simplll simpllr simplrr lgsquad2 cc neg1cn a1i neg1ne0 1zzd cprime 2prm nprmdvds1 mp1i syl22anc peano2zm dvdsval2 mp3an12i omoe 2ne0 mpbid adantr ad2antlr zmulcld expclzd mul01d lgsne0 gcdcom eqeq1d 2z bitrd syl2anr necon1bbid ad2ant2r biimpa syl2an 3eqtr4rd pm2.61dan ) ACD @@ -391468,7 +391468,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and $( Lemma 3 for ~ 2lgsoddprmlem3 . (Contributed by AV, 20-Jul-2021.) $) 2lgsoddprmlem3c $p |- ( ( ( 5 ^ 2 ) - 1 ) / 8 ) = 3 $= ( c5 c2 cexp co c1 cmin c8 cdiv c3 cmul caddc df-5 oveq1i 4cn eqtri 3cn 8cn - c4 ax-1cn 2cn wcel wceq binom21 ax-mp mulcli sq4e2t8 4t2e8 mulid2i mulcomli + c4 ax-1cn 2cn wcel wceq binom21 ax-mp mulcli sq4e2t8 4t2e8 mullidi mulcomli eqtr4i oveq12i adddiri 2p1e3 3eqtr2i mvrraddi cc0 0re 8pos gtneii divcan4i cc ) ABCDZEFDZGHDIGJDZGHDIVCVDGHVCRBCDZBRJDZKDZEKDZEFDVDVBVHEFVBREKDZBCDZVH AVIBCLMRVAUAVJVHUBNRUCUDOMVHVDEIGPQUESVGVDEKVGBGJDZEGJDZKDBEKDZGJDVDVEVKVFV @@ -391709,7 +391709,7 @@ minus one divided by eight (` ( 2 /L N ) ` = -1^(((N^2)-1)/8) ). Mario Carneiro, 20-Jun-2015.) $) 2sqlem6 $p |- ( ph -> A e. S ) $= ( vm cn wcel cmul wi wral cdvds cprime wa vx vy vz vn cv co wbr c1 wceq - breq2 imbi1d ralbidv oveq2 eleq1d imbi12d nncn mulid1d biimpd a1i breq1 + breq2 imbi1d ralbidv oveq2 eleq1d imbi12d nncn mulridd biimpd a1i breq1 rgen eleq1 rspcv cz prmz iddvds syl simprl simpll simprr simplr 2sqlem5 expr ralrimiva ex embantd syld c2 cuz cfv anim12 wo wb eluzelz ad2antrr simpr ad2antlr euclemma syl3anc jaob bitrdi ralbidva r19.26 cbvralvw cc @@ -391808,10 +391808,10 @@ minus one divided by eight (` ( 2 /L N ) ` = -1^(((N^2)-1)/8) ). cprime adantr zred cr rehalfcld nnsqcld 4sqlem7 letrd eluzelz zaddcld dvds2addd addsub4d adddid gcdcomd ax-1ne0 eqnetrd neneqd gcdeq0 mtbid 2sqlem8a dvdslegcd simpr necon3ai gcdn0cl syl21anc nnle1eq1 nn0addcld - 2nn rplpwr coprmdvds cin 2sqlem7 inss2 1cnd mulid1d 3eqtr2rd mulcanad + 2nn rplpwr coprmdvds cin 2sqlem7 inss2 1cnd mulridd 3eqtr2rd mulcanad mulgcd eqidd oveq2 oveq2d rspc2ev eqeq1 anbi2d 2rexbidv elab2 divgt0d ovex sylibr elnnz sylanbrc cfz prmnn peano2zm simprr prmz dvdsle 1red - nn0ge0d nnge1d lemul1ad mulid2d 3brtr3d le2addd 2halvesd crp rphalflt + nn0ge0d nnge1d lemul1ad mullidd 3brtr3d le2addd 2halvesd crp rphalflt recnd nnrpd lelttrd sqvald ltdivmul fznn mpbir2and jca dvdsmul2 breq1 zltlem1 eleq1w imbi12d breq2 imbi1d rspc2v syl3c expr ralrimiva inss1 eqeltrd 2sqlem6 ) AEMKUMUNUOZLUMUNUOZUPUOZMUQUOZJULRAMURUSZMUTVAZAMUM @@ -391986,7 +391986,7 @@ minus one divided by eight (` ( 2 /L N ) ` = -1^(((N^2)-1)/8) ). oveq12d negsubdi2 sylancr cgcd cabs cfv cle cr zred absresq resqcld prmnn clt cn nnred cn0 zsqcl2 nn0addge2 exp1d cuz prmuz2 eluz2gt1 1lt2 syl32anc 2z ltexp2a lelttrd eqbrtrd abscld absge0d nnnn0d nn0ge0d lt2sqd mpbird wi - eqbrtrrd ltnled cc0 sqnprm abs00ad eqeltrrd sq0i oveq2d syl5ibcom addid1d + eqbrtrrd ltnled cc0 sqnprm abs00ad eqeltrrd sq0i oveq2d syl5ibcom addridd eleq1d sylibd sylbid mtod wo nn0abscl elnn0 sylib ord mt3d eqtr3d wa cdif dvdsle dvdsabsb mtbird coprm mvrraddd negeqd dvdsmultr2 syl3anc mpd subsq sq1 oveq2i eqtr3id dvdsadd2b syl112anc subneg oveq1 breq2d rspcev eldifsn @@ -392047,7 +392047,7 @@ minus one divided by eight (` ( 2 /L N ) ` = -1^(((N^2)-1)/8) ). <-> ( A = 1 /\ B = 1 ) ) ) $= ( cn0 wcel wa c2 cexp co caddc c1 cle wbr adantr wb cc0 wi eqeq1d cq sylbid wceq cr nn0sqcl nn0red anim12ci nn0addge2 breq2 adantl ad2antlr nn0le2is012 - syl w3o ex oveq2 nn0cnd addid1d csqrt cfv nn0re sqge0d 2nn0 a1i 0le2 sqrt11 + syl w3o ex oveq2 nn0cnd addridd csqrt cfv nn0re sqge0d 2nn0 a1i 0le2 sqrt11 syl22anc nn0ge0 sqrtsqd wnel sqrt2irr wn df-nel id eqcoms eleq1d cz nn0z zq notbid pm2.24d com12 expd sylbi sylbird impancom cmin cc w3a 2cnd 1cnd 3jca ax-mp subadd2 bicomd sylan9bbr nn0sqeq1 2m1e1 eqcom bitrdi syl6 syld eqcomd @@ -392080,7 +392080,7 @@ minus one divided by eight (` ( 2 /L N ) ` = -1^(((N^2)-1)/8) ). (Contributed by Thierry Arnoux, 4-Feb-2020.) $) 2sqn0 $p |- ( ph -> A =/= 0 ) $= ( cc0 wceq c2 cexp co caddc cprime wcel eqeltrd adantr wa sq0i zcnd sqcld - oveq1d addid2d sylan9eqr wn cz sqnprm syl eqneltrd pm2.65da neqned ) ABIA + oveq1d addlidd sylan9eqr wn cz sqnprm syl eqneltrd pm2.65da neqned ) ABIA BIJZBKLMZCKLMZNMZOPZAUQUMAUPDOHEQRAUMSUPUOOUMAUPIUONMUOUMUNIUONBTUCAUOACA CGUAUBUDUEAUOOPUFZUMACUGPURGCUHUIRUJUKUL $. @@ -392240,8 +392240,8 @@ minus one divided by eight (` ( 2 /L N ) ` = -1^(((N^2)-1)/8) ). -> E. x e. NN E. y e. NN P = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) $= ( va vb cprime wcel co wceq wa c2 cexp caddc cn0 cc0 eqeq2d adantr adantl cn wi c4 cmo c1 cv 2sqnn0 wo elnn0 weq oveq1 oveq1d oveq2d rspc2ev 3expia - wrex a1d expcom sq0i nncn sqcld addid2d eqtrd wb eleq1 nnz sqnprm pm2.21d - cz wn syl sylbid com23 jaod addid1d oveqan12rd 00id eqtrdi pm2.21i syl6bi + wrex a1d expcom sq0i nncn sqcld addlidd eqtrd wb eleq1 nnz sqnprm pm2.21d + cz wn syl sylbid com23 jaod addridd oveqan12rd 00id eqtrdi pm2.21i syl6bi ex 0nprm jaoi sylbi com12 imp rexlimdvv mpd ) CFGZCUAUBHUCIZJZCDUDZKLHZEU DZKLHZMHZIZENUNDNUNCAUDZKLHZBUDZKLHZMHZIZBSUNASUNZDECUEWIWOXBDENNWGWJNGZW LNGZJZWOXBTZTWHXEWGXFXCXDWGXFTZXCWJSGZWJOIZUFZXDXGTWJUGXDXJXGXDWLSGZWLOIZ @@ -392375,7 +392375,7 @@ to the second component (see, for example, ~ 2sqreunnltb and c0ex sq0 1p0e1 negex 2ex sq2 cmin negsubi 3p1e4 subaddrii addcomli pm3.2i 4cn oveq1d eqeq1d 2nreu mp2 eqeq2 reubidv mtbiri a1d csqrt id 0cnd adantr opelxpd 1cnd peano2cnm sqrtcld animorrl opthneg sylan2 mpbird 3jca op1stg - wb mpdan op2ndg sq0id addid1 eqtrd syl2anc sqsqrtd pncan3d sylc pm2.61ine + wb mpdan op2ndg sq0id addrid eqtrd syl2anc sqsqrtd pncan3d sylc pm2.61ine jca expcom ) ACDZBUAZEFZXQGFZHIJZKJZALZBCCUBZUCZUDZUEAMAMLZYEXPYFYDYAMLZB YCUCZMNUFZYCDZOUOZHUFZYCDZYIYLPZUGYIEFZYIGFZHIJZKJZMLZYLEFZYLGFZHIJZKJZML ZUHYHUDYJYMYNMCDZNCDZYJUIUJMNCCUKULYKCDHCDYMOUMUNZUPYKHCCUKULYNMYKPZNHPZU @@ -392955,7 +392955,7 @@ to the second component (see, for example, ~ 2sqreunnltb and elrpd syl2anc cexp elrpii rpexpcl nnrpd rpdivcld epr rerpdivcl c3 egt2lt3 4pos simpri 3lt4 3re lttri mp2an ltled oveq1i nngt0d mpbird recnd breqtrd syl3anc cc recni rpcnd wceq wne lttrd leidi logdivlt loge breqtrdi pm3.2i - mpanl12 jca lt2mul2div syl22anc mulid2d ltmuldiv ltsub2dd subdird mulcomd + mpanl12 jca lt2mul2div syl22anc mullidd ltmuldiv ltsub2dd subdird mulcomd 2z zmulcl relogexp 2cnd nnnn0d cn0 2nn0 expmuld sq2 eqtrdi fveq2d 3eqtr2d divrec2d divcan5d eqtr3d oveq12d eqtrd relogdivd 3brtr4d cbc chebbnd1lem1 cif eqid ltmuldivd rpcnne0d divass flle lemuldiv2 ppiwordi lemul1d ltdiv1 @@ -393141,7 +393141,7 @@ to the second component (see, for example, ~ 2sqreunnltb and sqrtltd breqtrdi chtppilimlem2 wa adantr max1 wb 2re elicopnf ax-mp chtcl simplbi syl ppinncl sylbi nnrpd 1lt2 simprbi rplogcld rpmulcld rerpdivcld cn lelttr syl3anc mpand wceq recnd sqsqrtd oveq1d rpregt0d ltmuldiv bitrd - breq1d 2pos chtleppi mulid1d breqtrrd ledivmuld mpbird abssuble0d ltsub23 + breq1d 2pos chtleppi mulridd breqtrrd ledivmuld mpbird abssuble0d ltsub23 rpcnd 3imtr4d imim2d ralimdva reximdv mpd rgen ralrimiva ssriv 1cnd rlim2 cc wss mpbiri mptru ) ADUAUBEZAUCZUDFZYOUEFZYOUFFZGEZUGEZUHHUIIZJUUABUCYO KIZYTHUJEUKFZCUCZLIZUOZAYNULZBMUMZCNULUUHCNUUDNOZUUBHUUDUJEZHDUGEZKIZUUKU @@ -393243,7 +393243,7 @@ to the second component (see, for example, ~ 2sqreunnltb and eqidd 3eqtr2d mpteq2dva offval2 rpne0d syl211anc eqtrd co1 chto1lb rphalfcl dmdcan 1rp cxploglim rlimres2 o1rlimmul eqbrtrrd rlimadd 1p0e1 breqtrdi 1re readdcl chpcl chtcl readdcld letrd chpub lediv1dd rpcnd divdir divid oveq1d - breqtrd adantrr mulid2d chtlepsi eqbrtrd lemuldivd mpbid rlimsqz2 mptru + breqtrd adantrr mullidd chtlepsi eqbrtrd lemuldivd mpbid rlimsqz2 mptru 1le2 ) ABUAUBCZAUCZUDUEZYIUFUEZDCZEFGHIAYHFYIUGUEZYIUHUEZJCZYKDCZKCZYLFFIUI ZYRIAYHYQEFLKCFGIAYHFYPFLSIYIYHMZUJZUIZYTYOYKYTYMYNYTYIYSYISMZIYSUUBBYINHZB SMZYSUUBUUCUJUKULBYIUMUOZUNZUPZYTYIYSYIOMIYSYIUUFYSLBYIYSUQUUDYSULPUUFLBURH @@ -393311,7 +393311,7 @@ to the second component (see, for example, ~ 2sqreunnltb and fsumrecl rprege0 flge0nn0 faccl nnrpd relogcld rerpdivcl mpancom nnncan2d 3syl mpteq2ia eqtri cchp 1red chpo1ub rpre subcld adantr remulcld nndivre chpcl cabs syl2an reflcl resubcld vmage0 fracle1 lemul2ad subdid wne rpcn - rpcnne0 w3a divass eqtr3d syl3anc oveq1d eqtr4d mulid1d 3brtr3d fsummulc1 + rpcnne0 w3a divass eqtr3d syl3anc oveq1d eqtr4d mulridd 3brtr3d fsummulc1 div23 fsumle logfac2 oveq12d fsumsub chpval 3brtr4d clt wb rpregt0 lediv1 mpbid divsubdir rpne0 divcan4d eqtr2d fveq2d flle subge0d mulge0d breqtrd mpbird fsumge0 breqtrrd divge0 syl21anc absidd eqtrd chpge0 ad2antrl o1le @@ -393384,7 +393384,7 @@ to the second component (see, for example, ~ 2sqreunnltb and resubcld rpred nnred peano2rem remulcld caddc nncnd ax-1cn sylancl fveq2d npcan rpge0d syl2anc eqbrtrrd readdcld remulcl lem1d sqrtled mpbid mpbird loglesqrt subge0d leadd2dd times2d breqtrrd lemul1ad cexp sqsqrtd oveq12d - rpcnd subcl subsq nncan 3eqtr3d 2cn recnd mulassd 3brtr3d lemul1d mulid2d + rpcnd subcl subsq nncan 3eqtr3d 2cn recnd mulassd 3brtr3d lemul1d mullidd 1red mul32d remsqsqrt mul4d eqtr3d wne rpcnne0d divsubdiv syl22anc subdid mulcomd eqtr4d mulcld rpmulcld rerpdivcld rpne0d divcan2d 3eqtrd letrd wb clt nngt0d ledivmul syl112anc fsumle fvoveq1 oveq1 2m1e1 eqtrdi sqrt1 nnz @@ -393441,9 +393441,9 @@ to the second component (see, for example, ~ 2sqreunnltb and mpbid eqbrtrd 1re fveq2d nncnd elfzuz eluz2nn eleq2s syl2an sylan2 rpge0d nnmulcld nnrecred syl3anc nnne0d divrecd eqtr4d imbitrid imp simp3d lep1d leabsd prmuz2 nn0abscl nn0p1nn elfz5 ex ssrdv ssfid simprl vmappw adantrr - nnzd nnexpcl uz2m1nn nngt0d ltletrd mulid2d logled lemuldivd flge1nn nnuz + nnzd nnexpcl uz2m1nn nngt0d ltletrd mullidd logled lemuldivd flge1nn nnuz 2re eleqtrdi oveq2 eleq1d fsum1p exp1d iftrued subidd 1p1e2 oveq1i cq nnq - expnprm subid1d sumeq12dv nnnn0 addid2d 3eqtrd rpreccld peano2zd rpexpcld + expnprm subid1d sumeq12dv nnnn0 addlidd 3eqtrd rpreccld peano2zd rpexpcld flcld resqcld peano2nnd reexpcld subge02d dmdcan divsubdir divdiv1 sqvald 3eqtr3d breqtrrd resubcl recgt1 posdif ledivmul syl112anc lemul2d exprecd divid nnz sumeq2dv expcl fsummulc2 cfzo fzval3 ltned 2nn0 eluzp1p1 fveq2i @@ -393555,7 +393555,7 @@ to the second component (see, for example, ~ 2sqreunnltb and ( c1 co wcel cle cc0 vq vp crp cfl cfv cfz cvma cdiv csu clog cmin cmpt vk cv cmul co1 cvv cc cr rpssre a1i wa fzfid cn elfznn adantl vmacl syl nndivred recnd fsumcl adantr 1re wf cz nnnn0d elfzelz ffvelcdmd resubcl - eqid cn0 sylancr remulcld eqidd wceq 1cnd oveq1d mulid2d sumeq2dv eqtrd + eqid cn0 sylancr remulcld eqidd wceq 1cnd oveq1d mullidd sumeq2dv eqtrd eqtr3d cdvds wbr cprime crab wss cfn nnrpd relogcld cuz sylan2 fsumrecl 1red simpr 0re eqeltrdi wne ad3antrrr ad2antrr dchrn0 biimpa pm2.61dane adantlr dchr1 0le1 eqbrtrdi leidi mpbird clt vmage0 nnred nngt0d divge0 @@ -393723,12 +393723,12 @@ to the second component (see, for example, ~ 2sqreunnltb and cc ifnefalse eqid 3eqtrd wral reflcl lemuldiv2d fznn0 mpbir2and weq flle mul01d fzo0 sum0 oveq1 lep1d clt nnred nngt0d lemul2 syl112anc peano2re mpbid cuz nn0mulcl sylan nn0uz nn0p1nn nnmulcl elfz5 recnd - eleqtrdi adddid mulid1d peano2zm ad3antrrr elfzelz fsumshftm subidd + eleqtrdi adddid mulridd peano2zm ad3antrrr elfzelz fsumshftm subidd sub32d eqtr4d 3eqtr4d dvdsmul1 breqtrrd zndvds syl3anc sumeq2dv cbs nn0zd cbvsumv cres fveq2 nnne0d wf1o czrh reseq1i znf1o fvres dchrf eqeltrdi ffvelcdmda fsumf1o dchrsum 3eqtr3rd eqtr2di syl5ibr expcom cphi 00id a2d nn0ind impcom syldan crp modval nn0cnd pncan3d eqtr2d - zmodcl rspc2va syl21anc fsumcl addid2d breq1d zmodfzo eqbrtrd ) AHU + zmodcl rspc2va syl21anc fsumcl addlidd breq1d zmodfzo eqbrtrd ) AHU QURZUSZUTHVAVBZJVCZMVDZPVDZJVEZVFVDUTHOVGVBZVAVBZUYGJVEZVFVDZGVHUYC UYHUYKVFUYCUYHUTOHOVIVBZVJVDZVKVBZVAVBZUYGJVEZUYOHVAVBZUYGJVEZVLVBU TUYKVLVBUYKUYCUYPUYRUYGUYDJUYPUYRVMVNVOUYCUTUYOHVPVQUYCUYOUTHVRVBUR @@ -394112,7 +394112,7 @@ to the second component (see, for example, ~ 2sqreunnltb and cz dchrzrhcl elfznn mucl zred nndivre mpancom syl recnd mulcld fsumcl cn caddc cseq cli wbr climcl adantr cmin subcl 1red cc0 cle cpnf cico cabs adantlrr wf nnuz wne divcld 2fveq3 id oveq12d rpred syl2an nncnd - wceq mulid2d wb eqbrtrd flge1nn syl2anc sumeq2dv oveq1d nnne0d oveq2d + wceq mullidd wb eqbrtrd flge1nn syl2anc sumeq2dv oveq1d nnne0d oveq2d mpbid zcnd 3eqtr2d eleqtrdi abscld remulcld rerpdivcld absge0d nnnn0d cn0 eqtrd lemul12ad absmuld 3brtr4d syl3anc breqtrd letrd simpld 1zzd sylancr elrege0 sylib nnz nnne0 cbvmptv eqtri fmptd ffvelcdmda simprl @@ -394272,7 +394272,7 @@ to the second component (see, for example, ~ 2sqreunnltb and wa cz mucl syl cc adantr elfzelz adantl dchrzrhcl elfznn nncnd nnne0d zcnd divcld nnrpd rpdivcl syl2an relogcld mulcld adantrr dvdsflsumcom crp recnd fzfid wss fz1ssnn a1i cuz cle flge1nn syl2anc nnuz eleqtrdi - eluzfz1 musumsum dchrzrh1 oveq1d 1div1e1 eqtrdi rpcnd mulid2d 3eqtrrd + eluzfz1 musumsum dchrzrh1 oveq1d 1div1e1 eqtrdi rpcnd mullidd 3eqtrrd cr div1d wb fznnfl simprbda zred nndivred ad2antrr fsummulc2 rpcnne0d cc0 wne syl3anc rpne0d mulassd ad2antlr dchrzrhmul divmuldiv syl22anc div12 mul4d eqtr4d divdiv1 eqcomd mulcomd eqtrd 3eqtr4d sumeq2dv ) AU @@ -394325,7 +394325,7 @@ to the second component (see, for example, ~ 2sqreunnltb and rpdivcl syl2an fsumadd cmin relogdivd oveq2d cc pncan3d eqtr2d oveq1d rpdivcld nnne0d divdird adddid sumeq2dv dchrvmasumlem1 dchrvmasum2lem nncnd eqtrd oveq12d 3eqtr4rd iftrue fveq2d sumeq2sdv 3eqtr4d wn vmacl - addid1d iffalse eqcomd sylan9eqr pm2.61dan ) ABUDCUEUFZUGUHZGUIZIUFKU + addridd iffalse eqcomd sylan9eqr pm2.61dan ) ABUDCUEUFZUGUHZGUIZIUFKU FZXQUJUFZXQUKUHZULUHZGUMZBCUNUFZUOUPZUQUHZXPMUIZIUFKUFZYFURUFZYFUKUHZ ULUHZUDCYFUKUHZUEUFZUGUHZFUIZIUFKUFZBYKYNUPZUNUFZYNUKUHZULUHZFUMZULUH ZMUMZUSABUTYBYCUQUHZXPYJYMYOYKUNUFZYNUKUHZULUHZFUMZULUHZMUMZYEUUBAUUC @@ -394378,7 +394378,7 @@ to the second component (see, for example, ~ 2sqreunnltb and wb syl oveq2d ad3antrrr wne syl3anc readdcld nndivred divge0d fsumge0 o1add2 eleq1d absidd eqeltrd 1le3 1lt3 lbico1 simp1bi simp2bi ltletrd jctir mp3an 0lt1 elrpd r19.21bi biidd rspcv mpsyl jca log1 nncnd rpre - mulid2d fznnfl simplbda lemuldivd 1rp logled eqbrtrrid rpregt0 divge0 + mullidd fznnfl simplbda lemuldivd 1rp logled eqbrtrrid rpregt0 divge0 syl21anc mulge0 syl12anc fvoveq1d id oveq12d breq12d nndivre elicopnf absidm fveq2 sylanbrc rpcnne0 rpcnne0d divdiv2 mulassd eqtr4d breqtrd ax-mp breq1d syl3anbrc fsumharmonic fsummulc2 oveq1d breqtrrd adantrr @@ -394436,7 +394436,7 @@ to the second component (see, for example, ~ 2sqreunnltb and rspcdva elfzelz dchrzrhcl mucl syl zred mulcld fsumcl cle wbr fsumabs recnd nnrecred absge0d absmuld cbs eqid wf wfo cn0 nnnn0d znzrhfo fof ffvelcdmd dchrabs2 nncnd nnne0d cc0 rprege0d absid oveq2d eqtrd mule1 - absdivd lediv1dd eqbrtrd mulid2d breqtrd lemul1ad fsumle letrd leabsd + absdivd lediv1dd eqbrtrd mullidd breqtrd lemul1ad fsumle letrd leabsd lemul12ad divrec2d 3brtr4d adantrr o1le ) ABULUMBUNZUOUPZUQURZKFUSURZ UTUPZPUNZVAURZPVBZYRUUALUPZNUPZUUAVCUPZUUAVAURZVDURZYSVDURZPVBZUMVFAV GABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKVEAYPULVHZVIZYRUUBPUULUMYQ @@ -394783,10 +394783,10 @@ a multiplicative function (but not completely multiplicative). syl2anc cabs mpd eqtr3d pm2.61dane abscld leabsd eqbrtrd mpbid cmgp clt cmg ccnfld czring syl3anc cnfldexp fveq2d cmhm mgpbas mhmmulg prmz elfznn0 reexpcl fsumrecl 1le1 0le1 keephyp chash c0 cuz sylibr - cfn hashnncl nnge1d elfzelz sylan9eq ax-1cn fsumconst mulid1d caddc - fzn0 leidd mulid2d nn0p1nn 0expd expp1 prmnn nnexpcld sqsqrtd nnne0 + cfn hashnncl nnge1d elfzelz sylan9eq ax-1cn fsumconst mulridd caddc + fzn0 leidd mullidd nn0p1nn 0expd expp1 prmnn nnexpcld sqsqrtd nnne0 cq nnq 2z pcexp syl121anc pcid 3eqtr3rd pccld 2nn0 expmuld neg1sqe1 - oveq1i eqtrid mulid2i eqtrdi eqtrd negnegd cui dchrn0 dchrabs eqeq1 + oveq1i eqtrid mullidi eqtrdi eqtrd negnegd cui dchrn0 dchrabs eqeq1 biimpa syl5ibcom necon3ad absord ord negeqd 3brtr4d mul02d reexpcld peano2nn0 absexpd absge0d dchrabs2 exple1 syl31anc subge0 0re ifcli ifbothda necomd leneltd posdif lemuldiv syl112anc cfzo sumeq1d 0nn0 @@ -395044,12 +395044,12 @@ a multiplicative function (but not completely multiplicative). cr recnd anass1rs fsumcl ax-1cn neg1cn 0cn ifcli mulcl sylancl subdid fsummulc2 a1i ovif2 ad2antrr sylan9eqr fveq2d eqtrdi oveq1d ringidval fveq1 cmgp ccnfld eqtrd mul01d eqtrid oveq2d sumeq2dv wss ssfi syldan - oveq12d wral wo olcd sumss2 syl21anc wb 3eqtr3d eqtr3d 3eqtrd mulid1d + oveq12d wral wo olcd sumss2 syl21anc wb 3eqtr3d eqtr3d 3eqtrd mulridd csn caddc eqeltrd negeq neg0 ifsb mpteq2dva an32s cfz cvma clog chash cphi dchrfi cbs dchrf unitss adantrl wfo nnnn0d znzrhfo 3syl ffvelcdm fof elfzelz elfznn vmacl nndivred adantrr relogcl fsumsub mul12d cjre dchr1 1re ax-mp 1t1e1 wne df-ne cur ad5ant15 cmhm dchrmhm cnfld1 mhm0 - wn mulid2i ifeq1da ifeq2d sylan2br ifeq12da inss1 phicld nncnd sselda + wn mullidi ifeq1da ifeq2d sylan2br ifeq12da inss1 phicld nncnd sselda cuz ralrimiva elin baib ccnv cima eleq2i wfn ffnd baibd bitrid bitr2d fniniseg mul02d ifbieq2d ovif fsummulc1 sum2dchr mulass mul12 syl3anc w3a eqtr3id fsumcom cdif cabl cgrp dchrabl ablgrp grpidcl 4syl iftrue @@ -395167,7 +395167,7 @@ a multiplicative function (but not completely multiplicative). nnz dchrmusumlema simprrl simprrr dchrvmaeq0 breqtrd rexlimdvaa exlimdv wrex wex mpd seqex ovex simpl fsumcj cuz simpr eleqtrdi 3eqtr3rd climcj serf cj0 breqtrdi isumclim fveq1 sumeq2sdv eqeq1d elrab2 simplr nehash2 - suble0 df-2 oveq1i 1cnd addsubassd eqtrid adantrr adddid mulid1d 3eqtrd + suble0 df-2 oveq1i 1cnd addsubassd eqtrid adantrr adddid mulridd 3eqtrd lemul2ad mul01d 3brtr3d nn0ge0d vmage0 nngt0d syl22anc fsumge0 leaddsub clt divge0 mulge0d syl3anc 3brtr4d ex necon1bd mpi fveq1d eqtr3d cjrebd o1le ralrimiva ffnfv ) AJKUHUIZUJUAUKZJUIZULTZUAVVNUMVVNULJUNAVVNUOJAVV @@ -395458,7 +395458,7 @@ a multiplicative function (but not completely multiplicative). absid dchrabs2 lediv1dd eqbrtrd divsqrtsum2 sqrtdiv sqrtsq rpcnne0d mpdan rpcnne0 recdiv lemul12ad absmuld dmdcan reccl mulcomd 3brtr4d 1cnd eqtr3d fsumle chash flge0nn0 hashfz1 rpreccld fsumconst simprd - cfn divrecd flle mulid1d breqtrrd rpregt0 ledivmul adantrr eqeltrrd + cfn divrecd flle mulridd breqtrrd rpregt0 ledivmul adantrr eqeltrrd elo1d o1dif ) ABUPUQBURZUSUTZVAVBZIURZMUTZPUTZVUPVCUTZVDVBZVUMVJVEV BZVUPVDVBZLUTZVFVBZIVGZVHVKVIBUPVUOVUTIVGZGVFVBZVHVKVIABUPVVFGVLAVU MUPVIZVMZVUOVUTIVVIUQVUNVNZVVIAVUPVOVIZVUTVLVIZVUPVUOVIZAVVHVPZVUPV @@ -395607,7 +395607,7 @@ a multiplicative function (but not completely multiplicative). cz fsumcl syl3anc sumeq2dv rpred cuz ovexd ad2antrr elfzelz dchrzrhcl rpmulcl syl2an rpne0d divcld abscld sqrtmul oveq2d divdiv1 eqtr4d cin rpcnd c0 simpr reflcl ltp1d fzdisj cun cn0 flge0nn0 nn0p1nn 3syl nnuz - eleqtrdi rpexpcl sylancl recnd mulid1d simprr rpregt0 ad2antrl lemul2 + eleqtrdi rpexpcl sylancl recnd mulridd simprr rpregt0 ad2antrl lemul2 2z mpbid eqbrtrrd sqvald breqtrrd flword2 fzsplit2 eqeltrrd fsumsplit wb adantlrr eqtrd fveq2d eqled o1le ) ABULUMBUNZUOUPZUQURZUMUWKUSUTUR ZHUNZVAURUOUPZUQURZUWOKUPZNUPZUWOVBUPZVAURQUNZVBUPZVAURZQVCZHVCZUWLUM @@ -395753,7 +395753,7 @@ a multiplicative function (but not completely multiplicative). ( crp c1 cv cfl cfv cfz co cvma cdiv cmul csu cc0 wceq clog cif caddc cmpt co1 wa wne dchrisumn0 adantr ifnefalse syl oveq2d fzfid ad2antrr cz elfzelz adantl dchrzrhcl cn cr elfznn vmacl nndivre mpancom mulcld - wcel recnd fsumcl addid1d eqtrd mpteq2dva dchrvmasumif eqeltrrd ) ABU + wcel recnd fsumcl addridd eqtrd mpteq2dva dchrvmasumif eqeltrrd ) ABU HUIBUJZUKULZUMUNZHUJZKULMULZWQUOULZWQUPUNZUQUNZHURZFUSUTWNVAULZUSVBZV CUNZVDBUHXBVDVEABUHXEXBAWNUHWFZVFZXEXBUSVCUNXBXGXDUSXBVCXGFUSVGZXDUSU TAXHXFACDEFGIJKLMNOPQRSTUAUBUCUDUEUFUGVHVIFUSXCUSVJVKVLXGXBXGWPXAHXGU @@ -395806,7 +395806,7 @@ a multiplicative function (but not completely multiplicative). ( cfv c1 co cc0 cmin vm vy vf crp cphi cv cfl cfz cin cvma cdiv cmul clog csu cn wceq cdchr cbs c0g csn cdif crab chash cmpt wcel wa c0 adantr eqid co1 2fveq3 id oveq12d cbvsumv eqeq1i rabbii simpr dchrisum0 imnani eq0rdv - fveq2d hash0 eqtrdi oveq2d 1m0e1 cr relogcl recnd mulid1d eqtrd mpteq2dva + fveq2d hash0 eqtrdi oveq2d 1m0e1 cr relogcl recnd mulridd eqtrd mpteq2dva adantl cur pm2.21i rpvmasum2 eqeltrrd ) ABUDHUEPQBUFZUGPUHRDUIFUFZUJPWRUK RFUNULRZWQUMPZQUOUAUFZGPUBUFZPZXAUKRZUAUNZSUPZUBHUQPZURPZXGUSPZUTVAZVBZVC PZTRZULRZTRZVDBUDWSWTTRZVDVJABUDXOXPAWQUDVEZVFZXNWTWSTXRXNWTQULRWTXRXMQWT @@ -395932,7 +395932,7 @@ a multiplicative function (but not completely multiplicative). rpcn wne rprege0 absid oveq2d eqtrd abscld adantlr fsumabs fz1ssnn sselda fsumrecl absmuld absge0d simpl nnrpd rpdivcl sselid flle abssubge0d mule1 fracle1 eqbrtrd lemul12ad 1t1e1 breqtrdi chash cfn 1cnd fsumconst syl2anc - fsumle flge1nn sylan nnnn0d hashfz1 oveq1d mulid1d breqtrd letrd breqtrrd + fsumle flge1nn sylan nnnn0d hashfz1 oveq1d mulridd breqtrd letrd breqtrrd 3eqtrd ledivmuld mpbird elo1d crli ax-1cn divrcnv ax-mp rlimo1 mp1i o1add cn0 rpcnne0 w3a eqtr3d syl3anc 3eqtr3d sumeq2dv eqeltrrd nndivred fsumadd zred fsummulc2 npcand adddird div23 divass cdvds crab ssrab2 dvdsflsumcom @@ -395992,7 +395992,7 @@ a multiplicative function (but not completely multiplicative). adantr nnne0d absdivd absid eqtrd mule1 lediv1dd eqbrtrd harmonicbnd4 wne wceq rpcnne0 recdiv syl2anc breqtrd lemul12ad absmuld 1cnd dmdcan syl3anc rpcnd mulcomd 3brtr4d fsumle chash hashfz1 oveq1d fsumconst nn0cnd mpbird - cfn rpcn letrd rpne0 divrecd rpre flle mulid1d breqtrrd wb reflcl rpregt0 + cfn rpcn letrd rpne0 divrecd rpre flle mulridd breqtrrd wb reflcl rpregt0 clt ledivmul ad2antrl elo1d eqeltrrid o1dif mptru ) ADEAUAZUBFZUCGZCUAZUD FZUUTHGZEUUQUUTHGZUBFZUCGZEBUAZHGZBIZUVCUEFZJGZKGZCIZLUFMZNUVMADUUSUVBCIZ OKGZLUFMNADUVNOPUUQDMZUVNPMZNUVPUUSUVBCUVPEUURUGZUVPUUTUUSMZQZUVBUVTUVAUU @@ -396185,7 +396185,7 @@ a multiplicative function (but not completely multiplicative). cmpt co1 2cnd o1const mp1i recnd zred cexp subcld mulcld fsumcl adantrr cz wceq sylan2 nncnd eqtrd sumeq2dv fsummulc2 cc0 wne jca div23 syl3anc rpcnne0 divdiv1 fveq2d oveq12d oveq2d oveq2 rpred ere clt c3 ltleii 1rp - addge0d mulid2d letrd eqbrtrd mpbid wb logleb eqbrtrrid divge0 syl21anc + addge0d mullidd letrd eqbrtrd mpbid wb logleb eqbrtrrid divge0 syl21anc log1 ad3antrrr absidd breqtrdi lediv1 ad5ant15 adantlr logfacrlim2 mucl cmu simplr rerpdivcld rlimcl rlimo1 o1mul2 o1add2 sqcld ad2antrr addcld halfcld relogcl zcnd nnrp nnne0d div23d subdid fsumsub cdvds w3a divass @@ -396196,7 +396196,7 @@ a multiplicative function (but not completely multiplicative). rphalfcl emgt0 breqtrrdi 0le2 simplbda rpregt0 ad2antlr mulge0d absmuld fznnfl mule1 breqtrd mulog2sumlem1 abssubd oveq2i fveq2i eqtrdi mulassd lemul1ad divdiv2 3brtr4d abs2dif2d eqeltrrid abstrid sqge0d 2pos pm3.2i - resqcld rehalfcld wi ltle imp loge 0le1 le2sqd sq1 lemul2 recni mulid1i + resqcld rehalfcld wi ltle imp loge 0le1 le2sqd sq1 lemul2 recni mulridi leadd1dd le2addd eqbrtrid rpregt0d lemuldiv flcld 3re simpri lttrd fllt 2z 3z df-3 zleltp1 mpbird eluz2 syl3anbrc syldan logled fsumle fsumless fzss2 fsumharmonic df-2 oveq1i eqtr4i breqtrrd eqbrtrrd leabsd o1le ) A @@ -396384,8 +396384,8 @@ a multiplicative function (but not completely multiplicative). syl cdvds crab wceq vmasum wss rerpdivcld o1dif mpbird divge0d mpbid clo1 dvdsssfz1 ssfid ssrab2 simprr sselid fsumdivc oveq2 ad2antrl dvdsflsumcom anassrs 3eqtr4rd 3eqtr2d ioossre ax-1cn o1const mp2an dividd o1lo1d caddc - vmadivsum vmage0 rpred mulid2d wb simplbda eqbrtrd lemuldivd harmonicubnd - fznnfl syl2anc lesubadd2d lemul2ad mulid1d breqtrd lediv1dd adantrr lo1le + vmadivsum vmage0 rpred mullidd wb simplbda eqbrtrd lemuldivd harmonicubnd + fznnfl syl2anc lesubadd2d lemul2ad mulridd breqtrd lediv1dd adantrr lo1le fsumle 0red harmoniclbnd subge0d mulge0d fsumge0 o1lo12 mptru ) ADUCUDEZD AUEZUFFZUGEZBUEZUHFZUWKGEZUWHUWKGEZUIFZHEZBIZUWHUIFZGEZUWRUJGEZJEZKLMZNAU WGUWJUAUEZUIFZUXCGEZUAIZUWRGEZUWTJEZKZLMUXBNUXIAUWGUXFUWRUJUKEZUJGEZJEZDU @@ -396489,7 +396489,7 @@ a multiplicative function (but not completely multiplicative). wss oveq2d 3eqtr3d ssrdv vmadivsum o1res2 cc0 crli divlogrlim rlimo1 mp1i ex o1mul2 eqeltrrd o1dif mpbird divcld cabs abscld fsumabs absmuld vmage0 divge0d absidd cico fveq2 id oveq12d cbvsumv sumeq1d eqtrid fveq2d breq1d - wceq wral ad2antrr nncnd mulid2d wb eqbrtrd rpge0d fznnfl lemuldivd mpbid + wceq wral ad2antrr nncnd mullidd wb eqbrtrd rpge0d fznnfl lemuldivd mpbid simplbda 1re elicopnf sylanbrc rspcdva lemul2ad fsumle fsummulc1 breqtrrd ax-mp lediv1dd absdivd fsumge0 mulge0d div23d eqtr4d 3brtr4d adantrr o1le letrd ) ABJUAUBKZJBUFZUCLZUDKZGUFZUELZUVHMKZJUVEUVHMKZUCLZUDKZFUFZUELZUVN @@ -396627,14 +396627,14 @@ a multiplicative function (but not completely multiplicative). adantr resubcld recnd abscld sylancl negcli subcl absnegi wceq 0le2 absid 2cn mp2an eqtri oveq2i abs2dif eqbrtrrid oveq2d sumeq1d id oveq1d oveq12d fveq2 eqid ovex fvmpt3i 1rp cz 1z flid ax-mp eqtrdi 0cn sq0id fsum1 2t0e0 - log1 subid1i addid2i mulid2i df-neg eqtr4di mp1i fveq2d cpnf ioorp eqcomi + log1 subid1i addlidi mullidi df-neg eqtr4di mp1i fveq2d cpnf ioorp eqcomi cioo nnuz a1i 1red cxr pnfxr 1re 1nn0 nn0addge1i 0red simpr nnrp cdv cdiv sylan2 cpr reelprrecn crn ctg dvmptres mulcl mpteq2ia mpteq2dva eqtrd w3a 2cnd mpbid rpred letrd ccnfld ctopn recn dvmptid rpssre iooretop eqeltrri wss tgioo2 rerpdivcld cvv rpreccld rpcnd mulcld cnelprrecn sqcl cres wf1o wf relogf1o f1of feqmptd fvres dvrelog eqtr3di 2nn dvexp 2m1e1 exp1 oveq1 eqtrid oveq2 dvmptco ovexd dvmptc dvmptcmul dvmptsub dvmptadd subdird wne - rpne0 divrecd negsubd eqtr3id 3eqtr4rd dvmptmul mulid2d subsub2d divcan1d + rpne0 divrecd negsubd eqtr3id 3eqtr4rd dvmptmul mullidd subsub2d divcan1d eqtr4d npcand simp32 simp2l simp2r logled simp31 wb logleb logge0d le2sqd ad2antrl sqge0d simpl 1le1 rexrd pnfge syl dvfsum2 eqbrtrrd lesubaddd ) A DEZFAGHZUBZFAUCIZUDJZBUEZKIZLMJZBUFZAAKIZLMJZLLUXTNJZOJZPJZNJZOJZUGIZLOJZ @@ -396735,7 +396735,7 @@ a multiplicative function (but not completely multiplicative). wne syl3anc oveq1d npcand oveq2d adddid eqtr3d 3eqtr3d sumeq2dv mpteq2dva fsumadd 3eqtrrd 1red cfa crli 2cnd cn0 2nn0 logexprlim mp1i wss rlimconst eqtrd rlimadd rlimo1 readdcl sylancl abscld fsumabs absmuld absge0d mule1 - cabs lemul1ad mulid2d eqbrtrd 3syl wb mpbid letrd breqtrd subdid divcan2d + cabs lemul1ad mullidd eqbrtrd 3syl wb mpbid letrd breqtrd subdid divcan2d w3a oveq2i eqtr4di div23 oveq12d fveq2d rprege0 absid nncnd rpre simplbda fznnfl rpred lemuldivd log2sumbnd syl2anc eqbrtrrd fsumle chash fsumconst lemuldiv2d cfn flge0nn0 hashfz1 reflcl flle 2pos pm3.2i leadd2dd lediv1dd @@ -397273,7 +397273,7 @@ a multiplicative function (but not completely multiplicative). caddc subdid 3eqtrd eqtr4d divdiv1d epr logleb eqbrtrrid 1rp rpregt0 mp1i lediv2 syl3anc div1d elo1d o1mul2 eqeltrrd o1dif 2re rerpdivcl rerpdivcld breqtrd nndivre 2cnd subcld gt0ne0d fsumcl fsumabs vmage0 cico sumeq1d id - redivcld breq1d wral nncnd mulid2d fznnfl simplbda lemuldivd 1re elicopnf + redivcld breq1d wral nncnd mullidd fznnfl simplbda lemuldivd 1re elicopnf ax-mp sylanbrc rspcdva nnne0d absmuld div32d fsumle divcld addcld fsumsub eqbrtrrd lemul2ad relogdivd subsub3d oveq2 anasss fsumfldivdiag fsummulc2 mul32d mulcomd chpval 2timesd fsumadd adddid 3eqtr4rd lediv1dd divsubdird @@ -397457,7 +397457,7 @@ a multiplicative function (but not completely multiplicative). fsumdivc 3eqtr4rd rpcnd divrecd dividd 3eqtr3d ssrdv vmadivsum divlogrlim ex o1res2 crli rlimo1 mp1i fsumabs vmage0 divdiv2d div23d divcan1d eqtr2d mulassd subdird nnred 3eqtrd cico cbvsumv fvoveq1 sumeq12dv eqtrid breq1d - oveq2 id wral mulid2d fznnfl simplbda eqbrtrd 1re elicopnf ax-mp sylanbrc + oveq2 id wral mullidd fznnfl simplbda eqbrtrd 1re elicopnf ax-mp sylanbrc rspcdva eqbrtrrd lemul2ad div12d fsumle breqtrrd lediv1dd breqtrd absdivd letrd fsumge0 mulge0d adantrr o1le eqeltrd ) ABJUAUBKZJBUFZUCLZUGKZGUFZUD LZJUXAUXDMKZUCLZUGKZFUFZUDLZUXIUELZUXFUXIMKZUHLZUIKZNKZFUJZNKZGUJZUXAUXAU @@ -397666,7 +397666,7 @@ a multiplicative function (but not completely multiplicative). cn ax-1cn sylancl fzfid elfznn adantl fsumrecl cneg oveq2 fvoveq1 oveq12d oveq1 eqtrdi fveq2d nnrecred nnred peano2rem resubcld nncnd pncan eqeltrd eqcomd 1cnd subcld eqtr4d oveq2d 3eqtr3d mvrladdd peano2cn nnncan2d eqtrd - nnne0d divrec2d sumeq12rdv oveq1d mulcld divsubdird mulid1d eqtr3d negeqd + nnne0d divrec2d sumeq12rdv oveq1d mulcld divsubdird mulridd eqtr3d negeqd jca divcan5d nnrpd cof cvv eqidd offval2 mpteq2dva adantrr syl2anc mpbird cabs eqbrtrd sylancr eqeltrrd syl3anc rerpdivcld rpregt0d o1sub2 peano2nn subcl nnrp pntrf ffvelcdmi nnmulcl mpdan cfzo 1div1e1 1m1e0 c2 0re chpeq0 @@ -398186,7 +398186,7 @@ a multiplicative function (but not completely multiplicative). wbr crp 1rp rpne0d divdird mpteq2dva rerpdivcld divdiv1d divrecd eqtr3d a1i o1mul2 eqeltrd cc o1const sylancr 1cnd o1add2 remulcld nnrpd rpge0d cc0 divge0d addge0d fsumge0 absidd abscld ad2antrr oveq2 rpdivcld nncnd - wceq addcld nnne0d divassd divcld mulid2d oveq2d breqtrrd mulge0d letrd + wceq addcld nnne0d divassd divcld mullidd oveq2d breqtrrd mulge0d letrd lediv2ad breqtrd fsumle lemul2ad mpbird lediv1dd oveq1d eqbrtrd adantrr eqtr2d eqtr4d leabsd o1le ffvelcdmi subcld pntrval oveq12d eqtrd fveq2d abssuble0d cneg c2 negsubdi2d 3eqtr2d sumeq12rdv ancli rplogcld rpgecld @@ -398195,8 +398195,8 @@ a multiplicative function (but not completely multiplicative). mp1i fveq2 breq12d wral rspcdva leadd1dd adddird 3eqtrd nnred fsummulc2 lep1d reccld harmonicubnd syl2anc mul32d 3brtr3d pntrf absge0d resubcld ledivmul2d dividd sub4d abs2dif2d chpwordi syl3anc addsub4d fsumcl cfzo - negdi2d cn0 rprege0d flge0nn0 nn0p1nn 3syl 2re flltp1 mulid1d ltdivmuld - 1lt2 lttrd wb chpeq0 addid2d divcan2d div1d pncand negeqd fzval3 eqcomd + negdi2d cn0 rprege0d flge0nn0 nn0p1nn 3syl 2re flltp1 mulridd ltdivmuld + 1lt2 lttrd wb chpeq0 addlidd divcan2d div1d pncand negeqd fzval3 eqcomd cz flcld pncan2d cuz nnuz eleqtrdi fsumparts mulneg2d fsumneg neg11d ) ABNUBUCOZBUJZUDUEZNUYCUFUEZUGOZUYCHUJZNPOZQOZUDUEZUYIPOZHUHZPOZUYCUYCUI UEZROZQOZUYFUYGUYIEUEZUYCUYGQOZEUEZUKOZULUEZROZHUHZUYOQOZNUMAUNZABUYBUY @@ -398359,7 +398359,7 @@ a multiplicative function (but not completely multiplicative). subdid eqtr3d 3eqtr3rd fsumrecl rpne0d divdiv1d rerpdivcld divdird mulcld c0 3eqtr3d divsubdird mpteq2dva adantrr ello1d co1 mp1i mulge0d cioo cfzo elioore eliooord simpld rpgecld rprege0d flge0nn0 nn0p1nn nndivred flltp1 - cn0 pntrval mulid1d ltdivmuld 1lt2 lttrd wb chpeq0 abssuble0d 3eqtrd crab + cn0 pntrval mulridd ltdivmuld 1lt2 lttrd wb chpeq0 abssuble0d 3eqtrd crab cdvds nn0red nn0cnd flidm div1d pntrf mul01d pntsf relogcl remulcl sylan2 ifclda fmpti flcld rpaddcld negsubdi2d mulneg1d fzfid fsumneg oveq1 1m1e0 wn fzval3 flid fz10 sum0 rpne0 necon2bi iffalsed 2t0e0 0m0e0 cuz eleqtrdi @@ -398486,8 +398486,8 @@ a multiplicative function (but not completely multiplicative). cn fsumcl divdird subsubd subdid oveq2d eqtr3d oveq1d rerpdivcld fsumrecl 1cnd resubcld cif simpr 0red cle 2rp rpge0d wceq rpcnd cc 1re rpred mpbid sylancr rpre eleq1 id fveq2 oveq12d ifbieq1d ovex c0ex fvmpt iftrue eqtrd - ifex subdird mulid2d 3eqtrd fveq2d eqbrtrrd eqbrtrd eqtrdi ax-mp lemul2ad - letrd fsumle adantrr lo1le lo1const lo1add 3brtr4d subcld fsumsub mulid1d + ifex subdird mullidd 3eqtrd fveq2d eqbrtrrd eqbrtrd eqtrdi ax-mp lemul2ad + letrd fsumle adantrr lo1le lo1const lo1add 3brtr4d subcld fsumsub mulridd pncand 3eqtr3rd sumeq2dv mpteq2dva 2re pntrlog2bndlem4 nnred simpl ifclda wn fmpti divge0d absge0d difrp 3eqtr4d npcand logdifbnd lemuldiv2d mpbird subsub3d lesubadd2d eqeltrdi iftrued log1 ax-1cn mul01i oveq1 1m1e0 rpne0 @@ -398799,7 +398799,7 @@ a multiplicative function (but not completely multiplicative). elrpd rerpdivcl efgt0 breqtrrdi flge0nn0 syl2anc nnmulcld nndivred eqtr4d lttrd nn0p1nn pntrf fsumneg nnne0d divnegd breqtrrd 3eqtr4rd cico cxr wss nncnd 2rp rpaddcl rpdivcld rpred pnfxr icossre sseldd remulcld 1red efgt1 - mulid2d elicopnf simplbda mpdan ltletrd syl112anc flword2 syl3anc elfzuzb + mullidd elicopnf simplbda mpdan ltletrd syl112anc flword2 syl3anc elfzuzb ltmul1 uzid sylanbrc elfzle3 elfzel2 zred ltp1d peano2re ltnled rgen olci pm2.21dd 2a1i cfzo elfzofz imbitrid letrid oveq2d peano2zd fzsn sylan9eqr elfzp12 eluzfz2 wrex elfzle1 elfzelz zltp1le fllt elfzle2 simpr pntpbnd1a @@ -398897,7 +398897,7 @@ a multiplicative function (but not completely multiplicative). nndivred wn 2div2e1 2cn pncan2 oveq1d rpne0d divsubdird eqtr3d rerpdivcld eliooord rpaddcl cico elicopnf 0red 0lt1 reeflogd efle lesub1dd rerpdivcl ltletrd efgt0 rexrd elioopnf elfzuz eluznn syl2an peano2nnd 2cnd pnpcan2d - cuz cxr relogmuld mulid2d 1zzd nnrecre oveq2 1m1e0 zcnd cin reflcl fzdisj + cuz cxr relogmuld mullidd 1zzd nnrecre oveq2 1m1e0 zcnd cin reflcl fzdisj c0 cun flwordi syl3anc elfz2nn0 syl3anbrc fzsplit fsumsplit 3eqtrd fllep1 ltp1d rpmulcld resubcld 0re emgt0 simp2bi subge0d eqtrd simp3bi lesubaddd cicc peano2re rpmulcl flle le2addd 2timesd egt2lt3 simpli relogmul eqtrdi @@ -399069,10 +399069,10 @@ a multiplicative function (but not completely multiplicative). relogcld 1t1e1 breqtrdi ltadd2dd df-2 ltmul1dd rpcnne0 div23 lemuldiv2d ltmul12ad ltletrd pntibndlem2a simp1d simp2d rpgecld pntrf nndivred 3re abs2difd cn 4nn efgt1 ltaddrpd rplogcld peano2re chpcl renegcld abstrid - nnrp divsubdird dividd subdird dmdcan mulid2d 3eqtrd negsubdi2d npncand + nnrp divsubdird dividd subdird dmdcan mullidd 3eqtrd negsubdi2d npncand cneg 1cnd negeqd absnegd absmuld subge0d divge0d absdivd rprege0d absid absidd 3brtr3d pntrval subadd4 addsub4 3eqtr3d syl22anc eqtr2d chpwordi - sub4 an42s abssubge0d comraddd lediv1dd leadd2dd adddid mulid1d divdird + sub4 an42s abssubge0d comraddd lediv1dd leadd2dd adddid mulridd divdird abssuble0d 3eqtr4d weq oveq1 breq12d raleqbidv elioopnf sylanbrc simp3d 1xr ax-mp wi ltle mpd lemul1d w3a elicc2 mpbir3and divdir divassd div12 2cnd divcan3d 2cn mulcom oveq2i eqtrdi 3eqtr2d eqtr3d ledivmul2d ax-1cn @@ -399344,7 +399344,7 @@ a multiplicative function (but not completely multiplicative). ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) ) $= ( crp wcel cc0 c1 cioo co clt wbr cmin w3a cdiv pntlemd rpdivcld eqeltrid simp2d ce cfv rpred rpefcld cr rpgt0d caddc ltp1d breqtrrdi lelttrd rpcnd - cmul mulid1d breqtrrd 1red ltdivmuld mpbird eqbrtrid cxr wb 0xr 1xr mp2an + cmul mulridd breqtrrd 1red ltdivmuld mpbird eqbrtrid cxr wb 0xr 1xr mp2an elioo2 syl3anbrc efgt1 syl 1re ltaddrp sylancr cc wne wceq rpcnne0d divid wa ax-1cn addcom sylancl eqtrid 3brtr4d ltdiv23d difrp syl2anc mpbid 3jca ) AGUBUCIUBUCGUDUEUFUGUCZUEIUHUIZFGUJUGUBUCZUKAGFDULUGZUBTAFDRAJUBUCDUBUC @@ -399404,7 +399404,7 @@ a multiplicative function (but not completely multiplicative). 4z ltaddrpd ltletrd wceq rprege0d breqtrrd lt2sq syl2anc mpbird 3brtr3d ltled rerpdivcld 3jca relogcld nndivre relogexp recnd oveq2d oveq1d wne cc rpcnne0d 3eqtrd elico2 rpgecld rpsqrtcld 1lt2 simpli simpri 3lt4 3re - pnfxr pntlemd cioo pntlemc rpdivcl ltmul1dd mulid2d 4pos ltmul2 mulid1i + pnfxr pntlemd cioo pntlemc rpdivcl ltmul1dd mullidd 4pos ltmul2 mulridi jctir 4cn breqtrdi resqcld 3nn0 2nn decnncl 3rp breqtrrdi resqrtth 0le1 rpefcld syl21anc sq1 ltmul2dd remsqsqrt rplogcld readdcl addcomd eqtrid logltb eqbrtrrd ltmuldiv2 ltdiv1dd relogmuld eqtrd 2cnd mulcld divcan4d @@ -399630,7 +399630,7 @@ a multiplicative function (but not completely multiplicative). cc0 clt rpmulcld rpdivcl sylancl rpred caddc cexp rpdivcld remulcld c1 w3a c3 cfn cfl cfz syl recnd cr sylancr reflcl peano2re readdcld rphalfcld simp3d simp2d cz rpcnd cn syl2anc breqtrd ltled wa simprd - cuz letrd lemul2d wceq rprege0d lemuldivd mulid2d ltmul1dd breqtrrd + cuz letrd lemul2d wceq rprege0d lemuldivd mullidd ltmul1dd breqtrrd mpbid wne rpcnne0d rpcnne0 mp1i oveq2d syl3anc oveq1d eqtr3d mpbird cc wb 2z pntlemd cioo pntlemc 4re 4pos elrpii ceu csqrt cdc pntlemb cn0 fzfid eqeltrid hashcl nn0red rpaddcl 1cnd add32d rpsqrtcld cfzo @@ -399893,7 +399893,7 @@ a multiplicative function (but not completely multiplicative). wb jca cun peano2zd 1re ltle mpd uzid peano2uz leexp2ad lediv2d flword2 ssun1 flcld simprd elfzofz pntlemh sylan2 lttrd ltled flwordi syl3anbrc eluzp1p1 eluz2 fzsplit2 sseqtrrid sstrd adantlr ssun2 le2add mpand 1cnd - zcnd subcld adddid addcomd addsubd mulid1d 3eqtr3d cin c0 reflcl fzdisj + zcnd subcld adddid addcomd addsubd mulridd 3eqtr3d cin c0 reflcl fzdisj ltp1d fsumsplit sylibrd expcom a2d fzind2 mpcom fsumless letrd ) AJLVBV CZOLVDVEVCZVFVCZVGVDUUCZFVFVCZVHVCZUAVIVJZVDVEVCZVFVCZVFVCZVVOOLVFVCZVK VHVCZVWAVFVCZVFVCZQPVBVCZVFVCZVSUATVHVCZVLVJZVMVCZJKVTZVHVCZUAVWNVHVCZI @@ -399998,7 +399998,7 @@ a multiplicative function (but not completely multiplicative). 1rp adddird oveq2d 3brtr4d syl2anc logdivbnd lemuldiv2 syl112anc mpbird 2pos reflcl flle simprd lediv2d div1d logled leadd1dd 0red nnge1d lep1d log1 le2sqd loge rpge0d lemulge12d rprege0d epr leadd2dd binom21 sqvald - remsqsqrt df-3 oveq1i 2cnd 1cnd eqtrid mulid2d eqtr2d oveq12d sqcld 2cn + remsqsqrt df-3 oveq1i 2cnd 1cnd eqtrid mullidd eqtr2d oveq12d sqcld 2cn mulcl addassd 3cn 3eqtr4rd lemul2d adantr rpcnne0d div23 divass syl3anc wne eqtr3d sumeq2dv fsummulc2 eqtr4d mul12d eqtrd mulassd ) AJVBVCUATVD VEZVFVGZVHVEZKVIZVJVGZUWSVDVEZKVKZVLVEZVLVEZJUAVJVGZVMVNVEZUXEVLVEZVLVE @@ -400053,7 +400053,7 @@ a multiplicative function (but not completely multiplicative). wne oveq1d eqtr4d 3eqtr2d cfl cfz cv csu adantr fsumrecl mulcld remulcl eqtrd fveq2d letrd mpbird cdc ceu csqrt pntlemb pntlemc pntlemd decnncl pntrf 3nn0 2nn nnrp ax-mp rpmulcl sqdivd sqcld divass 3eqtrd oveq2i cn0 - df-3 2nn0 expp1 mulcomd rpreccld mulid2d divrec2d eqtr2id eqtr2d subcld + df-3 2nn0 expp1 mulcomd rpreccld mullidd divrec2d eqtr2id eqtr2d subcld 1cnd mul32d eqeltrrd rplogcld fzfid elfznn adantl nnrpd div23d rprege0d divsubdird absdivd absid divassd sumeq2dv fsumdivc 2fveq3 fveq2 cbvsumv cpnf oveq2 fvoveq1 simpl fvoveq1d sumeq12rdv id breq1d cxr 1re elioopnf @@ -400897,7 +400897,7 @@ a multiplicative function (but not completely multiplicative). nn0re simprr zmodfz rspcdva mulcomd abvge0 expge0d elfzle1 syl22anc divge0 flge0nn0 qabvle w3a simprl 0z elfzm11 simp3d expp1d ltdivmul breqtrd mpbird fllt ltled lemul1ad eqbrtrd nn0ge0d leexp1a syl32anc - nnnn0d max1 lemul2ad le2addd adddid mulid1d mulcld adddird breqtrrd + nnnn0d max1 lemul2ad le2addd adddid mulridd mulcld adddird breqtrrd nn0cn 1cnd max2 nn0z uzid peano2uz leexp2ad nnmulcld expr ralrimdva lemul2 biimtrid expcom a2d nn0ind impcom rspccv 3impia ) AMUJUKZNUL KMUMUNZUOUPUNZUQUNZUKZNHURZKMUSUNZGMUMUNZUSUNZUTVDZAUYDVAUGVBZHURZU @@ -400982,7 +400982,7 @@ a multiplicative function (but not completely multiplicative). cxpexp eqbrtrd mulgt0d mul12d divassd oveq2i oveq1d divcan1d eqtr3d cfz eqtr4di flltp1 ltmul1dd eflt nnz nn0zd 3brtr4d nnexpcl nnexpcld nnltlem1 nnnn0d nn0uz eleqtrdi nnzd peano2zm elfz5 wi 3expia syldan - ostth2lem2 mpd flle leadd1dd nnge1 leadd2dd adddid mulid1d breqtrrd + ostth2lem2 mpd flle leadd1dd nnge1 leadd2dd adddid mulridd breqtrrd nngt0d expgt0 syl3anc lemul1 expp1d remulcl mulcomd cxplead cxpmuld 3brtr3d lemul1d nngt0 rpgt0d ltp1d lttrd mul32d rpexpcld ledivmuld ) ANUHUIZUJZMIUKZGHULUMZUNUMNUOUMUWONUOUMZUWPNUOUMZUNUMZNLGUPUMZHUQ @@ -401045,7 +401045,7 @@ a multiplicative function (but not completely multiplicative). mpbid cdiv caddc cif ifcl eqeltrid cc0 0red a1i 0lt1 max2 breqtrrdi sylancl ltletrd elrpd clog nnrpd relogcld nnred rplogcld rerpdivcld rpcxpcld remulcld peano2re cv ostth2lem3 ostth2lem1 ledivmuld rpcnd - simprd mulid1d breqtrd adantr iftrue eqtrid oveq1d sylan9eqr mpbird + simprd mulridd breqtrd adantr iftrue eqtrid oveq1d sylan9eqr mpbird recnd 1cxpd mtand lttrd reeflogd wceq iffalse rpne0d cxpefd 3brtr4d ce eqtr2d efle div12d oveq2i eqtr4di mulcomd eqtrd breqtrrd 3brtr4g jca ) AUHLIUIZUJUKZEFULUKAYPYOUHULUKZUMZAYQMIUIZUHULUKZAUHYSUJUKZYT @@ -401079,7 +401079,7 @@ a multiplicative function (but not completely multiplicative). wrex clog crp cn c2 cuz wa eluz2b2 sylib simpld nnq syl qrngbas abvcl cdiv syl2anc rplogcld nnred simprd rpdivcld eqeltrid rpred rpgt0d cn0 ce nnnn0d qabvle wne nnne0d qrng0 abvgt0 syl3anc elrpd reeflogd nnrpd - cmul 3brtr4d wb relogcld efle mpbird rpcnd mulid1d breqtrrd ledivmuld + cmul 3brtr4d wb relogcld efle mpbird rpcnd mulridd breqtrrd ledivmuld 1red eqbrtrid cxr w3a 0xr elioc2 mp2an syl3anbrc qabsabv fvres oveq1d 1re cres mpteq2ia eqcomi abvcxp sylancr cz eluzelz zq fveq2 eqid ovex fvmpt 3syl adantl simpr nn0ge0d absidd recnd adantr cxpefd ostth2lem4 @@ -401144,7 +401144,7 @@ a multiplicative function (but not completely multiplicative). simpr rppwr cplusg cnfldadd ressplusg abvtri cnfldmul ressmulr abvmul ax-mp cmulr qabvexp reexpcld remulcld w3o elz simprbi abv0 syl5ibrcom 0le1 eqbrtrdi ralbidva rspccv cminusg abvneg qrngneg eqeltrd eqbrtrrd - lenlt expr ralrimiva rsp sylc expgt0 lemul2 syl112anc mulid1d breqtrd + lenlt expr ralrimiva rsp sylc expgt0 lemul2 syl112anc mulridd breqtrd 3jaod rpge0d max1 leexp1a syl32anc le2addd 2timesd anassrs rexlimdvva max2 sylbid rpregt0d ledivmul2 mp3an12i reexpcl pm2.21d breq2d notbid rexlimdva pm2.01d lttri3 mpbir2and 3eqtr4d eqtr2d pm2.61dne ostthlem2 @@ -406897,7 +406897,7 @@ property of surreals and will be used (via surreal cuts) to prove many ( co wcel va vb cv cadds wceq cleft cfv wrex cab cun cright cvv abrexex fvex a1i unexd csur wa wi c0s cslt wbr cbday cnadd wral adantr leftssno sseli adantl 0sno c0 bday0s oveq2i con0 bdayelon ax-mp eqtri uneq2i wss - naddid1 naddword1 mp2an ssequn2 mpbi leftssold wb oldbday sylancr mpbid + naddrid naddword1 mp2an ssequn2 mpbi leftssold wb oldbday sylancr mpbid cold naddel1 mp3an sylib elun1 syl addsproplem1 simpld eleq1a rexlimdva eqeltrid abssdv naddel2 unssd rightssno rightssold elun vex weq rexbidv wo eqeq1 elab orbi12i bitri reeanv csslt lltropt simprl simprr ad2antrl @@ -408801,7 +408801,7 @@ property of surreals and will be used (via surreal cuts) to prove many ( ( Q x.s R ) -s ( P x.s R ) ) ( A x.s Y ) e. No ) $= ( cmuls co csur wcel cslt wbr c0s wa csubs wi cfv cold oldssno sselid - cbday 0sno a1i cnadd c0 bday0s oveq12i con0 0elon naddid1 ax-mp eqtri + cbday 0sno a1i cnadd c0 bday0s oveq12i con0 0elon naddrid ax-mp eqtri wceq oveq2i bdayelon naddcl mp2an wss naddword1 oldbdayim syl naddel2 sstri wb mp3an sylib eqeltrid mulsproplem2 simpld ) ABNUCUDUEUFBNUGUH UIUIUGUHUJNUIUCUDBUIUCUDUKUDZWFUGUHULABCDEBNUIUIFGHIJKLMOPQRSTUAACUQU @@ -408846,7 +408846,7 @@ property of surreals and will be used (via surreal cuts) to prove many hypothesis. (Contributed by Scott Fenton, 20-Feb-2025.) $) mulsproplem4 $p |- ( ph -> ( X x.s B ) e. No ) $= ( cmuls co csur wcel cslt wbr c0s wa csubs wi cfv cold oldssno sselid - cbday 0sno a1i cnadd c0 bday0s oveq12i con0 0elon naddid1 ax-mp eqtri + cbday 0sno a1i cnadd c0 bday0s oveq12i con0 0elon naddrid ax-mp eqtri wceq oveq2i bdayelon naddcl mp2an wss naddword1 oldbdayim syl naddel1 sstri wb mp3an sylib eqeltrid mulsproplem2 simpld ) ANCUCUDUEUFNCUGUH UIUIUGUHUJCUIUCUDNUIUCUDUKUDZWFUGUHULABCDENCUIUIFGHIJKLMOPQRSTABUQUMZ @@ -408869,7 +408869,7 @@ property of surreals and will be used (via surreal cuts) to prove many Fenton, 20-Feb-2025.) $) mulsproplem5 $p |- ( ph -> ( X x.s Y ) e. No ) $= ( cmuls co csur wcel cslt wbr c0s wa csubs wi cfv cold oldssno sselid - cbday 0sno a1i cnadd c0 bday0s oveq12i con0 0elon naddid1 ax-mp eqtri + cbday 0sno a1i cnadd c0 bday0s oveq12i con0 0elon naddrid ax-mp eqtri wceq oveq2i bdayelon mp2an wss naddword1 sstri oldbdayim syl naddel12 naddcl syl2anc eqeltrid mulsproplem2 simpld ) ANOUDUEUFUGNOUHUIUJUJUH UIUKOUJUDUENUJUDUEULUEZWEUHUIUMABCDENOUJUJFGHIJKLMPQRSTUAABURUNZUOUNZ @@ -419257,7 +419257,7 @@ the identity of betweenness (Axiom A6). (Contributed by Thierry cle wbr remulcld wss 0re iccssre sseldd resubcld 1red recnd 1cnd subdid sylancr subcld necomd subne0d mulne0d eqnetrrd redivcld crp cxr iccgelb 0xr rexrd mp3an2i ne0gt0d elrpd clt iccleub leneltd difrp syl2anc mpbid - rpmulcld mpbird mulid1d breqtrrd mulcld ccvs wne eqid clmsubcl cvsdivcl + rpmulcld mpbird mulridd breqtrrd mulcld ccvs wne eqid clmsubcl cvsdivcl cfv syl3anc syl13anc syl grpsubcl clmvsass oveq12d eqtrd oveq1d mulcomd wb oveq2d 3eqtr3d lmodsubdir grpnnncan2 3eqtr2rd cvsmuleqdivd cvsdiveqd clmsub 3eqtr4d eqeltrd eqeltrrd subge0d mulge0d subdird breqtrd divge0d @@ -419792,7 +419792,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to ( wcel wbr c1 cmin co cmul caddc wceq cc0 cle wa cc oveq1d wb vk vt vp cv cfv wral wrex wi cr fveere 3ad2antl2 3ad2antl3 cneg resubcl 3adant3 recnd w3a oveq2d 3ad2ant3 1re mul4d recn 3ad2ant1 3ad2ant2 subdid ax-1cn subdir - sylancr mp3an2i mulid2d 3eqtr3d remulcld subsub4d 3eqtrd mulneg1d oveq12d + sylancr mp3an2i mullidd 3eqtr3d remulcld subsub4d 3eqtrd mulneg1d oveq12d simp1 eqtrd mpbird mulge0d mpbid eqbrtrd 3expa sylan an32s fveecn anim12i ralrimiva anandirs subcl ancoms mulcomd simp2r simp2l syl22anc simp3 mpan mulcld sub32d 3eqtr3rd 3eqtr2d syl2an weq eqeq12d rspccva breq1d ralbidva @@ -419804,11 +419804,11 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to mulneg2d simp3bi subge0 sqge0d le0neg2d anim12dan 3adantl1 simp1l mulsub2 ralrimivva syl oveqan12rd anandis 2ralbidva biimprcd rexlimdva mulsuble0b simp1r simpl1 letri3d pm4.25 breq2d anbi2d bitrid biimprd adantrd 0elunit - ad2antlr 3jca mulid2 mul02 addid1 ad2antll eqtr2d 1m0e1 eqtrdi necon3abid + ad2antlr 3jca mullid mul02 addrid ad2antll eqtr2d 1m0e1 eqtrdi necon3abid exp32 syldd df-ne rexbii rexnal bitri bitr4di rspcv ad2antrl lesub1d letr simpl simplrr ltlend mpbir2and posdifd divelunit div2subd lesub2d eqeltrd jaodan rspc2v simp11 simp12 simp13 simpl23 simpl21 simpl12 simpl22 simpl3 - imp simpl13 subne0d divdird npncan2 adddird mul02d simpl11 add32d addid2d + imp simpl13 subne0d divdird npncan2 adddird mul02d simpl11 add32d addlidd syld addsubd subdird eqtr3d addsubassd eqtr4d 3eqtr4d simpr adddid npcand nnncan2d addcld divmuld div23d divsubdird dividd syl331anc com23 ralrimdv 3expia anim12d syl6 rexlimdvaa sylbid pm2.61dne impbid ) AFUUAUEZGZBVVFGZ @@ -420033,7 +420033,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to cee cfv w3o caddc cfz wral relin01 adantl fveere adantlr adantll jca cexp c2 simprl resubcl ancoms adantr mulassd recn pncand oveq1d sqvald 3eqtr4d ad2antrr wo simprr sqge0d wb resqcld mulle0b syl2anc mpbird eqbrtrd sylan - orcd an32s ralrimiva ad2antlr sub32d ax-1cn subdir mp3an2i mulid2d eqtr2d + orcd an32s ralrimiva ad2antlr sub32d ax-1cn subdir mp3an2i mullidd eqtr2d expr wceq subsub4d 3eqtr3rd subidd eqtr4di oveq12d mpan ad2antrl mulneg2d df-neg mul4d negeqd 3eqtrd sylancr subge0 biimpar adantrl mulge0d breqtrd 1re simpl le0neg2d mpbid negsubdi2d simplr simpll peano2rem mul12d eqtr4d @@ -420257,8 +420257,8 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) ) $= ( vk wceq cfv wcel wa c1 cmin co cmul caddc wral cc0 cee cv cfz cexp cicc csu wrex w3a cmpt fveere 3ad2antl1 3ad2antl2 3ad2antl3 resubcld ralrimiva - c2 cr wb cn eleenn mptelee syl 3ad2ant1 mpbird fveecn 1m0e1 oveq1i mulid2 - cc eqtrid subcl sylan2 mul02d oveq12d addid1 eqtr2d syl3anc subcld nncand + c2 cr wb cn eleenn mptelee syl 3ad2ant1 mpbird fveecn 1m0e1 oveq1i mullid + cc eqtrid subcl sylan2 mul02d oveq12d addrid eqtr2d syl3anc subcld nncand 3impb oveq1d sumeq2dv 0elunit fveq1 fveq2 eqid ovex fvmpt sylan9eq oveq2d eqeq2d ralbidva eqeq1d anbi12d oveq2 oveq1 ralbidv anbi1d mp3an2 syl12anc weq rspc2ev 3expb adantll 2rexbidv syl5ibr imp ) CDJZCHUAKZLZDXILZMEXILZF @@ -420499,7 +420499,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to cicc caddc cfz w3a csu simpl2l fveecn sylancom simpl2r cr cle wbr elicc01 wral simp1bi recnd fveq2 oveq2d oveq12d eqeq12d rspccva adantll 3ad2antl3 3ad2ant3 oveq2 subdi 3coml ax-1cn subcl adantl simpl subdir mp3an2i nncan - mpan mulid2 3eqtr3rd 3adant2 simp1 mulcl ancoms 3adant1 subsub4d 3eqtr2rd + mpan mullid 3eqtr3rd 3adant2 simp1 mulcl ancoms 3adant1 subsub4d 3eqtr2rd sylan simp3 3adant3 sqmuld eqtrd syl31anc sumeq2dv fzfid resqcld 3adant2r sylan9eqr 3adant2l subcld sqcld 3expa 3adantl3 fsummulc2 eqtr4d ) GUAHZAG UBIZHZCXHHZJZDUCKUELHZEUDZBIZKDMLZXMAIZNLZDXMCIZNLZUFLZOZEKGUGLZURZJZUHZY @@ -420531,7 +420531,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to cicc csu simpl2l fveecn sylancom simpl2r cr cle wbr elicc01 simp1bi recnd wral adantr 3ad2ant3 fveq2 oveq2d oveq12d eqeq12d rspccva 3ad2antl3 oveq1 adantll oveq1d ax-1cn subcl mpan simp1 simp3 simp2 addsubassd subdi 3coml - mulcld syl3an1 subdir mp3an1 ancoms mulid2 3ad2ant2 eqtrd subsub2d 3eqtrd + mulcld syl3an1 subdir mp3an1 ancoms mullid 3ad2ant2 eqtrd subsub2d 3eqtrd 3adant1 eqtr4d 3adant3 sqmuld sylan9eqr syl31anc sumeq2dv resubcl sylancr fzfid 1re resqcld 3adant2r 3adant2l subcld sqcld 3expa 3adantl3 fsummulc2 ) GUAHZAGUBIZHZCXIHZJZDUCKUFLHZEUDZBIZKDMLZXNAIZNLZDXNCIZNLZOLZPZEKGUELZU @@ -420591,7 +420591,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to wb sqrtge0d resqrtth syl2anc oveq2d cc ax-1cn recnd sqmuld simp3r simp122 subcl simp123 simp132 simp133 brcgr syl22anc mpbid simp11 simp121 simp2ll simp2rl ax5seglem2 syl122anc simp131 simp2lr simp2rr 3eqtr3d sq11d simp3l - 3eqtr4d simp2bi oveq12d adddird npcan adantr oveq1d mulid2d eqtrd sqrt11 + 3eqtr4d simp2bi oveq12d adddird npcan adantr oveq1d mullidd eqtrd sqrt11 ax5seglem1 ) KUALZAKUBMZLZBYCLZCYCLZUCZDYCLZIYCLZJYCLZUCZUCZFUDNUEOZLZEYM LZUFZGUGZBMNFUHOZYQAMPOFYQCMPOUJOQGNKUIOZUKZYQIMNEUHOZYQDMPOEYQJMPOUJOQGY SUKZUFZUFZABULDIULUMRZBCULIJULUMRZUFZUCZYSHUGZAMZUUICMZUHOZSTOZHUNZUOMZYS @@ -420646,7 +420646,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to A =/= B ) -> T =/= 0 ) $= ( wcel cfv wa c1 cmin co cmul caddc wceq wral wne cc0 cc fveecn cn cee cv w3a cfz oveq2 1m0e1 eqtrdi oveq1d oveq1 oveq12d eqeq2d ralbidv biimpac wb - eqeefv 3adant1 3adant3r3 simplr1 sylancom simplr3 mulid2 oveqan12d addid1 + eqeefv 3adant1 3adant3r3 simplr1 sylancom simplr3 mullid oveqan12d addrid mul02 adantr eqtrd syl2anc eqcom bitr3di ralbidva syl5ibr expdimp necon3d eqeq1d bitrd 3impia ) FUAGZAFUBHZGZBVSGZCVSGZUDIZEUCZBHZJDKLZWDAHZMLZDWDC HZMLZNLZOZEJFUELZPZABQDRQWCWNIDRABWCWNDROZABOZWNWOIWPWCWEJWGMLZRWIMLZNLZO @@ -420673,7 +420673,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to ( wcel cfv wa wne cc0 c1 co cmul caddc wceq wral wb cc cn cee w3a cicc cv cmin cfz c2 cexp csu fveq1 oveq2d eqeq2d ralbidv biimparc simplr1 simplr2 wi eqeefv syl2anc fveecn sylan cr cle wbr elicc01 simp1bi ad2antlr ax-1cn - recnd npcan mpan oveq1d mulid2 sylan9eqr subcl adantl simpr simpl adddird + recnd npcan mpan oveq1d mullid sylan9eqr subcl adantl simpr simpl adddird eqtr3d eqeq1d eqcom bitrdi ralbidva bitrd syl5ibr expd impr necon3d com23 ex exp4a 3imp2 simplr3 eqeelen necon3bid mpbid ) GUAHZAGUBIZHZBWTHZCWTHZU CJZABKZDLMUDNHZEUEZBIZMDUFNZXGAIZONZDXGCIZONZPNZQZEMGUGNZRZUCZJZACKZXPFUE @@ -420743,10 +420743,10 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to ( ( T x. ( ( A - C ) ^ 2 ) ) - ( ( A - D ) ^ 2 ) ) ) ) $= ( cmin co c2 cexp cmul caddc oveq2i 2cn mulcli subcli cc wcel adddii wceq c1 binom2subi sqcli subdii oveq1i 3eqtri cc0 ax-1cn addcli subadd23 mp3an - binom2i addsubassi addsubi 3eqtr4i eqtri oveq12i addsub4i subdiri mulid2i + binom2i addsubassi addsubi 3eqtr4i eqtri oveq12i addsub4i subdiri mullidi eqtr4i subsub3 subsub4 3eqtr2i addassi add32i subsub2 cneg addcomi sqmuli addsub12 sqvali mulassi mul12i mulcomi eqtr3i eqeltrri 3eqtrri negsubdi2i - sub32 subsub mulneg2i negsubi subeq0i mpbir addid2i ) DBCIJKLJZMJZDBKLJZM + sub32 subsub mulneg2i negsubi subeq0i mpbir addlidi ) DBCIJKLJZMJZDBKLJZM JZDKBCMJZMJZMJZIJZDCKLJZMJZNJZUCDIJZAMJZDBMJZNJCIJZKLJZWTDABIJKLJZMJZACIJ KLJZIJZMJZNJZWJDWKWNIJZWQNJZMJDXKMJZWRNJWSWIXLDMBCGHUDODXKWQFWKWNBGUEZKWM PBCGHQZQZRCHUEZUAXMWPWRNDWKWNFXNXPUFUGUHXJXAKLJZKXAXBCIJZMJZMJZNJZXBKLJZW @@ -420927,7 +420927,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to ( vi vt wcel cfv wbr cv c1 co cmul caddc wceq wral cc0 wb cc cle wa cbtwn cee w3a cop cmin cfz cicc wrex simp2 simp3 brbtwn syl3anc elicc01 simp1bi cn wi recnd eqeefv 3adant1 adantr ax-1cn npcan mpan ad2antlr oveq1d subcl - cr simplr simpll3 fveecn sylancom adddird mulid2d 3eqtr3rd ralbidva bitrd + cr simplr simpll3 fveecn sylancom adddird mullidd 3eqtr3rd ralbidva bitrd eqeq2d biimprd sylan2 rexlimdva sylbid ) CUOFZACUBGZFZBWCFZUCZABBUDUAHZDI ZAGZJEIZUEKZWHBGZLKWJWLLKMKZNZDJCUFKZOZEPJUGKZUHZABNZWFWDWEWEWGWRQWBWDWEU IWBWDWEUJZWTEABBDCUKULWFWPWSEWQWJWQFZWFWJRFZWPWSUPXAWJXAWJVGFPWJSHWJJSHWJ @@ -420948,12 +420948,12 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) ) $= ( cc0 c1 co wcel cmin cmul wceq cr cle wbr recnd oveq2 oveq1d eqeq2d cdiv eqtrd cicc wa cv w3a wrex wi 1re elicc01 simp1bi ad2antrl resubcl sylancr - mul02d eqcomd ad2antll mulid2d adantr eqtrdi eqtr2d ax-1cn mul01i 1elunit + mul02d eqcomd ad2antll mullidd adantr eqtrdi eqtr2d ax-1cn mul01i 1elunit 1m0e1 eqtr4d 0elunit 1m1e0 eqeq1 eqeq1d 3anbi123d rspc2ev mp3an12 syl3anc ex wne caddc remulcld resubcld readdcld clt 1red simp2bi simp3bi lemul1ad breqtrd simpl ne0gt0d ltaddpos2d lelttrd posdifd gt0ne0d redivcld subge0d mpbid mpbird divge0 syl22anc ltled lesub1dd ledivmul2 syl112anc syl3anbrc - breqtrrd wb lemul2ad mulid1d addge01d div23d cc mp3an2 syl2anc divsubdird + breqtrrd wb lemul2ad mulridd addge01d div23d cc mp3an2 syl2anc divsubdird subdi nnncan2d pncand 3eqtr3d mulcomd oveq12d pncan2d syl113anc pm2.61ine dividd ) BEFUAGZHZAYBHZUBZDUCZFCUCZIGZFBIGZJGZKZYGFYFIGZFAIGZJGZKZYHBJGZY LAJGZKZUDZDYBUECYBUEZUFAEAEKZYEYTUUAYEUBZEEYIJGZKZFFYMJGZKZEBJGZFAJGZKZYT @@ -421371,7 +421371,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to cun cee impel ltned peano2zd eqeq1d imbitrid 1red letrd sylanbrc nnltp1le elnnz1 mpbid simpr1 simpr3 eqtri ltletrd gtned 0p1e1 simp1 mpbir2an uztrn eluz1i lttr mp3an12 mpani ltle mpan sylc jctil ltne necomd sumeq1i oveq2i - eleqtri fsumcl addid2d 3eqtrrd axlowdimlem7 ad2antrr neg1sqe1 zaddcl zrei + eleqtri fsumcl addlidd 3eqtrrd axlowdimlem7 ad2antrr neg1sqe1 zaddcl zrei 1lt3 ltp1i lttri mpbir ltnlei mpbi intnanr mtbiri eleq1 notbid syl5ibrcom 3nn necon2ad axlowdimlem9 3eqtr4rd syl3anc ) EHUAUBIZDJEKUCLZMLZIZNZHEMLZ CUEZAUBZJUDLZCOZUYNUYOBUBZJUDLZCOZPZUFEHEHPZUYLVUBUYIVUCDJJMLZIZHHMLZUYQC @@ -421963,9 +421963,9 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to syl simpr3 sylan an32s ralrimiva eleenn ad3antrrr mptelee 3adant3 simplrl cn wb mpbird wi cc fveecn eqcom ax-1cn simpr2 recnd subcl sylancr sylancl simpr1 mulcld addcld simplrr divmuld bitrid cneg negsubdi2 mulneg1d npcan - oveq1d adddird mulid2d 3eqtr3d oveq2d negsubd addcomd eqeq1d bitrdi add4d + oveq1d adddird mullidd 3eqtr3d oveq2d negsubd addcomd eqeq1d bitrdi add4d 3eqtr3rd bitr4d adddid oveq12d eqtr4d divdird divassd 3eqtrd eqeq2d fveq1 - addassd addid2d divcan2d fveq2 eqid fvmpt sylan9eq oveq2 ralbidva ralbidv + addassd addlidd divcan2d fveq2 eqid fvmpt sylan9eq oveq2 ralbidva ralbidv ovex oveq1 2rexbidv rexcom rexbii bitri subaddd 3bitr3rd biimpd mul02 jca npncan2 jctild df-3an syl6ibr ralimdva 3impia weq rspc2ev syl3anc 3anbi1d 3anbi13d 3anbi23d 3anbi2d 3anbi3d rspc3ev syl31anc 3bitri sylib ) DKUBUCZ @@ -422050,8 +422050,8 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to T Btwn <. x , y >. ) ) ) $= ( vi vp vq wcel wa c1 co cmul caddc wceq cc0 wrex vr vs vu cn cee cfv w3a cv cmin cfz wral wne cicc cop wbr simpl21 simpl22 simpl23 simpl3r simprll - cbtwn jca simprlr cc simp21 ad2antrr fveecn sylan simp3r mulid2 oveqan12d - mul02 addid1 adantr eqtrd syl2anc oveq2 1m0e1 eqtrdi oveq1d oveq1 oveq12d + cbtwn jca simprlr cc simp21 ad2antrr fveecn sylan simp3r mullid oveqan12d + mul02 addrid adantr eqtrd syl2anc oveq2 1m0e1 eqtrdi oveq1d oveq1 oveq12d eqeq1d ad2antlr mpbird eqeq2d eqcom bitrdi biimpd adantrd ralimdva simp3l wb impancom eqeefv sylibrd necon3d impr anasss eqtr2 ad2antll axeuclidlem ralimi syl231anc exp32 brbtwn syl3anc simp22 simp23 r19.26 2rexbii reeanv @@ -422133,13 +422133,13 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to cmin weu wreu wrex weq wi cop cbtwn wbr opeq2 breq2d breq1 orbi12d elrab2 wo simpll3 simpll2 simpr brbtwn syl3anc biimpa oveq2 oveq1d oveq1 oveq12d wb eqeq2d ralbidv adantr eqeefv syl2anc cc ad2antrr fveecn sylancom mul02 - mulid2 eqtrd ralbidva syl5ibr mpd cdiv cle elicc01 sylan clt 0le1 mpanl12 + mullid eqtrd ralbidva syl5ibr mpd cdiv cle elicc01 sylan clt 0le1 mpanl12 1re elrege0 sylanbrc adantll ad3antlr recnd simplr ad3antrrr ax-1cn reccl subcl sylancr mpan mulcld simprr adddird simpl subdi mp3an2 oveq2d recid2 - mulid1d mulassd 3eqtr3rd simpll simprl eqtr3d mulcl syl2an addassd adddid - cr ad2antll addid2 ralrimiva rspcev ralimi syl simplbi adantl sylib copab + mulridd mulassd 3eqtr3rd simpll simprl eqtr3d mulcl syl2an addassd adddid + cr ad2antll addlid ralrimiva rspcev ralimi syl simplbi adantl sylib copab mpbird cn cee cpnf cico wf1o cfz cicc simp-4r 1m0e1 eqtrdi eqcom ad4ant24 - biimpac oveqan12d addid1 expdimp necon3d simp1bi rereccl simp2bi ad5ant25 + biimpac oveqan12d addrid expdimp necon3d simp1bi rereccl simp2bi ad5ant25 bitr4d bitrid ne0gt0d divge0 addsubass subeq0bd 3eqtr2d sylan9eq ad2ant2r mp3an3 addcld npcan eqtr4d ad2antrl syl22anc syl5ibrcom impancom r19.29an ralbi rexbidv syldan wss 3simpa 3imtr4i ssriv ssrexv mpsyl jaodan sylan2b @@ -422296,12 +422296,12 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to wrex oveq1 wb syl2anc cc fveecn ad2antrr sylancom eqtrd mpd simp2l cle cr elicc01 simp1bi syl cdiv clt simp2bi 0red simp3 ne0gt0d ltletrd divelunit simp13 recnd simplr divcl adantl ax-1cn simpr2 subcl simpll mulcld adddid - syl22anc sylancr oveq2d addassd adddird subdi mp3an2 mulid1d simp1 npncan + syl22anc sylancr oveq2d addassd adddird subdi mp3an2 mulridd simp1 npncan divcan1 mp3an2i mulassd eqtr3d 3eqtr2rd ralrimiva eqeq2d rspcev ex r19.43 syl23anc id ralbi orbi12d 3expia brbtwn syl3anc bitri bitr4di ralimdv jca breq2d rspcv 3anim123d anim2d sseldd simpr3 syl2an sselda jca32 an4 sylib simplr2 cfz cicc simpl2r simpl eqcom 1m0e1 eqtrdi biimpac simpl2l simpl3l - bitrid eqeefv simp1r mulid2 mul02 oveqan12d addid1 bitr4d syl5ibr expdimp + bitrid eqeefv simp1r mullid mul02 oveqan12d addrid bitr4d syl5ibr expdimp ralbidva sylan2 necon3d simp1l simp2r letrid simpr mpbird simpll3 gt0ne0d 3jca simp12 3eqtr3rd biimpar simp112 simp113 simp12r simp12l eqtr2d 3expa 3eqtrrd orim12d syl6ibr eqeqan12d eqeqan12rd rexbidv syl5ibrcom com23 imp @@ -422467,12 +422467,12 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to id oveq2d eqeqan12d ralimi ralbi syl rexbidv fveecn sylan simpll3 elicc01 cc simp1bi recnd adantr elrege0 simplbi adantl ax-1cn simpr1 simpr3 subcl mulcld sylancr mpan 3ad2ant2 simpll subdird simpr2 nnncan1 mp3an2i oveq1d - subdi mp3an2 mulid1 eqtrd syl2anc npncan eqtr2d 3ad2ant1 3ad2ant3 adddird + subdi mp3an2 mulrid eqtrd syl2anc npncan eqtr2d 3ad2ant1 3ad2ant3 adddird cr mulassd 3eqtrd 3eqtr3d simplr eqeq12d addcld addsubeq4d addassd adddid eqtr4d eqeq2d 3bitr2rd syl23anc ralbidva subcld mulcan1g r19.32v ad2antlr wn wi oveq1 rspceeqv bitrd neneqd bitrdi subeq0ad eqeefv 3adant1 3bitr3rd biorf orcom orbi2d bitrid 3bitrd anassrs rexbidva biimpi simp3bi syl31anc - lemul1a mulid2d breqtrd breq1 syl5ibrcom rexlimdva 0elunit mul02d adantrl + lemul1a mullidd breqtrd breq1 syl5ibrcom rexlimdva 0elunit mul02d adantrl 1red simpl a1d ex cdiv simp3 simprbi 0red simp1 ne0gt0d ltletrd divelunit syl22anc mpbird gt0ne0d divcan1d eqcomd pm2.61ine impbid sylan9bbr anasss clt 3exp sylan2b syldan ) IUANZJIUBUCZNZFUWLNZUDZJFUEZOZDCNZECNZOZOZDJEUF @@ -422548,10 +422548,10 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to cv cfz wral cc0 cicc wrex cpnf cico axcontlem6 ex 3anim123d imp adantr wi 3an6 0elunit cc cr simplr1 ad2antlr elrege0 simplbi recnd simprrl simprrr syl simpl breqtrrd simplr2 letri3d simpll simpll2 fveecn sylancom simpll3 - mpbir2and ax-1cn subcl sylancr simprl mulcld mulcl adantrl addcld mulid2d - mul02d oveq12d addid1d eqtr2d 3adant2 oveq2 oveq1d oveq1 oveq2d eqeqan12d + mpbir2and ax-1cn subcl sylancr simprl mulcld mulcl adantrl addcld mullidd + mul02d oveq12d addridd eqtr2d 3adant2 oveq2 oveq1d oveq1 oveq2d eqeqan12d 3ad2ant2 mpbid syl122anc ralrimiva 1m0e1 eqeq2d ralbidv rspcev adantl clt - cdiv resubcld mpbird necomd ad2antrl sylan subcld divdird mulid1d mulcomd + cdiv resubcld mpbird necomd ad2antrl sylan subcld divdird mulridd mulcomd eqtrd subdi mp3an2 syl2anc subdird divmuld div23d 3eqtr3d joinlmuladdmuld wb 3eqtr4rd mulassd adddid sseli simplr3 lesub1dd subge0d leneltd posdifd eqtrdi letrd syl22anc simp2r simp2l simp1l simp1r subne0d subdid addsub4d @@ -423006,11 +423006,11 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to wb cbs 3ad2ant1 eleq2d biimpa 3adant3 wi biimpcd syl11 a1d brbtwn syl3anc 3imp raleqi rexbii bitr4di bitr3d cun cxr cle 0xr 1xr 0le1 snunico eqcomi mp3an a1i rexeqdv wo rexun wn wne eldifsn weq wfn wf syl6bi sylbid 3imp31 - elee ffn eqcom imp biimpd ffvelcdmda recnd mulid2d oveq12d eqtrd ralbidva + elee ffn eqcom imp biimpd ffvelcdmda recnd mullidd oveq12d eqtrd ralbidva eqeq2d mtbird oveq2 oveq1d oveq1i eqtrdi oveq1 ralbidv rexsng ax-mp biorf sylnibr bitrid syl 3bitrd eqfnfv syl2anc biimprd syl6ibr 3exp com24 sylbi - necon3ad mul02d mpbidi addid2d 1m1e0 bitr2di snunioc ralbii sylnib sylibd - 1re orcom addid1d 0re 1m0e1 bitr4id 3orbi123d rabbidva mpoeq3dva ) HUAMZH + necon3ad mul02d mpbidi addlidd 1m1e0 bitr2di snunioc ralbii sylnib sylibd + 1re orcom addridd 0re 1m0e1 bitr4id 3orbi123d rabbidva mpoeq3dva ) HUAMZH UBUCZUDUCABCCAUEZUFZUGZIUEZUVIBUEZUVHUHUCZNMZUVIUVLUVMUVNNMZUVMUVIUVLUVNN MZUIZICUJZUKABCUVKDUEZUVLUCZOEUEZULNUVTUVIUCZPNUWBUVTUVMUCZPNUMNQZDGUNZER OUONZSZUWCOJUEZULNZUWAPNZUWIUWDPNZUMNZQZDGUNZJROUPNZSZUWDOFUEZULNZUWCPNZU @@ -430169,7 +430169,7 @@ property of a complete graph (see also ~ cplgr0v ), but cannot be an ( chash cfv crab wcel wceq cxad cc0 c0 cvv c2 cv cle wbr cpw wf wa csn co cvtxdg fveq1i vtxdgval adantl eqtrid lfgrnloop adantr fveq2d hash0 eqtrdi eqid oveq2d cxr cxnn0 ciedg dmeqi eqtri fvex dmex eqeltri hashxnn0 xnn0xr - cdm rabex mp2b xaddid1 mp1i 3eqtrd ) BUAAUBZLMUCUDAGUENZFUFZDGOZUGZDCMZDV + cdm rabex mp2b xaddrid mp1i 3eqtrd ) BUAAUBZLMUCUDAGUENZFUFZDGOZUGZDCMZDV RFMZOZABNZLMZWDDUHPABNZLMZQUIZWGRQUIZWGWBWCDEUJMZMZWJDCWLKUKWAWMWJPVTABDE FGHIJULUMUNWBWIRWGQWBWISLMRWBWHSLVTWHSPWAABDVSEFGIJVSUTUOUPUQURUSVAWGVBOZ WKWGPWBWFTOWGVCOWNWEABBEVDMZVLZTBFVLWPJFWOIVEVFWOEVDVGVHVIVMWFTVJWGVKVNWG @@ -430583,7 +430583,7 @@ property of a complete graph (see also ~ cplgr0v ), but cannot be an = ( ( VtxDeg ` G ) ` U ) ) $= ( cvtxdg cfv cop csn cxad co cc0 p1evtxdeqlem cvv wcel ciedg wceq fvexi wa cvtx snex pm3.2i opiedgfv opvtxfv 1hevtxdg0 oveq2d cxnn0 vtxdgelxnn0 - mp1i cxr xnn0xr 3syl xaddid1d 3eqtrd ) ABDUAUBUBBEUAUBUBZBHGCUCZUDZUCZU + mp1i cxr xnn0xr 3syl xaddridd 3eqtrd ) ABDUAUBUBBEUAUBUBZBHGCUCZUDZUCZU AUBUBZUEUFVJUGUEUFVJABCDEFGHIJKLMNOPQRSUHAVNUGVJUEAGBCVMHIJHUIUJZVLUIUJ ZUNZVMUKUBVLULAVOVPHEUOKUMVKUPUQZVLHUIUIURVDVQVMUOUBHULAVRVLHUIUIUSVDPR STUTVAAVJABHUJVJVBUJVJVEUJREHBKVCVJVFVGVHVI $. @@ -430978,7 +430978,7 @@ a loop (see ~ uspgr1v1eop ) is a singleton of a singleton. (Contributed cdif eqcomi rabeqi rabun2 eqtri fveq2i cvv cin fvexi dmex rab2ex wnel wss ssrab2 ss2in mp2an elneldisj sseq2i ss0 sylbi ax-mp hashunx mp3an oveq12i c0 cxnn0 hashxnn0 a1i xnn0add4d cxr xnn0xaddcl xaddcom vtxdginducedm1lem4 - xnn0xr cc0 oveq2d xaddid1 eqtrdi eleq2d cbvrabv eqtrid vtxdginducedm1lem2 + xnn0xr cc0 oveq2d xaddrid eqtrdi eleq2d cbvrabv eqtrid vtxdginducedm1lem2 fveq2 eqtrd vtxdginducedm1lem3 rabbiia eqeq1d oveq1i eldifi eqid vtxdgval syl cvtx cop difexg eqeltrid resexg opvtxfvi eleq2i sylbbr oveq1d 3eqtr4d cres rgen ) AUAZFUBUCUCZXSCUBUCUCZXSLUAZEUCZUDZLHUEZUFUCZUGUHZUIAKJUJZUOZ @@ -435429,7 +435429,7 @@ According to Wikipedia ("Cycle (graph theory)", ( cc0 co wceq wa ccsh oveq2 cdm cword wcel ccrcts cfv wbr cwlks crctiswlk wlkf 3syl cshw0 syl sylan9eqr eqtrid cfz cv cmin cle cif cmpt crctcshlem1 caddc wb nn0cnd subid1d breq2d adantr adantl elfzelz zcnd sylan9eq fveq2d - addid1d fvoveq1d ifbieq12d mpteq2dva elfzle2 iftrued chash wf wlkp eqcomi + addridd fvoveq1d ifbieq12d mpteq2dva elfzle2 iftrued chash wf wlkp eqcomi wfn ffn oveq2i fneq2i sylib dffn5 eqcomd mpdan eqtrd jca ) AESUAZUBZHFUAD CUAWRHFEUCTZFQWQAWSFSUCTZFESFUCUDAFIUEZUFUGZWTFUAAFCGUHUIUJZFCGUKUIUJZXBN CFGULZCFGIMUMUNXAFUOUPUQURWRDBSJUSTZBUTZJEVATZVBUJZXGEVFTZCUIZXJJVATCUIZV @@ -443003,7 +443003,7 @@ edge remains odd if it was odd before (regarding the subgraphs induced ( cfv c1 caddc co wne wa w3a c2 cvtxdg cdvds wbr wn cc0 wceq cpr cif wcel c0 cvv ciedg cop csn 3ad2ant1 cvtx fvexd simpl adantl simpr nelprd df-nel wnel wi sylibr neleq2 syl5ibr expd 1hevtxdg0 oveq2d eupth2lem3lem1 nn0cnd - 3imp addid1d eqtrd breq2d notbid wb cv crab fveq2 elrab3 eleq2d bitr3d wo + 3imp addridd eqtrd breq2d notbid wb cv crab fveq2 elrab3 eleq2d bitr3d wo 3ad2ant3 2thd neeq1 bibi12d syl5ibcom pm5.32rd neneqd biorf bitrdi anbi2d syl orcom 3bitr3d eupth2lem1 3bitr4d 3bitrd ) AHCUGZHUHUIUJCUGZUKZDXPUKZD XQUKZULZUMZUNDJUOUGZUGZDKUOUGUGZUIUJZUPUQZURUNYDUPUQZURZDUSCUGZXPUTVDYJXP @@ -443358,7 +443358,7 @@ edge remains odd if it was odd before (regarding the subgraphs induced cc 3ad2ant2 sylbi npcan1 sylan9eq oveq2d cdm cword ccrcts cwlks crctiswlk cn 3syl wlkf syl cshwn eqtrd eqtr3id csn cun crctcshlem1 fz0sn0fz1 eleq2d eqid elun bitrdi elsni 0le0 eqbrtrdi iftrued fveq2d ctrls crctprop eqcomd - wo simpr adantr addid2d sylan9eqr fveq2 ex wn sylbid eqtrid breq2d fveq2i + wo simpr adantr addlidd sylan9eqr fveq2 ex wn sylbid eqtrid breq2d fveq2i 3eqtr4d oveq1d mpteq2dv 3brtr4d cz cima 3jca eqtrdi reseq2d eucrct2eupth1 imp a1i sseqtrid resmptd wi 3ad2ant1 eqcom cr wb ad2antlr sylan2 iffalsed elfznn nnnle0 nncnd pncand jaod mpteq2dva subidd oveq1i ifbieq12d wf wlkp @@ -445563,7 +445563,7 @@ that the number n of vertices in G is exactly k(k-1)+1.". Variant of eqid c1 eleq2d mpbird pfxcctswrd syl2an2r impcom 3jca sylbid rspceov syl6 sylan caddc eluzelcn 2cnd npcand biimpd clwwlknonccat impel cedg clwwlkn2 ex cpr cfzo cn 2nn lbfzo0 mpbir eleqtrrid ccatval3 subcld sylan9eq fveq2d - addid2d 3eqtr4d exp53 com24 com13 3adant3 mpbir2and syl5ibrcom rexlimdvva + addlidd 3eqtr4d exp53 com24 com13 3adant3 mpbir2and syl5ibrcom rexlimdvva imp eleq1 impbid ) IGMZFUGUANMZOZHIFCPZMZHJUHZKUHZUBPZQZKIREUCNZPZUDJIFRU EPZUUBPZUDZYOYQHUUDUFPZUUEMZHUUDFUIZUJPZUUCMZHUUGUUJUBPZQZUSZUUFYOYQHIFUU BPZMZUUDHNZIQZOZUUNYNYMFRUANMZYQUUSUKFULZABCDEFGHILUMUNYOUUSUUNYOUUSOZUUH @@ -450153,7 +450153,7 @@ Norman Megill (2007) section 1.1.3. Megill then states, "A number of Carneiro, 18-Jun-2015.) $) ex-fl $p |- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 ) $= ( c3 c2 co c1 wceq cneg wbr clt 1re 3re 2cn eqbrtri wb 2re mpbi cr wa mp2an - wcel cz cdiv cfl cfv caddc rehalfcli cmul mulid2i 2lt3 2pos ltmuldivi ax-mp + wcel cz cdiv cfl cfv caddc rehalfcli cmul mullidi 2lt3 2pos ltmuldivi ax-mp cle cc0 ltleii 3lt4 2t2e4 breqtrri pm3.2i ltdivmul mp3an mpbir df-2 breqtri c4 1z flbi mpbir2an renegcli ltnegi cmin negcli ax-1cn negdi2 negnegi eqtri cc oveq1i 2m1e1 readdcli ltnegcon1i 2z znegcl ) ABUACZUBUCDEZWCFZUBUCBFZEZW @@ -450197,7 +450197,7 @@ Norman Megill (2007) section 1.1.3. Megill then states, "A number of ex-fac $p |- ( ! ` 5 ) = ; ; 1 2 0 $= ( c5 cfa cfv c4 c1 caddc co cmul c2 cdc cc0 df-5 fveq2i 4nn0 eqtri 2nn0 5cn 0nn0 2cn mulcomli wcel wceq facp1 ax-mp fac4 4p1e5 oveq12i 5nn0 eqid 5t2e10 - cn0 1nn0 addid2i decaddi 4cn 5t4e20 decmul1c ) ABCZDBCZDEFGZHGZEIJZKJZURUTB + cn0 1nn0 addlidi decaddi 4cn 5t4e20 decmul1c ) ABCZDBCZDEFGZHGZEIJZKJZURUTB CZVAAUTBLMDUKUAVDVAUBNDUCUDOVAIDJZAHGVCUSVEUTAHUEUFUGIDVBKAIVEUHPNVEUIRPEKI IAHGIULRPAIEKJQSUJTISUMUNADIKJQUOUPTUQOO $. @@ -450374,7 +450374,7 @@ Norman Megill (2007) section 1.1.3. Megill then states, "A number of avril1 $p |- -. ( A ~P RR ( _i ` 1 ) /\ F (/) ( 0 x. 1 ) ) $= ( vy vx vz c1 ci cfv cr cpw wbr cc0 c0 wa wn cv c0r cop wcel wceq cmul co c1r cio cnr csn cxp cvv weq equid dfnul2 eqabi con2bii mpbi eleq1 vtocleg - mtbii elex con3i pm2.61i df-br 0cn mulid1i opeq2i eleq1i bitri mtbir df-i + mtbii elex con3i pm2.61i df-br 0cn mulridi opeq2i eleq1i bitri mtbir df-i intnan fveq1i df-fv eqtri breq2i wss cab sseq2 abbidv df-pw 3eqtr4g ax-mp df-r breqi anbi1i notbii mpbir ) AFGHZIJZKZBLFUAUBZMKZNZOAFCPQUCRZKCUDZUE QUFUGZJZKZWJNZOWJWPWJBLRZMSZWRUHSZWSOZXADWRUHDPZWRTXBMSZWSDDUIZXCODUJXCXD @@ -451905,7 +451905,7 @@ which is an Abelian group (i.e. the vectors, with the operation of group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.) $) cnaddabloOLD $p |- + e. AbelOp $= - ( vx vy vz caddc cc cc0 cv cneg cnex ax-addf addass 0cn addid2 negcl wcel + ( vx vy vz caddc cc cc0 cv cneg cnex ax-addf addass 0cn addlid negcl wcel co wceq addcom mpdan negid eqtr3d isgrpoi cxp fdmi isabloi ) ABDEABCFDAGZ HZEIJUFBGZCGKLUFMUFNZUFEOZUFUGDPZUGUFDPZFUJUGEOUKULQUIUFUGRSUFTUAUBEEUCED JUDUFUHRUE $. @@ -451916,7 +451916,7 @@ which is an Abelian group (i.e. the vectors, with the operation of (Proof modification is discouraged.) $) cnidOLD $p |- 0 = ( GId ` + ) $= ( vy vx caddc cgi cfv cv co wceq cc wral crio cc0 wcel cablo cnaddabloOLD - cgr ablogrpo ax-mp cxp ax-addf fdmi grporn eqid grpoidval addid2 rgen 0cn + cgr ablogrpo ax-mp cxp ax-addf fdmi grporn eqid grpoidval addlid rgen 0cn wreu wb grpoideu oveq1 eqeq1d ralbidv riota2 mp2an mpbi eqtr2i ) CDEZAFZB FZCGZUTHZBIJZAIKZLCPMZURVDHCNMVEOCQRZBAURCICIVFIISICTUAUBZURUCUDRLUTCGZUT HZBIJZVDLHZVIBIUTUEUFLIMVCAIUHZVJVKUIUGVEVLVFBACIVGUJRVCVJAILUSLHZVBVIBIV @@ -451927,7 +451927,7 @@ which is an Abelian group (i.e. the vectors, with the operation of ` x. ` . (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.) $) cncvcOLD $p |- <. + , x. >. e. CVecOLD $= - ( vx vy vz cmul caddc cop cc cnaddabloOLD cxp ax-addf fdmi ax-mulf mulid2 + ( vx vy vz cmul caddc cop cc cnaddabloOLD cxp ax-addf fdmi ax-mulf mullid cv adddi adddir mulass eqid isvciOLD ) ABCDEEDFZGHGGIGEJKLANZMBNZUACNZOUB UCUAPUBUCUAQTRS $. $} @@ -452807,7 +452807,7 @@ which is an Abelian group (i.e. the vectors, with the operation of nvm1 $p |- ( ( U e. NrmCVec /\ A e. X ) -> ( N ` ( -u 1 S A ) ) = ( N ` A ) ) $= ( cnv wcel wa c1 cneg co cfv cabs cmul cc wceq neg1cn mp3an2 absnegi abs1 - nvs ax-1cn eqtri oveq1i nvcl recnd mulid2d eqtrid eqtrd ) CIJZAEJZKZLMZAB + nvs ax-1cn eqtri oveq1i nvcl recnd mullidd eqtrid eqtrd ) CIJZAEJZKZLMZAB NDOZUPPOZADOZQNZUSUMUPRJUNUQUTSTUPABCDEFGHUDUAUOUTLUSQNUSURLUSQURLPOLLUEU BUCUFUGUOUSUOUSACDEFHUHUIUJUKUL $. $} @@ -452835,7 +452835,7 @@ which is an Abelian group (i.e. the vectors, with the operation of nvpi $p |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A G ( _i S B ) ) ) = ( N ` ( B G ( -u _i S A ) ) ) ) $= ( wcel c1 ci co cfv cmul cneg ax-icn wceq cnv w3a cr simp1 mp3an2 3adant2 - nvscl nvgcl syld3an3 nvcl syl2anc recnd mulid2d cabs absnegi eqtri oveq1i + nvscl nvgcl syld3an3 nvcl syl2anc recnd mullidd cabs absnegi eqtri oveq1i absi negicn nvs simp2 nvdi mp3anr1 syl12anc mulneg1i ixi negeqi negneg1e1 cc nvsass mpanr1 nvsid 3eqtr3a oveq2d nvcom syld3an2 3eqtrd fveq2d eqtr3d wa 3adant3 eqtr3id ) DUALZAGLZBGLZUBZMANBCOZEOZFPZQOZWIBNRZACOZEOZFPZWFWI @@ -452909,7 +452909,7 @@ which is an Abelian group (i.e. the vectors, with the operation of ( N ` ( A M B ) ) <_ ( ( N ` A ) + ( N ` B ) ) ) $= ( wcel c1 cfv co caddc cle neg1cn eqid mp3an2 3adant2 cmul cnv w3a cns cc cneg cpv wbr nvscl syld3an3 nvmval fveq2d wceq wa cabs nvs ax-1cn absnegi - nvtri abs1 eqtri oveq1i nvcl recnd mulid2d eqtrid eqtr2d oveq2d 3brtr4d ) + nvtri abs1 eqtri oveq1i nvcl recnd mullidd eqtrid eqtr2d oveq2d 3brtr4d ) CUAJZAFJZBFJZUBZAKUEZBCUCLZMZCUFLZMZELZAELZVOELZNMZABDMZELVSBELZNMOVIVJVK VOFJZVRWAOUGVIVKWDVJVIVMUDJZVKWDPVMBVNCFGVNQZUHRSAVOCVPEFGVPQZIURUIVLWBVQ EABVNCVPDFGWGWFHUJUKVLWCVTVSNVIVKWCVTULVJVIVKUMZVTVMUNLZWCTMZWCVIWEVKVTWJ @@ -453577,7 +453577,7 @@ which is an Abelian group (i.e. the vectors, with the operation of ( wcel co c1 ci cexp c2 cmul vk cnv w3a c4 cfz cv cfv csu cdiv cneg caddc cmin ipval cc ax-icn ipval2lem4 mpan2 mulcl sylancr neg1cn subcld negsubd negicn mulm1d oveq2d mulneg1 oveq12d mp3an1 syl2anc oveq1d sub32d 3eqtr4d - eqtrd wceq subdi wa nvsid fveq2d 3adant2 ipval2lem3 recnd mulid2d cn nnuz + eqtrd wceq subdi wa nvsid fveq2d 3adant2 ipval2lem3 recnd mullidd cn nnuz c3 df-4 oveq2 i4 eqtrdi cn0 nnnn0 expcl adantl sylan2 mulcld df-3 i3 df-2 i2 cz 1z exp1 ax-mp fsum1 1nn jctil eqidd fsump1i simprd subadd23d eqtr4d addcomd ) EUBNZAHNZBHNZUCZABCOPUDUEOQUAUFZROZAXRBDOZFOZGUGZSROZTOZUAUHZUD @@ -453659,10 +453659,10 @@ which is an Abelian group (i.e. the vectors, with the operation of cmin cdiv eqid ipval2 3anidm23 nv2 fveq2d cr cle wbr pm3.2i nvsge0 mp3an2 2re 0le2 eqtrd oveq1d nvcl recnd cn0 2cn mulexp mp3an13 syl oveq1i eqtrdi 2nn0 sq2 cn0v nvrinv adantr sq0id oveq12d 4cn sqcld mulcl sylancr subid1d - nvz0 csqrt 1re neg1rr absreim mp2an ax-icn ax-1cn mulneg2i mulid1i negeqi + nvz0 csqrt 1re neg1rr absreim mp2an ax-icn ax-1cn mulneg2i mulridi negeqi eqtri oveq2i fveq2i sqneg ax-mp 3eqtr3i 3eqtr2i negicn addcli nvs 3eqtr4a cabs nvdir mp3anr1 mpanr1 nvsid 3eqtr3d oveq2d w3a ipval2lem4 mpan2 it0e0 - subidd addid1d eqtr2d wne 4ne0 divcan3 mp3an23 3eqtr2d ) CUAIZAEIZUBZAABJ + subidd addridd eqtr2d wne 4ne0 divcan3 mp3an23 3eqtr2d ) CUAIZAEIZUBZAABJ ZAACUCKZJZDKZLMJZANUDZACUEKZJYJJZDKZLMJZUGJZOAOAYOJZYJJZDKZLMJZAOUDZAYOJZ YJJZDKZLMJZUGJZPJZQJZUFUHJZUFADKZLMJZPJZUFUHJZUUNYFYGYIUULRAABYOCYJDEFYJU IZYOUIZGHUJUKYHUUOUUKUFUHYHUUKUUOSQJUUOYHYSUUOUUJSQYHYSUUOSUGJUUOYHYMUUOY @@ -454660,7 +454660,7 @@ a normed complex vector space (normally a Hilbert space). @) ( M ` ( T ` x ) ) <_ ( A x. ( L ` x ) ) ) -> ( N ` T ) <_ ( abs ` A ) ) $= ( wcel cle wbr wf cr cv cfv cmul co wral cabs wa c1 cnv nvcl mpan remulcl wi sylan2 adantr recn abscld syl2an ad2antrr cc0 simpl nvge0 adantl leabs - jca lemul1a syl31anc 1red absge0d 3jca lemul2a sylan wceq mulid1d breqtrd + jca lemul1a syl31anc 1red absge0d 3jca lemul2a sylan wceq mulridd breqtrd w3a recnd adantlll ffvelcdm sylancr adantlr adantll ad2antlr letr syl3anc letrd mpan2d ex com23 ralimdva imp cxr rexrd nmoubi biimpar syldan 3impa wb ) IJCUAZBUBRZAUCZCUDZFUDZBXCEUDZUEUFZSTZAIUGZCGUDBUHUDZSTZXAXBUIZXIXFU @@ -455724,7 +455724,7 @@ Inner product (pre-Hilbert) spaces 26-Apr-2007.) (New usage is discouraged.) $) ip2i $p |- ( ( 2 S A ) P B ) = ( 2 x. ( A P B ) ) $= ( co c1 caddc cc0 wcel wceq c2 cneg cmul cnv cc phnvi nvgcl mp3an dipcl - addid1i cn0v cfv eqid nvrinv mp2an oveq1i dip0l eqtri oveq2i w3a ax-1cn + addridi cn0v cfv eqid nvrinv mp2an oveq1i dip0l eqtri oveq2i w3a ax-1cn df-2 3pm3.2i nvdir nvsid oveq12i 3eqtr4ri ip1i ) UAADOZBCOZAAFOZBCOZAPU BADOFOZBCOZQOZUAABCOUCOVLRQOVLVOVJVLEUDSZVKGSZBGSZVLUESELUFZVPAGSZVTVQV SMMAAEFGHIUGUHNVKBCEGHKUIUHUJVNRVLQVNEUKULZBCOZRVMWABCVPVTVMWATVSMADEFG @@ -455797,7 +455797,7 @@ Inner product (pre-Hilbert) spaces ( wcel co cmul wceq cc0 oveq1 vj vk cn0 cv wi c1 caddc wa cc ax-1cn cnv nn0cn w3a phnvi nvdir mpan mp3an2 sylan nvsid adantl oveq2d eqtrd dipcl oveq1d mp3an13 nvscl mp3an1 ipdiri mp3an3 sylancom eqtr4d adddir syl2an - mulid2d sylan9eq adantr exp31 a2d cn0v cfv eqid dip0l mp2an nv0 3eqtr4a + mullidd sylan9eq adantr exp31 a2d cn0v cfv eqid dip0l mp2an nv0 3eqtr4a mul02d eqeq12d imbi2d nn0indALT imp ) GUCOAHOZGADPZBCPZGABCPZQPZRZWKUAU DZADPZBCPZWQWNQPZRZUEWKSADPZBCPZSWNQPZRZUEWKUBUDZADPZBCPZXFWNQPZRZUEWKX FUFUGPZADPZBCPZXKWNQPZRZUEWKWPUEUAUBGXFUCOZWKXJXOXPWKXJXOXPWKUHZXJUHXMX @@ -455820,7 +455820,7 @@ Inner product (pre-Hilbert) spaces ( ( -u N S A ) P B ) = ( -u N x. ( A P B ) ) ) $= ( wcel co cmul cc cc0 wceq cn0 wa cneg nn0cn negcld phnvi dipcl mp3an13 cnv mulcl syl2an nvscl mp3an1 sylan syl cmin caddc ax-1cn mulneg2 mpan2 - c1 mulid1 negeqd eqtr2d adantr oveq1d neg1cn nvsass mpan mp3an2 mp3an12 + c1 mulrid negeqd eqtr2d adantr oveq1d neg1cn nvsass mpan mp3an2 mp3an12 w3a ipasslem1 sylan2 oveq2d negsubd mulneg1 adantl adddid ipdiri mp3an3 eqtrd mpdan cn0v eqid nvrinv dip0l mp2an eqtrdi eqtr3d mul01d sylan9eqr cfv 3eqtr2d subeq0d eqcomd ) GUAOZAHOZUBZGUCZABCPZQPZWTADPZBCPZWSXBXDWQ @@ -455855,7 +455855,7 @@ Inner product (pre-Hilbert) spaces ipasslem4 $p |- ( ( N e. NN /\ A e. X ) -> ( ( ( 1 / N ) S A ) P B ) = ( ( 1 / N ) x. ( A P B ) ) ) $= ( wcel c1 co cmul cc adantr cn wa cdiv nnrecre recnd phnvi nvscl mp3an1 - cnv sylan dipcl mp3an13 syl mulcl syl2an nncn cc0 recidd oveq1d mulid2d + cnv sylan dipcl mp3an13 syl mulcl syl2an nncn cc0 recidd oveq1d mullidd wne nnne0 sylan9eq wceq nvsid simpr w3a nvsass syl3anc eqtr3d cn0 nnnn0 mpan ipasslem1 syl2anc 3eqtrd adantl mulassd mulcanad ) GUAOZAHOZUBZPGU CQZADQZBCQZWCABCQZRQZGWBWDHOZWESOZVTWCSOZWAWHVTWCGUDUEZEUIOZWJWAWHEMUFZ @@ -455970,7 +455970,7 @@ Inner product (pre-Hilbert) spaces phnvi cc nvscl mp3an 4ipval2 4cn negicn dipcl mul12i nvgcl nvcli neg1cn recni sqcli subcli mulcli addcomi adddii w3a nvsass mp2an oveq1i eqtr3i 3pm3.2i ixi oveq2i fveq2i negeqi negneg1e1 3eqtri nvsid 3eqtr3i oveq12i - mulneg1i mulm1i negsubdi2i eqtr2i mulassi eqtr4i mulcomli mulid2i eqtri + mulneg1i mulm1i negsubdi2i eqtr2i mulassi eqtr4i mulcomli mullidi eqtri 4ne0 mulcani mpbi dipcj cjmuli cr neg1rr cjrebi cji ax-1cn mul2negi ) B QADRZCRZUAUBZQUCZBACRZSRZUAUBZXFBCRZQABCRZSRZXGXKUAUDXGSRZUDXKSRZUEXGXK UEXPBXFFRZGUBZUFUGRZBUHUCZXFDRZFRZGUBZUFUGRZUIRZQBQXFDRZFRZGUBZUFUGRZBX @@ -456191,7 +456191,7 @@ orthogonal vectors (i.e. whose inner product is 0) is the sum of the pythi $p |- ( ( A P B ) = 0 -> ( ( N ` ( A G B ) ) ^ 2 ) = ( ( ( N ` A ) ^ 2 ) + ( ( N ` B ) ^ 2 ) ) ) $= ( co cc0 wceq caddc wcel mp3an cfv c2 cexp ip2dii id cnv phnvi diporthcom - wb biimpi oveq12d 00id eqtrdi oveq2d cc dipcl addcli addid1i eqtrid nvgcl + wb biimpi oveq12d 00id eqtrdi oveq2d cc dipcl addcli addridi eqtrid nvgcl ipidsq mp2an oveq12i 3eqtr3g ) ABCOZPQZABEOZVGCOZAACOZBBCOZROZVGFUAUBUCOZ AFUAUBUCOZBFUAUBUCOZROVFVHVKVEBACOZROZROZVKABABCDEGHIKLMNMNUDVFVQVKPROVKV FVPPVKRVFVPPPROPVFVEPVOPRVFUEVFVOPQZDUFSZAGSZBGSZVFVRUIDLUGZMNABCDGHKUHTU @@ -456665,7 +456665,7 @@ orthogonal vectors (i.e. whose inner product is 0) is the sum of the crab wss rabss sylib ad2antrr cnv ccbn bnnv ax-mp a1i crp rpcnd simpr cc eqid nvscl syl3anc nvgcl rspcdva cmul cnsb cxmet ccmet cmet cbncms cmetmet metxmet mp2b xmetsym imsdval nvpncan2 fveq2d 3eqtrd cc0 eqtrd - rprege0d nvsge0 mulid1d eqcomd breq12d nvcl mpan adantl lemul2d wb cn + rprege0d nvsge0 mulridd eqcomd breq12d nvcl mpan adantl lemul2d wb cn 1red ralbidv cba 2fveq3 elrab sylancr adantr ccld mopntopon ffvelcdmd syl ctopon syl32anc syld ralrimiva syl2anc bitr4d breq2 rabbidv fvexi rabex fvmpt eleq2d bitrdi 3imtr3d com12 ad2antlr xmet0 rpge0d eqbrtrd @@ -457142,7 +457142,7 @@ orthogonal vectors (i.e. whose inner product is 0) is the sum of the ( vw cv wcel wa co cexp cc0 caddc cle wbr cfv wral cnv wceq ccphlo phnv c2 syl adantr css wss ccbn cin inss1 sselid eqid syl2anc sselda imsdval sspba syl3anc oveq1d cr clt cinf wne wrex w3a minvecolem1 simp1d simp2d - 0red simp3d breq1 ralbidv rspcev infrecl eqeltrid resqcld recnd addid1d + 0red simp3d breq1 ralbidv rspcev infrecl eqeltrid resqcld recnd addridd c0 breq12d nvmcl nvcl nvge0 wb infregelb mpbird breqtrrdi le2sqd breq2i syl31anc bitrid 3bitr2d cmpt crn raleqi cvv rgenw breq2 ralrnmptw ax-mp fvex bitri bitrdi ) ABUGZNUHZUIZDYBEUJZVBUKUJZGVBUKUJZULUMUJZUNUOZDYBJU @@ -457543,7 +457543,7 @@ orthogonal vectors (i.e. whose inner product is 0) is the sum of the adantr id syl2an ex weq fveq2 breq1d elrab cba oveq2d mpteq2dv mptfvmpt fveq1d oveq1 ovex fvmpt sylan9eqr ad2ant2lr rsp2 impl eqtrd fveq2d cmul adantrr simpl dipcl mp3an1 abscld mpan ad2antrl remulcld sii 1red nvge0 - cc0 jca simprr lemul2a syl31anc recnd mulid1d breqtrd eqbrtrd syl5ibcom + cc0 jca simprr lemul2a syl31anc recnd mulridd breqtrd eqbrtrd syl5ibcom cc letrd syld syl2anc wb mpbid ad2ant2r mpd expr sylan2b fveq1 ralrimiv rexlimdva brralrspcev ralrimiva wss crn imassrn eqsstri frnd ccbn hlobn sstrid cnnv cnnvnm ubth simpr sylibr cdm dmmptd eleq2d biimpar funfvima @@ -465495,7 +465495,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). $( Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) $) - hvaddid2 $p |- ( A e. ~H -> ( 0h +h A ) = A ) $= + hvaddlid $p |- ( A e. ~H -> ( 0h +h A ) = A ) $= ( chba wcel c0v cva co wceq ax-hv0cl ax-hvcom mpan2 ax-hvaddid eqtr3d ) ABC ZADEFZDAEFZAMDBCNOGHADIJAKL $. @@ -465539,15 +465539,15 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). 23-May-2005.) (New usage is discouraged.) $) hv2neg $p |- ( A e. ~H -> ( 0h -h A ) = ( -u 1 .h A ) ) $= ( chba wcel c0v cmv cneg csm cva wceq ax-hv0cl hvsubval mpan neg1cn hvmulcl - co c1 cc hvaddid2 syl eqtrd ) ABCZDAEOZDPFZAGOZHOZUDDBCUAUBUEIJDAKLUAUDBCZU + co c1 cc hvaddlid syl eqtrd ) ABCZDAEOZDPFZAGOZHOZUDDBCUAUBUEIJDAKLUAUDBCZU EUDIUCQCUAUFMUCANLUDRST $. ${ - hvaddid2.1 $e |- A e. ~H $. + hvaddlid.1 $e |- A e. ~H $. $( Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) $) - hvaddid2i $p |- ( 0h +h A ) = A $= - ( chba wcel c0v cva co wceq hvaddid2 ax-mp ) ACDEAFGAHBAIJ $. + hvaddlidi $p |- ( 0h +h A ) = A $= + ( chba wcel c0v cva co wceq hvaddlid ax-mp ) ACDEAFGAHBAIJ $. $( Addition of negative of a vector to itself. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) $) @@ -465841,7 +465841,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) $) hvsubeq0i $p |- ( ( A -h B ) = 0h <-> A = B ) $= ( cmv co c0v wceq c1 cneg csm cva hvsubvali eqeq1i oveq1 sylbi chba ax-mp - wcel eqtri neg1cn hvmulcli hvadd32i hvnegidi ax-hvaddid hvaddid2i 3eqtr3g + wcel eqtri neg1cn hvmulcli hvadd32i hvnegidi ax-hvaddid hvaddlidi 3eqtr3g hvassi oveq2i hvsubid eqtrdi impbii ) ABEFZGHZABHZUNAIJZBKFZLFZBLFZGBLFZA BUNURGHUSUTHUMURGABCDMNURGBLOPUSABLFUQLFZAAUQBCUPBUADUBZDUCVAABUQLFZLFZAA BUQCDVBUHVDAGLFZAVCGALBDUDUIAQSVEAHCAUERTTTBDUFUGUOUMBBEFZGABBEOBQSVFGHDB @@ -465861,7 +465861,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). $( Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.) $) hvaddcani $p |- ( ( A +h B ) = ( A +h C ) <-> B = C ) $= - ( cva co wceq c1 cneg csm c0v neg1cn hvmulcli hvadd32i oveq1i hvaddid2i + ( cva co wceq c1 cneg csm c0v neg1cn hvmulcli hvadd32i oveq1i hvaddlidi oveq1 3eqtri hvnegidi 3eqtr3g oveq2 impbii ) ABGHZACGHZIZBCIUGUEJKZALHZ GHZUFUIGHZBCUEUFUIGSUJAUIGHZBGHMBGHBABUIDEUHANDOZPULMBGADUAZQBERTUKULCG HMCGHCACUIDFUMPULMCGUNQCFRTUBBCAGUCUD $. @@ -465990,7 +465990,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). ( chba wcel wa cva co cmv wceq hvaddcl ancoms hvsub4 syldan hvsubid hvsubcl c0v syl 3eqtrd wb adantr adantl ad2ant2lr hvsubcan2 syl3anc anim2i ad2antlr simpr oveq2d ax-hvaddid adantlr adantrr simpl anim1i oveq1d adantll eqeq12d - ad2antrr hvaddid2 bitr3d ) AEFZBEFZGZCEFZDEFZGZGZABHIZCBHIZJIZCDHIZVJJIZKZV + ad2antrr hvaddlid bitr3d ) AEFZBEFZGZCEFZDEFZGZGZABHIZCBHIZJIZCDHIZVJJIZKZV IVLKZACJIZDBJIZKVHVIEFZVLEFZVJEFZVNVOUAVDVRVGABLUBVGVSVDCDLUCVCVEVTVBVFVEVC VTCBLMUDVIVLVJUEUFVHVKVPVMVQVDVEVKVPKVFVDVEGZVKVPBBJIZHIZVPRHIZVPVDVEVEVCGZ VKWCKVEVDWEVDVCVEVBVCUIUGMABCBNOWAWBRVPHVCWBRKVBVEBPUHUJVBVEWDVPKZVCVBVEGVP @@ -466244,7 +466244,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). ( ( B -h A ) .ih ( D -h C ) ) $= ( c1 cneg cmv co csm csp hvnegdii cmul neg1cn hvsubcli ax-1cn 3eqtri wcel oveq12i ccj his35i cr wceq neg1rr cjre ax-mp oveq2i mul2negi 1t1e1 oveq1i - cfv hicli mulid2i eqtr3i ) IJZBAKLZMLZURDCKLZMLZNLZABKLZCDKLZNLUSVANLZUTV + cfv hicli mullidi eqtr3i ) IJZBAKLZMLZURDCKLZMLZNLZABKLZCDKLZNLUSVANLZUTV DVBVENBAFEODCHGOUBVCURURUCUNZPLZVFPLIVFPLVFURURUSVAQQBAFERZDCHGRZUDVHIVFP VHURURPLIIPLIVGURURPURUEUAVGURUFUGURUHUIUJIISSUKULTUMVFUSVAVIVJUOUPTUQ $. $} @@ -466325,7 +466325,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). ( ( ( S x. -u R ) x. ( G .ih F ) ) + ( ( R ^ 2 ) x. ( G .ih G ) ) ) ) $= ( cmul co csp cfv cneg caddc c2 cexp oveq2i eqtri c1 csm cmv ccj mulcli recni normlem0 cjmuli cr wcel wceq cjrebi negeqi mulneg2i eqtr4i oveq1i - mpbi cjcli eqcomi mul4i cabs absvalsqi 3eqtr3i oveq12i mulid2i sqvali + mpbi cjcli eqcomi mul4i cabs absvalsqi 3eqtr3i oveq12i mullidi sqvali sq1 ) CBAJKZDUAKUBKZVHLKCCLKZVGUCMZNZCDLKZJKZOKZVGNZDCLKZJKZVGVJJKZDDLK ZJKZOKZOKVIBUCMZANZJKZVLJKZOKZBWCJKZVPJKZAPQKZVSJKZOKZOKVGCDBAEAHUEZUDF GUFVNWFWAWKOVMWEVIOVKWDVLJVKWBAJKZNWDVJWMVJWBAUCMZJKZWMBAEWLUGZWNAWBJAU @@ -466406,7 +466406,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). ( co cmul cle wbr cc0 cr wcel vx c2 cexp csqrt cfv cabs caddc cmin wtru c4 csp chba hiidrcl ax-mp eqeltri a1i normlem2 cv cif wceq oveq1 oveq2d oveq2 oveq12d oveq1d breq2d 0re elimel dedth adantl discr mptru resqcli - normlem5 4re remulcli lesubadd2i mpbi recni addid1i breqtri sqge0i 4pos + normlem5 4re remulcli lesubadd2i mpbi recni addridi breqtri sqge0i 4pos wb ltleii hiidge0 breqtrri mulge0i mp2an sqrtlei absrei sqrtmulii sqrt4 oveq12i eqtr2i 3brtr4i ) BUBUCNZUDUEZUJACONZONZUDUEZBUFUEUBAUDUECUDUEON ZONZPWQWTPQZWRXAPQZWQWTRUGNZWTPWQWTUHNRPQZWQXFPQXGUIUAABCASTUIAFFUKNZSK @@ -466455,7 +466455,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). ( ( ( A .ih C ) + ( B .ih D ) ) - ( ( A .ih D ) + ( B .ih C ) ) ) $= ( co csp c1 cneg caddc neg1cn cmul wcel chba wceq mp3an 3eqtri cmv csm cc cva cmin hvsubvali oveq12i hvmulcli normlem8 ccj ax-his3 oveq2i cr neg1rr - cfv his5 ax-mp ax-1cn mul2negi mulid2i oveq1i cjcli hicli mulassi 3eqtr3i + cfv his5 ax-mp ax-1cn mul2negi mullidi oveq1i cjcli hicli mulassi 3eqtr3i cjre mulm1i eqtri negdii eqtr4i addcli negsubi ) ABUAIZCDUAIZJIAKLZBUBIZU DIZCVODUBIZUDIZJIACJIZVPVRJIZMIZAVRJIZVPCJIZMIZMIZVTBDJIZMIZADJIZBCJIZMIZ UEIZVMVQVNVSJABEFUFCDGHUFUGAVPCVREVOBNFUHGVODNHUHZUIWFWHWKLZMIWLWBWHWEWNM @@ -466650,7 +466650,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). norm-ii-i $p |- ( normh ` ( A +h B ) ) <_ ( ( normh ` A ) + ( normh ` B ) ) $= ( co csp csqrt cfv caddc cle c2 wbr cmul c1 cr wcel wceq ax-1cn cc0 ax-mp - cva cno cexp ccj 1re cjrebi mpbi oveq1i hicli mulid2i eqtri abs1 normlem7 + cva cno cexp ccj 1re cjrebi mpbi oveq1i hicli mullidi eqtri abs1 normlem7 oveq12i eqbrtrri cneg eqid normlem2 cjcli mulcli addcli eqeltrri 2re chba negrebi hiidge0 hiidrcl sqrtcli remulcli readdcli leadd2i normlem8 oveq2i addcomi recni binom2i sqcli 2cn add32i sqsqrti 3eqtri 3brtr4i wb hvaddcli @@ -466711,7 +466711,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.) $) normsubi $p |- ( normh ` ( A -h B ) ) = ( normh ` ( B -h A ) ) $= ( c1 cneg cmv csm cno cfv cabs neg1cn hvsubcli norm-iii-i hvnegdii fveq2i - co cmul ax-1cn eqtri absnegi abs1 oveq1i normcli recni mulid2i 3eqtr3i ) + co cmul ax-1cn eqtri absnegi abs1 oveq1i normcli recni mullidi 3eqtr3i ) EFZBAGQZHQZIJUHKJZUIIJZRQZABGQZIJULUHUILBADCMZNUJUNIBADCOPUMEULRQULUKEULR UKEKJEESUAUBTUCULULUIUOUDUEUFTUG $. @@ -466721,7 +466721,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). normpythi $p |- ( ( A .ih B ) = 0 -> ( ( normh ` ( A +h B ) ) ^ 2 ) = ( ( ( normh ` A ) ^ 2 ) + ( ( normh ` B ) ^ 2 ) ) ) $= ( csp co cc0 wceq cva caddc cno c2 cexp normlem8 chba wcel eqtrdi normsqi - cfv hicli id wb orthcom mp2an biimpi oveq12d oveq2d addcli addid1i eqtrid + cfv hicli id wb orthcom mp2an biimpi oveq12d oveq2d addcli addridi eqtrid 00id hvaddcli oveq12i 3eqtr4g ) ABEFZGHZABIFZUQEFZAAEFZBBEFZJFZUQKSLMFAKS LMFZBKSLMFZJFUPURVAUOBAEFZJFZJFZVAABABCDCDNUPVFVAGJFVAUPVEGVAJUPVEGGJFGUP UOGVDGJUPUAUPVDGHZAOPBOPUPVGUBCDABUCUDUEUFUKQUGVAUSUTAACCTBBDDTUHUIQUJUQA @@ -466765,7 +466765,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). normpyc $p |- ( ( A e. ~H /\ B e. ~H ) -> ( ( A .ih B ) = 0 -> ( normh ` A ) <_ ( normh ` ( A +h B ) ) ) ) $= ( chba wcel wa co cc0 wceq cno cfv cexp cle wbr caddc normcl resqcld adantr - c2 cr syl csp cva recnd addid1d sqge0d adantl wb 0re leadd2 mp3an1 eqbrtrrd + c2 cr syl csp cva recnd addridd sqge0d adantl wb 0re leadd2 mp3an1 eqbrtrrd syl2anr mpbid normpyth imp breqtrrd ex hvaddcl normge0 le2sqd sylibrd ) ACD ZBCDZEZABUAFGHZAIJZRKFZABUBFZIJZRKFZLMZVFVILMVDVEVKVDVEEVGVGBIJZRKFZNFZVJLV DVGVNLMVEVDVGGNFZVGVNLVBVOVGHVCVBVGVBVGVBVFAOZPZUCUDQVDGVMLMZVOVNLMZVCVRVBV @@ -466882,7 +466882,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). ( 2 x. ( ( normh ` B ) ^ 2 ) ) ) $= ( co cno cfv c2 cexp caddc csp normsqi oveq12i oveq2i hicli 2timesi eqtri cmul addcli eqtr4i cmv cva hvsubcli hvaddcli cneg normlem9 negsubi negcli - cmin normlem8 add42i cc0 negidi addid1i add4i 3eqtri ) ABUAEZFGHIEZABUBEZ + cmin normlem8 add42i cc0 negidi addridi add4i 3eqtri ) ABUAEZFGHIEZABUBEZ FGHIEZJEUQUQKEZUSUSKEZJEZHAFGHIEZREZHBFGHIEZREZJEZURVAUTVBJUQABCDUCLUSABC DUDLMVHAAKEZVIJEZBBKEZVKJEZJEZVCVEVJVGVLJVEHVIREVJVDVIHRACLNVIAACCOZPQVGH VKREVLVFVKHRBDLNVKBBDDOZPQMVCVIVKJEZABKEZBAKEZJEZUEZJEZVPVSJEZJEVPVPJEZVS @@ -466954,7 +466954,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). cva cmv csm 4cn 4ne0 2cn adddii wceq ppncan 2timesi eqtr4i oveq2i mulassi cc mp3an 2t2e4 oveq1i 3eqtr2ri pnncani normlem8 normlem9 3eqtr4i hvmulcli cneg ccj cfv chba cji eqtri ax-his3 eqtr3i 3eqtri negicn mulcli mul12i c1 - his5 mulneg2i ixi negeqi negneg1e1 mulid2i 3eqtr3i mulm1i negsubi 3eqtr2i + his5 mulneg2i ixi negeqi negneg1e1 mullidi 3eqtr3i mulm1i negsubi 3eqtr2i mvllmuli ) UAABIJZACUCJDBUCJIJZACUDJDBUDJIJZKJZLALCUEJZUCJDLBUEJZUCJIJZAW NUDJDWOUDJIJZKJZMJZNJZUFABEFOZUGPWJCDIJZNJZWJXBKJZNJZMJZPXCMJZPXDMJZNJUAW JMJZWTPXCXDUHWJXBXACDGHOZQZWJXBXAXJUBUIXFPPWJMJZMJPPMJZWJMJXIXEXLPMXEWJWJ @@ -467025,7 +467025,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.) $) hilablo $p |- +h e. AbelOp $= ( vx vy vz cva chba c0v c1 cv csm co ax-hilex ax-hfvadd ax-hvass ax-hv0cl - cneg hvaddid2 cc wcel neg1cn ax-hvcom hvmulcl mpan wceq mpancom eqtrd cxp + cneg hvaddlid cc wcel neg1cn ax-hvcom hvmulcl mpan wceq mpancom eqtrd cxp hvnegid isgrpoi fdmi isabloi ) ABDEABCFDGOZAHZIJZEKLULBHZCHMNULPUKQRULERZ UMERZSUKULUAUBZUOUMULDJZULUMDJZFUPUOURUSUCUQUMULTUDULUGUEUHEEUFEDLUIULUNT UJ $. @@ -467035,7 +467035,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). (New usage is discouraged.) $) hilid $p |- ( GId ` +h ) = 0h $= ( vy vx cva cgi cfv cv co wceq chba wral crio c0v cgr wcel cablo ablogrpo - hilablo ax-mp cxp ax-hfvadd fdmi grporn eqid grpoidval hvaddid2 rgen wreu + hilablo ax-mp cxp ax-hfvadd fdmi grporn eqid grpoidval hvaddlid rgen wreu wb ax-hv0cl grpoideu oveq1 eqeq1d ralbidv riota2 mp2an mpbi eqtri ) CDEZA FZBFZCGZUTHZBIJZAIKZLCMNZURVDHCONVEQCPRZBAURCICIVFIISICTUAUBZURUCUDRLUTCG ZUTHZBIJZVDLHZVIBIUTUEUFLINVCAIUGZVJVKUHUIVEVLVFBACIVGUJRVCVJAILUSLHZVBVI @@ -467250,7 +467250,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). ( abs ` ( A .ih B ) ) <_ ( normh ` B ) ) $= ( chba wcel cno cfv c1 cle wbr w3a co cabs cmul cr wa hicl 3adant3 3ad2ant2 csp normcl abscld remulcl syl2an bcs cc0 3ad2ant1 normge0 jca simp3 lemul1a - 1re mp3anl2 syl21anc wceq recnd mulid2d breqtrd letrd ) ACDZBCDZAEFZGHIZJZA + 1re mp3anl2 syl21anc wceq recnd mullidd breqtrd letrd ) ACDZBCDZAEFZGHIZJZA BSKZLFZVABEFZMKZVFUSUTVENDVBUSUTOVDABPUAQUSUTVGNDZVBUSVANDZVFNDZVHUTATZBTZV AVFUBUCQUTUSVJVBVLRZUSUTVEVGHIVBABUDQVCVGGVFMKZVFHVCVIVJUEVFHIZOZVBVGVNHIZU SUTVIVBVKUFVCVJVOVMUTUSVOVBBUGRUHUSUTVBUIVIGNDVPVBVQUKVAGVFUJULUMUTUSVNVFUN @@ -467949,7 +467949,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] (New usage is discouraged.) $) hsn0elch $p |- { 0h } e. CH $= ( vf vx vy c0v wcel cn cv chli wbr wa wal chba cva co csm ax-hv0cl pm3.2i - wral cc wceq csn cch csh wf wi wss snssi ax-mp elexi snid velsn hvaddid2i + wral cc wceq csn cch csh wf wi wss snssi ax-mp elexi snid velsn hvaddlidi oveq12 eqtrdi ovex elsn sylibr syl2anb rgen2 oveq2 hvmul0 sylan9eqr issh2 sylan2b mpbir2an wb fconst2 hlim0 breq1 mpbiri sylbi hlimuni eleq1d sylan cxp mpbii gen2 isch2 ) DUAZUBEVSUCEZFVSAGZUDZWABGZHIZJZWCVSEZUEZBKAKVTVSL @@ -468144,7 +468144,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] hhssnv $p |- W e. NrmCVec $= ( vx vy csm cc cva cno c0v wcel wss wceq chba mp2an cfv co ovres eqtrd cxp cres cablo cgr hhssabloi ablogrpo ax-mp cdm shssii xpss12 ax-hfvadd - vz fdmi sseqtrri ssdmres grporn cgi csh sh0 ax-hv0cl hvaddid2i eqtri wb + vz fdmi sseqtrri ssdmres grporn cgi csh sh0 ax-hv0cl hvaddlidi eqtri wb mpbi eqid grpoid mpbir cop wf wfn crn ax-hfvmul ffn ssid fnssres ovelrn cv wrex wa shmulcl mp3an1 eqeltrd eleq1 syl5ibrcom rexlimivv sylbi df-f ssriv mpbir2an c1 ax-1cn sheli ax-hvmulid syl w3a id ax-hvdistr1 syl3an @@ -468779,7 +468779,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) $) shscli $p |- ( A +H B ) e. SH $= ( vx vy vf vg vv vu co wcel chba c0v wa cv cva csm wceq wrex vz cph csh - vw wss cc shsss mp2an sh0 ax-mp ax-hv0cl hvaddid2i eqcomi rspceov mp3an + vw wss cc shsss mp2an sh0 ax-mp ax-hv0cl hvaddlidi eqcomi rspceov mp3an wral shseli mpbir pm3.2i shaddcl mp3an1 ad2ant2r ad2ant2l sheli anim12i wi oveq12 hvadd4 syl2an eqtr4d syl3anc ancoms exp43 rexlimivv com3l imp an4s syl2anb sylibr rgen2 shmulcl adantrr adantrl oveq2 adantl ad2antll @@ -469209,7 +469209,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] (New usage is discouraged.) $) shunssi $p |- ( A u. B ) C_ ( A +H B ) $= ( vx vy vz co cv wcel cva wceq wrex c0v sheli eqcomd syl csh sh0 ax-mp wo - cun chba ax-hvaddid rspceov mp3an2 mpdan hvaddid2 mp3an1 jaoi elun shseli + cun chba ax-hvaddid rspceov mp3an2 mpdan hvaddlid mp3an1 jaoi elun shseli cph 3imtr4i ssriv ) EABUBZABUMHZEIZAJZURBJZUAURFIGIKHLGBMFAMZURUPJURUQJUS VAUTUSURURNKHZLZVAUSURUCJZVCURACOVDVBURURUDPQUSNBJZVCVABRJVEDBSTFGABURNUR KUEUFUGUTURNURKHZLZVAUTVDVGURBDOVDVFURURUHPQNAJZUTVGVAARJVHCASTFGABNURURK @@ -471231,7 +471231,7 @@ equals the join of their closures (double orthocomplements). adantr sylan syl3anc c0v wne w3a csn cspn cpjh cva cno cexp cort spansnch cv wrex simp2 eqid pjeq mpbii simprd oveq1 ad2antll pjhcl choccl syl chel simpl1 ax-his2 csh spansnsh spansnid simpr shocorth 3impib orthcom syldan - c2 wb mpbid 3ad2antl1 oveq2d cc hicl addid1d 3eqtrd adantrr oveq1d simpl3 + c2 wb mpbid 3ad2antl1 oveq2d cc hicl addridd 3eqtrd adantrr oveq1d simpl3 eqtrd axpjcl normcan eqtr2d rexlimddv ) ADEZBDEZAUAUBZUCZBBAUDUEFZUFFFZCU LZUGGZHZWQBAIGZAUHFVOUIGZJGZAKGZHCWPUJFZWOWPLEZWMWTCXEUMZWLWMXFWNAUKZMZWL WMWNUNZXFWMNZWQWPEZXGXKWQWQHXLXGNWQUOCBWQWPUPUQUROWOWRXEEZWTNZNZXDWQAIGZX @@ -471276,7 +471276,7 @@ equals the join of their closures (double orthocomplements). hvmulcl jca 3expb syl2an eqtr4d rspceov syl3anc sylibr oveq2 eqeq2d eleq1 biimpa biimparc exp43 rexlimdv biimtrid rexlimiv sylbi cneg negcl syl3an2 syl ancoms c0v c1 hvm1neg hvnegid sylancl ax-hvcom syl2anc 3eqtr3d oveq1d - hvaddcl hvaddid2 anim12i hvadd4 impbii eqriv chssii spanuni spanid oveq1i + hvaddcl hvaddlid anim12i hvadd4 impbii eqriv chssii spanuni spanid oveq1i ax-mp eqtri 3eqtr4i ) ABUBZUCJZKLZABAUDJZUEJJZUBZUCJZKLZAYBUFUCJZAYGUFUCJ ZUAYDYIUAMZYDNZYLYINZYMYLEMZFMZOLZPZFYCQZEAQYNEFAYCYLACUGZBRNZYBRUHZYCUIN DBRUJZYBUKULZUMYSYNEAYOANZYRYNFYCYPYCNYPGMZBSLZPZGTQUUEYRYNUNZGBYPDUOUUEU @@ -472264,7 +472264,7 @@ Note that the (countable) Axiom of Choice is used for this proof via ( cv wcel wa cva co wceq cmv chba shsvsi ad2ant2r adantr ancoms eleqtrrdi shscomi ad2ant2l wb sheli anim12i oveq1 adantl simpr anim2i hvsub4 syldan cph c0v hvsubid oveq2d ad2antlr hvsubcl ax-hvaddid adantlr 3eqtrd adantrr - syl simpl anim1i syl2anc oveq1d ad2antrl hvaddid2 adantrl adantll 3eqtr3d + syl simpl anim1i syl2anc oveq1d ad2antrl hvaddlid adantrl adantll 3eqtr3d eleq1d sylan mpbird elind ) AMZENZBMZFNZOZCMZGNZDMZHNZOZOZWAWCPQZWFWHPQZR ZOZEGUQQZFHUQQZWAWFSQZWKWRWPNZWNWBWGWSWDWIEGWAWFIKUAUBUCWOWRWQNZWHWCSQZWQ NZWKXBWNWDWIXBWBWGWDWIOXAHFUQQZWQWIWDXAXCNHFWHWCLJUAUDFHJLUFUEUGUCWKWATNZ @@ -472512,7 +472512,7 @@ Note that the (countable) Axiom of Choice is used for this proof via v e. ( B +H ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) ) ) $= ( wcel wa cva co wceq cmv c0v cph simplll simpllr chba 3oalem1 hvaddsub12 cin 3anidm23 hvsubid oveq2d ax-hvaddid sylan9eqr ad2ant2l adantlr simprlr - cv eqtr3d syl eqtr2 oveq1d anim1i hvsub4 syldan ad2antrr hvsubcl hvaddid2 + cv eqtr3d syl eqtr2 oveq1d anim1i hvsub4 syldan ad2antrr hvsubcl hvaddlid simpl adantll 3eqtrd simpr anim2i syl2anc ad2antll ancoms adantrr 3eqtr3d ad2ant2rl simpll chshii shsvsi shscomi syl2an eqeltrd simplr elind shscli eleqtrrdi shincli shsvai eqeltrrd wb eleq1 ad2antlr mpbird ) AUPZFNZBUPZH @@ -473174,7 +473174,7 @@ Note that the (countable) Axiom of Choice is used for this proof via ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) ) $= ( cno cfv cpjh cle wbr wcel wceq c2 cexp caddc normcli chba normge0 ax-mp co cc0 wne wa wn clt pjnormi biantrur cort c0v pjoc1i pjpythi sq0 pjhclii - oveq2i resqcli recni addid1i eqtr2i eqeq12i choccli sqcli addcani wb 0le0 + oveq2i resqcli recni addridi eqtr2i eqeq12i choccli sqcli addcani wb 0le0 0cn 0re sq11i mp2an norm-i-i 3bitri bitr2i necon3bbii ltleni 3bitr4i ) AE FZABGFFZEFZUAZVPVNHIZVQUBABJZUCVPVNUDIVRVQABCDUEUFVSVNVPVSABUGFZGFFZUHKZV NLMSZVPLMSZKZVNVPKZABCDUIWEWDWAEFZLMSZNSZWDTLMSZNSZKZWBWCWIWDWKABCDUJWKWD @@ -473845,10 +473845,10 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. ${ $d x T $. - hoaddid1.1 $e |- T : ~H --> ~H $. + hoaddrid.1 $e |- T : ~H --> ~H $. $( Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.) $) - hoaddid1i $p |- ( T +op 0hop ) = T $= + hoaddridi $p |- ( T +op 0hop ) = T $= ( vx cv ch0o chos co cfv wceq chba wral wcel cva wf hosval mp3an12 ho0val c0v ho0f oveq2d ffvelcdmi ax-hvaddid syl 3eqtrd rgen hoaddcli hoeqi mpbi ) CDZAEFGZHZUIAHZIZCJKUJAIUMCJUIJLZUKULUIEHZMGZULRMGZULJJANJJENUNUKUPIBSU @@ -473857,7 +473857,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. $( Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) $) hodidi $p |- ( T -op T ) = 0hop $= - ( chod co ch0o wceq chos hoaddid1i ho0f hodsi mpbir ) AACDEFAEGDAFABHAAEB + ( chod co ch0o wceq chos hoaddridi ho0f hodsi mpbir ) AACDEFAEGDAFABHAAEB BIJK $. $( Composition of the zero operator and a Hilbert space operator. @@ -473886,8 +473886,8 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. $( Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) $) - hoaddid1 $p |- ( T : ~H --> ~H -> ( T +op 0hop ) = T ) $= - ( chba wf ch0o chos co wceq cif oveq1 id eqeq12d ho0f elimf hoaddid1i dedth + hoaddrid $p |- ( T : ~H --> ~H -> ( T +op 0hop ) = T ) $= + ( chba wf ch0o chos co wceq cif oveq1 id eqeq12d ho0f elimf hoaddridi dedth ) BBACZADEFZAGPADHZDEFZRGADARGZQSARARDEITJKRBBADLMNO $. $( Difference of a Hilbert space operator from itself. (Contributed by NM, @@ -473916,7 +473916,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) $) ho0subi $p |- ( S -op T ) = ( S +op ( 0hop -op T ) ) $= - ( chod co ch0o chos wceq ho0f hosubcli hoadd12i oveq2i hoaddid1i hoaddcli + ( chod co ch0o chos wceq ho0f hosubcli hoadd12i oveq2i hoaddridi hoaddcli hodseqi 3eqtri hodsi mpbir ) ABEFAGBEFZHFZIBUAHFZAIUBABTHFZHFAGHFABATDCGB JDKZLUCGAHBGDJPMACNQABUACDATCUDORS $. @@ -473957,7 +473957,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. $( The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) $) hosubid1 $p |- ( T : ~H --> ~H -> ( T -op 0hop ) = T ) $= - ( chba wf ch0o chod co chos wceq ho0sub mpan2 hodidi oveq2i hoaddid1 eqtrid + ( chba wf ch0o chod co chos wceq ho0sub mpan2 hodidi oveq2i hoaddrid eqtrid ho0f eqtrd ) BBACZADEFZADDEFZGFZAQBBDCRTHOADIJQTADGFASDAGDOKLAMNP $. $( Relationship between Hilbert space operator addition and subtraction. @@ -473973,7 +473973,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. $d x A $. $d x B $. $d x T $. $d x U $. $( An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) $) - homulid2 $p |- ( T : ~H --> ~H -> ( 1 .op T ) = T ) $= + homullid $p |- ( T : ~H --> ~H -> ( 1 .op T ) = T ) $= ( vx chba wf cv c1 chot co cfv wceq wral wcel wa csm ax-1cn homval mp3an1 cc ffvelcdm ax-hvmulid eqtrd ralrimiva wb homulcl mpan hoeq mpancom mpbid syl ) CCADZBEZFAGHZIZUKAIZJZBCKZULAJZUJUOBCUJUKCLZMZUMFUNNHZUNFRLZUJURUMU @@ -474055,7 +474055,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. 24-Aug-2006.) (New usage is discouraged.) $) honegneg $p |- ( T : ~H --> ~H -> ( -u 1 .op ( -u 1 .op T ) ) = T ) $= ( chba wf c1 cneg cmul co chot neg1mulneg1e1 oveq1i cc wcel neg1cn homulass - wceq mp3an12 homulid2 3eqtr3a ) BBACZDEZTFGZAHGZDAHGTTAHGHGZAUADAHIJTKLZUDS + wceq mp3an12 homullid 3eqtr3a ) BBACZDEZTFGZAHGZDAHGTTAHGHGZAUADAHIJTKLZUDS UBUCOMMTTANPAQR $. $( Relationship between operator subtraction and negative. (Contributed by @@ -474173,7 +474173,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. (New usage is discouraged.) $) ho2times $p |- ( T : ~H --> ~H -> ( 2 .op T ) = ( T +op T ) ) $= ( chba wf c2 chot co c1 chos caddc df-2 oveq1i cc wcel wceq ax-1cn hoadddir - mp3an12 eqtrid hoadddi anidms mp3an1 hoaddcl homulid2 syl 3eqtr2d ) BBACZDA + mp3an12 eqtrid hoadddi anidms mp3an1 hoaddcl homullid syl 3eqtr2d ) BBACZDA EFZGAEFZUHHFZGAAHFZEFZUJUFUGGGIFZAEFZUIDULAEJKGLMZUNUFUMUINOOGGAPQRUFUKUINZ UNUFUFUOOGAASUATUFBBUJCZUKUJNUFUPAAUBTUJUCUDUE $. @@ -474200,7 +474200,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. $( Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) $) hosd1i $p |- ( T +op U ) = ( T -op ( 0hop -op U ) ) $= - ( ch0o chod co chos wceq hosubcli hoaddcomi hoaddsubassi eqtr4i hoaddid1i + ( ch0o chod co chos wceq hosubcli hoaddcomi hoaddsubassi eqtr4i hoaddridi ho0f hoaddcli oveq1i hoaddsubi hodseqi 3eqtri hodsi mpbir eqcomi ) AEBFGZ FGZABHGZUEUFIUDUFHGZAIUGUFEHGZBFGZUFBFGZAUGUFUDHGUIUDUFEBODJZABCDPZKUFEBU LODLMUHUFBFUFULNQUJABFGZBHGBUMHGAABBCDDRUMBABCDJDKBADCSTTAUDUFCUKULUAUBUC @@ -474215,7 +474215,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. $( Hilbert space operator cancellation law. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) $) hopncani $p |- ( ( T +op U ) -op U ) = T $= - ( chos co chod ch0o hoaddsubassi hodidi oveq2i hoaddid1i 3eqtri ) ABEFBGF + ( chos co chod ch0o hoaddsubassi hodidi oveq2i hoaddridi 3eqtri ) ABEFBGF ABBGFZEFAHEFAABBCDDINHAEBDJKACLM $. $( Hilbert space operator cancellation law. (Contributed by NM, @@ -474228,7 +474228,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) $) hosubeq0i $p |- ( ( T -op U ) = 0hop <-> T = U ) $= ( chod co ch0o wceq c1 cneg chot chos honegsubi eqeq1i oveq1 sylbir eqtri - chba wf hoaddid1i cc wcel neg1cn homulcl hoadd32i hoaddassi hodidi oveq2i + chba wf hoaddridi cc wcel neg1cn homulcl hoadd32i hoaddassi hodidi oveq2i mp2an ho0f hoaddcomi 3eqtr3g eqtrdi impbii ) ABEFZGHZABHZUPAIJZBKFZLFZBLF ZGBLFZABUPUTGHVAVBHUTUOGABCDMNUTGBLOPVAABLFUSLFZAAUSBCURUAUBRRBSRRUSSUCDU RBUDUIZDUEVCABUSLFZLFZAABUSCDVDUFVFAGLFAVEGALVEBBEFZGBBDDMBDUGZQUHACTQQQV @@ -476469,7 +476469,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. oveq12i hisubcomi lnopsubi hvmulcli hvsubdistr1i cneg hvsubvali hvmulassi hvsubcli his35i ixi ax-1cn mul2negi 1t1e1 3eqtri oveq1i neg1cn ax-hvmulid mulcli 3eqtr3i eqtr3i eqtri fveq2i cc lnopmuli lnopaddmuli mp3an mulneg2i - cji negeqi negneg1e1 recni mulid2i hvdistr1i 3eqtr4i hvaddcli lnopmulsubi + cji negeqi negneg1e1 recni mullidi hvdistr1i 3eqtr4i hvaddcli lnopmulsubi lnophmlem1 cc0 wne 4ne0 resubcli addcli 4re cjdivi cr negsubi lnopsubmuli cjreim negsubdi2i cjre 3eqtrri his1i ) BCIJZBDKZCDKZIJZLJZBCUAJZXJXKUAJZL JZMJZNBNCOJZIJZXJNXKOJZIJZLJZBXRUAJZXJXTUAJZLJZMJZPJZUBJZUCUDJZCXJLJZUEKZ @@ -476643,7 +476643,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. rspcev c0ex eqeq1 anbi2d rexbidv elab mpbir ne0ii wi crp cdiv wal rpdivcl c2 co 2rp mpan rpred adantr csm cc rehalfcld recnd simprl hvmulcl syl2anc rpre normcl syl cmul simprr ad2antrl 1red rphalfcl lemul2d mpbid norm-iii - sylan rpge0 absidd oveq1d eqtr2d mulid1d 3brtr3d rphalflt lelttrd imbi12d + sylan rpge0 absidd oveq1d eqtr2d mulridd 3brtr3d rphalflt lelttrd imbi12d cabs rpcn rspcv mpid ltmuldiv2d rprecred ltle sylbid rpne0 recdiv mpanr12 2cn 2ne0 breq2d 3imtr3d syld an32s anassrs breq1 mp3an ralab breq2 imbi2d expimpd rexlimdva alrimiv albidv bitrid ax-mp supxrre suprcl eqeltri @@ -476904,7 +476904,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. lnfn0i $p |- ( T ` 0h ) = 0 $= ( c0v cfv caddc co cmin chba wcel cc ax-hv0cl lnfnfi ffvelcdmi ax-mp wceq cc0 c1 ax-1cn eqtr3i oveq1i pncan3oi cmul csm cva lnfnli mp3an ax-hvaddid - hvmulcli ax-hvmulid eqtri fveq2i mulid2i subidi ) CADZUNEFZUNGFZUNPUNUNCH + hvmulcli ax-hvmulid eqtri fveq2i mullidi subidi ) CADZUNEFZUNGFZUNPUNUNCH IZUNJIKHJCAABLMNZURUAUNUNGFUPPUNUOUNGQUNUBFZUNEFZUNUOQCUCFZCUDFZADZUTUNQJ IUQUQVCUTORKKQCCABUEUFVBCAVBVACVAHIVBVAOQCRKUHVAUGNUQVACOKCUINUJUKSUSUNUN EUNURULTSTUNURUMSS $. @@ -476914,7 +476914,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. lnfnaddi $p |- ( ( A e. ~H /\ B e. ~H ) -> ( T ` ( A +h B ) ) = ( ( T ` A ) + ( T ` B ) ) ) $= ( chba wcel wa c1 csm cva cfv cmul caddc wceq ax-1cn lnfnli mp3an1 adantr - co cc ax-hvmulid fvoveq1d lnfnfi ffvelcdmi mulid2d oveq1d 3eqtr3d ) AEFZB + co cc ax-hvmulid fvoveq1d lnfnfi ffvelcdmi mullidd oveq1d 3eqtr3d ) AEFZB EFZGZHAISZBJSCKZHACKZLSZBCKZMSZABJSCKZUMUOMSHTFUHUIULUPNOHABCDPQUHULUQNUI UHUKABCJAUAUBRUJUNUMUOMUHUNUMNUIUHUMETACCDUCUDUERUFUG $. @@ -476924,7 +476924,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. ( T ` ( A .h B ) ) = ( A x. ( T ` B ) ) ) $= ( cc wcel chba wa csm co c0v cva cfv cmul wceq ax-hv0cl lnfnli mp3an3 cc0 caddc hvmulcl ax-hvaddid syl fveq2d lnfn0i oveq2i lnfnfi ffvelcdmi sylan2 - mulcl addid1d eqtrid 3eqtr3d ) AEFZBGFZHZABIJZKLJZCMZABCMZNJZKCMZTJZUQCMV + mulcl addridd eqtrid 3eqtr3d ) AEFZBGFZHZABIJZKLJZCMZABCMZNJZKCMZTJZUQCMV AUNUOKGFUSVCOPABKCDQRUPURUQCUPUQGFURUQOABUAUQUBUCUDUPVCVASTJVAVBSVATCDUEU FUPVAUOUNUTEFVAEFGEBCCDUGUHAUTUJUIUKULUM $. @@ -477643,7 +477643,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. ( vy cfv cnop cle wbr cno c1 chba wf wcel bdopf mp2b wa cmul co cr adantr cbo cado cv wi cxr wb adjbdln nmopxr nmopub mp2an ffvelcdmi normcl nmopre wral syl ax-mp remulcl sylancr 1re remulcli a1i nmopadjlei nmopge0 pm3.2i - cc0 lemul2a mp3anl3 mpanl2 sylan letrd recni mulid1i breqtrdi ex mprgbir + cc0 lemul2a mp3anl3 mpanl2 sylan letrd recni mulridi breqtrdi ex mprgbir ) AUADZEDAEDZFGZCUBZHDZIFGZVRVODZHDZVPFGZUCZCJJJVOKZVPUDLZVQWDCJUMUEATLZV OTLWEBAUFVOMNZWGJJAKZWFBAMZAUGNCVPVOUHUIVRJLZVTWCWKVTOZWBVPIPQZVPFWLWBVPV SPQZWMWKWBRLZVTWKWAJLWOJJVRVOWHUJWAUKUNSWKWNRLZVTWKVPRLZVSRLZWPWGWQBAULUO @@ -477745,7 +477745,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. adantr wn wne df-ne cdiv csm cc ffvelcdmi normcl syl recnd divrec2 mp3an2 cabs ancoms rerecclzi clt bdopf nmopgt0 recgt0i 0re ltle mpan sylc absidd sylbi oveq1d recclzi norm-iii syl2an lnopmuli hocoi oveq2d adantl hvmulcl - nmoplb mp3an1 mulid2i breqtrrdi biimpi ledivmul2 syl112anc mpbird eqbrtrd + nmoplb mp3an1 mullidi breqtrrdi biimpi ledivmul2 syl112anc mpbird eqbrtrd 1red syl2anc ad2antrl jctil syl3anc mpbid sylanbr pm2.61ian ex mprgbir ) ABUBZUCEZAUCEZBUCEZFGZHIZUAUDZJEKHIZYRYLEZJEZYPHIZUEZUALLLYLUFYPUGMYQUUCU ALUHUIYLABAUJMZAUKMCAULNZBUJMZBUKMDBULNZUMZUNZYPYNYOUUDYNOMZCAUONZUUFYOOM @@ -477835,9 +477835,9 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. mp2b cado ccom cnop c2 cexp cv wi wf cxr wral wb cbo adjbdlnb mpbi hocofi bdopf nmopre resqcli rexr nmopub mp2an hocoi fveq2d ffvelcdmi normcl 3syl remulcl sylancr remulcli a1i nmbdoplbi 1re nmopge0 pm3.2i lemul2a mp3anl3 - mpanl2 sylan recni mulid1i breqtrdi letrd syl21anc nmopadji oveq1i sqvali + mpanl2 sylan recni mulridi breqtrdi letrd syl21anc nmopadji oveq1i sqvali eqbrtrd eqtr4i mprgbir csqrt bdopcoi sqrtcli csp cabs hicl mpancom abscld - ex cc remulcld bcs hococli normge0 jca simpr mp3anl2 recnd mulid1d eqtr4d + ex cc remulcld bcs hococli normge0 jca simpr mp3anl2 recnd mulridd eqtr4d breqtrd resqcl absidd normsq cdm bdopadj adj2 mp3an1 adjadj fveq1i oveq2i sqge0 eqtr2di eqtrd eqtr3d sqsqrti 3brtr4d sqrtge0i le2sq mpanr12 syl2anc mpbird le2sqi breqtri letri3i mpbir2an ) AUADZAUBZUCDZAUCDZUDUEEZFYRYTGHZ @@ -477916,7 +477916,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. chod wceq cuo unopbd bdopf hocofi hosubcli nmop0h mpan2 0le0 00id oveq12d breqtrri breqtrrid eqbrtrd wne cmul chos honpncani fveq2i bdopcoi bdophdi nmoptrii hocsubdiri nmopcoi eqbrtrri bdopln hoddii remulcli le2addi mp2an - clo bdophsi readdcli letri c1 nmopun oveq2d mulid1i eqtrdi oveq1d mulid2i + clo bdophsi readdcli letri c1 nmopun oveq2d mulridi eqtrdi oveq1d mullidi recni breqtrid pm2.61ine ) CDUAZABUAZUEIZJKZCAUEIZJKZDBUEIZJKZUBIZLUCMUDM UDUFZWMNWRLWSMMWLOWMNUFWJWKCDCPQZMMCOCUGQWTECUHRZCUIRZDPQZMMDODUGQZXCFDUH RZDUIRZUJZABAPQZMMAOAUGQZXHGAUHRZAUIRZBPQZMMBOBUGQXLHBUHRZBUIRZUJZUKWLULU @@ -477953,7 +477953,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. recnd reccld simpl hvmulcl syl2anc 1le1 eqbrtrdi ax-his3 syl3anc rereccld csm hiidrcl remulcld eqeltrd normgt0 biimpa recgt0d 0re ltle mpan hiidge0 norm1 sylc mulge0d breqtrrd absidd recid2d oveq2d mul12d c2 sqvald normsq - cexp eqtr3d eqtrd mulid1d 3eqtr3rd 3eqtr4rd breq1d fvoveq1 eqeq2d anbi12d + cexp eqtr3d eqtrd mulridd 3eqtr3rd 3eqtr4rd breq1d fvoveq1 eqeq2d anbi12d rspcev syl12anc wb rexbidva mpbird elabd breq2 sylan adantlr supxr2 nmfn0 ex csn cxp bra0 fveq2i norm0 3eqtr4i a1i pm2.61ne ) AEFZAUBGZUCGZAHGZIJUB GZUCGZJHGZIZAJAJIUUOUURUUPUUSAJUCUBUDAJHUEUFUUMAJUGZKZUUOBUHZHGZLMNZCUHZU @@ -478327,7 +478327,7 @@ Positive operators (cont.) 0re wceq w3a 3simpa 3impia mp3an1 anim1i leopmuli syl2anc cdiv wne gt0ne0 rereccl syldan hmopm 3adant3 recgt0 wi sylancr mpd jca31 anassrs sylan cc cmul recn recid2d oveq1d chba wf 3ad2ant1 hmopf 3ad2ant2 homulass syl3anc - reccld homulid2 syl 3eqtr3d breqtrd impbida ) ACDZBEDZFAGHZUAZIBJHZIABKLZ + reccld homullid syl 3eqtr3d breqtrd impbida ) ACDZBEDZFAGHZUAZIBJHZIABKLZ JHZWCWDMVTWAMZFANHZWDMWFWCWGWDVTWAWBUBOWCWHWDVTWBWHWAFCDZVTWBWHSWIVTWBWHF APUCUDQUEABUFUGWCWFMIRAUHLZWEKLZBJWCWJCDZWEEDZMZFWJNHZMWFIWKJHZWCWLWMWOVT WBWLWAVTWBAFUIWLAUJZAUKULZQVTWAWMWBABUMUNVTWBWOWAVTWBMZFWJGHZWOAUOWSWIWLW @@ -478408,7 +478408,7 @@ Positive operators (cont.) wa clt wf wceq cnop cfv ch0o clo hmoplin nmlnopne0 biimpar rereccl simpll cdiv cle idhmop hmopm mpan2 ad2antlr simplr nmopgt0 biimpa recgt0d wi 0re hmopf ltle sylancr mpd leopnmid adantr leopmul2i syl32anc recn cmul reccl - simpl wf1o hoif f1of ax-mp homulass mp3an3 syl2anc recid2 oveq1d homulid2 + simpl wf1o hoif f1of ax-mp homulass mp3an3 syl2anc recid2 oveq1d homullid cc eqtr3d eqtrdi breqtrd syldan 3impa ) ABCZAUAUBZDCZAUCEZFWKUJGZAHGZIJKZ WJWLQZWMWKLEZWPWJWMWRWLWJAUDCZWMWRAUEWSWRWMAUFUGMNWQWRQZWOWNWKIHGZHGZIJWT WNDCZWJXABCZLWNUKKZAXAJKZWOXBJKWLWRXCWJWKUHZOWJWLWRUIWLXDWJWRWLIBCXDULWKI @@ -478436,7 +478436,7 @@ Positive operators (cont.) cle hmopm mpanr1 mpanl1 ad2ant2lr cc recni homulcl homco1 mp3an1 syl2anc2 clo hmoplin homco2 syl2anc oveq2d fco homulass mp3an12 syl sqrtthi oveq1i cmul eqtr3di 3eqtrd cdiv eqtrdi nmlnopne0 recidzi sylbir oveq1d rerecclzi - c1 id recnd mp3an13 homulid2 mp1i 3eqtr3d sylan9eqr adantlr eqtrd adantrl + c1 id recnd mp3an13 homullid mp1i 3eqtr3d sylan9eqr adantlr eqtrd adantrl wb breq2 coeq1 coeq2 eqeq1d anbi12d rspcev syl12anc r19.29a ) DJUAZJBUBZU CKZXKXKLZCMZNZJAUBZUCKZXPXPLZDMZNZAOUDZBOXJXKOPZNZXONDUEUFZUGUFZXKQRZOPZJ YFUCKZYFYFLZDMZYAYBYGXJXOYEUHPZYBYGUIYDURKZYKSSDUJZYLDOPZYMEDUKTZDULTZYDF @@ -478512,7 +478512,7 @@ Positive operators (cont.) leopsq mpbi wa leopadd mpanl2 mpanr2 cdiv wf cc 2cn hmopf homulcl ax-mp fco hosubcl hosubsub4 mp3an1 hoadd32 mp3an13 oveq1i eqtr4di hoaddsubass ho2times eqtr4d oveq1d hoaddcl mp3an23 3eqtr3d hosubadd4 mpanr1 hoadddi - mpanl1 halfcn cmul 2ne0 recidi homulass mp3an12 homulid2 3eqtr3a oveq2d + mpanl1 halfcn cmul 2ne0 recidi homulass mp3an12 homullid 3eqtr3a oveq2d eqtrd 3eqtr4d hosubdi hocsubdir clo hmoplin hoddi hoid1i hoico2 oveq12d a1i mp3an2 hoico1 jctil syl12anc 3eqtr2d opsqrlem5 breqtrd peano2nn cc0 cr clt 2re 2pos leopmul mpbird sylancl nn1suc ) UAUCZEUDZKLUGUREUDZKLUG @@ -479213,7 +479213,7 @@ Positive operators (cont.) -> ( ( projh ` G ) o. ( projh ` H ) ) = ( projh ` ( G i^i H ) ) ) $= ( cpjh cfv ccom cort chos co cin ch0o chj wceq sylbi chincli ax-mp eqeq2i wss pjfi ccm wbr cmbri fveq2 inss2 choccli chub2i chdmm3i sseqtrri pjscji - sstri coeq2 pjsdii pjss1coi mpbi pjorthcoi oveq12i hoaddid1i 3eqtri inss1 + sstri coeq2 pjsdii pjss1coi mpbi pjorthcoi oveq12i hoaddridi 3eqtri inss1 eqtrdi syl cmcm3i bitri chdmm4i chub1i chdmm2i oveq12d chba df-iop coeq2i chio hoid1i eqtr3i pjtoi hocofi eqtr2i 3eqtr3g ) ABUAUBZAEFZBEFZVTGZGZVTW AAHFZEFZGZGZIJZABKZEFZLIJZVTWAGZWJVSWCWJWGLIVSVTWIABHFZKZMJZEFZNZWCWJNZVS @@ -479700,7 +479700,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) = 0 ) -> ( S ` A ) = 0h ) $= ( chst wcel cch cfv chba csp co cc0 wceq wa cort caddc hstcl syl recnd wb 3adant3 mpbid w3a cno c0v c2 cexp cva choccl sylan2 syl3anc normsq eqcomd - his7 wss ococ eqimss2 jca adantl hstorth mpdan oveq12d cr resqcld addid1d + his7 wss ococ eqimss2 jca adantl hstorth mpdan oveq12d cr resqcld addridd normcl 3eqtrrd hstoc oveq2d eqtrd id sylan9eq 3impa cc sqeq0 hst0h ) BCDZ AEDZABFZGBFZHIZJKZUAZVQUBFZJKZVQUCKZWAWBUDUEIZJKZWCVOVPVTWFVOVPLZVTWEVSJW GWEVQVQAMFZBFZUFIZHIZVSWGWKVQVQHIZVQWIHIZNIZWEJNIWEWGVQGDZWOWIGDZWKWNKABO @@ -482989,7 +482989,7 @@ SUPPLEMENTARY MATERIAL (USERS' MATHBOXES) ( cr wcel wa c2 clt wbr c1 cmin co caddc cmul a1i syl3anc ax-1cn syl adantr wb wceq simpr 2re simpl 1re ltsub1 df-2 eqcomi subaddrii breq1i bitrd mpbid 2cn anim12i an4s wi peano2rem anim1i mulgt1 ex cc recn mulsub breq2d biimpd - jca mulid2i eqcom biimpi mp1i oveq2d mulid1 oveq12d readdcl remulcl syl2anc + jca mullidi eqcom biimpi mp1i oveq2d mulrid oveq12d readdcl remulcl syl2anc adantl ltaddsub2 ltadd1 bicomd sylbird 3syld mpd ) ACDZBCDZEZFAGHZFBGHZEZEZ IAIJKZGHZIBIJKZGHZEZABLKZABMKZGHZWCWFWDWGWNWCWFEZWKWDWGEZWMWRWFWKWCWFUAWRWF FIJKZWJGHZWKWRFCDZWCICDZWFXASXBWRUBNWCWFUCXCWRUDNFAIUEOXAWKSWRWTIWJGFIIULPP @@ -487493,7 +487493,7 @@ its graph has a given second element (that is, function value). $( Multiplying by a positive integer ` M ` yields greater than or equal nonnegative integers. (Contributed by Thierry Arnoux, 13-Dec-2021.) $) nnmulge $p |- ( ( M e. NN /\ N e. NN0 ) -> N <_ ( M x. N ) ) $= - ( cn wcel cn0 wa c1 cmul co simpr nn0cnd mulid2d 1red cr nnre adantr nn0red + ( cn wcel cn0 wa c1 cmul co simpr nn0cnd mullidd 1red cr nnre adantr nn0red cle nn0ge0d wbr nnge1 lemul1ad eqbrtrrd ) ACDZBEDZFZGBHIBABHIRUFBUFBUDUEJZK LUFGABUFMUDANDUEAOPUFBUGQUFBUGSUDGARTUEAUAPUBUC $. @@ -487547,7 +487547,7 @@ its graph has a given second element (that is, function value). xaddeq0 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A +e B ) = 0 <-> A = -e B ) ) $= ( cxr wcel wa cxad cc0 wceq cxne cpnf cmnf syl simplr syl2anc oveq1d eqtr3d - co simpr ex wne cr w3o wi elxr simpll rexrd xnegneg xnegcld xaddid2 xaddcom + co simpr ex wne cr w3o wi elxr simpll rexrd xnegneg xnegcld xaddlid xaddcom xpncan ancoms adantr 3eqtr3d xnegeq wn renepnf mp1i eqnetrd neneqd xaddpnf2 0re stoic1a nne sylib xnegmnf eqtr2di eqtrd renemnf xaddmnf2 xnegpnf sylanb 3jaoian xnegcl ad2antlr xnegid 3eqtrd impbid ) ACDZBCDZEZABFQZGHZABIZHZVSAU @@ -487585,7 +487585,7 @@ its graph has a given second element (that is, function value). xraddge02 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ B -> A <_ ( A +e B ) ) ) $= ( cxr wcel wa cc0 cle wbr cxad co xrleid adantr simpl jctir xle2add mpancom - wi 0xr mpand wb xaddid1 breq1d sylibd ) ACDZBCDZEZFBGHZAFIJZABIJZGHZAUIGHZU + wi 0xr mpand wb xaddrid breq1d sylibd ) ACDZBCDZEZFBGHZAFIJZABIJZGHZAUIGHZU FAAGHZUGUJUDULUEAKLUDFCDZEUFULUGEUJQUFUDUMUDUEMRNAFABOPSUDUJUKTUEUDUHAUIGAU AUBLUC $. @@ -487610,7 +487610,7 @@ its graph has a given second element (that is, function value). xlt2addrd $p |- ( ph -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) ) $= ( cxad co wceq clt cxr wcel ad2antrr cr cv wbr w3a wrex cpnf wa cc0 rexrd - 0xr a1i xaddid1 eqcomd ltpnf simplr breqtrrd 0ltpnf simpr breqtrrid oveq1 + 0xr a1i xaddrid eqcomd ltpnf simplr breqtrrd 0ltpnf simpr breqtrrid oveq1 syl eqeq2d breq1 3anbi12d oveq2 3anbi13d rspc2ev syl113anc wne c1 xnegcld cxne xaddcld cmnf renemnfd cneg cmin wo wn xrnepnf biimpi sylancom orcomd 1xr neneqd pm2.53 sylc 1re rexsub sylancl resubcl eqeltrd rexneg renegcld @@ -488218,7 +488218,7 @@ its graph has a given second element (that is, function value). nnne0d caddc bcp1n nnz adantl npcand oveq1d oveq12d oveq2d eqeq12d imbitrid zcnd 1cnd 3impia 3anidm13 wne crp cn0 cle elfznn0 simpr nnnn0d elfzelz zred wb cz elfzle2 zltlem1 syl2an ltled elfz2nn0 syl3anbrc bcrpcl syl rpcnd cneg - subcld negsubdi2d resubcld recnd addid2d breqtrrd ltsubaddd lt0ne0d negne0d + subcld negsubdi2d resubcld recnd addlidd breqtrrd ltsubaddd lt0ne0d negne0d 0red eqnetrrd divcld rpcnne0d divmul2 syl3anc bccl2 recdivd 3eqtr3d ) ACBDE FZGFHZBIHZJZDBAKFZWNAKFZLFZLFDBBAEFZLFZLFWSWRLFXABLFWQWTXBDLWQWTXBMZWRWSXBN FZMZWOWPXEWOWPWOXEWOWNDUAFZAKFZWSXFXFAEFZLFZNFZMWQXEAWNUBWQXGWRXJXDWQXFBAKW @@ -488888,7 +488888,7 @@ its graph has a given second element (that is, function value). 16-Dec-2021.) $) dp20u $p |- _ A 0 = A $= ( cc0 cdp2 c1 cdc cdiv co caddc df-dp2 cc wcel wne 10nn0 nn0rei recni 0re - wceq 10pos gtneii div0 mp2an oveq2i nn0cni addid1i 3eqtri ) ACDACECFZGHZI + wceq 10pos gtneii div0 mp2an oveq2i nn0cni addridi 3eqtri ) ACDACECFZGHZI HACIHAACJUHCAIUGKLUGCMUHCRUGUGNOPCUGQSTUGUAUBUCAABUDUEUF $. $} @@ -488898,7 +488898,7 @@ its graph has a given second element (that is, function value). 16-Dec-2021.) $) dp20h $p |- _ 0 A = ( A / ; 1 0 ) $= ( cc0 cdp2 c1 cdc cdiv co caddc df-dp2 crp wcel cc ax-mp 10nn0 nn0cni 0re - rpcn 10pos gtneii divcli addid2i eqtri ) CADCAECFZGHZIHUECAJUEAUDAKLAMLBA + rpcn 10pos gtneii divcli addlidi eqtri ) CADCAECFZGHZIHUECAJUEAUDAKLAMLBA RNUDOPCUDQSTUAUBUC $. $} @@ -489295,7 +489295,7 @@ its graph has a given second element (that is, function value). (Contributed by Thierry Arnoux, 16-Dec-2021.) $) 0dp2dp $p |- ( ( 0 . _ A B ) x. ; 1 0 ) = ( A . B ) $= ( cc0 cdp2 cdp co c1 cdc cmul cexp 0p1e1 0z 1z cc wcel wceq ax-mp oveq2i - dpexpp1 10nn0 nn0cni exp0 exp1 3eqtr3ri crp rpdpcl rpcn mulid1 eqtri ) EA + dpexpp1 10nn0 nn0cni exp0 exp1 3eqtr3ri crp rpdpcl rpcn mulrid eqtri ) EA BFGHZIEJZKHZABGHZIKHZUOUOUMELHZKHULUMILHZKHUPUNABEICDMNOUAUQIUOKUMPQZUQIR UMUBUCZUMUDSTURUMULKUSURUMRUTUMUESTUFUOPQZUPUORUOUGQVAABCDUHUOUISUOUJSUK $. @@ -489433,8 +489433,8 @@ its graph has a given second element (that is, function value). oveq2i cc eqeltrri dp2cl 0nn0 nn0cni dpmul100 sq10 eqtr4i dfdec100 ax-1cn mulcomi addcomi eqtr3i eqid addassi 3eqtr4i subcli remulcli readdcli mpbi mp3an dpmul1000 divcan4i cexp 2nn0 nncni sqvali decsuc mvlladdi karatsuba - addid2i 1nn0 subdiri 3eqtrri wceq subsub 3eqtr4ri crp 1re resubcli adddii - sqcli 3decltc ltadd2i dfdec10 mulid2i 3brtr4i eqbrtri posdifi mpbir2an wa + addlidi 1nn0 subdiri 3eqtrri wceq subsub 3eqtr4ri crp 1re resubcli adddii + sqcli 3decltc ltadd2i dfdec10 mullidi 3brtr4i eqbrtri posdifi mpbir2an wa ltsubrp wb decnncl2 nngt0i pm3.2i ltdiv1 gt0ne0ii div23i 3brtr3i divassi elrp ) ABCDVSZVSZVTWAZKLWBZMWBZNWBZWCWAZWDWEWBZWEWBZWEWBZWFWAZUBUCWBZUDWB ZUEWBZUWIWFWAZUWBKLMNVSZVSZVTWAZWCWAZUBUCUDUEVSZVSZVTWAZWGUWFUWMWGWHZUWJU @@ -489521,7 +489521,7 @@ its graph has a given second element (that is, function value). $( Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.) $) 1mhdrd $p |- ( ( 0 . _ 9 9 ) + ( 0 . _ 0 1 ) ) = 1 $= - ( cc0 c9 cdp2 cdp co c1 caddc 0nn0 9nn0 1nn0 cdc eqcomi deceq1i 9cn addid1i + ( cc0 c9 cdp2 cdp co c1 caddc 0nn0 9nn0 1nn0 cdc eqcomi deceq1i 9cn addridi dec0h oveq1i 9p1e10 eqtri decaddc dpadd3 dp20u oveq2i dp0u 3eqtri ) ABBCDEA AFCDEGEFAACZDEFADEFABBAAFFAAHIIHHJJHHBBAFFAKZAABKZBKAAKZFKIIHJUHBBBUHBIPLMU IAFAUIAHPLMBAGEZFGEBFGEUGUJBFGBNOQRSHRTUAUFAFDAHUBUCFJUDUE $. @@ -489581,7 +489581,7 @@ its graph has a given second element (that is, function value). Arnoux, 17-Dec-2016.) $) xmulcand $p |- ( ph -> ( ( C *e A ) = ( C *e B ) <-> A = B ) ) $= ( vx cxmu co wceq c1 cr wcel syl2anc wa oveq2 cxr adantr cv wi cc0 xrecex - wne wrex simprl rexrd xmulcom simprr eqtrd oveq1d xmulass syl3anc xmulid2 + wne wrex simprl rexrd xmulcom simprr eqtrd oveq1d xmulass syl3anc xmullid syl 3eqtr3d eqeq12d imbitrid rexlimddv impbid1 ) ADBJKZDCJKZLZBCLZADIUAZJ KZMLZVDVEUBINADNOZDUCUEVHINUFGHIDUDPVDVFVBJKZVFVCJKZLAVFNOZVHQZQZVEVBVCVF JRVNVJBVKCVNVFDJKZBJKZMBJKZVJBVNVOMBJVNVOVGMVNVFSOZDSOZVOVGLVNVFAVLVHUGUH @@ -489598,7 +489598,7 @@ its graph has a given second element (that is, function value). E! x e. RR* ( B *e x ) = A ) $= ( vy cxr wcel cr w3a cv cxmu co wceq wrex wa wi wral 3adant1 oveq2 eqeq1d c1 cc0 wne wreu ressxr xrecex ssrexv mpsyl simprl simpll xmulcld ad2antll - wss oveq1 simplr rexrd xmulass syl3anc xmulid2 syl 3eqtr3d rspcev syl2anc + wss oveq1 simplr rexrd xmulass syl3anc xmullid syl 3eqtr3d rspcev syl2anc rexlimdvaa 3adant3 mpd eqtr3 simp1 simp3l simp3r xmulcand imbitrid expcom simp2 3expa ralrimivv reu4 sylanbrc ) BEFZCGFZCUAUBZHZCAIZJKZBLZAEMZWDCDI ZJKZBLZNZWBWFLZOZDEPAEPWDAEUCWAWGTLZDEMZWEGEULWAWLDGMZWMUDVSVTWNVRDCUEQWL @@ -489666,7 +489666,7 @@ its graph has a given second element (that is, function value). ( A /e B ) = ( A *e ( 1 /e B ) ) ) $= ( cxr wcel cr cc0 wne w3a cxdiv co c1 cxmu simp2 xmulcom syl2anc wb xdivmul wceq syl112anc eqtrd rexrd simp1 1xr a1i simp3 xdivcld xmulcld xmulass eqid - syl3anc mpbii oveq2d 3eqtrd xmulid1 syl mpbird ) ACDZBEDZBFGZHZABIJAKBIJZLJ + syl3anc mpbii oveq2d 3eqtrd xmulrid syl mpbird ) ACDZBEDZBFGZHZABIJAKBIJZLJ ZRZBVBLJZARZUTVDAKLJZAUTVDVBBLJZAVABLJZLJZVFUTBCDZVBCDZVDVGRUTBUQURUSMZUAZU TAVAUQURUSUBZUTKBKCDZUTUCUDZVLUQURUSUEZUFZUGZBVBNOUTUQVACDZVJVGVIRVNVRVMAVA BUHUJUTVHKALUTVHBVALJZKUTVTVJVHWARVRVMVABNOUTVAVARZWAKRZVAUIUTVOVTURUSWBWCP @@ -490050,7 +490050,7 @@ its graph has a given second element (that is, function value). ltsubposd mpbii ltled syl2anc adantlr sbcie mpd biimpa eqbrtrd nn0p1elfzo syl3anc wrdsymbcl leidd sbceq1d id eqidd s2eqd imbi2d wb eqbrtrrid eqcomd 0red adantl elfzd pfxlen eqtr2d vex dfsbcq bitr3id pfxcl ad2antrl rspcdva - simplr cfzo 2nn0 addid2d eqeltrrd leadd1dd eqbrtrrd nn0sub syl21anc recnd + simplr cfzo 2nn0 addlidd eqeltrrd leadd1dd eqbrtrrd nn0sub syl21anc recnd subsubd 2m1e1 eqtr3d lem1d nn0ge2m1nn0 npcand oveq1 ovex 3imtr4g vtocl3ga w3a 1red readdcld 0p1e1 nn0ge0d le2addd lemul2ad breqtrd eqid pfxlsw2ccat simpll sbceq1a mpbird expr ralrimiva biimtrid nn0ind rspcdv adantllr wrex @@ -491237,7 +491237,7 @@ real number multiplication operation (this has to be defined in the main however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.) $) xrs0 $p |- 0 = ( 0g ` RR*s ) $= ( cc0 cxrs c0g cfv wceq wtru cxr cxad cbs xrsbas a1i cplusg xrsadd wcel 0xr - vx cv co xaddid2 adantl xaddid1 grpidd mptru ) ABCDEFPGHBAGBIDEFJKHBLDEFMKA + vx cv co xaddlid adantl xaddrid grpidd mptru ) ABCDEFPGHBAGBIDEFJKHBLDEFMKA GNFOKPQZGNZAUDHRUDEFUDSTUEUDAHRUDEFUDUATUBUC $. $( The "strictly less than" relation for the extended real structure. @@ -491266,9 +491266,9 @@ real number multiplication operation (this has to be defined in the main ( A ( .g ` RR*s ) B ) = ( A *e B ) ) $= ( cxr wcel cxrs co cxmu wceq cv cc0 c1 oveq1 eqeq12d xrsbas wa cxad simpr oveq1d adantl cr vn vm vx vy cz cmg cfv cneg caddc xrs0 eqid mulg0 xmul02 - eqtr4d cn simpll xrsadd mulgnnp1 syl2anc xaddid2 simpl eqtrd 0p1e1 eqtrdi + eqtr4d cn simpll xrsadd mulgnnp1 syl2anc xaddlid simpl eqtrd 0p1e1 eqtrdi cn0 mulg1 3eqtr4rd wo elnn0 mpjaodan adantr cle wbr nn0ssre ressxr sselid - sylib sstri nn0ge0 ad2antlr 1xr a1i xadddi2r syl221anc 1re rexadd xmulid2 + sylib sstri nn0ge0 ad2antlr 1xr a1i xadddi2r syl221anc 1re rexadd xmullid 0le1 syl oveq2d 3eqtr3d 3eqtr4d exp31 xnegeq cminusg mulgnegnn ancoms cvv cxne xrsex ssidd simp2 simp3 xaddcld mulgnnsubcl 3anidm12 xrsinvgval nnre w3a rexneg nnssre xmulneg1 eqtr3d zindd impcom ) BCDZAUEDABEUFUGZFZABGFZH @@ -491432,7 +491432,7 @@ real number multiplication operation (this has to be defined in the main (Contributed by Thierry Arnoux, 6-Jul-2017.) $) xrge0addgt0 $p |- ( ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) ) /\ 0 < A ) -> 0 < ( A +e B ) ) $= - ( cc0 cpnf cicc co wcel wa clt wbr cxad wceq cxr 0xr xaddid1 simplr simplll + ( cc0 cpnf cicc co wcel wa clt wbr cxad wceq cxr 0xr xaddrid simplr simplll a1i sselid breq2d ax-mp simpr wi iccssxr simpllr xlt2add syl22anc eqbrtrrid mp2and oveq2 adantl syl bitr3d mpbird cle wo iccgelb syl3anc xrleloe biimpa pnfxr syl21anc mpjaodan ) ACDEFZGZBVDGZHZCAIJZHZCBIJZCABKFZIJZCBLZVIVJHZCCC @@ -491450,7 +491450,7 @@ real number multiplication operation (this has to be defined in the main syl oveq1d cicc w3a iccssxr simpl2 rge0ssre xadddir syl3anc simpll1 simpll2 cico clt xaddcld xrge0addgt0 syl21anc xmulpnf1 oveq2 ad2antlr wne ge0xmulcl cmnf simpll3 xrge0neqmnf xaddpnf2 3eqtr4d eqtrd eqtr4d xmul02 oveq1 xmulcld - adantl xaddid2 3eqtr3d 3eqtr2rd cle wo 0xr a1i pnfxr iccgelb xrleloe biimpa + adantl xaddlid 3eqtr3d 3eqtr2rd cle wo 0xr a1i pnfxr iccgelb xrleloe biimpa mpjaodan wb 0lepnf eliccelico mp3an 3anbi3i simp3bi ) ADEUAFZGZBWIGZCWIGZUB ZCDEUJFZGZABHFZCIFZACIFZBCIFZHFZJZCEJZWMWOKZALGZBLGZCMGXAXCWILADEUCZWJWKWLW ONOXCWILBXFWJWKWLWOUDOXCWNMCUEWMWOPOABCUFUGWMXBKZDAUKQZXADAJZXGXHKZWQEWSHFZ @@ -491480,9 +491480,9 @@ real number multiplication operation (this has to be defined in the main ( ( A +e -e B ) +e B ) = A ) $= ( cc0 cpnf co wcel cle wbr wceq cxne wa cxr simpl1 sselid simpr syl2anc syl cxad cmnf wne cicc w3a iccssxr simpl3 eqbrtrrd xgepnf biimpa xnegeq oveq12d - pnfxr xnegid eqtrdi oveq1d oveq2d xaddid2 mp1i 3eqtrd eqtr4d wn xrge0neqmnf + pnfxr xnegid eqtrdi oveq1d oveq2d xaddlid mp1i 3eqtrd eqtr4d wn xrge0neqmnf ax-mp simpl2 xnegcld xnegneg xnegmnf stoic1a neqned xaddass xaddcom mpancom - sylan9req syl222anc xnegcl eqtrd xaddid1 sylan9eqr pm2.61dan ) ACDUAEZFZBVR + sylan9req syl222anc xnegcl eqtrd xaddrid sylan9eqr pm2.61dan ) ACDUAEZFZBVR FZBAGHZUBZBDIZABJZREZBREZAIWBWCKZWFDAWGWFCBRECDREZDWGWECBRWGWEDDJZREZCWGADW DWIRWGALFZDAGHZADIZWGVRLACDUCZVSVTWAWCMNWGBDAGWBWCOZVSVTWAWCUDUEWKWLWMAUFUG PZWGWCWDWIIWOBDUHQUIDLFZWJCIUJDUKVAULUMWGBDCRWOUNWQWHDIWGUJDUOUPUQWPURWBWCU @@ -495059,7 +495059,7 @@ Formula in property (b) of [Lang] p. 32. (Contributed by Thierry w3a eqcomd wb negcld cvv biimpar archirng cplusg ogrpaddlt grplid addcomd syl131anc 1p1e2 eqtrdi eqtr3d mulgdir syl13anc grpinvcl posasymb syl32anc simplrr grpinvinv 3eqtr2rd 2m1e1 eqtr2di negeqd negsubdid simp-4l adantrr - addsubassd 3anassrs negdi sylancl addid1d 3eqtrd wi ovexd pltle sylc tlt2 + addsubassd 3anassrs negdi sylancl addridd 3eqtrd wi ovexd pltle sylc tlt2 mpjaodan carchi syldan r19.29a wss nn0ssz ssrexv mpsyl w3o tlt3 mpjao3dan wo ) AIJUCZEUDZHDUEZICUFZIUWGUGUHUEZHDUEZFUFZUIZEUJUKZIJCUFZJICUFZAUWFUIZ UGULZUJUMUWRHDUEZICUFZIUWRUGUHUEZHDUEZFUFZUIZUWNUNUWQUWTUXCUWQUWSJICAUWSJ @@ -495774,7 +495774,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a slmd0vlid.v $e |- V = ( Base ` W ) $. slmd0vlid.a $e |- .+ = ( +g ` W ) $. slmd0vlid.z $e |- .0. = ( 0g ` W ) $. - $( Left identity law for the zero vector. ( ~ hvaddid2 analog.) + $( Left identity law for the zero vector. ( ~ hvaddlid analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) $) slmd0vlid $p |- ( ( W e. SLMod /\ X e. V ) -> ( .0. .+ X ) = X ) $= @@ -497092,8 +497092,8 @@ then this is also a division ring (see ~ fldgensdrg ). If the base $( ` .r ` is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.) $) resvmulr $p |- ( A e. V -> .x. = ( .r ` H ) ) $= - ( cmulr mulrid cnx csca cfv scandxnmulrndx necomi resvlem ) ABDHECFGIJK - LJHLMNO $. + ( cmulr mulridx cnx csca cfv scandxnmulrndx necomi resvlem ) ABDHECFGIJ + KLJHLMNO $. $( Obsolete proof of ~ resvmulr as of 31-Oct-2024. ` .s ` is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) @@ -497194,7 +497194,7 @@ then this is also a division ring (see ~ fldgensdrg ). If the base ( vx vn crefld carchi wcel cv czrh cfv clt wbr cn wrex cr cofld wral eqid wb c1 co wceq reofld rebase relt isarchiofld ax-mp arch cz nnz cmg cfield cmul crg cdr refld ccrg isfld simplbi drngring mp2b re1r zrhmulg mpan 1re - remulg mpan2 zcn mulid1d 3eqtrd breq2d syl rexbiia sylibr mprgbir ) CDEZA + remulg mpan2 zcn mulridd 3eqtrd breq2d syl rexbiia sylibr mprgbir ) CDEZA FZBFZCGHZHZIJZBKLZAMCNEVNVTAMOQUAAMIBVQCUBVQPZUCUDUEVOMEVOVPIJZBKLVTVOBUF VSWBBKVPKEVPUGEZVSWBQVPUHWCVRVPVOIWCVRVPRCUIHZSZVPRUKSZVPCULEZWCVRWETCUJE ZCUMEZWGUNWHWICUOECUPUQCURUSCWDRVQVPWAWDPUTVAVBWCRMEWEWFTVCRVPVDVEWCVPVPV @@ -497223,7 +497223,7 @@ then this is also a division ring (see ~ fldgensdrg ). If the base cfv vr vw vx vq cslmd ccmn cico csrg cv cicc caddc w3a cmul wral xrge0cmn wa cxrs eqeltri wb ovex resvcmn rge0srg icossicc simplr ge0xmulcl syl2anc simprr simprl xrge0adddi rge0ssre simpll rexadd oveq1d xrge0adddir eqtr3d - mpbi syl3anc 3jca rexmul rexrd iccssxr xmulass xmulid2 syl jca ralrimivva + mpbi syl3anc 3jca rexmul rexrd iccssxr xmulass xmullid syl jca ralrimivva xmul02 rgen2 xrge0base fveq2i eqtr4i resvbas cplusg xrge0plusg ax-xrsvsca cbs resvplusg cvsca ressvsca resvvsca c0g xrge00 resv0g cin csca df-refld crefld oveq1i ressress mp2an eqtri ax-xrssca resssca rebase resvsca incom @@ -499886,11 +499886,11 @@ The sumset operation can be used for both group (additive) operations and ( ( RSpan ` R ) ` ( i .(x) j ) ) ) = ( .r ` S ) ) $= ( wcel cv cfv cnx cop cmulr wss cpr co crsp cmpo cbs cplusg clsm ctp crab cts wn cmpt crn cple wa copab cun cvv wceq clidl fvexi mpoex cc0 cdc eqid - c1 idlsrgstr mulrid csn snsstp3 ssun1 sstri strfv ax-mp cidlsrg idlsrgval - eqtrid fveq2d eqtr4id ) BHMZEFAAENZFNZDUABUBOOZUCZPUDOAQZPUEOBUFOZQZPROWC - QZUGZPUIOEAVTWASZUJFAUHUKULZQPUMOVTWATASWIUNEFUOZQTZUPZROZCROWCUQMWCWNURE - FAAWBABUSJUTZWOVAWCWMRUQVEVEVBVCQAWEWCWJWKWMWMVDVFVGWGVHWHWMWDWFWGVIWHWLV - JVKVLVMVSCWMRVSCBVNOWMIWEBDEFGAHJWEVDKLVOVPVQVR $. + c1 idlsrgstr mulridx csn snsstp3 ssun1 sstri strfv ax-mp idlsrgval eqtrid + cidlsrg fveq2d eqtr4id ) BHMZEFAAENZFNZDUABUBOOZUCZPUDOAQZPUEOBUFOZQZPROW + CQZUGZPUIOEAVTWASZUJFAUHUKULZQPUMOVTWATASWIUNEFUOZQTZUPZROZCROWCUQMWCWNUR + EFAAWBABUSJUTZWOVAWCWMRUQVEVEVBVCQAWEWCWJWKWMWMVDVFVGWGVHWHWMWDWFWGVIWHWL + VJVKVLVMVSCWMRVSCBVPOWMIWEBDEFGAHJWEVDKLVNVOVQVR $. $} ${ @@ -502557,7 +502557,7 @@ such that every prime ideal contains a prime element (this is a cfv mptru eqtri csra clvec clmod cdr cgrp crg w3a c1 wss cnring ax-resscn eqidd cnfldbas sraring mp2an ringgrp ax-mp ccrg cfield refld isfld simpli mpbi drngring simpr1 recnd simpr3 mulcld simpr2 adddid simpl adddird 3jca - mulassd mulid2d jca32 ralrimivvva cbs sseqtri a1i srabase cplusg cnfldadd + mulassd mullidd jca32 ralrimivvva cbs sseqtri a1i srabase cplusg cnfldadd sraaddg cmulr cvsca cnfldmul sravsca cress df-refld srasca rebase replusg rgen csca remulr re1r islmod mpbir3an islvec mpbir2an ) EFUARRZUBGXBUCGZH UDGZXCXBUEGZHUFGZAIZBIZJKZLGZXGXHCIZMKJKXIXGXKJKMKNZDIZXGMKXHJKXMXHJKXIMK @@ -502579,7 +502579,7 @@ such that every prime ideal contains a prime element (this is a clbs df-pr c0g eqidd cnfld0 wss ax-resscn cnfldbas sseqtri sralmod0 mptru eqid wne ax-1cn ax-1ne0 srabase eqtri lindssn mp3an cin cv clmod lveclmod ine0 wb cress csca df-refld srasca cmulr cnfldmul sravsca lspsnel anbi12i - cvsca mp2an reeanv simprl simpll recnd mulid1d eqtrd negeqd simprr simplr + cvsca mp2an reeanv simprl simpll recnd mulridd eqtrd negeqd simprr simplr cneg mulcomd oveq12d cmin eqeltrd subidd negsubdid 3eqtr3d renegcld creq0 neg0 eqtr3d syl2anc mpbird simpld negcon1ad 3eqtr2d rexlimivv 0red oveq1d ex simpr eqeq2d anbi1d rexbidv mul02i eqeq2i rspcedvd impbii eqriv cplusg @@ -503981,7 +503981,7 @@ elements in a zero ring (see ~ 0ringirng ). (Contributed by Thierry wf fvco fzto1stfv1 3eqtrd crg ccrg minmar1cl psgnco syl3anc psgnfzto1st crngring psgninv oveq12d sselid nnnn0d cn0 a1i nn0addcld expcld expaddd eqtrd c2 nncnd oveq2i eqtrdi madjusmdetlem3 madjusmdetlem1 1cnd fz1ssnn - syl22anc negcld 1nn0 mul4d add4d 1p1e2 2nn0 neg1sqe1 mulid1d eqtr3d cpr + syl22anc negcld 1nn0 mul4d add4d 1p1e2 2nn0 neg1sqe1 mulridd eqtr3d cpr cz nn0zd m1expcl2 1neg1t1neg1 ) AONQPURUSEHUTZVAZVJRVBUSZVCURZURZFIUTZV AZUVCURZVDUSZSURZNOQVEURUSMURZJUSVJVFZNOVGUSZVKUSZSURZUVJJUSABCDUVAUVFG UVCJNOQUVBGVHUSURUSZKLMUVBVIURZVLURZNOPQRUPUQUVBUVBUPWCZUVAURZUQWCZUVFU @@ -505426,7 +505426,7 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). cpsmet cfv cmetid simpl cdm metidss dmxpid sseqtrdi wrel xpss sstrdi df-rel cxp cvv sylibr simprl releldm simprr syl3anc fovcdmda syl12anc eqeltrd rnss psmetf crn rnxpid eqeltrrd psmettri2 syl13anc metidv biimpa syl21anc eqtr3d - relelrn jca oveq1d xaddid2 breqtrd oveq2d xaddid1 3eqtrd xrletrd 3eqtr4d wb + relelrn jca oveq1d xaddlid breqtrd oveq2d xaddrid 3eqtrd xrletrd 3eqtr4d wb eqtrd xrletri3 mpbir2and ) CFUBUCGZABCUDUCZHZDEWJHZIZIZADCJZBECJZKZWOWPLHZW PWOLHZWNWOBDCJZWPWNWODACJZMWNWIAFGZDFGZWOXAKWIWMUEZWNWJUFZFAWNWIXEFNXDWIXEF FUNZUFZFWIWJXFNZXEXGNCFUGZWJXFUAOFUHUIOZWNWJUJZWKAXEGWNWIXKXDWIWJUOUOUNZNXK @@ -505453,7 +505453,7 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). ( vx wcel cv wbr wa ssbrd imp brxp sylib co cc0 wceq metidv wb cle syldan cxad simpld cpsmet cfv cmetid cvv cxp wss wrel metidss xpss sstrdi df-rel vy sylibr psmetsym 3expb eqeq1d ancom2s 3bitr4d biimpd impancom mpd simpl - vz simprr simprl simprd psmettri2 syl13anc eqtr3d oveq12d cxr 0xr xaddid1 + vz simprr simprl simprd psmettri2 syl13anc eqtr3d oveq12d cxr 0xr xaddrid mpbid eqtrdi breqtrd psmetge0 syl3anc psmetcl xrletri3 mpbir2and syl12anc ax-mp sylancl mpbird psmet0 anabsan2 impbida iserd ) ABUAUBDZCULVCBAUCUBZ WJWKUDUDUEZUFWKUGWJWKBBUEZWLABUHZBBUIUJWKUKUMWJCEZULEZWKFZGZWOBDZWPBDZGZW @@ -505721,7 +505721,7 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). mpbid simprd sqrtltd rpre recnd 2timesd fveq2d sqrtsqd oveq2d 0le2 sqrtmuld rpge0 2cnd sqrtcld rpcn wne 2ne0 div32d 3eqtr4d eqtr3d 2lt4 wb 4re 0re 4pos ltleii sqrtlt mp4an mpbi 2pos sqrtpclii ltdiv1ii sqrtsq mp2an oveq1i fveq2i - wceq sq2 2div2e1 3eqtr3i breqtri 1red ltmul1d mpbii mulid2d breqtrd eqbrtrd + wceq sq2 2div2e1 3eqtr3i breqtri 1red ltmul1d mpbii mullidd breqtrd eqbrtrd id syl lttrd ex ) BDEZFBGHZIZCDEZFCGHZIZIZAULEZIZBAJKLZMHZCYCMHZIZBJUBLZCJU BLZUCLZNOZAMHYBYFIZYJYCJUBLZYLUCLZNOZAYKYIYKYGYHYKBXNXOXSYAYFUDZUEZYKCYKXQX RXPXSYAYFUFZUAZUEZUGZYKYGYHYPYSYKBYOUHYKCYRUHUIZUJYKYMYKYLYLYKYCYKAYKAXTYAY @@ -507164,7 +507164,7 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). ( wcel cnzr wa cfv c1 cmul co wceq syl eqid cnlm cnrg w3a cz cur simpl3 cabs cmg crg nzrring simpr zrhmulg fveq2d syl2anc simpl1 nmmulg syl3anc ringidcl cnm zlmnm fveq1d simpl2 c0g wne nzrnz zlm1 zlm0 isnzr sylanbrc - nrgring nm1 eqtrd oveq2d 3eqtrd cc zcnd abscl recnd mulid1 3syl ) FUAKZ + nrgring nm1 eqtrd oveq2d 3eqtrd cc zcnd abscl recnd mulrid 3syl ) FUAKZ FUBKZBLKZUCZDUDKZMZDCNZENZDUGNZOPQZWIWFWHDBUENZBUHNZQZENZWIWKENZPQZWJWF BUIKZWEWHWNRWFWCWQWAWBWCWEUFZBUJSZWDWEUKZWQWEMWGWMEBWLWKCDJWLTZWKTZULUM UNWFWAWEWKAKZWNWPRWAWBWCWEUOWTWFWQXCWSABWKGXBURSABWLDEWKFGHIXAUPUQWFWOO @@ -508356,7 +508356,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be = sum_ x e. A B ) $= ( cfv cmul co csu wcel wa cc0 c1 cfn syldan wceq adantr cv cind cc sselda cpr cr pr01ssre wf indf syl2anc ffvelcdmda sselid recnd mulcld cdif simpr - wss ind0 syl3anc oveq1d difssd mul02d fsumss ind1 mulid2d sumeq2dv eqtr3d + wss ind0 syl3anc oveq1d difssd mul02d fsumss ind1 mullidd sumeq2dv eqtr3d eqtrd ) ACBUAZCEUBIIZIZDJKZBLEVLBLCDBLACEVLBGAVICMZVIEMZVLUCMACEVIGUDZAVN NZVKDVPVKVPOPUEZUFVKUGAEVQVIVJAEQMZCEUQZEVQVJUHFGCEQUIUJUKULUMHUNRAVIECUO ZMZNZVLODJKZOWBVKODJWBVRVSWAVKOSAVRWAFTAVSWAGTAWAUPCEQVIURUSUTAWAVNWCOSAV @@ -508379,8 +508379,8 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be ( cmul co csu cc0 wceq wcel adantr sselda cind cfv cin cdif caddc inindif cv c0 a1i cun inundif eqcomi wa c1 cpr cc cr pr01ssre ax-resscn sstri wss wf indf syl2anc ffvelcdmd sselid mulcld fsumsplit inss2 ind1 oveq1d inss1 - syl3anc syldan mulid2d eqtrd sumeq2dv ssdifd ind0 difssd mul02d cfn diffi - syl cuz sumz olcs oveq12d infi fsumcl addid1d 3eqtrd ) ABEUGZCFUAUBUBZUBZ + syl3anc syldan mullidd eqtrd sumeq2dv ssdifd ind0 difssd mul02d cfn diffi + syl cuz sumz olcs oveq12d infi fsumcl addridd 3eqtrd ) ABEUGZCFUAUBUBZUBZ DMNZEOBCUCZWPEOZBCUDZWPEOZUENWQDEOZPUENXAAWQWSWPBEWQWSUCUHQABCUFUIBWQWSUJ ZQAXBBBCUKULUIIAWMBRZUMZWODXDPUNUOZUPWOXEUQUPURUSUTXDFXEWMWNAFXEWNVBZXCAF GRZCFVAZXFHKCFGVCVDSABFWMJTVEVFLVGVHAWRXAWTPUEAWQWPDEAWMWQRZUMZWPUNDMNDXJ @@ -508893,7 +508893,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be ( cun cesum cxad co wcel cvv syl wceq cc0 cdif nfv nfcv difexd c0 disjdif elex cin a1i cv cpnf difssd sselda wa 0e0iccpnf eqeltrdi syldan esumsplit cicc undif2 esumeq1 ax-mp ralrimiva esumeq2d esum0 eqtrd cxr iccssxr wral - oveq2d esumcl syl2anc sselid xaddid1 3eqtr3d ) ABCBUAZLZDEMZBDEMZVPDEMZNO + oveq2d esumcl syl2anc sselid xaddrid 3eqtr3d ) ABCBUAZLZDEMZBDEMZVPDEMZNO ZBCLZDEMZVSABVPDEAEUBZEBUCZEVPUCZABFPZBQPHBFUGRACBGIUDZBVPUHUESABCUFUIJAE UJZVPPZWICPZDTUKUSOZPZAVPCWIACBULUMZAWKUNDTWLKUOUPUQURVRWCSZAVQWBSWOBCUTV QWBDEVAVBUIAWAVSTNOZVSAVTTVSNAVTVPTEMZTAVPDTEWDADTSZEVPAWJWKWRWNKUQVCVDAV @@ -509201,7 +509201,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be ~ esumpr . (Contributed by Thierry Arnoux, 2-Jan-2017.) $) esumpr2 $p |- ( ph -> sum* k e. { A , B } C = ( D +e E ) ) $= ( cxad wceq adantr cpnf cpr cesum co wa csn simpr dfsn2 eqtr2id esumeq1 - preq2 3syl esumsn eqtrd cc0 oveq2 cxr wcel 0xr eleq1 mpbiri xaddid1 syl + preq2 3syl esumsn eqtrd cc0 oveq2 cxr wcel 0xr eleq1 mpbiri xaddrid syl wo cmnf wne pnfxr pnfnemnf neeq1 xaddpnf1 syl2anc 3eqtr4d jaoi syl6 imp id cv simpll wi eqeq2 biimprd cicc eqtr3d oveq2d 3eqtr2d adantlr esumpr pm2.61dane ) ABCUAZDFUBZEGQUCZRBCABCRZUDZWIEEEQUCZWJWLWIBUEZDFUBZEWLWKW @@ -509235,7 +509235,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be vex eqid ax-mp r19.29af ex ssrdv adantr esummono 0ex esumsn breqtrd rabn0 nfn wne necon1bi mpteq12df mpt0 eqtrdi rn0 esumeq1d esumnul 0le0 eqbrtrdi rneqd wral mptexgf rnexg simplll adantlr syl2anc eqeltrd ralrimiva esumcl - 3syl cxr sselid oveq1d xaddid2 3eqtr4d cin esumsplit eqtri elxrge0 sselii + 3syl cxr sselid oveq1d xaddlid 3eqtr4d cin esumsplit eqtri elxrge0 sselii pm2.61dan jca iccssxr xrletri3 mpbird simpl esumeq2d ssrind neqned necomd esum0 incom neneqd nrex mtbir disjsn mpbir eqtr3i sseqtrdi rabnc mpteq12i ss0 rabxm mptun rneqi rnun ) AGDPQZGCUAZDUBZUCZGUVIUKZGCUAZDUBZUCZUDZEBUE @@ -509426,9 +509426,9 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be cnfldbas ressbas2 ax-mp cxr icossxr xrsbas eqtr3i cplusg simprl eleqtrrdi eqid simprr ge0addcl ovex cnfldadd ressplusg oveqi 3eltr3g syl2anc sselid cv simpl simpr rexadd eqcomd xrsadd 3eqtr3g ffund crn sseqtrdi gsumpropd2 - frnd cnfldex 0e0icopnf addid2d jca gsumress xrge0base xrge0plusg ressress - addid1d cin mp2an icossicc dfss mpbi eqtr4i oveq2i eqtr2i iccssxr xaddid2 - incom syl xaddid1 3eqtr4d ) ACEZAFGUBHZBUCZIZJXJKHZBLHMXJKHZBLHJBLHMFGUDH + frnd cnfldex 0e0icopnf addlidd jca gsumress xrge0base xrge0plusg ressress + addridd cin mp2an icossicc dfss mpbi eqtr4i oveq2i eqtr2i iccssxr xaddlid + incom syl xaddrid 3eqtr4d ) ACEZAFGUBHZBUCZIZJXJKHZBLHMXJKHZBLHJBLHMFGUDH ZKHZBLHXLUABXMXNNNNDXKXIBNEAXJCBUEUFXLJXJKUGXLMXJKUGXMUHUIZXNUHUIZOXLXJXQ XRXJPUJZXJXQOXJQPUKULUMZXJPXMJXMVDZUNUOUPZXJUQUJXJXROFGURXJUQXNMXNVDZUSUO UPUTTXLDVNZXQEZUAVNZXQEZIIZYDXJEZYFXJEZYDYFXMVAUIZHZXQEYHYDXQXJXLYEYGVBYB @@ -509613,7 +509613,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be ( c1 cfz co cesum cc0 cle wbr cpnf cxr cvv wcel cn cxad cicc iccssxr wral caddc ovexd cv elfznn wa cico sselid sylan2 ralrimiva nfcv esumcl syl2anc icossicc xrleidd cuz cfv adantr peano2nn nnuz eleqtrdi fzss1 simpr sseldd - wss syl syldan elxrge0 simprbi wi 0xr a1i xle2add syl22anc mp2and xaddid1 + wss syl syldan elxrge0 simprbi wi 0xr a1i xle2add syl22anc mp2and xaddrid 3syl wceq eqcomd cun eluzfz fzsplit esumeq1 nfv clt cin nnre ltp1d fzdisj c0 esumsplit eqtrd 3brtr4d ) AIDJKZBCLZMUAKZWRDIUEKZEJKZBCLZUAKZWRIEJKZBC LZNAWRWRNOZMXBNOZWSXCNOZAWRAMPUBKZQWRMPUCZAWQRSBXISZCWQUDWRXISAIDJUFZAXKC @@ -512393,7 +512393,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cbvrabv eqeq1i ancri sylbir df-rex biimpri syl2an adantlr eqtr4d eqtrdi elrab csiga eximi sylibr adantll r19.29a elpwi sspwuni eluni2 nfre1 exbii nfn neq0 3bitr4i con1i esumeq1d esumnul con3i pm2.61dan simprbi esumeq2dv - notbid rabex esum0 oveq12d vuniex elpw iccssxr xaddid1 3eqtrrd adantrl ex + notbid rabex esum0 oveq12d vuniex elpw iccssxr xaddrid 3eqtrrd adantrl ex sselid 4syl rgen crn reex pwsiga elrnsiga ismeas mp2b ) EFUAZUBGHZUVTIJUC UDZEUEZKEGILZAUFZUGUHUIZBUWEBUFZUJZMZUWEUKZEGZUWEUWGEGZBNZLZUPZAUVTUAZULZ ICUFZHZOIUMZUWBHZCUVTULUWCUXACUVTUWSOIUWBOUWBHZOPHZIOUNUIZOJUNUIZUQUOUXCU @@ -512511,7 +512511,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry wa cmeas crn cuni cdm w3a wo unrab ianor rabbii eqtr4i fveq2i measbasedom cun eqeq1i biimpi simp2 csiga dmmeas unelsiga syl3an1 wss ssun1 a1i simpr measssd measle0 simp3 jca measbase syl cxad measunl simprl simprr oveq12d - ssun2 co cxr 0xr xaddid1 ax-mp eqtrdi impbida bitr3id wb anbi12d 3bitr4d + ssun2 co cxr 0xr xaddrid ax-mp eqtrdi impbida bitr3id wb anbi12d 3bitr4d braew ) DUAUBUCGZAHZCEIZDUDZGZBHZCEIZWLGZUEZABTZHZCEIZDJZKLZWKDJZKLZWODJZ KLZTZWRCEIDMNZACEIDMNZBCEIDMNZTZXBWKWOUMZDJZKLZWQXGXMXAKXLWTDXLWJWNUFZCEI WTWJWNCEUGWSXOCEABUHUIUJUKUNWQXNXGWQXNTZXDXFXPDWLUAJGZWMXCKONXDWQXQXNWIWM @@ -513725,7 +513725,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry 1xr elxrge0 nnex esummono csu oveq2 cico ioossico eqidd fvmptd cseq cli ovexd ax-1cn geo2lim 1re esumcvgsum geoihalfsum eqtrdi breqtrd xlemul2a syl31anc cmul recnd wne 2cn 2ne0 expne0d divrecd rexmul eqtr4d esumeq2d - 1rp esummulc2 cab feqmptd cnveqd funeqd fnrnfv 3eqtr2rd xmulid1 3brtr3d + 1rp esummulc2 cab feqmptd cnveqd funeqd fnrnfv 3eqtr2rd xmulrid 3brtr3d esumc xleadd2a eqbrtrd rpred anasss exlimdv xralrple xrge0nre pm2.61dan pnfge breqtrrd ) AHCFUJZBUKZULPZBHCUNZFUJZWUCUMQZAWUDRZHUOUAUPZUUAZUAUQ ZWUGAWUKWUDAHUOURQZWUKAHUSURQZUSUOUUCQWULMUOUSUUDUUEHUSUOUUFUTHUOUAUUGV @@ -513902,7 +513902,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry baselcarsg $p |- ( ph -> O e. ( toCaraSiga ` M ) ) $= ( ve cfv wcel wss cxad co wceq wa cc0 sylib fveq2d c0 adantr ccarsg cin cv cdif cpw wral ssidd elpwi adantl df-ss ssdif0 eqtrd oveq12d cxr cpnf - cicc iccssxr wf simpr ffvelcdmd sselid xaddid1 syl ralrimiva jca mpbird + cicc iccssxr wf simpr ffvelcdmd sselid xaddrid syl ralrimiva jca mpbird elcarsg ) ACBUAIJCCKZHUCZCUBZBIZVICUDZBIZLMZVIBIZNZHCUEZUFZOAVHVRACUGAV PHVQAVIVQJZOZVNVOPLMZVOVTVKVOVMPLVTVJVIBVTVICKZVJVINVSWBAVICUHUIZVICUJQ RVTVMSBIZPVTVLSBVTWBVLSNWCVICUKQRAWDPNVSGTULUMVTVOUNJWAVONVTPUOUPMZUNVO @@ -513913,7 +513913,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry 0elcarsg $p |- ( ph -> (/) e. ( toCaraSiga ` M ) ) $= ( ve c0 ccarsg cfv wcel cxad co wceq a1i cc0 fveq2i cxr cpnf wss cv cin cdif cpw wral 0ss wa eqtrid dif0 oveq12d adantr cicc iccssxr ffvelcdmda - in0 sselid xaddid2 syl eqtrd ralrimiva elcarsg mpbir2and ) AIBJKLICUAZH + in0 sselid xaddlid syl eqtrd ralrimiva elcarsg mpbir2and ) AIBJKLICUAZH UBZIUCZBKZVEIUDZBKZMNZVEBKZOZHCUEZUFVDACUGPAVLHVMAVEVMLZUHZVJQVKMNZVKAV JVPOVNAVGQVIVKMAVGIBKQVFIBVEUPRGUIVIVKOAVHVEBVEUJRPUKULVOVKSLVPVKOVOQTU MNZSVKQTUNAVMVQVEBFUOUQVKURUSUTVAAIHBCDEFVBVC $. @@ -514221,7 +514221,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cfz xnegneg xnegmnf ccarsg simpll fz1ssnn sselda fzfi mptfi fiunelcarsg rnfi eqeltrd elcarsg mpbid simprd ineq1 difeq1 oveq12d eqeq12d xaddpnf1 rspcv 3eqtr3d neneqd pm2.65da neqned xaddass xnegid oveq1d ineq2d mptss - syl222anc xaddid1 disjrnmpt carsgclctunlem1 ineq2 elexi inss2 sseq2 ss0 + syl222anc xaddrid disjrnmpt carsgclctunlem1 ineq2 elexi inss2 sseq2 ss0 rnss mpbii 3eqtr3rd iunss1 mp1i sscond xleneg syl21anc xleadd2a xrletrd biimpa esumgect eqbrtrrid xleadd1a xrge0npcan syl3anc ) AFESDUCZUDZGUEZ FUWSUFZGUEZUGUHZFGUEZUXCUOZUGUHZUXCUGUHZUXEUIAUXAUJTUXGUJTUXCUJTZUXAUXG @@ -514286,7 +514286,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry ( vk wcel c0 cn vf vn ve vg vz cuni cin cfv cdif cxad co cle wbr cpnf wceq wa cxr cc0 cicc iccssxr cpw elpwincl1 ffvelcdmd sselid elpwdifcl xaddcld adantr pnfge syl simpr breqtrrd wne clt wn uni0 eqtrdi ineq2d - unieq in0 fveq2d difeq2d dif0 oveq12d adantl oveq1d xaddid2 3eqtrd wb + unieq in0 fveq2d difeq2d dif0 oveq12d adantl oveq1d xaddlid 3eqtrd wb eqeltrd xeqlelt syl2anc mpbid simpld adantlr wfo csdm cdom wex ccarsg cv wss cvv fvex ssex 0sdomg biimpar com nnenom ensymi domentr sylancl 3syl cen ad2antrr fodomr cfzo ciun fveq2 iundisj crn wfn fofn fniunfv @@ -515327,7 +515327,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cmnmnd sitg0 eqeltrid xmetf sylib xmetge0 syl3anc elxrge0 eqsstrd xmstopn rneq methaus mndidcl eqtrdi adantl eqeltrrd ax-xrsvsca ressvsca ge0xaddcl cmopn rgen2a xaddf mpbi reseq1i resmpo fnmpo 0le0 pnfge elicc1 mpbir3an - w3a pnfxr xaddid1 xrge0le comnd xrge0omnd xmstri2 syl13anc ofrfval mpbird + w3a pnfxr xaddrid xrge0le comnd xrge0omnd xmstri2 syl13anc ofrfval mpbird cofr sitgle xrge0plusg sitgadd breqtrd oveq12d 3brtr4d ispsmet mp2b ) ACB UFIZUDZUXBUEZJCBUGIZKZUAUHZUXFUXDIZLMZUXFUBUHZUXDIZUCUHZUXFUXDIZUXKUXIUXD IZNIZUIUJZUCUXBUKUBUXBUKZOZUAUXBUKZOZUXDUXBULPQZAUXEUXRAUXCLUMUTIZUXDKUYA @@ -516335,7 +516335,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cbits c1 cc0 cif csn cxp ciun c2nd c1st cdm cnvimass cpr cmap wf wf1o wa eulerpartgbij f1of ax-mp ffvelcdmi elin simpld elmapi fdm sseqtrid sylib 3syl sselda eulerpartlemgvv oveq1d syldan sumeq2dv weq 2rexbidv - crab eqeq2 elrab simprbi iftrued elrabi nncnd mulid2d sumeq2i cres id + crab eqeq2 elrab simprbi iftrued elrabi nncnd mullidd sumeq2i cres id eqtrd cfn cen wss adantl syl adantr ssfi syl2anc cvv wral sylancl cn0 inss2 simpr sselid cc mulcld eqtr3d inss1 vex oveq12d eulerpartlemsv2 wn sseli wbr eulerpartlemgf 1nn eqeltrdi wb wfn elpreima mpbir2and ex @@ -517912,8 +517912,8 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry elssuni sylib cxad cin disjdif itgmunTMP c1 csn cxp itgmresTMP unveldom cmul indval2 eqtrd reseq1d resundir ssid xpssres ax-mp eqeq1i xpdisjres incom mpbi uneq12i un0 3eqtri eqtrdi itgmeq2dTMP 3eqtr3rd cr probvalrnd - inidm 1re itgmcst recnd mulid2 3eqtrd uncom eqtri csiga crn domprobsiga - cc difelsiga syl3anc 0re mul02 oveq12d cxr rexrd xaddid1 3eqtr2d ) ABEJ + inidm 1re itgmcst recnd mullid 3eqtrd uncom eqtri csiga crn domprobsiga + cc difelsiga syl3anc 0re mul02 oveq12d cxr rexrd xaddrid 3eqtr2d ) ABEJ ZDJCUEZUAZXHCKBXJBUBZLZXHCKZBCJZACDXHFGAXHCUCJZMBXJUFJZJZXOMZACUGMZBXIM ZXRFHBCUHUDAXHXQXOABEXPIUIZUJUKULAXLXJXHCABXJUMZXLXJNAXTYBHBXIUPOZBXJUN UQUOAXMBXHCKZXKXHCKZURPXNQURPZXNABXKXHCBXKUSZRNZABXJUTZSVAAYDXNYEQURAYD @@ -517978,7 +517978,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry sylibr ralrimiva crrv cind cprb orvcgteel indrrv fveq1d eleq1d mpbird eqbrtrd syl2anc rrvfn csiga domprobsiga elex uniexb sylib ofcfn inidm crn 3syl c1 cmpt wss wceq elssuni syl indval eqtrd 1re ifclda fvmpt2d - 0re ofcval ovif rpcnd mulid2d mul02d ifeq12d eqtrid eqidd ofrfval ) A + 0re ofcval ovif rpcnd mullidd mul02d ifeq12d eqtrid eqidd ofrfval ) A FBOUAZUBUCZEUDZBPUEUCZFOUFUGGUHZXSQZBRUIZYBFUDZOUGZGCUJZUKZULAYFGYHAY BYHQZUMZYDYEBXRUGZBRUIZYEOYJYCYKBRAGBCXRUNFHJKUOUPYJYKBYEOUGZUMZYKUQZ RYEOUGZUMZURZYLYEOUGZYJYKYOURZYRYTYJYKUSUTYJYKYNYOYQYKYKYKUMYJYNYKVAY @@ -518247,7 +518247,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry vy exptval coinfliplem itgmeq123dTMP eqtri itgmdiv3TMP cv prex itgmcntTMP cdm cesum oveq1i cxad fveq2 adantl cpnf cicc cop fveq1i wne 1ex fvpr1 cxr cle wbr cr ressxr 1re sselii 0le1 elxrge0 mpbir2an eqeltri c0ex fvpr2 0xr - 0le0 esumpr oveq12i xaddid1 3eqtri 2re 2ne0 rexdiv mp3an ) DAUAJZJZCBUBZD + 0le0 esumpr oveq12i xaddrid 3eqtri 2re 2ne0 rexdiv mp3an ) DAUAJZJZCBUBZD UCXCUDUEZKLUFMZUGZXCDXDUGZKLMZNKUHMZXBAVEUIZDAUGZXFCOPZXBXKQEXLAXADAUJPXL ABCDEFGHIUKRXLXAULDAUMJPXLABCDEFGHIUNRUQSXLXKXFQEXLXJXCXEDDAXJXCQXLABCDEF GHIUORXLDULAXEQXLABCDEFGHIURRUSSUTXCKDXDVAXHXCUPVBZDJZUPVFZKLMNKLMZXIXGXO @@ -519530,7 +519530,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry eqnetri dividi 3eqtri wn ballotlem8 cun rabxm fveq2i c0 sstri rabnc eqtri hashun elpw2 mpbir ballotlem2 wi nfrab1 dfss2f ballotlem4 imdistani rabid cin eldif 3imtr4i simprbi sylanbrc oveq2i divassi 2timesi 3eqtr2i nncni - 2cn mpgbir rabss2 3eqtr3i 3eqtr4ri mulcli addsub4i subidi addid1i oveq12i + 2cn mpgbir rabss2 3eqtr3i 3eqtr4ri mulcli addsub4i subidi addridi oveq12i eqssi subcli 3eqtr3ri ) GBUDZUELGUFZUGUDZLUGUDZUHUIZUJUIZUEUTKJKUKUIZUHUI ZULUIZUJUIZJKUJUIZUUSUHUIZUUMUUPUUOUJUIZUUPUHUIZUUPUUPUHUIZUUQUJUIZUURUUM GUGUDZUUPUHUIZUVFGLUMZUUMUVJUNZGUOEVNMVNZHUDUDUPUQEUEUUSURUIZUSZMLVALSUVO @@ -519640,7 +519640,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry mul0ord wn cc adantr gt0ne0d mulne0bad neneqd pm2.21dd cxr cle 0red simplll ltled simplr prodgt0 syl12anc syl2anc 1t1e1 eqtr2di renegcld lt0neg1d mpbid sgnp wb mul2negd breqtrrd syl22anc mpbird sgnn neg1mulneg1e1 sgn3da lt0ne0d - mulcomd breq1d mul2lt0rgt0 neg1cn mulid2i mul2lt0rlt0 mulid1i ) ACDZBCDZEZA + mulcomd breq1d mul2lt0rgt0 neg1cn mullidi mul2lt0rlt0 mulridi ) ACDZBCDZEZA BFGZHIZAHIZBHIZFGZJKXGJZLXGJZLUAZXGJZXCXBXCABUBMXDKXGUCXDLXGUCXDXJXGUCXBXCK JZEZAKJZXHBKJZXMXNEXGKXFFGZKXNXGXPJXMXNXEKXFFXNXEKHIZKAKHUDUJUEUFUGXAXPKJWT XLXNXAXFXAXFBUHNUIUKULXMXOEXGXEKFGZKXOXGXRJXMXOXFKXEFXOXFXQKBKHUDUJUEOUGWTX @@ -519809,7 +519809,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry ccatmulgnn0dir $p |- ( ph -> ( A ++ B ) = C ) $= ( cc0 chash co cfzo wcel wceq vi cfv caddc cmin cif cmpt cconcat cmul csn cv c1 cxp fveq2i cfn fzofi snfi hashxp mp2an cn0 hashfzo0 hashsng oveq12d - eqtri syl eqtrid nn0cnd mulid1d eqtrd oveq2d simpll simpr eleqtrd fconstg + eqtri syl eqtrid nn0cnd mulridd eqtrd oveq2d simpll simpr eleqtrd fconstg wa wf a1i feq1d mpbird fvconst sylan syl2anc wn cz eqeltrd nn0zd fzocatel simplr syl22anc ifeqda mpteq12dva ovex snex xpex eqeltri ccatfval 3eqtr4g cvv fconstmpt ) AUAOBPUBZCPUBZUCQZRQZUAUJZOWSRQZSZXCBUBZXCWSUDQZCUBZUEZUF @@ -519837,7 +519837,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry = ( ( F oFC R K ) ++ ( G oFC R K ) ) ) $= ( cconcat co cc0 chash cfv cfzo wcel cmul wceq syl csn cxp cof cofc cword wf fconst6g iswrdi 3syl cfn fzofi snfi hashxp mp2an c1 cn0 lencl hashfzo0 - hashsng oveq12d nn0cnd mulid1d eqtrd eqtr2id ofccat cvv ccatcl wrdf ovexd + hashsng oveq12d nn0cnd mulridd eqtrd eqtr2id ofccat cvv ccatcl wrdf ovexd syl2anc ofcof caddc ccatlen oveq2d xpeq1d ccatmulgnn0dir 3eqtr4a 3eqtr4d eqid ) AEFKLZMENOZPLZGUAZUBZMFNOZPLZWCUBZKLZBUCZLZEWDWILZFWGWILZKLVTGBUDZ LZEGWMLZFGWMLZKLABCDEFWDWGHIAGDQZWBDWDUFWDDUEZQJWBGDUGDWAWDUHUIAWQWFDWGUF @@ -519906,7 +519906,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cmin mp2an simprl simprr readdcld remulcld 0re eqid coef2 sylancl feqmptd plymul cvv cnex plyf syl ax-mp mulcomd caofcom fveq2d fveq1d adantr simpr coemul syl3anc elfznn0 coeidp oveq1d ovif eqtrdi sumeq2dv wb velsn bicomi - ad2antrr fznn0sub ffvelcdmd recnd mulid2d mul02d ifbieq12d eqeq2 oveq2 cz + ad2antrr fznn0sub ffvelcdmd recnd mullidd mul02d ifbieq12d eqeq2 oveq2 cz cfz 0z fzsn wn elsni ax-1ne0 nesymi eqeq1 mtbiri notbii biimpi sumeq12rdv iffalse 3syl cuz cfn wo snfi olci sumz simpll wne neqned elnnne0 sylanbrc cn wral cle wbr 1nn0 nnnn0d nnge1d elfz2nn0 syl3anbrc snssd sylbi nnm1nn0 @@ -519990,7 +519990,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry sylancr zsubcld cxpexpzd oveq2d expnegd zcnd negsubdi2d breqtrd eqbrtrd nn0cnd nn0red 3eqtr2d expsubd clt elfzolt2 difrp biimpa syl21anc cxplim eqbrtrrd rlimmul mul01d fsumrlim wo olcd sumz fveq2 oveq2 oveq12d sumsn - oveq1d syl2anc divcan4d rlimadd addid2d eqtr4di ) ELUAUBMZEFNUCOZAPQDUD + oveq1d syl2anc divcan4d rlimadd addlidd eqtr4di ) ELUAUBMZEFNUCOZAPQDUD OZKUEZCUBZAUEZUVIUFOZUGOZUVKDUFOZNOZKUHZDUIZUVOKUHZUJOZRZBUKUVFUVGAPQDU LOZUVMKUHZUVNNOZRUVTUVFASPUWBUVNNPEFTTUVFSSELEUMUNUVNTMZAPUOFPUPUVFUWDA PUVKDUFUQZURAPUVNFTJVCUSSTMUVFUTVAPTMUVFPLVBVDVEVAPSVFZSPVNPVGPLSVDVHVI @@ -520238,7 +520238,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cc mp2an sselid mul01d eqtrd mtbiri 2falsed simplr tpcomb eleqtrdi neqned jca cdif eldifsn neg1ne0 ax-1ne0 diftpsn3 eleq2i bitr3i sylib 0le1 lenlti cle 1re mpbi neg1mulneg1e1 breq1i mtbir 2false oveq2 mpbiri adantl neg1rr - bibi12d neg1lt0 0lt1 lttri gtneii mulid1i 2th elpri mpjaodan adantr neeq2 + bibi12d neg1lt0 0lt1 lttri gtneii mulridi 2th elpri mpjaodan adantr neeq2 eqbrtri oveq1 mpbird necomi mulcomi wo ad2antrr ifbothda bitrd ) CIUFZIUA ZJZDYCKIUBZJZLZCDAMZCNDKOZCDUCZCNZCDPMZKQRZYHYIYKCYECYFJYGYIYKOYDYFCYDYCU DZIUDZUGYFYCIUEYOYPYFYCKIUHYCKIUIUJUKULACDEFGUMUNUOYJCCNZYNSDCNZYNSZYLYNS @@ -520777,7 +520777,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry c0 cfzo wf eldifad signstf wrdf 3syl cn eldifsn sylib lennncl fzo0end signstlen syl oveq2d eleqtrrd ffvelcdmd remulcld simpr eqtri eqeltrid oveq1i fveq2i mulcomd breqtrrd breqtrdi ltnsymd iffalsed cn0 signsvvf - recnd a1i nn0cnd addid1d 3eqtrd ) AUGBCUHUIZUJUKZULZKMUMZJMUMZJUNUMZU + recnd a1i nn0cnd addridd 3eqtrd ) AUGBCUHUIZUJUKZULZKMUMZJMUMZJUNUMZU OUPUIZJEUMZUMZBUHUIZUGUJUKZUOUGUQZURUIZXMUGURUIXMAXLYAUSXJAXLJBUTVAUI ZMUMZYAAKYBMUCVBAJVCVDZVNVEZVFVGZUGJUMUGVHBVCVGZYCYAUSUAUBUDDEFGHIJBM NOPQRSTVIVJVKVLXKXTUGXMURXKXSUOUGXKUGXRXKVMXKXQBXKUGXPUNUMZVOUIZVCXOX @@ -520889,7 +520889,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cmul clt wbr cif caddc wceq 0re signsvfn mpan2 ltnri cneg cpr cc neg1cn wss ax-1cn prssi mp2an cfzo cn simpl eldifsn sylib lennncl fzo0end 3syl signstfvcl mpdan sselid mul01d breq1d mtbiri iffalsed oveq2d cn0 wf a1i - signsvvf eldifad ffvelcdmd nn0cnd addid1d 3eqtrd ) GUAUBZUCUDZUEPZQGRQU + signsvvf eldifad ffvelcdmd nn0cnd addridd 3eqtrd ) GUAUBZUCUDZUEPZQGRQU FZUGZGQUHUISHRZGHRZGUJRZTUKSZGBRRZQULSZQUMUNZTQUOZUPSZXAQUPSXAWSQUAPWTX HUQURABCDEFGQHIJKLMNOUSUTWSXGQXAUPWSXFTQWSXFQQUMUNQURVAWSXEQQUMWSXDWSTV BZTVCZVDXDXIVDPTVDPXJVDVFVEVGXITVDVHVIWSXCQXBVJSPZXDXJPWSGWOPGUCUFUGZXB @@ -521712,10 +521712,10 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry adantr vz cz cif cioo ci c2 cpi cv citg wral oveq1 oveq2d ax-resscn sstri ioossre sseli mul02d ax-icn 2cn picn mulcli mul01i ef0 sylan9eq ralrimiva eqtrdi itgeq2 syl cdm ioombl 0re 1re ioovolcl mp2an ax-1cn itgconst mp3an - cvol cmin cle wbr 0le1 volioo subid1i oveq2i mulid1i 3eqtri adantl eqcomd + cvol cmin cle wbr 0le1 volioo subid1i oveq2i mulridi 3eqtri adantl eqcomd eqtri wn cdv cmnf cpnf ioomax eqcomi 0red 1red wss sselda 2cnd simpl zcnd simpr efcld syldan ine0 pipos gtneii mulne0i neqned mulne0d divcld fmpttd - wne ccncf reelprrecn cnelprrecn dvmptid dvmptcmul mulid1d mpteq2dva eqtrd + wne ccncf reelprrecn cnelprrecn dvmptid dvmptcmul mulridd mpteq2dva eqtrd 2ne0 cpr dvef eff feqmptd 3eqtr3a dvmptdivc fveq2 oveq1d dvmptco divcan1d wf efcn cres mp1i eqeltrrd fvmptd resmpt eqid mulc1cncf rescncf cncfmpt1f wi mpd eqeltrd ftc2re fveq1d cbvmptv fvmpt2d mulassd sylan2 ef2kpi sselid @@ -521775,10 +521775,10 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry fzss1 cin c0 wceq nnnn0d nn0addcld nn0red ltp1d fzdisj syl cun zaddcld cr cle nnred nn0addge2 syl2anc elfzle2 leadd2dd elfzd fzsplit fzfid fz2ssnn0 adantl sseldd cmin eleq1d simplr an32s ralrimiva nnsscn nn0cnd negsubdi2d - wral eluzmn eqeltrrd rspcdva fsumsplit zcnd addid2d oveq1d eqcomd sumeq1d + wral eluzmn eqeltrrd rspcdva fsumsplit zcnd addlidd oveq1d eqcomd sumeq1d syl21anc sumeq2dv cfn fzfi sumz olcs ax-mp eqtrdi oveq12d elfzuz3 eluzadd zsscn addcomd fveq2d fsumcl 3eqtrrd cfzo zred letrd 3eqtr4d eqtrd fsumcom - fzval3 3eltr3d fzss2 eleqtrd addid1d ineq2d fzodisj peano2zd lep1d uneq2d + fzval3 3eltr3d fzss2 eleqtrd addridd ineq2d fzodisj peano2zd lep1d uneq2d nn0ge0d fzosplit simpl adantrl fz0ssnn0 simprl anass1rs anasss ancom2s fzofi ) AUAHUBOZPGUBOZBDQZEQUUTPGHROZUBOZCFQZEQUVAUUTBEQDQUVDUUTCEQFQAUUT UVBUVEEAEUCZUUTSZUDZUVBPUVFROZGUVFROZUBOZCFQZUVEUVHBCDFUVFPGUVHUVFUAHAUVG @@ -522024,7 +522024,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry csu crab c0 reprval wn wral wa zred nn0red cfn fzofi wss sstrd ad2antrr nnssre nnex ssexd elexi simpr elmapg biimpa syl21anc ffvelcdmd fsumrecl wf sseldd clt wbr cle chash cmul cc ax-1cn fsumconst mp2an hashcl ax-mp - cn0 nn0cni mulid1i eqtri hashfzo0 syl eqtrid 1red nnge1 fsumle eqbrtrrd + cn0 nn0cni mulridi eqtri hashfzo0 syl eqtrid 1red nnge1 fsumle eqbrtrrd ltletrd ltned necomd neneqd ralrimiva rabeq0 sylibr eqtrd ) ABDCUAKLUBC UCLZIUDZJUDZKZIUFZDMZJBXBUELZUGZUHABCDIJEFGUIAXGUJZJXHUKXIUHMAXJJXHAXDX HNZULZXFDXLDXFXLDXFADONXKADFUMPZXLDCXFXMACONXKACGUNPXLXBXEIXBUONZXLUBCU @@ -522050,7 +522050,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry csu cv cfn cc wceq reprfi inss2 a1i reprss ssfid 1cnd fsumconst syl2anc cif wral cuz wo ralrimivw olcd sumss2 syl21anc wa crn wb reprinrn incom wi oveq1i eleq2i bibi1i imbi2i mpbi baibd ifbid cvv nnex r19.21bi fzofi - cz cn0 simpr reprf prodindf eqtr4d sumeq2dv eqtrd hashcl nn0cnd mulid1d + cz cn0 simpr reprf prodindf eqtr4d sumeq2dv eqtrd hashcl nn0cnd mulridd fssd syl 3eqtr3rd ) ABCUAZEDUBMZNZOGUHZXBUCMZOUDNZCEXANZPDUENZFUIGUIZMB QUFMMMFUGZGUHZXDAXBUJRZOUKRZXCXEULAXFXBACDELIJKUMZACWTDELIJWTCSABCUNUOU PZUQZAURZXBOGUSUTAXCXFXHXBRZOPVAZGUHZXJAXBXFSXLGXBVBXFPVCMSZXFUJRZVDXCX @@ -522535,7 +522535,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cv cexp crepr wa cle wbr cr wss nn0ssre a1i sselda leid syl caddc breq1 oveq2 prodeq1d oveq1 oveq2d fveq2 oveqd oveq1d adantr sumeq12dv eqeq12d wi imbi12d csn c0 cc 0nn0 cfn cif fz1ssnn 0zd repr0 eqid iftruei eqtrdi - cn snfi eqeltrdi fzo0 prodeq1i prod0 exp0 oveq12d ax-1cn mulid1i fsumcl + cn snfi eqeltrdi fzo0 prodeq1i prod0 exp0 oveq12d ax-1cn mulridi fsumcl eqtri simpl sumsn sylancr sumeq1d cvv 0ex mulcld fveq1 fveq2d ralrimivw prodeq2d 3eqtr2d eqtrd 3eqtrd nn0cnd mul02d 3eqtr4rd a1d simpll cbvsumv fz0sn simplr eqeq2i fveq1d sumeq2dv cbvprodv prodeq2i fveq12d prodeq2dv @@ -522739,7 +522739,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry negsubd cvol ioombl sumex adantllr subcld an32s anasss fvex cicc ioossicc cvv cdm ccncf 0red unitsscn syl3anc itgfsum simprd cif oveq2 nn0zd eqtr4d 1red sumeq1d ssidd cncfmptc cncfmptid efmul2picn cniccibl iblmulc2 simpld - mulcncf iblss mulid1d mul01d ifeq3da velsn subeq0ad bitr4id ifbid zsubcld + mulcncf iblss mulridd mul01d ifeq3da velsn subeq0ad bitr4id ifbid zsubcld csn itgexpif syl 1cnd 0cnd ifcld 3eqtr4rd wral cuz wo 0zd zmulcld nn0ge0d cle wbr nnmulge elfzd snssd syldan ralrimiva olcd sumss2 syl21anc 3eqtr2d sumsn itgmulc2 reprfz1 3eqtr4d 3eqtrrd ) ABLMUANZLCUBNZBUCZFUCZDUDZEUENUD @@ -522988,7 +522988,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cz recnd fsumcl eqcomi fveq2 cdc c7 cexp 10nn0 7nn0 nn0expcli nn0rei cdiv w3a 3pm3.2i 1lt10 ltexp2a cc wne 10pos 4z mp3an oveq2i 4nn0 1rp oveq1i c5 nn0zi dpexpp1 rpdp2cl 6nn0 2nn0 deccl cle eqtr3i expgt0 ltleii lttrd 2prm - 3rp addid1i eqid dpadd2 chtvalz inss2 sumeq2dv eqtr4d infi inss1 sstri c0 + 3rp addridi eqid dpadd2 chtvalz inss2 sumeq2dv eqtr4d infi inss1 sstri c0 inindif inundif fsumsplit eqtr2d oveq2d breq12d ax-ros336 rspcdva eqbrtrd mvrladdd wral log2le1 0z 3z 3pos numexp0 recni gtneii expm1 4m1e3 divrec2 nn0cni 3eqtr3ri 3brtr4i dp0h breqtrri 4p1e5 5p1e6 6p1e7 3eqtrri breqtrrdi @@ -523090,11 +523090,11 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry wb mpbird eqbrtrd ovex ccnfld ctopn ctx ccn addcn difss cncfmptid divcncf ssid cmnf cioc ax-1 jca ellogdm sylibr cncfss relogcn eqeltrrdi cncfmpt2f wi mp2an rereccld rpge0 halfre renegcli resubcli relogcl cncfcdm syl21anc - rpre biimpar eqeltrd cxpadd syl211anc mulid2d negsubd cxpneg cxp1d eqtr2d + rpre biimpar eqeltrd cxpadd syl211anc mullidd negsubd cxpneg cxp1d eqtr2d mulcomd 3eqtr4rd mul32d oveq1d cicc ioossicc fct2relem sstrd sselda ovexd - adddird 0red rpcxpcl rpge0d mulid2i reefcld eliooord simpld ltled reeflog + adddird 0red rpcxpcl rpge0d mullidi reefcld eliooord simpld ltled reeflog 2cn letrd efle sylancr eqbrtrid lemuldivd mpbid divrec2d mulneg1d subnegd - 2rp addid2d 3eqtrd leaddsub lemul1ad mul02d fdvnegge 3brtr3d ) ACHIHUFZUB + 2rp addlidd 3eqtrd leaddsub lemul1ad mul02d fdvnegge 3brtr3d ) ACHIHUFZUB JZUXRUCJZUDKZUEZJBUYBJCUBJZCUCJZUDKZBUBJZBUCJZUDKZLAUABCMUGIUYBMUGURKIUHU IZDEAHIUYANAUXRIOZUJZUXSUXTUYKUXRAUYJUKZULUYKUXTUYKUXRUYLUMUNUYJUXTMUOZAU YJUXTIOUYMUXRUPUXTUQUSUTZVAVBANUYBVJKZHIPUXRUDKZUXRPQUDKZVCZVDKZRKZUYRUXR @@ -523158,10 +523158,10 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry 1lt10 ltexp2a ltmul1i recni oveq2i eqtr3i expgt0 decltc 6nn nngt0i 6nn0 2rp cz c6 breqtri simpli 4pos ltmul2i 4cn ltdiv1i dpexpp1 3re nn0re adantr 0lt1 1lt9 expge1 simpr ltletrd elrpd relogcld rpge0d resqrtcld sqrtgt0d redivcld - gtned sqrtpclii sqrtgt0ii nn0ssq 8pos rpdpcl ce 2cn mulid2i lemul1i lt2addi + gtned sqrtpclii sqrtgt0ii nn0ssq 8pos rpdpcl ce 2cn mullidi lemul1i lt2addi 2pos 3ne0 logmul2 3t3e9 9lt10 ltlei decltdi lemul2i eqeltrri relogef rpefcl letri 3brtr4i logdivsqrle lemul2ad 3lt10 7p1e8 dplti numexp0 loggt0b expmul - 0z divgt0i 7t2e14 sqrtsq 1lt2 sqrtlt ltdiv2 declt 2exp4 mulid1i 7lt10 2lt10 + 0z divgt0i 7t2e14 sqrtsq 1lt2 sqrtlt ltdiv2 declt 2exp4 mulridi 7lt10 2lt10 declti 10nn decnncl2 nnrei cn 8nn dpgti mulcli nnne0i divdiv1 div23i oveq1i nnrp mulne0i divassi cmin expp1 sq10 mulcomi 3eqtrri 2p1e3 eqtri expsub 7cn 4p3e7 addcomli subaddrii 3eqtr2i numexp1 nnzi 3p1e4 3eqtri 3eqtr3i breqtrri @@ -523238,16 +523238,16 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cdc cdp2 cmul clt wbr cr wa cle cn0 0re 7re 9re 5re pm3.2i dp2cl ax-mp dpcl mp2an crp cn nnrp rpdp2cl rpdpcl rpre remulcli 7nn0 9nn0 5nn0 5lt10 dp2lt10 6re 7p1e8 dp2ltsuc 8nn0 dp2lt dplt rpge0 mpbi recni 6nn0 deccl 10pos nn0cni - 8re dec0h addid1i 6cn addid2i decadd 4cn 1t1e1 oveq12i oveq2i eqtri 3eqtr4i + 8re dec0h addridi 6cn addlidi decadd 4cn 1t1e1 oveq12i oveq2i eqtri 3eqtr4i dp0u 10nn0 dec10p ax-1cn addcomi 6p1e7 oveq1i 8cn 8p1e9 3eqtri dpmul4 lttri - decaddc 3nn0 3lt10 9cn 2cn eqtr3i mulid1i 4p1e5 dpmul 7cn addcomli decaddci + decaddc 3nn0 3lt10 9cn 2cn eqtr3i mulridi 4p1e5 dpmul 7cn addcomli decaddci 2p1e3 decaddi 6p3e9 deceq1i 5p1e6 mulcomli 5cn 1p1e2 dpadd decsuc 4p2e6 4re dpadd3 2re 3re 3rp mul01i 3cn 3p1e4 6p4e10 0cn 2p2e4 resqcli 4nn 1re sqge0i rpgt0 ltleii mulge0i 5nn 8nn rpdp2cl2 9lt10 dp20u breqtrri wb lt2sqi sqvali cexp 4lt10 8t8e64 7p4e11 9t9e81 3eqtr4ri ltmul1ii 1lt10 8lt10 9p1e10 9p2e11 eqbrtri addcli 6t4e24 7t4e28 7t2e14 8t7e56 8t2e16 6p6e12 6p2e8 4p4e8 9p5e14 mulcomi 3eqtr3i 3eqtr2i 4p3e7 8p6e14 7nn ltmul12a 6lt10 3t2e6 9t2e18 7t3e21 - 9t7e63 9p6e15 eqeltrri cc 4t4e16 9t4e36 5p5e10 8t5e40 5t3e15 mulid2i 8p5e13 + 9t7e63 9p6e15 eqeltrri cc 4t4e16 9t4e36 5p5e10 8t5e40 5t3e15 mullidi 8p5e13 9t6e54 7t6e42 7t7e49 3p2e5 3pos ltmul2i dp2eq2i eqtr2i 4t3e12 mul02i eqtr4i 9nn 5p4e9 eqeltri 9p4e13 3t3e9 wceq eqeq1i 7p2e9 00id ) ABCDEEFFUAZUAZUAZUA ZUAZGHZIUUQHZBJBJUAZUAZGHZUBHZBJIKAUAZUAZUAZGHZBCALLAUAZUAZUAZUAZGHZUBHZUBH @@ -524410,7 +524410,7 @@ conditions of the Five Segment Axiom ( ~ axtg5seg ). See ~ df-ofs . = ( L - ( # ` W ) ) ) $= ( cc0 chash cfv co cmul c1 wceq cfn wcel cn0 syl cmin cfzo csn cxp snfi fzofi hashxp mp2an a1i cle cword lencl nn0sub2 syl3anc hashfzo0 hashsng - wbr oveq12d nn0cnd mulid1d 3eqtrd ) AJDEKLZUAMZUBMZBUCZUDKLZVDKLZVEKLZN + wbr oveq12d nn0cnd mulridd 3eqtrd ) AJDEKLZUAMZUBMZBUCZUDKLZVDKLZVEKLZN MZVCONMVCVFVIPZAVDQRVEQRVJJVCUFBUEVDVEUGUHUIAVGVCVHONAVCSRZVGVCPAVBSRZD SRVBDUJUQVKAECUKRVLGCEULTFIVBDUMUNZVCUOTABCRVHOPHBCUPTURAVCAVCVMUSUTVA $. @@ -532463,7 +532463,7 @@ have become an indirect lemma of the theorem in question (i.e. a lemma cn0 nn0fz0 swrdlen syl3an 3anidm13 cc nn0cnd elfzelz adantl nncand cv w3a zcnd simp1 simp3 revfv sylan syl2anc fveq2d 3ad2ant2 elfzoelz 1cnd sub32d 3ad2ant3 ubmelm1fzo eqeltrrd pfxfv syld3an3 3eqtrd eleq2d biimp3ar swrdfv - caddc id syl3anl1 syl3anl3 stoic3 syl3an2 elfzuz3 addid2d eluzsub mp3an2i + caddc id syl3anl1 syl3anl3 stoic3 syl3an2 elfzuz3 addlidd eluzsub mp3an2i wss fzoss1 nn0zd 3ad2ant1 zsubcld fzo0addel pncan3d eleqtrd sseldd subcld 0z subsub4d 3eqtr3d eqtr4d 3expa eqfnfvd ) CBUBZDZAECFGZHIZDZUCZUAEAJIZCA UDIZUEGZCUEGZYCAKIZYCUFUGIZYFYIEYIFGZJIZLZYIYGLYBYOYEYBYHYADZYIYADYOBCAUH @@ -534191,7 +534191,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by ffvelcdm elfz1end 1z fzsn fvex bitrdi 1m1e0 hash0 rspcv adantld subfacf df-ne nnnn0 nnm1nn0 nn0addcld mul02d 3eqtr4a a1d simplr sylancom imim1d peano2fzr elfzp1 fzfi eqeltri fzp1disj inelcm sylan2br necon2bi addsubd - deranglem a1i subcl ad2antrr adddird mulid2d exmidne orcom biantru andi + deranglem a1i subcl ad2antrr adddird mullidd exmidne orcom biantru andi bitri rabbii eqtr4i simpr necon3ai adantl imnan cid cdif cres cpr nnne0 simpll eqeq2i 0cn addcan2 mp3an23 necon3bbid mpbird elfzp12 biimpa df-2 0p1e1 ord oveq1i eleqtrrdi ovex subfacp1lem5 subfacp1lem3 eqtrid eqtr4d @@ -534281,7 +534281,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by cn sylancl fveq2d eqtrd peano2nn0 nncnd fzfid elfznn0 fsumcl expcl nnne0d syl sylancr divcld adddid cuz eleqtrdi fsump1 facp1 mulcomd oveq1d nn0cnd mulassd 3eqtrd div12d divcan3d mulcld negsub eqtr3d expp1 3eqtr4d addassd - id divcan2d add32d eqcomd mulid1d adddird syl5ibr jcad nn0ind simpld ) HU + id divcan2d add32d eqcomd mulridd adddird syl5ibr jcad nn0ind simpld ) HU BKHDLZHMLZNHOPZQUCZFUDZUEPZUUMMLZUFPZFUGZRPZSZHQTPZDLZUUTMLZNUUTOPZUUPFUG ZRPZSZAUDZDLZUVGMLZNUVGOPZUUPFUGZRPZSZUVGQTPZDLZUVNMLZNUVNOPZUUPFUGZRPZSZ UHQQNNOPZUUPFUGZRPZSZNQNQOPZUUPFUGZRPZSZUHUAUDZDLZUWIMLZNUWIOPZUUPFUGZRPZ @@ -534347,7 +534347,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by gt0ne0ii subcld abscld peano2nn peano2nnd nnred nnmulcld nndivred nnrecre cuz cneg cv cexp csu cle wbr cmpt eqid neg1cn a1i absnegi abs1 eqtri 1le1 ax-1cn eqbrtri eftlub wa wceq nnnn0d eluznn0 sylan eftval sumeq2dv fveq2d - oveq1i cz nnzd 1exp eqtrid oveq1d recnd mulid2d eqtrd 3brtr3d cr wb eftcl + oveq1i cz nnzd 1exp eqtrid oveq1d recnd mullidd eqtrd 3brtr3d cr wb eftcl clt mpan cli cdm eftlcvg sylancr isumcl nngt0d lemul2 syl112anc mpbid cfz cseq subfacval2 nncn pncan sylancl oveq2d sumeq1d eqtr4d divrec ce oveq2i df-e efneg ax-mp adantl 3eqtrd mulcld nnne0d adddird nnre nngt0 jca efval @@ -535735,7 +535735,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by cnmptid adantr simprr sseldd cnmptc ctx mulcn cnmpt12f subcn simprl addcn 1cnd cnmpt1res crn wb 3exp2 com23 imp42 fmpttd frnd cnrest2 syl3anc mpbid wi oveq2i eleqtrrdi 0elunit oveq1 oveq2 1m0e1 eqtrdi oveq1d oveq12d fvmpt - eqid ovex ax-mp mul02d mulid2d addid2d eqtrd eqtrid 1elunit 1m1e0 addid1d + eqid ovex ax-mp mul02d mullidd addlidd eqtrd eqtrid 1elunit 1m1e0 addridd fveq1 eqeq1d anbi12d syl12anc ralrimivva resttopon toponuni syl raleqbidv rspcev raleqdv ispconn sylanbrc ) AGUBLMUAUCZUDZCUCZNZOYJUDZBUCZNZUIZUAUE GUFPZUGZBGUHZUJZCYTUJZGUKLAGFEULPZUBKAFUBLEUMLZUUCUBLFJUNAEQUOZQUMLUUDHUP @@ -535768,9 +535768,9 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by cnmpt2nd eqeltrid toponuni eleqtrrd sseldd mulcn cnmpt22f ax-1cn cnmpt1st cnmpt2c subcn cnfldtop cnrest2r ax-mp oveq2i eleqtrdi sselid cnmpt21f crn addcn wb ffvelcdmd 3exp2 imp42 an32s ralrimivva oveq2 oveq1d eleq1d simpr - oveq2d weq eqtrdi simpl fveq2d oveq12d ovex ovmpoa mul02d mulid2d 1elunit + oveq2d weq eqtrdi simpl fveq2d oveq12d ovex ovmpoa mul02d mullidd 1elunit 3eqtrd 3eqtr3d eqtrd ad2ant2rl rspc2va syl21anc fmpo frnd cnrest2 syl3anc - mpbid eleqtrrdi 1m0e1 toponunii ffvelcdmda addid2d 1m1e0 addid1d fvconst2 + mpbid eleqtrrdi 1m0e1 toponunii ffvelcdmda addlidd 1m1e0 addridd fvconst2 fvex adantl eqtr4d subcl adddird simplrr isphtpy2d ne0d isphtpc syl3anbrc pncan3 expr ralrimiva issconn sylanbrc ) AGUCMZNUAUDZUEZOUVMUEZUFZUVMNOUI PZUVNUGUJZGUHUEUKZULZUAQGUMPZUNGUOMABCDEFGHIJKUPZAUVTUAUWAAUVMUWAMZUVPUVS @@ -535861,7 +535861,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by w3a cmin df-3an weq oveqan12d eleq1d ralbidv unitssre sstri sselid simpr2 simpr sseldd mulcld ax-1cn subcl sylancr simpr1 nncan oveq1d oveq2d iirev addcomd eqtr4d adantl eleq1i reconn bitrid biimpa r19.21bi anasss simplll - 3adantr3 remulcld resubcl readdcld pncan3 sylancl adddird mulid2d 3eqtr3d + 3adantr3 remulcld resubcl readdcld pncan3 sylancl adddird mullidd 3eqtr3d 1re recnd elicc01 sylib simp3d wb subge0 mpbird simplr3 lemul2ad leadd2dd eqbrtrrd simp2d leadd1dd breqtrd elicc2 syl2anc mpbir3and ralrimiva oveq1 oveq12d rspcv sylc eqeltrd cbvralvw wloglei sylan2b cvxsconn eqeltrrd ex @@ -536628,7 +536628,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by cvmliftlem2 $p |- ( ( ph /\ ps ) -> W C_ ( 0 [,] 1 ) ) $= ( wa c1 cmin co cdiv cicc cc0 cr wcel cle wbr wss 0red clt cfz elfznn 1red cn syl nnred peano2rem cn0 nnm1nn0 adantr nngt0d divge0 syl22anc - nn0ge0d cmul elfzle2 nncnd mulid1d breqtrrd ledivmul syl112anc mpbird + nn0ge0d cmul elfzle2 nncnd mulridd breqtrrd ledivmul syl112anc mpbird wb iccss eqsstrid ) ABUNZRPUOUPUQZQURUQZPQURUQZUSUQZUTUOUSUQZUMWMUTVA VBUOVAVBZUTWOVCVDZWPUOVCVDZWQWRVEWMVFWMVJZWMWNVAVBZUTWNVCVDQVAVBZUTQV GVDZWTWMPVAVBZXCWMPWMPUOQVHUQVBZPVKVBZULPQVIVLZVMZPVNVLWMWNWMXHWNVOVB @@ -538280,7 +538280,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by ( cn cc0 c1 co cv cmul cfv wceq wcel cr cle wbr cfn cicc cexp cmo cfl cfz crab chash cdiv cn0 wss ssrab2 ssfi sylancl hashcl nn0red nndivre mpancom fzfid syl clt nn0ge0d nnre nngt0 divge0 syl22anc ssdomg mpisyl wb hashdom - cdom syl2anc mpbird nnnn0 hashfz1 breqtrd nncn mulid1d breqtrrd syl112anc + cdom syl2anc mpbird nnnn0 hashfz1 breqtrd nncn mulridd breqtrrd syl112anc 1red ledivmul elicc01 syl3anbrc fmpti ) EHIJUAKZACDLUBKMKCUCKUDNBOZDJELZU EKZUFZUGNZWGUHKZFGWGHPZWKQPZIWKRSZWKJRSZWKWEPWJQPZWLWMWLWJWLWITPZWJUIPWLW HTPZWIWHUJZWQWLJWGURZWFDWHUKZWHWIULUMZWIUNUSZUOZWJWGUPUQWLWPIWJRSWGQPZIWG @@ -541579,7 +541579,7 @@ proper pair (of ordinal numbers) as model for a Godel-set of membership sylan2 cmhm cvrmd cmcn fvexi cmvar unex frmdmnd a1i eleqtrd fmpttd cplusg ax-mp simpr1 feq23d mpbid simpr3 simprl simprr cbs frmdbas eqcomi frmdadd wb syl2anc ffvelcdm ad2ant2lr ad2ant2l 2ralbidva chash caddc fveq2 eqtrdi - cc0 raleqdv raleqbidv bitr3d 3ad2antr1 cn0 wrd0 lencl nn0cnd 0cnd addid1d + cc0 raleqdv raleqbidv bitr3d 3ad2antr1 cn0 wrd0 lencl nn0cnd 0cnd addridd eleqtrrid fvoveq1 oveq1d oveq2 ccatidid rspc2va syl21anc ccatlen addcanad 3eqtrrd hasheq0 sylib pm3.2i frmd0 ismhm mpbiran syl3anbrc fcompt vrmdval syl vrmdf mpan mpteq2ia frmdup3lem syl32anc 3eqtr4rd cpm wfn cmap mrsubff @@ -543551,12 +543551,12 @@ I hope someone will enjoy solving (proving) the simple equations, problem4.4 $e |- ( ( 3 x. A ) + ( 2 x. B ) ) = 7 $. $( Practice problem 4. Clues: ~ pm3.2i ~ eqcomi ~ eqtri ~ subaddrii ~ recni ~ 7re ~ 6re ~ ax-1cn ~ df-7 ~ ax-mp ~ oveq1i ~ 3cn ~ 2cn ~ df-3 - ~ mulid2i ~ subdiri ~ mp3an ~ mulcli ~ subadd23 ~ oveq2i ~ oveq12i + ~ mullidi ~ subdiri ~ mp3an ~ mulcli ~ subadd23 ~ oveq2i ~ oveq12i ~ 3t2e6 ~ mulcomi ~ subcli ~ biimpri ~ subadd2i . (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) $) problem4 $p |- ( A = 1 /\ B = 2 ) $= ( c1 wceq c2 c7 c6 cmin co caddc eqcomi c3 cmul 3cn 2cn eqtri ax-1cn df-7 - 7re recni 6re subaddrii df-3 oveq1i mulid2i subdiri mulcli subadd23 mp3an + 7re recni 6re subaddrii df-3 oveq1i mullidi subdiri mulcli subadd23 mp3an cc wcel 3t2e6 mulcomi oveq12i subcli oveq2i subadd2i biimpri ax-mp pm3.2i ) AGHBIHGAGJKLMZAVEGJKGJUCUDZKUEUDZUAJKGNMUBOUFOAKNMZJHZVEAHZVHPAQMZIBQMZ NMZJVHVKIAQMZLMZKNMZVMAVOKNAPILMZAQMZVOVRAVRGAQMAVQGAQPIGRSUAPIGNMZUGOZUF @@ -543602,7 +543602,7 @@ I hope someone will enjoy solving (proving) the simple equations, divcan3i oveq12i eqtr2i mulcomi divcan2i divassi cc cc0 wa pm3.2i divdiv1 wcel wne mp3an eqtri 3eqtr2i addassi eqcomi pncan3oi df-neg eqtr4i negcli 3eqtr3i addcomi sqdivi 4cn 4ne0 divmuldivi c1 dividi eqtr3i mulm1i neg1cn - mulid2i mulassi 3eqtri 2t2e4 sqvali eqnetri negsubi subcli eqsqrtor ax-mp + mullidi mulassi 3eqtri 2t2e4 sqvali eqnetri negsubi subcli eqsqrtor ax-mp wb mpbi sqrtcl divmuli eqcom bitr3i subadd2i divneg eqeq2i 3bitri orbi12i ) KALMZDBXKNMZOMZLMZBKUAMZUBACLMZLMZPMZUCUFZQZXNXSUDZQZUEZDBUDZXSOMXKNMZQ ZDYDXSPMZXKNMZQZUEXNKUAMZXRQZYCYJXKKUAMZXMKUAMZLMYLXRYLNMZLMXRXKXMKARFUGZ @@ -543718,10 +543718,10 @@ Real and complex numbers (cont.) 3eltr4i fvco3 syldan climcn1lem 0cn oveq1d eqtrdi oveq2d fvmpt csin eqtrd ovex 1re eqeltrd fveq2 id oveq12d resincld simprd cmul eqbrtrd cneg ltled cle sq0i div0i 1m0e1 1ex breqtrdi resqcld nndivre resubcl redivcld abscld - ax-mp 3nn subdird mulid2d caddc df-3 oveq2i cn0 2nn0 expp1 absresq eqtrid + ax-mp 3nn subdird mullidd caddc df-3 oveq2i cn0 2nn0 expp1 absresq eqtrid div23d eqtr2d cioc absrpcld rpgt0d ltle mpd cxr w3a wb syl3anbrc sin01bnd 0xr elioc2 ltmuldivd mpbid sinneg eqeq1d syl5ibrcom absord mpjaod breqtrd - div2negd 3brtr4d mulid1d breqtrrd ltdivmuld mpbird eqbrtrrd climsqz ) ANC + div2negd 3brtr4d mulridd breqtrrd ltdivmuld mpbird eqbrtrrd climsqz ) ANC FDUDZEDUDZGUEGUFUGZUWOUHZAGLUIZAUWMOFUGZNUJAUAUBUCOCDUWMFGUEUWOUWPIAFUMDU EPZUWMUEPBQNBUKZULUNRZSTRZUORZFKUPAUQUROUSUTZDVAZUQUEPUWSHVBUQUXDUEDVGVCZ FDUEVDVEUWQACUKZUWOPZVFZUXGDUGZUXIUXJURPZUXJOVHZUXIUXJUXDPZUXKUXLVFUXIUQU @@ -543793,7 +543793,7 @@ Real and complex numbers (cont.) a1i resincld eldifsni redivcld fco mp2an mptru feq1i mpbi ffvelcdmi recnd eldifi nncn nnne0 divassd oveq1d simpr nndivre 2ne0 divcan3d eqtrd fveq2d rpne0d divcan2d eqtr4d oveq2d oveq2 ovex fvmpt mulassd mulcl mul4d mul32d - eqeltrrd 3eqtr2d eqtrid 3eqtr4d climmulc2 mulid1i breqtri ) BHIJKZCJKZUCJ + eqeltrrd 3eqtr2d eqtrid 3eqtr4d climmulc2 mulridi breqtri ) BHIJKZCJKZUCJ KZYFUDBYGUDUELUCYFUADMIDUFZNKZOPZYINKZUGZBUCUHMUILUJLUBQRUKZULZUBUFZOPZYO NKZUGZDMYIUGZUMZYLUCUDLDUBMYNYIYQYKYSYRYHMSZYIYNSZLUUAYIQSYIRUNUOUUBUUAYI UUAIUPSZYHUPSYIUPSUQYHVAIYHURUSUTYIQRVBVCZVDLYSVELYRVEYOYIVFZYPYJYOYINYOY @@ -544439,7 +544439,7 @@ Real and complex numbers (cont.) c2 wcel cn cdiv nncnd vm vn vj cv cc0 caddc oveq1 1m1e0 eqtrdi fz10 oveq2 c0 eqeq12d weq prod0 eqtr4i wa cfa cfv nncn 1cnd pncand cuz elnnuz biimpi simpr cn0 nnnn0 elfzelz bccl syl2an nn0cnd fprodm1 bcnn syl fzfid fprodcl - cz mulid1d fz1ssfz0 bcm1nt sylan2 prodeq2dv nnm1nn0 clt wbr elfznn adantl + cz mulridd fz1ssfz0 bcm1nt sylan2 prodeq2dv nnm1nn0 clt wbr elfznn adantl sseli cc nnred nn0red cr cle elfzle2 ltm1d lelttrd wb simpl nnsub syl2anc nnre mpbid nnne0d divcld fprodmul fproddiv chash cfn fzfi sylancr hashfz1 fprodconst eqtr2d fprodfac nnz 1zzd nn0zd fprodrev nncand prodeq1d 3eqtrd @@ -544760,7 +544760,7 @@ Real and complex numbers (cont.) nncnd 1cnd addcomd oveq1d cc ax-1cn addcli nnaddcld nnne0d divassd eqtrid a1i cuz seqp1 nnuz eleq2s adantr adantl peano2nn syl nnrpd simpl rpaddcld rpdivcld crp cr nn0re nndivred recnd cc0 cle wbr divge0d ge0p1rpd eqeltrd - nn0ge0 1rp rpreccld rpmulcld mulassd rpne0d divcan5d mul12d mulid1d simpr + nn0ge0 1rp rpreccld rpmulcld mulassd rpne0d divcan5d mul12d mulridd simpr adddid nn0cnd divcan2d addassd eqtrd eqtr3d divcan1d 3eqtr3rd exp31 nnind a2d impcom eqtr4d ralrimiva wfn wb seqfn fneq2i mpbir fnmpti eqfnfv mp2an sylibr ) CUCEZDFZGBHICBFZJKZLKZIIYQJKZLKZGKZICILKZYQJKZLKZJKZUDZIUEZMZYPA @@ -544817,7 +544817,7 @@ Real and complex numbers (cont.) ( vm vk wcel cmul cn c1 cv cdiv co caddc cmpt cli cvv nnuz nnex mptex a1i adantl cn0 cseq faclimlem1 1zzd 1cnd nn0p1nn nnzd cfv wceq weq oveq1 eqid oveq12d ovex fvmpt divcnvlin nncnd cc wa peano2nn nnred nnaddcld nndivred - simpr adantr recnd fmpttd oveq2d eqtr4d climmulc2 mulid1d breqtrd eqbrtrd + simpr adantr recnd fmpttd oveq2d eqtr4d climmulc2 mulridd breqtrd eqbrtrd ffvelcdmda ) BUAEZFAGHBAIZJKLKHHVPJKLKFKHBHLKZVPJKLKJKMHUBCGVQCIZHLKZVRVQ LKZJKZFKZMZVQNCABUCVOWCVQHFKVQNVOHVQDCGWAMZWCHOGPVOUDZVOHVQDWDHOGPWEVOUEV OVQBUFZUGWDOEVOCGWAQRSDIZGEZWGWDUHZWGHLKZWGVQLKZJKZUIVOCWGWAWLGWDCDUJZVSW @@ -544921,7 +544921,7 @@ Real and complex numbers (cont.) ( wcel cn cfa cfv c1 caddc co cexp cmul cdiv cli wceq oveq2 oveq12d nncnd cc0 va vm vk cn0 cmpt wbr oveq2d fveq2d mpteq2dv breq1d weq wtru cvv nnuz cv 1zzd nnex mptex a1i 1cnd fveq2 oveq1 oveq1d fvoveq1 eqid ovex peano2nn - fvmpt exp0d nnnn0 faccl mulid1d eqtrd addid1d nnne0d dividd 3eqtrd adantl + fvmpt exp0d nnnn0 faccl mulridd eqtrd addridd nnne0d dividd 3eqtrd adantl syl nncn climconst mptru wa simpr nn0p1nn nnzd divcnvlin adantr cc nnnn0d nnexpcl sylan ancoms nnmulcld nnred nnnn0addcl nndivred fmpttd ffvelcdmda recnd adantlr nnaddcld simpl expp1d mulassd eqtr4d nn0addcld facp1 nn0cnd @@ -551467,8 +551467,8 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and fwddifn0 $p |- ( ph -> ( ( 0 _/_\^n F ) ` X ) = ( F ` X ) ) $= ( vk cc0 co cfv cbc c1 cmin cexp cmul wcel cc wceq eqtrd cfwddifn cv cneg cfz caddc csu cn0 0nn0 a1i sseldd csn cz 0z fzsn ax-mp eleq2i velsn bitri - wa oveq2 adantl addid1d eqeltrd adantr fwddifnval fveq2d oveq2d ffvelcdmd - sylan2b mulid2d bcnn eqtrdi 0m0e0 neg1cn exp0 oveq12d fsum1 sylancr ) ADI + wa oveq2 adantl addridd eqeltrd adantr fwddifnval fveq2d oveq2d ffvelcdmd + sylan2b mullidd bcnn eqtrdi 0m0e0 neg1cn exp0 oveq12d fsum1 sylancr ) ADI CUAJKIIUDJZIHUBZLJZMUCZIVTNJZOJZDVTUEJZCKZPJZPJZHUFZDCKZABHCIDIUGQZAUHUIE FABRDEGUJZVTVSQZAVTISZWEBQWMVTIUKZQWNVSWOVTIULQZVSWOSUMIUNUOUPHIUQURAWNUS WEDIUEJZBWNWEWQSAVTIDUEUTZVAAWQBQWNAWQDBADWLVBZGVCVDVCVIVEAWIMMWQCKZPJZPJ @@ -551497,10 +551497,10 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and 1cnd wne mulcomd mulm1d 3eqtrd mulneg1d mulneg2d oveq12d negsubd sumeq2dv eqtr3d fzfid fsumsub cuz nn0uz eleqtrdi oveq1 fveq2d fsum1p df-neg oveq2i oveq2 bcneg1 eqtr3id 0z 1z mp2an zsubcld eluzfz1 ralrimiva eleq1d syl2anc - wral rspcva mul02d wo olc wb elfzp12 biimpar sylan2 syldan fsumcl addid2d + wral rspcva mul02d wo olc wb elfzp12 biimpar sylan2 syldan fsumcl addlidd ppncand 1zzd 0zd addassd fzp1elp1 rspccv imp eqeltrd fsumshft weq cbvsumv wi eqtr3di cfa fwddifnval 3eqtr2d fsump1 cdiv cif fzp1nel iffalsei eqtrdi - bcval eluzfz2 sylc fzelp1 addid1d peano2cn anbi2d imbi12d chvarvv 3eqtr4d + bcval eluzfz2 sylc fzelp1 addridd peano2cn anbi2d imbi12d chvarvv 3eqtr4d rspcv ) ALEMNOZUAOZUUSCUSZUBOZMUCZUUSUVAPOZUDOZFUVANOZDUEZQOZQOZCUHZLEUAO ZEUVAUBOZUVCEUVAPOZUDOZFMNOZUVANOZDUEZQOZQOZCUHZUVKUVLUVNUVGQOZQOZCUHZPOZ FUUSDUFOUEUVOEDUFOZUEZFUWEUEZPOAUVJUUTEUVAMPOZUBOZUVCEUWHPOZUDOZUVGQOZQOZ @@ -552127,7 +552127,7 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and anbi2d rexbidv dvdsmultr2 simp-4r bitrdi coprmdvds mpan2d impbid divcan2d cgcd breq2d bitrd ex com23 pm2.61d embantd syld biimpar eluz1i zre ltletr 2pos 0re 2re mp3an12 mpani imp sylbi ancld syl6ibr sylbid ancoms eluzelcn - a1d mulid2d prmgt1 1red nnred ltmul1 eqbrtrrd mpbird ltdiv1 simprll caddc + a1d mullidd prmgt1 1red nnred ltmul1 eqbrtrrd mpbird ltdiv1 simprll caddc ltdivmul peano2nn nnnn0 ad2antll nncnd expp1d eqcomd mpbiri com12 gcdcomd simplr simprl prmdvdsexpb equcom con3d coprm eqtrd exp32 rexlimdvv 3impia cn0 nncn 3exp2 com24 imp32 3syld simpl2 1nn a1i exp1d 3adant1 mpid biimpr @@ -554302,7 +554302,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= ( cfv c1 co caddc cmin cabs cc0 wcel wceq syl halfre a1i cxr eqtrd c2 cfl cdiv cr zred dnival cz wa jca flzadd cle wbr clt w3a rexri halfgt0 ltleii cico 0re halflt1 3pm3.2i wb 0xr pm3.2i elico1 ax-mp mpbir ico01fl0 oveq2d - 1xr recnd addid1d oveq1d subidd fveq2d abs0 ) ACDGZCHUAUCIZJIUBGZCKIZLGZM + 1xr recnd addridd oveq1d subidd fveq2d abs0 ) ACDGZCHUAUCIZJIUBGZCKIZLGZM ACUDNVQWAOACFUEZBCDEUFPAWAMLGZMAVTMLAVTCCKIMAVSCCKAVSCVRUBGZJIZCACUGNZVRU DNZUHVSWEOAWFWGFWGAQRUIVRCUJPAWECMJICAWDMCJAVRMHURINZWDMOWHAWHVRSNZMVRUKU LZVRHUMULZUNZWIWJWKVRQUOMVRUSQUPUQUTVAMSNZHSNZUHWHWLVBWMWNVCVJVDMHVRVEVFV @@ -555300,7 +555300,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= cabs wbr simpr knoppcnlem3 recnd cseq cli cdm knoppndvlem4 fvex isumsplit seqex breldm nn0cnd 1cnd pncand oveq2d sumeq1d oveq1d eluznn0 knoppcnlem1 3eqtrd sylan cz cle eluzle adantl jca zltp1le mpbird knoppndvlem2 dnizeq0 - wb cc expcld mul01d sumeq2dv wss cfn ssidd orcd sumz knoppndvlem5 addid1d + wb cc expcld mul01d sumeq2dv wss cfn ssidd orcd sumz knoppndvlem5 addridd wo eqtrd ) AENUCZUDKUEUFZHUGZEJUCZUCZHUHZKUIUJUFZUKUCZYMHUHZUJUFZYNAYIULY MHUHZUDYOUIUMUFZUEUFZYMHUHZYQUJUFYRADEULYKDUGZJUCZUCZHUHYSUNNUOQUUCEUPZUL UUEYMHUUFYKUUDYLUUCEJUQUSURAEUTMVAUFZKVBVCUFUTVDUFLVAUFZUNEUUHUPZARVEAKLM @@ -555389,7 +555389,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= wbr wn wcel wb odd2np1 mpbid wa eqcom biimpi oveq1d adantl 2cnd cc mulcld syl zcn 1cnd cc0 wne 2ne0 a1i divdird divcan3d eqtrd fveq2d dnizphlfeqhlf adantrr id rexlimddv oveq2d cr cabs clt knoppndvlem3 simpld expcld div12d - recnd divcld mulid2d 3eqtrd ) AIDHUBUBEIUCUDZJTUEUDZFUBZUFUDWPUGTUEUDZUFU + recnd divcld mullidd 3eqtrd ) AIDHUBUBEIUCUDZJTUEUDZFUBZUFUDWPUGTUEUDZUFU DZWPTUEUDZABCDEFGHIJKLMNPQRUHAWRWSWPUFATUAUIZUFUDZUGUJUDZJUKZWRWSUKUAULAT JUMUOUPZXEUAULUNZSAJULUQXFXGURQUAJUSVIUTAXBULUQZXEVAZVAZWRXBWSUJUDZFUBZWS XJWQXKFXJWQXDTUEUDZXKXIWQXMUKZAXEXNXHXEJXDTUEXEJXDUKXDJVBVCVDVEVEAXHXMXKU @@ -555548,12 +555548,12 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= wcel simpld resubcld recnd abscld fzfid wa 2re cn syl remulcld adantr cn0 nnre elfznn0 reexpcld fsumrecl 2ne0 redivcld 1red 0red 0lt1 knoppndvlem12 adantl wne lttrd jca ltne knoppndvlem11 cle oveq12d nnge1 ltletrd mulne0d - simprd znegcld reexpclzd zcnd subdid eqcomd pncan2d oveq2d mulid1d 3eqtrd + simprd znegcld reexpclzd zcnd subdid eqcomd pncan2d oveq2d mulridd 3eqtrd 1cnd eqtrd fveq2d absdivd cc w3a mulcld 3jca absexpz absmuld pm3.2i absid 0le2 ax-mp ltled absidd oveq1d geoser expcld necomd div2subd eqeltrrd crp cz 2rp rpgt0d mulgt0d mulassd expgt0 divge0d elrpd lem1d lediv1dd divrecd lemul2ad div23d knoppndvlem13 absne0d mulexpz jca32 expaddz nn0cnd negidd - addcomd exp0d mulid2d breqtrd eqbrtrd letrd ) AUCKUDUEUFZUGUFZHUHZEJUIUIH + addcomd exp0d mullidd breqtrd eqbrtrd letrd ) AUCKUDUEUFZUGUFZHUHZEJUIUIH UJZUVCUVDDJUIUIHUJZUEUFZUKUIEDUEUFZUKUIZUVCULMUMUFZFUKUIZUMUFZUVDUNUFZHUJ ZUMUFZUVKKUNUFZULUOUFZUDUVLUDUEUFZUOUFZUMUFZAUVGAUVGAUVEUVFABCEFGHIJUVBMN OAEUVJKUPZUNUFZULUOUFZLUDUQUFZUMUFZUREUWEUSAQUTZAKUWDMUAAKSVAZALTVBZVCVDZ @@ -555616,7 +555616,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= cv cr clt wbr simpld recnd abscld reexpcld 2re a1i wne 2ne0 redivcld 1red nnred remulcld resubcld wa 0red knoppndvlem12 simprd lttrd jca gt0ne0 syl 0lt1 cneg wceq nn0zd knoppndvlem1 knoppcnlem3 caddc peano2zd knoppndvlem5 - eqeltrd subcld remulcl resubcl 1cnd subdid mulid1d oveq1d eqbrtrd abssubd + eqeltrd subcld remulcl resubcl 1cnd subdid mulridd oveq1d eqbrtrd abssubd leidd knoppndvlem10 eqtrd eqcomd breqtrd knoppndvlem14 letrd knoppndvlem6 le2subd abs2difd cn0 cuz elnn0uz sylib adantr elfznn0 adantl fveq2 fsumm1 cn oveq12d subadd4d fveq2d ) AGUFUGZLUHUIZUJUKUIZULULUJNUMUIZYHUMUIZULUNU @@ -555653,7 +555653,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= knoppndvlem16 $p |- ( ph -> ( B - A ) = ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) $= ( cmin co c2 cmul cneg cexp c1 wceq a1i cdiv caddc oveq12d 2cnd nncnd cc0 mulcld wne 2ne0 nnne0d mulne0d znegcld expclzd mulne0bad divcld zcnd 1cnd - nn0zd addcld subdid eqcomd pncan2d oveq2d mulid1d eqtrd 3eqtrd ) ACBLMNFO + nn0zd addcld subdid eqcomd pncan2d oveq2d mulridd eqtrd 3eqtrd ) ACBLMNFO MZDPZQMZNUAMZERUBMZOMZVJEOMZLMZVJVKELMZOMZVJACVLBVMLCVLSAHTBVMSAGTUCAVPVN AVJVKEAVINAVGVHANFAUDZAFKUEZUGANFVQVRNUFUHAUITAFKUJUKZADADIURULUMVQANFVQV RVSUNUOZAERAEJUPZAUQZUSWAUTVAAVPVJROMVJAVORVJOAERWAWBVBVCAVJVTVDVEVF $. @@ -555726,7 +555726,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= cr gt0ne0d cz nnz adantl expnegd adantrr crp 2rp jca rpmulcl syl rpexpcld elrpd rprecred knoppndvlem3 simpld abscld nnnn0 reexpcld rpred ifcld max1 rpne0d redivcld simprr lelttrd cc mulexpd rpge0d w3a absge0d simprd ltled - 1red exple1 lemul2ad mulid1d breqtrd eqbrtrd ltletrd ltrec1d wb reexpclzd + 1red exple1 lemul2ad mulridd breqtrd eqbrtrd ltletrd ltrec1d wb reexpclzd 3jca nnnegz ltdivmuld mpbird max2 letrd ledivmul2d mpbid eqcomi 0le1 1lt2 1t1e1 ltmul12ad mulassd eqcomd expnbnd reximddv wss nnssnn0 ssrexv ax-mp wi ) ANGUAOZDUBZUCZUDOZNUEOCUFPZEYKBUGUKZUAOZYLUDOZFUAOQPZUHZDUIUJZYTDULU @@ -555967,7 +555967,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= cnndvlem1 $p |- ( W e. ( RR -cn-> RR ) /\ dom ( RR _D W ) = (/) ) $= ( co wcel wtru c1 c2 c3 clt wbr cc0 cr ccncf cdv cdm c0 wceq cdiv 3nn a1i cn cneg cxr w3a wa neg1rr rexri 1re halfre 3pm3.2i neg1lt0 halfgt0 pm3.2i - cioo 0re lttri ax-mp halflt1 elioo3g mpbir knoppcn2 cabs cfv cmul mulid2i + cioo 0re lttri ax-mp halflt1 elioo3g mpbir knoppcn2 cabs cfv cmul mullidi mptru 2cn 2lt3 eqbrtri wb 2pos nnrei 2re ltmuldivi mpbi cle ltleii absidi oveq2i nncni 2ne0 divreci eqcomi eqtri breqtrri knoppndv ) HUAUAUBLMZUAHU CLUDUEUFZWPNABCOPUGLZDEFGQHIJKQUJMNUHUIZWROUKZOVCLMZNXAWTULMZOULMZWRULMZU @@ -564869,7 +564869,7 @@ Complex numbers (supplements) bj-bary1lem $p |- ( ph -> X = ( ( ( ( B - X ) / ( B - A ) ) x. A ) + ( ( ( X - A ) / ( B - A ) ) x. B ) ) ) $= ( cmin co cmul cdiv caddc cc0 mulcld subcld oveq1d eqtrd subdird oveq12d - addsub12d sub32d bj-subcom oveq2d 0cnd addsubassd addid1d subdid 3eqtr4rd + addsub12d sub32d bj-subcom oveq2d 0cnd addsubassd addridd subdid 3eqtr4rd 3eqtr2d necomd subne0d divdird divcan4d div23d 3eqtr3d ) ADCBIJZKJZUQLJZC DIJZBKJZUQLJZDBIJZCKJZUQLJZMJZDUTUQLJBKJZVCUQLJCKJZMJAUSVAVDMJZUQLJVFAURV IUQLACBKJZDBKJZIJZDCKJZBCKJZIJZMJZVMVKIJZVIURAVPVMNVKIJZMJZVMNMJZVKIJVQAV @@ -564888,7 +564888,7 @@ coordinates of a barycenter of two points in one dimension (complex ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) -> T = ( ( X - A ) / ( B - A ) ) ) ) $= ( cmul co caddc wceq c1 wa cmin oveq1 cdiv wi pncand pm5.31 sylancl eqtr2 - eqcomd syl6 oveq1d eqtr sylan2 subdird mulid2d eqtrd sylan9eqr ex sylan2d + eqcomd syl6 oveq1d eqtr sylan2 subdird mullidd eqtrd sylan9eqr ex sylan2d 1cnd mulcld subadd23d subdid oveq2d eqeq2d sylibd subcld pncan2d imbitrid eqcom mulcomd eqeq1d necomd subne0d rdiv biimpd sylbid biimtrid 3syld ) A FDBMNZECMNZONZPZDEONZQPZRZFBECBSNZMNZONZPZFBSNZWFPZEWIWEUANPZAWDFBEBMNZSN @@ -569295,7 +569295,7 @@ sides of the biconditional correlate (they are the same), if they exist at cneg wb resqrtcld w3a elicc2i wa clt pm3.2i ledivmul mp3an23 bicomd anbi12d 2pos cxr rehalfcl rexrd 0xr rexri elicc4 mp3an12 bitr4d biimpd 3impib sylbi sinq12ge0 sqrtge0d cexp recnd ax-1cn coscl subcl halfcld halfcl caddc sqcld - sqsqrtd mulcom sylancl oveq2d mulid2i df-2 eqtri oveq1i eqtrdi subdir mulcl + sqsqrtd mulcom sylancl oveq2d mullidi df-2 eqtri oveq1i eqtrdi subdir mulcl mp3an13 subsub3 3eqtr4d sincl pncand sincossq oveq1d cos2t wne 2ne0 divcan2 eqtr3d fveq2d eqtrd sincld divcan4 3eqtr2rd sq11d ) ABCDEFZUAFZGZACHFZUBIZJ AKIZLFZCHFZUCIZYIYJYIAYHMABMGYGMGYHMUDUECDUFUGUHZBYGUIULUMZUJUKYIAMGZYOMGYQ @@ -569359,7 +569359,7 @@ sides of the biconditional correlate (they are the same), if they exist at sylancl coscld ax-1cn eqtr4di a1i cexp sqcld simpld subneg addcom breq2d wo recn csn cun snunioo eleq2i elun bitr3i elsni fveq2 cos0 eqtrdi oveq2d df-2 eqnetrd sinq12gt0 ltne mpan elioore oveq1 df-neg eqeq1i coscl subadd bitrid - sincl 0cnd addcan2d sincossq neg1sqe1 addid2d eqeq12d sqeq0 3bitr3d 3imtr3d + sincl 0cnd addcan2d sincossq neg1sqe1 addlidd eqeq12d sqeq0 3bitr3d 3imtr3d 0cn necon3d syl5 jaoi ne0gt0d elrpd rphalfcld sqrtdivd subcl 2cnne0 divcan7 mpd addcl mp3an3 syl12anc fveq2d 3eqtr2d ) ABCUADZEZAFGDZUBHZIAUCHZUDDZFGDZ UEHZIUXAUFDZFGDZUEHZGDZUXCUXFGDZUEHUXBUXEGDZUEHUWRUWTUWSUOHZUWSUCHZGDZUXHUW @@ -570015,7 +570015,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= imass2 cr neeq12d sylib wi anbi12d bitri xchbinxr ffvelcdmda ifbieq1d cof id elfzelz ltm1d iftrued elfzm1b adantl csbied ffnd ccnv wfun wfo wb dff1o3 simprbi ima0 elfzuz eqeltrd elfzuz3 eluzp1p1 fzsplit2 f1ofo - uneq2d foima inidm eqidd fvun2 addid1d 3eqtrd ifbieq2d ltnrd iffalsed + uneq2d foima inidm eqidd fvun2 addridd 3eqtrd ifbieq2d ltnrd iffalsed uzss ovex csbie fz1ssfz0 sselid fzsplit fzss1 fvun1 3netr4d ralrimiva peano2re fzpr f1ofn fnimapr syl3anc ralpr wf1 f1of1 f1veqaeq syl12anc raleqdv necon3d mpd anbi1d neeq1 anbi2d rspc2ev syl112anc dfrex2 rmo4 @@ -570549,7 +570549,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= ima0 eqtri uneq1i uncom un0 3eqtri oveq2d csbie elrabi eleq2s xp1st wfo crab xp2nd fvex f1oeq1 elab sylib f1ofo foima eqtrid adantr cn0 nnm1nn0 0elfz ovexd fvmptd wfn elmapfn fnconstg mp1i eqidd fvconst2 - cn adantl wf elmapi ffvelcdmda elfzonn0 nn0cnd addid1d offveq ) AOH + cn adantl wf elmapi ffvelcdmda elfzonn0 nn0cnd addridd offveq ) AOH PEUBPZUBPZQJUCRZOUDZUEZUFUGZRZYSABOGBVNZEUHPZUIUJZUUEUUEQUFRZUKZYSY RUHPZQGVNZUCRZULZQUDZUEZUUJUUKQUFRZJUCRZULZUUAUEZUMZUUCRZUNZUUDOJQU ORZUCRZHUPAEDSZHBUVDUVBUQZURZMUVEEOIUSRZYTUTRZYTYTFVNZVAZFVBZUEZOJU @@ -570602,7 +570602,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= mpteq2dv eqeq2d elrab2 breq1 id ifbieq1d lelttrd oveq2 adantl oveq1 iftrued oveq2d csbied 1red lesub1dd elfz2nn0 fvmptd fz1ssfz0 sstrdi 3eqtr4d expr necon1d wi elmapi elfzonn0 ltned fzss1 eluzfz1 fnfvima - syl3anc mp3an12 fvconst2 mpdan nn0cnd addid1d 3eqtrd elfz1end fvun1 + syl3anc mp3an12 fvconst2 mpdan nn0cnd addridd 3eqtrd elfz1end fvun1 3netr4d neeq12d syl5ibrcom impbid riota5 ) AHUAZKRUBUCZIUDZUDZVUGKI UDZUDZUEZHRLUFUCZKEUGUDZUHUDZUDZAVUNVUNKVUPAVUNVUNVUPUIZVUNVUNVUPUJ AVUPVUNVUNFUAZUIZFUUAZSZVURAVUOUKJUUBUCZVUNUUCUCZVVAULZSZVVBAEVVEUK @@ -570728,7 +570728,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= 2re mp3an2 nnm1nn0 nn0red lenltd bitrd iffalsed oveq2 oveq2d csbied lem1d letrd lesub1dd elfz2nn0 syl3anbrc fvmptd lep1d nnnn0d 3eqtr4d 1re expr necon1d wi elmapi elfzonn0 eluzfz1 fnfvima syl3anc mp3an12 - ltned fvconst2 mpdan nn0cnd addid1d 3eqtrd elfz1end 3netr4d neeq12d + ltned fvconst2 mpdan nn0cnd addridd 3eqtrd elfz1end 3netr4d neeq12d fzss2 fvun1 syl5ibrcom impbid riota5 ) AHUAZKUBUCUDZIUEZUEZVVFKRUCU DZIUEZUEZUFZHRLUGUDZKEUHUEZUIUEZUEZAVVNVVNKVVPAVVNVVNVVPUJZVVNVVNVV PUNAVVPVVNVVNFUAZUJZFUUDZSZVVRAVVOUKJUUAUDZVVNUUBUDZVWAULZSZVWBAEVW @@ -571083,7 +571083,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= cn eqtrid 3eqtr4a sseqtrd sylan2br npcan1 sylan9eqr elfzm1b syl2anc nncnd cle elfzle1 0red nnm1nn0 lenltd iffalsed oveq2 oveq1d sylancr imaundi eqtr3id fneq2d inidm eqidd fvun2 sylan fvconst2 ofval mpdan - mp3an12 elmapi ffvelcdmda elfzonn0 nn0cnd addid1d fvun1 poimirlem10 + mp3an12 elmapi ffvelcdmda elfzonn0 nn0cnd addridd fvun1 poimirlem10 3eqtrd adantrr 3eqtr3d wne gtned neneqd pm2.65da iman sylibr ssrdv ) ABEUAUBZUCUBZUDKUEUFZUGZFUAUBZUCUBZUXHUGZABUHZUXIUIZUXMUXLUIZUJZU KZUJUXNUXOUUAAUXQUXMUXFUAUBZUBZUDUPUFZUXSULZAUXQUKUXMKUDUMUFZIUBZUB @@ -571180,7 +571180,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= peano2uz eqeltrrd fzsplit2 syl2anc uncom difeq1d incom fzdisj 3eqtr4a eqtrid disj3 sseqtrd sylan2br nnred peano2rem cle elfzle2 ltm1d oveq2 id lelttrd oveq1 oveq1d eqtr3id mpbid inidm eqidd fvun2 mp3an12 sylan - fneq2d fvconst2 ofval mpdan elmapi ffvelcdmda elfzonn0 nn0cnd addid1d + fneq2d fvconst2 ofval mpdan elmapi ffvelcdmda elfzonn0 nn0cnd addridd 3eqtrd fvun1 adantrr nngt0d poimirlem5 wne gtned neneqd pm2.65da iman 3eqtr3d sylibr ssrdv ) ABEUAUBZUCUBZUDKUEUFZUGZFUAUBZUCUBZUXOUGZABUHZ UXPUIZUXTUXSUIZUJZUPZUJUYAUYBUUAAUYDUXTUXMUAUBZUBZUDUKUFZUYFULZAUYDUP @@ -571569,7 +571569,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= rneqd jctl eleq1d rspceeqv ex impbid bitrid eqrdv uneq12i eqtri sneqd ibir sylan9ssr con2d iffalsed anim12ci bitr4di 3eqtr2rd unass 3eqtr4d eqtr3i snssi 0cn unssi fconst feq2d ffvelcdmda sselid ifbothda elmapi - fun addid2d elfzonn0 nn0cnd ifcli addassd eqbrtrd cr eqeltrdi csbeq1d + fun addlidd elfzonn0 nn0cnd ifcli addassd eqbrtrd cr eqeltrdi csbeq1d lenlt ovex oveq2 csbie fvexd ffnd nfmpt1 nfco nfima nfxp nfun offval2 0re dffn5f 3eqtr4rd ) AIBUAKSUBUCZUDUCZGBUEZEUFUGZUHUIZWUGWUGSUJUCZUK ZEULUGZULUGZWULUFUGZSGUEZUDUCZUMZSUNZUOZWUNWUOSUJUCZKUDUCZUMZUAUNZUOZ @@ -571741,7 +571741,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= mp3an12 adantllr ifbid csbeq12dv mpteq2dv elrab2 eleq2d eqid elrnmpti rneqd csbex bitrdi elfzle1 0re eqeltrdi lenlt iffalsed rexbidva bitrd eqbrtrd biimpa r19.29a eqtr3 necon3d rexlimdva mpd leneltd ffvelcdmda - syl3anbrc nn0cnd addid1d ifbothda fmpttd elmap wreu simpr jctil jctir + syl3anbrc nn0cnd addridd ifbothda fmpttd elmap wreu simpr jctil jctir elfzelz fzaddel eleqtrd peano2z fzsubel pncan3oi oveq1i elfznn mp3an2 zcnd subadd2 bicomd eqcom 3bitr4g reu6i f1ompt sylanbrc f1osng ltnled sylancl nsyl disjsn 1re ltp1i zrei mpbi mto mpbir f1oun syl22anc uzid @@ -572117,7 +572117,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= fnun nncnd npcan1 elfzuz3 f1ofo foima eqtr3d fneq2d inidm eqidd f1ofn fzss1 eluzp1p1 uzss nnzd uzidd fveq2d eleqtrrd sseldd eluzfz2 fnfvima syl3anc fvun2 mp3an12 fvconst2 ofval mpidan sylan9eqr adantllr elrab2 - addid1d peano2zm nnred elfzle2 ltm1d lelttrd breqtrrd iftrued csbeq1d + addridd peano2zm nnred elfzle2 ltm1d lelttrd breqtrrd iftrued csbeq1d zred vex weq oveq1 csbie mpteq2dva eleq2d eqid elrnmpti bitrdi biimpa rneqd r19.29a biimpd rexlimdva elnnne0 nnm1nn0 elfzo0 simp2d elfzolt2 mpd w3a lttrd syl3anbrc adantlr ffvelcdmda ifbothda fmpttd elmap wreu @@ -572308,7 +572308,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= poimirlem1 biimpri poimirlem2 necon1bd biimparc anim2i anassrs ifbieq1d eleq1 breq1 nngt0d iftrued sylan9eqr oveq2 fz10 oveq1 ima0 xpeq1i csbie 0xp f1ofo foima ovexd fnconstg mp1i fvconst2 ffvelcdmda elfzonn0 nn0cnd - wfo eqidd addid1d offveq 0elfz ad2antrr fvexd fvmptd an32s mpdan eleq1d + wfo eqidd addridd offveq 0elfz ad2antrr fvexd fvmptd an32s mpdan eleq1d nnm1nn0 anbi2d imbi12d expcom mpcom eqtr3d biimpd mpan2d imp poimirlem9 vtoclga mpand jca31 ltp1d fveq1 cr ltp1 ovex fpr f1oi disjdif fun mpan2 prid1 elun1 fvco3 fnresi pm3.2i fvun1 mp3an23 fvpr1 fveq2d wf1 f1of1 cc @@ -572557,7 +572557,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= c2nd fveq2 eqeq2d vex eqid ax-mp nfeq2 zred necomd eldifsn sylanbrc cr elfzelzd ad2antrr cle syl2an ad2antlr syl2anr mpbir2and breq1 id oveq1 ifbieq12d cof cdif nncnd nnzd peano2zm uzid peano2uz eqeltrrd - cop cfzo wo elun fzofzp1 elfzoelz zcnd addid1d elfzofz eqeltrd jaod + cop cfzo wo elun fzofzp1 elfzoelz zcnd addridd elfzofz eqeltrd jaod fzss2 biimtrid imp 1ex c0ex pm3.2i wf1o ccnv wfun wfo simprbi imain dff1o3 elfznn0 fzdisj ima0 eqtrdi sylan9req fun sylancr cn0 nn0p1nn ltp1d nnuz eleqtrdi elfzuz3 imaundi eqtr2di f1ofo foima feq2d inidm @@ -572671,7 +572671,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= crab neq0 iddvds df-2 breqtri rabfi neldifsn hashunsng difsnid fveq2d mp2 eqtr3id ralrimiva nfne neeq2d rspc mpan9 nesym cmap cab f1oexrnex elmap mpan2 ibir opelxpi imbi2d c1st cof elun fzofzp1 elfzonn0 nn0cnd - elabg addid1d elfzofz eqeltrd jaod imp xp1st xp2nd fvex elab 1ex c0ex + elabg addridd elfzofz eqeltrd jaod imp xp1st xp2nd fvex elab 1ex c0ex elmapi wfo ccnv wfun dff1o3 imain simplbiim clt elfznn0 nn0red fzdisj ltp1d ima0 eqtrdi sylan9req fun nn0p1nn nnuz elfzuz3 fzsplit2 imaundi f1ofo foima sylan9eqr feq2d fzfid inidm off feq1 vtocl an12s vtoclgaf @@ -572799,7 +572799,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= nfne cbvcsbw eqtrid impcom adantll 1st2nd2 ad3antlr neeqtrd poimirlem25 xp1st cbvrexw bitr2id iba sylan9bb rabbidv breqtrd dvds0 hash0 breqtrri rspc ex simpr ralrimivw rabeq0 breqtrrid pm2.61d1 hashxp hashsng oveq1i - con3i nn0cni mulid2i 3eqtri breqtrrdi fsumdvds cuz nncnd npcan1 nnm1nn0 + con3i nn0cni mullidi 3eqtri breqtrrdi fsumdvds cuz nncnd npcan1 nnm1nn0 cc nn0zd uzid peano2uz 3syl eqeltrrd fzss2 ssralv wnel raldifb biimparc nnel velsn cle nnred ltm1d nn0red ltnled mpbid elfzle2 syl5ibrcom con2d clt nsyl imp syl5 expdimp an32s biimtrid idd jad biimtrrid con3d dfrex2 @@ -573403,10 +573403,10 @@ curry M LIndF ( R freeLMod I ) ) ) $= nncn mpd simpll anim1i renegcld ltleadd syl21anc abstrid lelttr mpand jaod 3syld jctild adantld cxr rpxrd cxmet rexmet elbl mp3an1 fnconstg elmapfn mp1i fzfid inidm fvconst2 remetdval eqtrd anbi2d rpxr fzofzp1 - vex cfzo elfzonn0 addid1d elfzofz eqeltrd biimtrid dffn3 fssd divdird + vex cfzo elfzonn0 addridd elfzofz eqeltrd biimtrid dffn3 fssd divdird nn0cnd subnegd eqtr4d adantllr negcld subsubd fveq2d 2halvesd breq12d 3eqtrd eqcomd anbi12d 3imtr4d ralimdva nndivre rpregt0d divge0 syl2an - off elfzle1 elfzle2 cmul 1red ledivmuld mulid1d breq2d mpbird elicc01 + off elfzle1 elfzle2 cmul 1red ledivmuld mulridd breq2d mpbird elicc01 syl3anbrc impr a1i eleq2i elmap bitri sylibr jctird elin elixp anbi1i syl6ibr ssel com12 syl6 impd ralrimdva expd 3exp2 imp43 r19.29 fveq1d eqtr4di rspcev rexlimivw expcom ralrimdvva syl6d rexlimdva syld com23 @@ -573545,7 +573545,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= fveq1 ralrimdva reximdv ccmp ctg cixp wtru ctop fconst6 mp2an eqeltri cpt cuni cha cin c0 wo c1st cof cabs cmin cbl wel cpr elfzonn0 nn0red ccom nndivre clt elfzole1 jca rpregt0d divge0 elfzo0le cmul ledivmuld - 1red nncn mulid1d breq2d bitrd mpbird elicc01 syl3anbrc ancoms oveq2d + 1red nncn mulridd breq2d bitrd mpbird elicc01 syl3anbrc ancoms oveq2d elsni impr wf1o cab xp1st elmapfn 3syl df-f fconst inidm elmap sylibr off fmpttd frnd ominf cen nnenom enfi iunid imaiun dffn3 mpbi fimacnv fnmpti 3eqtr3ri eleq1i rexbii rexnal ralbii elnnuz fzouzsplit difeq1d @@ -573690,8 +573690,8 @@ curry M LIndF ( R freeLMod I ) ) ) $= anbi12d ad2antlr crest ccn cuni cioo crn ctg retop fconst6 pttop cicc cpt eqeltri reex unitssre mapss eqsstri uniretop restuni cnf ad2antrr ptuniconst simplr nnre redivcld jca nnrp rpregt0d divge0 elfzle2 cmul - nnne0 1red crp ledivmuld nncn mulid1d mpbird elicc01 syl3anbrc ancoms - bitrd impr c1st c2nd cfzo fzofzp1 elfzonn0 nn0cnd addid1d elfzofz wfn + nnne0 1red crp ledivmuld nncn mulridd mpbird elicc01 syl3anbrc ancoms + bitrd impr c1st c2nd cfzo fzofzp1 elfzonn0 nn0cnd addridd elfzofz wfn sylan wf1o cab ffvelcdmda xp1st elmapfn df-f sylanbrc cin pm3.2i wfun 1ex xp2nd fvex f1oeq1 elab dff1o3 simprbi imain elfznn0 nn0red fzdisj wfo ltp1d imaeq2d ima0 eqtrdi sylan9req imaundi nn0p1nn nnuz eleqtrdi @@ -573818,7 +573818,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= wor lenlt rspcev expcom sylan9 an32s rexbii rexnal bitr3i syl6ib sylan2 nne con4d syl5 ralunb ralrab anbi12i sylancr 3adant2 cmin zred 3ad2ant1 rspcva simpr1 simpll simplr nndivre elfzle1 jca rpregt0d divge0 elfzle2 - cicc nnrp cmul 1red crp ledivmuld nncn mulid1d mpbird elicc01 syl3anbrc + cicc nnrp cmul 1red crp ledivmuld nncn mulridd mpbird elicc01 syl3anbrc bitrd ancoms elsni oveq2d eleq1d syl5ibrcom impr simprr vex fzfid inidm fconst off eleq2i elmap sylibr 3adantr3 ancom wfn ffn ad2antrl fnconstg 3anass mp1i simplrr fvconst2 ofval anasss sylan2b nnne0 dividd ad2antlr @@ -574395,7 +574395,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= eluz mpbird ad3antlr seqsplit ad2antrr csn elfzelz nnred ad3antrrr nngt0d 0red zred elfzle1 elnnz sylanbrc nnre ltp1d ltnled breq1 equcoms sylan9bb elfzle2 nsyl ad4ant14 fvco3 fveq2i abs0 elfzuz c0ex fvconst2 ser0 oveq12d - 0m0e0 recnd addid1d ad5ant15 leidd eqbrtrd simpr simplr letrd elfz mp3an2 + 0m0e0 recnd addridd ad5ant15 leidd eqbrtrd simpr simplr letrd elfz mp3an2 ad2antlr ad5ant2345 eqeltrd biimpar absge0d breqtrrd syldan sermono ralrn mpbir2and ltlecasei r19.21bi lensymd supmax 3eqtr3rd c1st 1st2nd2 eqtr4di c2nd xp1st xp2nd iccssre eqsstrd iunss uzid iunconst iccid snssi eqsstrdi @@ -574588,7 +574588,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= jca eqtri ad6antr syl21anc eleq1w vtoclga sylan9eq oveq2d adantll simp-6r subaddd simpll simp-4l simprlr ltsub23d eqbrtrrd ad5antlr simp-4r npncand letrd simplrl sylan9ssr eqtr4d 3eqtrd ad3antlr simprr lesubadd2d leltaddd - eqtrd simprrr lt2addd df-3 2cn ax-1cn addcomi oveq1i rpcnd adddir mulid2d + eqtrd simprrr lt2addd df-3 2cn ax-1cn addcomi oveq1i rpcnd adddir mullidd 2timesd oveq12d 3cn 3ne0 divcan2 mp3an23 eqtr3d lelttrd eqbrtrid ltsub13d subadd2d 3eqtr4rd breqtrrd fvex rexlimdvva syld exp31 com34 3imp1 eqsupd exlimddv ) BFGZBHIZFJZKZCFGZCHIZFJZKZWUFDUHZBGZAUHZWUMUAIZLZKZDUIUJZUUAIZ @@ -575264,8 +575264,8 @@ curry M LIndF ( R freeLMod I ) ) ) $= cico rabbidva cxr clt 0xr imaeq2i eqid mptpreima 3eqtr3ri eqtrdi mbfimasn imaundi mp3an3 mbfima unmbl syl2anc eqeltrd inmbl iffalse recnd mpteq2dva mbfres sylan2b eqtrid cmnf wb elioomnf ltnle bitr3d bitrid eqeltrrd rabxm - a1i mbfres2 pnfxr 0ltpnf snunioo eqeltrid anandi 3bitr4i ad2antll addid2d - mp3an adantrr eqtrd ssid dfrab3ss ineq1i 3eqtrri oveq2d addid1d sylan9eqr + a1i mbfres2 pnfxr 0ltpnf snunioo eqeltrid anandi 3bitr4i ad2antll addlidd + mp3an adantrr eqtrd ssid dfrab3ss ineq1i 3eqtrri oveq2d addridd sylan9eqr indir anasss mbfpos eqcomi ) ACMEUAUBZBCUCZUVCUDZBCUCZBCMDUAUBZDMUEZUVCEM UEZUFUHZNZABCUVJOABUIZCPZQZUVHUVIUVNDOPZMOPZUVHOPGUGUVGDMOUJUKZUVNEOPZUVP UVIOPIUGUVCEMOUJUKULUMZAUVDUVGBCUCZUVDUNZUVGUDZBCUCZUVDUNZUVKUVDRZACOUVKU @@ -575462,7 +575462,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= cnvimarndm difeq1i difeq12d mpbi wfn ffn fniniseg syl6bi itg10a breqtrd dfss4 difeq1d biantrurd eldif bitr4di bitr4d con2bid 3syl 0m0e0 ifeq2da cof simpll 1ex fconstmpt resubcl 2re cvol i1fima mblvol i1fima2 remulcl - neldifsn 2cnd 2ne0 mulne0d redivcld mulid1d mul01d ifeq12d i1f1 i1fmulc + neldifsn 2cnd 2ne0 mulne0d redivcld mulridd mul01d ifeq12d i1f1 i1fmulc i1fsub sylan9eqr ianor ifbid ifboth sylancl pm2.61d1 jaoi sylbi pm2.61i eleq1w fvmpt eqtr4id mpteq2ia ad2antrl posdifd mblss ovolge0 ltlen 2pos wo i1fpos mulgt0 mpanl12 divgt0d elrpd inidm anim2i breq1 divcld npcand @@ -575808,7 +575808,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= elrege0 sylib simprd adantr wn neeq1 oveq1 breq1d imbi12d rge0ssre sselid ad2antlr rerpdivcld reflcl peano2rem 3syl remulcld lesub1dd lemul1d mpbid syl leadd1dd cc recnd ax-1cn sylancl a1i rpcnd adddird oveq2d eqtrd bitri - mulid2d readdcld ifbothda imp eqbrtrd cvv reex c0ex eqidd ofrfval2 mpbird + mullidd readdcld ifbothda imp eqbrtrd cvv reex c0ex eqidd ofrfval2 mpbird feqmptd oveq2 ifeq2d mpteq2dv rspcev oveq12d resubcld cneg eqtr3d offval2 wb c2 eqeq2d anbi1d 2exbidv rexcom4 cofr cdm itg2addnclem2 simplr rpdivcl i1fsub 3rp mpan2 weq fvoveq1d neeq1d ifbieq12d eqid fvmpt cpnf cico df-ne @@ -575957,8 +575957,8 @@ curry M LIndF ( R freeLMod I ) ) ) $= bitr3d mpan2 3syl eqeltrrd ressxr rgen brralrspcev sylancl supadd oveq12d sstri eqtrd ge0addcl sselid inidm off itg2addnclem3 simpl i1fadd ad3antlr reeanv biimpri ad2ant2r ifcl anbi1d oveq12 eqtrdi adantll simplbi addge0d - 00id simpll a1d wn oveq1 simplrr i1ff addid2d rpre min2 leadd2dd addge02d - syl2an mpbid letrd adantld oveq2 simplrl addid1d addge01d adantrd addassd + 00id simpll a1d wn oveq1 simplrr i1ff addlidd rpre min2 leadd2dd addge02d + syl2an mpbid letrd adantld oveq2 simplrl addridd addge01d adantrd addassd ifboth ltaddrpd ltled le2add syl22anc syl2and ralimdva wb anbi12d bitr4di min1 r19.26 ovexd wfn eqeq1d ifbieq2d offval 3imtr4d ifeq2d mpteq2dv syl5 ofval a1dd imp31 ad2ant2l itg1add eqcomd eqtr ancoms eqeq2d syl12anc impd @@ -576307,7 +576307,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= eqtrd 00id wne ovif2 wn wb negne0bd biimpa le0neg2d leloe bitrd df-ne clt wo biorf sylbi orcom bitr2di sylan9bb syldan ltnle adantr addcomd negidd ifbieq12d ifnot eqtrid pm2.61dane 3eqtrd eqeltrrd itgaddnclem1 - addid1d eqeltrd le0neg1d bitr3d 3eqtr3d itgcl addcld itgreval 3eqtr4d + addridd eqeltrd le0neg1d bitr3d 3eqtr3d itgcl addcld itgreval 3eqtr4d cc ) ABCNDEOPZQUAZYJNUBZUCZBCNYJUDZQUAZYNNUBZUCZUEPZBCNDQUAZDNUBZUCZB CNDUDZQUAZUUBNUBZUCZUEPZBCNEQUAZENUBZUCZBCNEUDZQUAZUUJNUBZUCZUEPZOPZB CYJUCBCDUCZBCEUCZOPAYRBCYTUUHOPZUCZBCUUDUULOPZUCZUEPZUUAUUIOPZUUEUUMO @@ -576489,7 +576489,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= wn elrege0 0e0icopnf imcld eqidd offval2 wceq iftrue oveq12d 00id iffalse eqtr4d 3eqtr4a pm2.61i mpteq2i fveq2d eqid iblcn mpbid simpld iblabsnclem eqtr2di simprd itg2addnc eqtrd readdcld eqeltrd cofr addge0d wral cmul cc - ax-icn mulcl sylancr abstrid adantl replimd absmul c1 absi oveq1i mulid2d + ax-icn mulcl sylancr abstrid adantl replimd absmul c1 absi oveq1i mullidd eqtrid eqtr2d oveq2d 3brtr4d ex pm2.61d1 ralrimivw ofrfval2 mpbird itg2le 0le0 syl3anc itg2lecl iblpos mpbir2and ) ABCDIJZKZUALYJUBLBMBUEZCLZYINUFZ KZOJZMLZHAMNUCUDPZYNUGZBMYLDUHJZIJZDUIJZIJZQPZNUFZKZOJZMLYOUUFRSZYPABMYMY @@ -576534,7 +576534,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= itg2mulc fconstmpt oveq2d adantl remulcld c1 oveq1i 3brtr4d breq1 syl3anc absi 3eqtr4d pm2.61dan mpteq2dv 3eqtr3d mulge0d ad2antrr releabsd absdivd mul01d cofr elfznn0 absexp 1exp eqtrid div1d 3eqtrd absmuld breqtrd mulcl - cn0 abstrid replimd absmul eqtrdi mulid2d eqtr2d lemul2ad letrd ifboth ex + cn0 abstrid replimd absmul eqtrdi mullidd eqtr2d lemul2ad letrd ifboth ex syl2anc 0le0 pm2.61d1 eqbrtrid ralrimivw ofrfval2 mpbird itg2le ralrimiva itg2lecl isibl2 mpbir2and ) ABCEDKLZMZUBNUWDUCNBOBUDZCNZPUWCUHUAUDZUELZUF LZUGQZRSZUIUWJPUJZMZUKQZONZUAPULUMLZUNJAUWOUAUWPAUWGUWPNZUIZOPUOUPLZUWMUQ @@ -577132,7 +577132,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= c0p cof subcl anassrs weq eleq1w negeqd ifbieq12d negex oveq12d ccnv c1 ovex wo crn frn ref ax-mp fnco mp3an1 elpreima 3syl fco biantrurd fvco3 bitr4di pm5.32da eldif baibr 0le0 breqtrri biantru bitr3di orbi12d elun - bitr3d elimif 3bitr4g offval2 ovif12 0cn addid1d mulm1d addid2d ifeq12d + bitr3d elimif 3bitr4g offval2 ovif12 0cn addridd mulm1d addlidd ifeq12d ifbid eqtrid sylan9eq pnfxr 0ltpnf snunioo mp3an imaeq2i imaundi eqtr3i cioo 0xr ismbfcn mbfimasn mp3an3 mbfima unmbl fdmd difmbl eqcomd ifeq1d mbfdm i1fres neg1rr i1fmulc cmmbl ifnot tru fconst inidm fvconst2 ofval @@ -577274,7 +577274,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= eldifn iffalsed iftrue iblss2 iblcn simpld ftc1anclem4 ancoms eqeltrrid cdif 3expb simprd imcld anim12dan rpred lt2add an32s cofr i1ff adantlrr cof subcl adantlrl addcld elxrge0 cico icossicc ge0addcl sselid elrege0 - reex inidm off abstrid ralrimiva ovexd eqidd absmul absi oveq1i mulid2d + reex inidm off abstrid ralrimiva ovexd eqidd absmul absi oveq1i mullidd eqtr2d offval2 ofrfval2 itg2le syl3anc ccom absf cofmpt resubcl crn csn fvexd iunin2 imaiun iunid imaeq2i eqtr3i ineq2i cnvimass dmmpti sseqtri cnvimarndm fdmd sseqtrrid df-ss sylib frnd ad2antrr sylan 2thd ltaddsub @@ -577430,7 +577430,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= i1f0rn fimaxre2 suprcl 0red adantrr absgt0 biimpa 3jca ltletrd rexlimdvaa anasss elrpd imp rpmulcl rerpdivcld simp-4l iccssre anandis rpdivcl rpred adantlll sseld idd 3anim123d ifcl 2timesd readdcld abs0 eqeltrrid addge0d - readdcl abstri c1 absmul absi oveq1i eqtrdi mulid2d ffnd fnfvelrn le2addd + readdcl abstri c1 absmul absi oveq1i eqtrdi mullidd ffnd fnfvelrn le2addd eqtrd elun2 letrd 3brtr4d iffalse pm2.61d1 ovex cico covol mblvol ovolioo eqtrid resubcl ancoms 3adant3 elrege0 ge0addcl itg2const ledivmuld eqtr4d ex abssubge0 simpr lelttrd ltdiv1d lt2addd ) AJUCZUDUEZUFZKUCZVWRUFZUGZUG @@ -577556,9 +577556,9 @@ curry M LIndF ( R freeLMod I ) ) ) $= simplll 3simpa cof ioossre cvol fvex cdif wn eldifn resmpt eqtr4i csn cxp rembl i1ff mulcl addcl fconstmpt fveq2 ftc1anclem3 i1fmbf ioombl eqeltrid mbfres cico elxrge0 c0p cofr frn ax-resscn sstrdi fnco mp3an1 inidm fvco3 - offval addid1d mpteq2dva it0e0 mpteq2i coeq2d i1f0 mpan2 weq coeq2 eleq1d + offval addridd mpteq2dva it0e0 mpteq2i coeq2d i1f0 mpan2 weq coeq2 eleq1d 3eqtr4d vtoclga i1fadd ge0addcl 0plef simprd itg2itg1 itg1cl icossicc fss - sylib eqeltrd 0re readdcl leidd ifboth abstri absmul absi mulid2d breqtrd + sylib eqeltrd 0re readdcl leidd ifboth abstri absmul absi mullidd breqtrd oveq1i ralrimiva elioore sylan2 jca df-icc elixx3g simprbi simpld anim12i letrd ioossioo sstrd sselda breqtrrd 0le0 ralrimivw subcld cibl cpr ccncf wb recncf prid1g ftc1anclem2 mp3an3 imcncf prid2g recld off 0xr 0red cneg @@ -577958,7 +577958,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= sqcl rere breq2d biimpac ne0gt0d ltnle mpbid orcomd 3ianor 3orrot 3bitrri imor olcd 3orass xchbinxr eldifi mnflt0 mpbiri necon3bi ccnfld cnfldtopon ctopn dvmptid toponrestid crest recld2 iocmnfcld icopnfcld tgioo2 eleqtri - ctg uncld fveq2i restcldr toponunii cldopn eqeltri mulid1i sqcld asindmre + ctg uncld fveq2i restcldr toponunii cldopn eqeltri mulridi sqcld asindmre elin eqimssi sylbir incom df-ioc df-ioo xrltnle ixxdisj crp elioore rexri resqcld elioo2 cabs recn abscld absge0d lt2sq mpanr12 abslt resqcl posdif absresq breq12d sylancl bitrd 3bitr3d biimpd 3impib sylbi elrpd eleqtrrdi @@ -578128,7 +578128,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= cmin elioore recnd rpcn wne rpne0 divcld asincl 1cnd sqcld subcld sqrtcld syl mulcld addcld ovexd w3a cxr wb rpre renegcld rexrd rpxr syl2anc simpr elioo2 redivcld a1d mulm1d breq1d neg1rr simpl ltmuldivd bitr3d ltdivmuld - biimpd 1red mulid1d exp4b 3impd sylbid imp 1xr mp2an recn dvmptid ioossre + biimpd 1red mulridd exp4b 3impd sylbid imp 1xr mp2an recn dvmptid ioossre id iooretop dvmptres ax-resscn oveq2d resqcld resubcld cabs absltd abscld wss absge0d cle lt2sqd absresq breq12d resqcl posdifd elrpd 0red mpteq2ia 3bitrd dvmptco 2cnd mulneg2d oveq1d negcld 3eqtr3rd eqtrdi mulcomd sqvald @@ -578138,7 +578138,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= wf 0le1 dvmptc ctopon cnfldtopon toponmax mp1i cin df-ss mpbi dvexp ax-mp cn 2nn 2m1e1 oveq2i exp1 eqtrid dvmptres3 dvmptsub df-neg mpteq2i eqtr4di eqtri dvsqrt gt0ne0d sqr00d ex necon3d mpd 2ne0 divcan5d mulne0d divrec2d - dvmptmul dvmptadd mulid2d divassd addcomd 2timesd negsubd sqsqrtd divdird + dvmptmul dvmptadd mullidd divassd addcomd 2timesd negsubd sqsqrtd divdird negeqd eqtr4d addassd 3eqtrrd oveq1 mulassd divrecd 3eqtr3d mul12d sqge0d dvmptcmul rpge0 eqcomd rpgt0 0le0 sq0 bitrd mpbid 3bitr3rd ltled sqrtmuld subdid sqne0 mpbird divcan2d sqrtsqd 3eqtrd mpteq2dva ) BUBDZEABUCZBUDFZB @@ -578275,7 +578275,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= wa syl ax-icn mulcld 1cnd subcld sqrtcld addcld wn wo w3a clt 0lt1 oveq1d simp3 div0d 3ad2ant1 eqtrd oveq2d it0e0 eqtrdi fveq2d oveq2i 1m0e1 fveq2i sq0 eqtri sqrt1 oveq12d 0p1e1 breq2d 0red 1red eqeltrd ltnled mpbii 3expa - bitr3d olcd inelr 3adant3 mulassd mulid1d simpr redivcld resqcld resubcld + bitr3d olcd inelr 3adant3 mulassd mulridd simpr redivcld resqcld resubcld wi adantl sqdivd breq1d bitrd mp2an eqtr4d mpteq2dva crest ctopon mulcncf wb divrec2d mpbird cncfmpt2f cncfcn pncand divcan6d 3eqtrrd elicc2 resqcl subge0d recn rpgt0 0le0 rpge0 lt2sqd mpbid elrpd ledivmuld absresq eqcomd @@ -578421,7 +578421,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= rpne0 dvmptntr weq 3eqtr4rd eqtr4d ccncf sstri ssid 3pm3.2i cncfmptc mp1i 2cn cres resmpt areacirclem2 rescncf mpsyl eqeltrrid mulcncf ioombl iblss ioossicc areacirclem4 ftc2nc eqidd ubicc2 dividd asin1 mul01d 2ne0 divcli - addid1d lbicc2 divnegd ax-1cn asinneg neg1sqe1 negcli pidiv2halves subdid + addridd lbicc2 divnegd ax-1cn asinneg neg1sqe1 negcli pidiv2halves subdid sq1 subnegi mulcomd 3eqtr3d pm2.61dane ) CHIZJCKLZMZDUENZFHDFUAZUBUCZUDNZ UFZUGCOUHPZQPZVXGDUEUIIZVXHVXLRVXGDHHUUAZUJZVXJUDUUBHUCZIZFHUUCFHVXKUKZUL IVXOVXQVXGDAUAZHIBUAZHIMVYAOUHPVYBOUHPUMPVXMKLZMABUUJVXPEVYCABHHUUDUUEUNV @@ -580683,9 +580683,9 @@ counterexample is the discrete extended metric (assigning distinct ad2antlr adantr wn letrd absled elfz syl3anc eqeq2d adantl frnd cnex cmet syl2an fmpttd elpw2 sylibr bnd2lem mpan sselda adantrl recld simprl imcld gzreim rpne0d divcld subcld addsub4d redivd imdivd oveq12d 3eqtr4d elrpii - absreimsq resqcli mp1i sqvali halflt1 ltmul1ii mulid2i breqtri lt2halvesd + absreimsq resqcli mp1i sqvali halflt1 ltmul1ii mullidi breqtri lt2halvesd eqbrtri eqbrtrd sq1 breqtrrdi 0le1 lt2sqd ltmul2dd mulcld divcan2d eqtr3d - rpge0 3eqtrrd mulid1d cxmet rpxr ad2antrl elbl2 syl22anc expr eliun rgenw + rpge0 3eqtrrd mulridd cxmet rpxr ad2antrl elbl2 syl22anc expr eliun rgenw 3brtr3d rexrnmptw ax-mp bitri ssrdv simpl ssbnd sylib cneg cmpo fzfi xpfi cfz cxp wfo wfn dffn4 fofi elrnmpti elgz simp2bi simpllr simplrl readdcld fnmpoi peano2zd absrele cxad simplrr sslin adantlr rpxrd rexrd w3a bldisj @@ -581753,9 +581753,9 @@ the next (since the empty set has a finite subcover, the uzid sylibd jcad cr ad2antrr wf algrf ffvelcdmda syl3anc ffvelcdmd rpre metcl ad2antlr readdcld mettri2 syl13anc cmul rpred remulcld ralrimivva lt2halves jca oveq1 oveq2d breq12d oveq2 rspc2v sylc metge0 1re sylancl - 1red ltle mpd lemul1ad recnd mulid2d breqtrd letrd leadd1dd mpand 3syld + 1red ltle mpd lemul1ad recnd mullidd breqtrd letrd leadd1dd mpand 3syld lelttr syl5 rexlimdva wb 0re syl2an breqtrrd ralrimiva alrple mpbir2and - addid2d mpbird letri3 meteq0 mpbid id eqeq12d rspcev ) AHIUCUDZUDZKUEZU + addlidd mpbird letri3 meteq0 mpbid id eqeq12d rspcev ) AHIUCUDZUDZKUEZU UOGUDZUUOUFZDUGZGUDZUUSUFZDKUHAIKUIUDUEZHUUOUUNUJZUUPAFKUKUDUEZFKULUDUE ZUVBAFKUOUDUEUVDLFKUMUPZFKUNZFIKRUQURABCEFGHIJKLMNOPQRSTUSZUUOHIKUTVAZA UUQUUOFVBZVCUFZUURAUVKUVJVCVDUJZVCUVJVDUJZAUVLUVJVCBUGZVEVBZVDUJZBVFVGZ @@ -581802,7 +581802,7 @@ the next (since the empty set has a finite subcover, the wreu sylib c1st ccom csn cxp cseq cmopn ccmet adantr crp clt cmul adantlr cn wf eqid simpr bfplem2 exlimddv cmin oveq12 adantl eqbrtrrd cmet cr syl cmetmet ad2antrr simplrl simplrr metcl syl3anc rpred remulcld mpbird 1cnd - suble0d recnd subdird mulid2d oveq1d eqtrd resubcl sylancr mul01d 3brtr4d + suble0d recnd subdird mullidd oveq1d eqtrd resubcl sylancr mul01d 3brtr4d 1re wb 0red posdif sylancl mpbid lemul2 syl112anc metge0 letri3 mpbir2and 0re meteq0 ex ralrimivva fveq2 id eqeq12d anbi1d equequ1 imbi12d cbvralvw ralbidv reu4 sylanbrc ) ADUBZFUCZYFUDZDHUEZYHCUBZFUCZYJUDZUFZDCUGZUHZCHUI @@ -582133,7 +582133,7 @@ the next (since the empty set has a finite subcover, the cress cds cres chash csqrt caddc crrn repwsmet rrnmet hashcl nn0re nn0ge0 cxp cn0 resqrtcld cc0 sqrtge0d ge0p1rpd crp 1rp cv cmet metcl 3expb sylan a1i simprl simprr w3a metge0 syl3anc simpld remulcld peano2re id rrnequiv - jca simprd lep1d lemul1a syl31anc letrd recnd mulid2d breqtrrd cc ctotbnd + jca simprd lep1d lemul1a syl31anc letrd recnd mullidd breqtrrd cc ctotbnd wss cbnd wb ax-resscn cnpwstotbnd mpan equivbnd2 ) AUAIZGHUBJUDKAUCKZUELZ DDUPUFZBAUGLZUHLZMUIKZMWTAUJLZCDWTACWSWSNZWTNZEUKZACEULZWRXCWRXBUQIZXCJIZ AUMZXJXBXBUNZXBUOZUROZWRXJUSXCPQXLXJXBXMXNUTOVAMVBIWRVCVIWRGVDZCIZHVDZCIZ @@ -655274,8 +655274,8 @@ fixed reference functional determined by this vector (corresponding to (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.) $) hlhilsmul $p |- ( ph -> .x. = ( .r ` R ) ) $= - ( cmulr mulrid cnx cstv cfv starvndxnmulrndx necomi hlhilslem ) ACBDEOF - GHIJKLMPQRSQOSTUANUB $. + ( cmulr mulridx cnx cstv cfv starvndxnmulrndx necomi hlhilslem ) ACBDEO + FGHIJKLMPQRSQOSTUANUB $. $( Obsolete version of ~ hlhilsmul as of 6-Nov-2024. The scalar multiplication for the final constructed Hilbert space. (Contributed @@ -656071,7 +656071,7 @@ fixed reference functional determined by this vector (corresponding to ( c1 c5 c2 cdc cgcd co clt wbr cr wcel wb mp2an mpbir cprime cmul 5nn 2nn caddc cc0 eqtr3i wceq wne wo 2lt5 olci 5re 2re 5prm 2prm gcdaddmzz2nncomi lttri2 prmrp nnzi mulcomnni 5t2e10 oveq1i 1nn0 0nn0 nnnn0i eqid dec0h 2cn - 1p0e1 addid2i decadd eqtri oveq2i decnncl gcdcomnni eqtr2i ) ABACDZEFZVKB + 1p0e1 addlidi decadd eqtri oveq2i decnncl gcdcomnni eqtr2i ) ABACDZEFZVKB EFBCEFZAVLVMAUAZBCUBZVOBCGHZCBGHZUCZVQVPUDUEBIJCIJVOVRKUFUGBCUKLMBNJCNJVN VOKUHUIBCULLMVMBCBOFZCRFZEFVLCBCPQCQUMUJVTVKBEVTASDZCRFVKVSWACRBCOFVSWABC PQUNUOTUPASSCACWACUQURURCQUSZWAUTCWBVAVCCVBVDVEVFVGVFTBVKPACUQQVHVIVJ $. @@ -656081,7 +656081,7 @@ fixed reference functional determined by this vector (corresponding to $( The gcd of 60 and 6 is 6. (Contributed by metakunt, 25-Apr-2024.) $) 60gcd6e6 $p |- ( ; 6 0 gcd 6 ) = 6 $= ( c6 cc0 cdc cgcd co 6nn decnncl2 gcdcomnni c1 cmul nnnn0i 1nn0 0nn0 eqid - 6cn mulid2i mul02i decmul1 10nn eqtr3i mulcomnni oveq2i gcdmultiplei + 6cn mullidi mul02i decmul1 10nn eqtr3i mulcomnni oveq2i gcdmultiplei eqtri ) AABCZDEZUEADEAAUEFAFGHUFAAIBCZJEZDEAUEUHADUGAJEUEUHIBABAUGAFKLMUG NAOPAOQRUGASFUATUBAUGFSUCUDT $. $} @@ -656105,7 +656105,7 @@ fixed reference functional determined by this vector (corresponding to $( The gcd of 420 and 8 is 4. (Contributed by metakunt, 25-Apr-2024.) $) 420gcd8e4 $p |- ( ; ; 4 2 0 gcd 8 ) = 4 $= ( c8 c4 cgcd co c2 cdc cc0 c5 cmul caddc 8nn 4nn 5nn0 2nn decnncl c1 4nn0 - c6 1nn0 eqtr3i nnzi gcdaddmzz2nncomi deccl 6nn0 0nn0 dec0h nn0cni addid1i + c6 1nn0 eqtr3i nnzi gcdaddmzz2nncomi deccl 6nn0 0nn0 dec0h nn0cni addridi eqid 1p1e2 decsuc 6p4e10 decaddc2 8nn0 2nn0 0p1e1 8t5e40 8t2e16 mulcomnni decmul2c oveq1i oveq2i eqtr4i gcdcomnni 4t2e8 gcdmultiplei eqtri decnncl2 3eqtr3ri ) ABCDZABEFZGFZCDZBVLACDVJAHEFZAIDZBJDZCDVMVNABKLVNHEMNOZUAUBVLV @@ -656137,7 +656137,7 @@ fixed reference functional determined by this vector (corresponding to $( The lcm of 12 and 5 is 60. (Contributed by metakunt, 25-Apr-2024.) $) 12lcm5e60 $p |- ( ; 1 2 lcm 5 ) = ; 6 0 $= ( c6 cc0 cdc c1 c2 c5 1nn0 2nn decnncl 5nn 1nn 6nn decnncl2 12gcd5e1 6nn0 - 0nn0 mulid2i co caddc 5cn deccl nn0cni 5nn0 2nn0 eqid oveq1i 5p1e6 5t2e10 + 0nn0 mullidi co caddc 5cn deccl nn0cni 5nn0 2nn0 eqid oveq1i 5p1e6 5t2e10 cmul eqtri 2cn mulcomli decmul1c lcmeprodgcdi ) ABCZDUODECZFDEGHIJKALMNUO UOABOPUAUBQDEABFDUPUCGUDUPUEPGDFUIRZDSRFDSRAUQFDSFTQUFUGUJFEDBCTUKUHULUMU N $. @@ -656154,7 +656154,7 @@ fixed reference functional determined by this vector (corresponding to $( The lcm of 60 and 7 is 420. (Contributed by metakunt, 25-Apr-2024.) $) 60lcm7e420 $p |- ( ; 6 0 lcm 7 ) = ; ; 4 2 0 $= ( c4 c2 cdc cc0 c1 c6 c7 6nn decnncl2 7nn 1nn 4nn0 2nn 2nn0 deccl 0nn0 co - cmul 7cn mulcomli decnncl 60gcd7e1 nn0cni mulid2i 7nn0 6cn 7t6e42 addid1i + cmul 7cn mulcomli decnncl 60gcd7e1 nn0cni mullidi 7nn0 6cn 7t6e42 addridi 6nn0 eqid 2cn decaddi mul01i dec0h eqcomi eqtr4i decmul1c lcmeprodgcdi 0cn ) ABCZDCZEVAFDCZGFHIJKUTABLMUAIUBVAVAUTDABLNOPOUCUDFDUTDGDVBUEUIPVBUJ PPABBFGRQDLNPGFUTSUFUGTBUKUHULGDDDCZSUSGDRQDVCGSUMDVCDPUNUOUPTUQUR $. @@ -656165,7 +656165,7 @@ fixed reference functional determined by this vector (corresponding to 420lcm8e840 $p |- ( ; ; 4 2 0 lcm 8 ) = ; ; 8 4 0 $= ( c4 c8 cdc cc0 cmul co 4nn0 2nn decnncl decnncl2 8nn 4nn 8nn0 eqid 4t2e8 c2 eqtr4i eqtri 0nn0 caddc 420gcd8e4 mulcomnni oveq1i nnnn0i 4cn mulcomli - mulassnni 2cn 8cn addid1i 2t2e4 dec0h decmul2c decaddi 2t0e0 lcmeprodgcdi + mulassnni 2cn 8cn addridi 2t2e4 dec0h decmul2c decaddi 2t0e0 lcmeprodgcdi eqcomi oveq2i ) ABACZDCZEFZAUTAPCZDCZBVBAPGHIZJZKLUSBAMLIJUAVANVCBEFZAPVC EFZEFZVAVFAPEFZVCEFZVHVFBVCEFVJVCBVEKUBVIBVCEOUCQAPVCLHVEUGRVGUTAEVBDUSDP DVCPHUDZVBVDUDSVCNSSBAAPVBEFDMGSAPBAPDVBVKGVKVBNGSPAEFZDTFBDTFBVLBDTAPBUE @@ -656442,7 +656442,7 @@ fixed reference functional determined by this vector (corresponding to cv wa elunitcn caddc ax-1cn negsub mpan oveq1d adantl negcl wi cle wbr wb 1cnd nnnn0d nn0sub syl2anc mpbid binom 3com23 3expia syl5 imp w3a elfzelz eqtr3d nnzd zsubcl sylan 1exp syl 3adant2 3ad2ant2 elfznn0 3ad2ant3 expcl - sylan2 mulid2d eqtrd mulm1 eqtr4d neg1cn mulexp mp3an1 oveq2d bccl syl2an + sylan2 mullidd eqtrd mulm1 eqtr4d neg1cn mulexp mp3an1 oveq2d bccl syl2an 3adant1 nn0cnd sylancr mulassd mulcomd 3expa sumeq2dv itgeq2dv eqtrid ) A DBUAKUBLZBUHZEKMLNLZKXFMLZFEMLZNLZOLZUCBXEXGUAXIUDLZKUEZCUHZNLZXIXNUFLZOL ZXFXNNLZOLZCUGZOLZUCGABXEXKYAAXFXEPZUIXJXTXGOYBAXFQPZXJXTRXFUJAYCUIZXJXLX @@ -656532,7 +656532,7 @@ fixed reference functional determined by this vector (corresponding to lcmineqlem4 $p |- ( ph -> ( ( _lcm ` ( 1 ... N ) ) / ( M + K ) ) e. ZZ ) $= ( vk c1 cfz co caddc cdvds cn wcel wa syl wb wceq clcmf cfv cdiv cv breq1 wbr cz wss cfn wral fzssz fzfi pm3.2i a1i dvdslcmf cmin cc0 1zzd nnzd 0zd - zsubcld cle nnred leidd fznn mpbir2and 1cnd addid1d eqcomd wi nncnd eqcom + zsubcld cle nnred leidd fznn mpbir2and 1cnd addridd eqcomd wi nncnd eqcom npcand jca subcl addcom eqeq2 bitrd pm5.74i mpbi fzadd2d rspcdva lcmfnncl cc fz1ssnn ax-mp cn0 elfznn0 nnnn0addcl syl2anc nndivdvds mpbid ) AJDKLZU AUBZCBMLZUCLZAWOWNNUFZWPOPZAIUDZWNNUFZWQIWMWOWSWOWNNUEAWMUGUHZWMUIPZQZWTI @@ -656715,7 +656715,7 @@ fixed reference functional determined by this vector (corresponding to 1nn0 nnzd mpbid bccl2d div1d cfz w3a peano2zd peano2nnd nnge1d 3jca elfz1 1z mpan syl mpbird bcm1k pncand oveq2d oveq1d oveq12d nnred ltled divassd subcld eqcomd divmul2d mulcomd divcan3d mul12d cc0 wne 0ne1 mulne0d gtned - necomd subne0d recbothd mulid1d divmuldivd ) AGBGUAHZCWNIHZJHZKHZBGJHZCBL + necomd subne0d recbothd mulridd divmuldivd ) AGBGUAHZCWNIHZJHZKHZBGJHZCBL HZBCBIHZJHZJHZKHZBWSKHGXAKHJHAWQBXBKHZXCAWQXDMWPGKHZXBBKHZMAXEBWSWTJHZJHZ BKHZXFAXEXGXIAXEWPXGAWPAWNWOABGABDNZAUBZUCZAWOAWNCEABGABDUDZGUEOAUNUFUGAB CUHUIZWNCUJUIZFABPOCPOZXNXOUKABDUOZACEUOZBCULUMUPZUQZNZQZURAWPWTWSJHZXGAY @@ -656742,7 +656742,7 @@ fixed reference functional determined by this vector (corresponding to = ( 1 / ( 1 x. ( N _C 1 ) ) ) ) $= ( vx vy cc0 c1 co cmin cexp cmul wcel cc wceq syl adantr eqtrd cmpt a1i cicc cv citg cdiv elunitcn wa 1m1e0 oveq2i simpr exp0d eqtrid oveq1d 1cnd - cbc subcld cn0 cn nnm1nn0 expcld mulid2d sylan2 itgeq2dv cioo 0red itgioo + cbc subcld cn0 cn nnm1nn0 expcld mullidd sylan2 itgeq2dv cioo 0red itgioo 1red cfv eqidd oveq2 adantl adantlr cr elioore recn 3syl fvmptd cneg wral cdv cpr cnelprrecn nnnn0 nn0cnd mulcld negcld 0cnd dvmptc dvmptsub df-neg dvmptid mpteq2dv eqtr4d dvexp oveq1 oveq2d dvmptco nncnd nnne0d dvmptcmul @@ -656751,7 +656751,7 @@ fixed reference functional determined by this vector (corresponding to ax-1cn ssid mp3an cncfmptid mp2an subcncf ssidd cncfcompt2 resopunitintvd expcncf eleq1d mpbird cibl ioossicc cvol cdm ioombl resclunitintvd eqtr3d w3a 3jca cnicciblnc iblss eqeltrd mp3an23 mulcncf ftc2 0exp 1elunit 1m0e1 - mul01d cz 1exp mulid1d 0elunit oveq12d divnegd eqtr2d reccld negnegd bcn1 + mul01d cz 1exp mulridd 0elunit oveq12d divnegd eqtr2d reccld negnegd bcn1 nn0zd ) ABGHUAIZBUBZHHJIZKIZHUVDJIZCHJIZKIZLIZUCBUVCUVIUCZHHCHUNIZLIZUDIZ ABUVCUVJUVIUVDUVCMZAUVDNMZUVJUVIOUVDUEZAUVPUFZUVJHUVILIUVIUVRUVFHUVILUVRU VFUVDGKIHUVEGUVDKUGUHUVRUVDAUVPUIZUJUKULUVRUVIUVRUVGUVHUVRHUVDUVRUMUVSUOA @@ -656905,9 +656905,9 @@ fixed reference functional determined by this vector (corresponding to ( c1 caddc co c2 cmul cfa cfv cdiv cmin cc0 wcel a1i cle oveq1d eqtrd syl wceq cbc cfz 0zd cz 2z nnzd zmulcld peano2zd 1red nnnn0d nn0ge0d wbr 0le1 nnred addge0d readdcld addge01d mpbid recnd add32d 2timesd eqcomd breqtrd - 1cnd elfzd bcval2 addsub4d pncand 1m1e0 oveq12d addid1d fveq2d oveq2d cn0 + 1cnd elfzd bcval2 addsub4d pncand 1m1e0 oveq12d addridd fveq2d oveq2d cn0 zcnd cn faccl nncnd 1nn0 nn0addcld mulcomd facp1 addcld mulassd nn0mulcld - 2nn0 mulcld peano2nnd nnne0d mulne0d divassd dividd divcld mulid2d breq2d + 2nn0 mulcld peano2nnd nnne0d mulne0d divassd dividd divcld mullidd breq2d divmuldivd bitr4d ) ABDEFZGBHFZDEFZWRUAFZHFZWTWSIJZBIJZXDHFZKFZHFZWTWSBUA FZHFZAXBWTXCHFZXEKFZXGAXBWRWRKFZXKHFZXKAXBWRXJHFWRXEHFZKFZXMAXBWRXJXNKFZH FZXOAXAXPWRHAXAWTIJZXNKFZXPAXAXRXDWRIJZHFZKFZXSAXAXRWTWRLFZIJZXTHFZKFZYBA @@ -657087,7 +657087,7 @@ fixed reference functional determined by this vector (corresponding to 3exp7 $p |- ( 3 ^ 7 ) = ; ; ; 2 1 8 7 $= ( c3 c2 c1 cdc c8 c7 c6 3nn0 c9 co 7nn0 2nn0 1nn0 8nn0 0nn0 cmul caddc c4 cc0 4nn0 6nn0 6p1e7 cexp deccl 9nn0 3cn 3t2e6 mulcomli 3exp3 dec0h nn0cni - 2cn eqid mul02i 6cn ax-1cn addcomli oveq12i 7cn addid2i eqtri 2t2e4 4p2e6 + 2cn eqid mul02i 6cn ax-1cn addcomli oveq12i 7cn addlidi eqtri 2t2e4 4p2e6 7t2e14 1p1e2 8cn 4cn 8p4e12 decaddci decma2c decmac 4p4e8 7t7e49 decmul2c decaddi decmul1c numexp2x 7t3e21 1p0e1 oveq1i 6p2e8 decma 9t3e27 numexpp1 ) ABCDZEDZFDGFHUAUBFBDZIWFFABAGUCJHFBKLUDUEAWGIDBFDZAGHHABGUFULUGUHZUIBFW @@ -657106,8 +657106,8 @@ fixed reference functional determined by this vector (corresponding to cn0 7cn nn0addcli pm3.2i expp1 ax-mp w3a 3pm3.2i expadd sqvali oveq12i 9nn0 7t7e49 6nn0 4t4e16 1p1e2 4p4e8 8cn 6cn 8p6e14 addcomli decaddci 3nn0 9t4e36 c3 3p1e4 6p4e10 decaddci2 decmac 8nn0 9cn mulcomli 9t9e81 decmul1c decmul2c - 4cn 3eqtri 7nn0 2cn 7t2e14 4p2e6 decaddi 7t4e28 addid1i mul01i dec0h eqcomi - 00id ax-1cn mulid1i nn0cni mulcomi 5nn0 mulid2i addid2i 9p6e15 9t6e54 5p1e6 + 4cn 3eqtri 7nn0 2cn 7t2e14 4p2e6 decaddi 7t4e28 addridi mul01i dec0h eqcomi + 00id ax-1cn mulridi nn0cni mulcomi 5nn0 mullidi addlidi 9p6e15 9t6e54 5p1e6 7p4e11 9t8e72 mul02i decma 9t7e63 3lt10 6lt10 2lt10 1lt10 5lt6 declt decltc 0cn 6nn eqbrtri decsuc decadd 5cn 2t1e2 2t2e4 decma2c 0p1e1 breqtri cr 7pos 7re reexpcl wtru a1i 6p1e7 wb cz 5nn expgt0 9re 1nn decnncl nnrei ltmuldivi @@ -657272,12 +657272,12 @@ fixed reference functional determined by this vector (corresponding to 1lt2 cn0 reexpcld nn0red gt0ne0d redivcld cn 1nn0 decnncl nnred rehalfcld 5nn 0red divgt0d 3lexlogpow2ineq1 simpli lttrd ltled simpri leexp1ad df-5 oveq2d recnd 4nn0 expp1d eqtrd c6 c8 6nn0 deccl 7nn0 9nn0 9re mptru 2lt10 - c7 5lt9 6lt7 decltc decleh 8nn0 eqid 0nn0 9cn 8cn 9t8e72 mulcomli addid1i - 2cn decaddi ax-1cn mulid1i dec0h eqcomi eqtr4i decmul1c eqcomd breqtrd cc + c7 5lt9 6lt7 decltc decleh 8nn0 eqid 0nn0 9cn 8cn 9t8e72 mulcomli addridi + 2cn decaddi ax-1cn mulridi dec0h eqcomi eqtr4i decmul1c eqcomd breqtrd cc 2p2e4 expaddd sqvali 5t5e25 oveq12d 3eqtrd nn0cni mul02i addcomli oveq12i - nncni eqtri 5p1e6 addid2i 2t2e4 0p1e1 4p1e5 5t2e10 decmac decmul2c eqtr2d + nncni eqtri 5p1e6 addlidi 2t2e4 0p1e1 4p1e5 5t2e10 decmac decmul2c eqtr2d 6cn decma2c 3cn 3t3e9 9t9e81 oveq1d 3brtr3d mpbird crp 3rp cz 4z rpexpcld - ledivmuld expdivd nngt0i divdiv2d 9t5e45 3nn0 mulid2i oveq1i 3p1e4 5t3e15 + ledivmuld expdivd nngt0i divdiv2d 9t5e45 3nn0 mullidi oveq1i 3p1e4 5t3e15 5cn mulcld divmuld elrpd rpdivcld lemuldivd eqbrtrd letrd ) ABUACZDECZFDG ZHIJUVEDBUBCZDECZUVFJUVDDJABAUCKJUDLZMAUEIJUFLZBUCKJUGLZMBUEIJUHLZJFAJFAJ UIFAUEIJUMLUJUKULZDUNKJNLZUOJUVGDJDBJDUVNUPZUVKJBUVLUQZURZUVNUOJUVFUVFUSK @@ -657668,7 +657668,7 @@ fixed reference functional determined by this vector (corresponding to cfz cc0 cle wbr wa cn0 2re a1i clt 2pos cceil nngt0d 1red ltned relogbcld 1lt2 necomd 5nn0 reexpcld ceilcl syl zred wceq eleq1d mpbird c7 0red 7pos 3lexlogpow5ineq3 ltled ceilge breqtrrd letrd ltletrd flcld caddc readdcld - 7re leidd 1cnd addid2d wne recnd gtned logbid1 syl3anc eqcomd breq12d 5re + 7re leidd 1cnd addlidd wne recnd gtned logbid1 syl3anc eqcomd breq12d 5re cc nn0addge1i recni 5cn addcomi 5p2e7 eqtri eqbrtrd cuz wb 2z uzidd elrpd crp 2rp logbleb mpbid fllep1 leadd1d jca elnn0z sylibr adantr cn remulcld fzfid c4 oveq2d cdif eldifd wi syl2anc oveq1d eqtrd eqeltrd cdiv breqtrd @@ -657878,18 +657878,18 @@ fixed reference functional determined by this vector (corresponding to 5nn0 2re elrpd relogcld reexpcld nn0zd expne0d redivcld readdcld remulcld 1red expcld divcld 1cnd rplogcld rpexpcld 3nn0 df-4 letrd expge0d divge0d mulne0d 5re resqcld syl mpbird rpmulcld lemul1ad lediv2ad lemul2ad div23d - cz divmuldivd oveq12d mulcld mulassd divassd mulid1d mulcomd expsubd wceq + cz divmuldivd oveq12d mulcld mulassd divassd mulridd mulcomd expsubd wceq leadd1dd recni cc jca syl2anc mpbid divdiv1d dividd eqeltrd cdc 10nn0 ceu wa wn ltled lenltd gtned rpne0d exp1d eqeltrrd eqbrtrd c6 2nn0 c7 3brtr3d c9 expaddd adddird lemul1d eqidd div32d addcld nn0addge2i breqtrrdi ltp1d 1re logge0d lelttrd 4nn0 2z 1nn0 1lt4 divne0d 4z nn0ge0d mulge0d rpdivcld 0le2 sylib ge0p1rpd 0le1 rpred rpge0d lep1d sqcld divdiv2d ax-1cn subaddi cneg 4p1e5 subid1d eqtr4d jctir 0cnd subeqrev df-neg eqtr4di 1zzd expnegd - 5t2e10 nn0cni mulid2d 10re 3z ere nn0red egt2lt3 simpri 3lt4 lttrd mtbird + 5t2e10 nn0cni mullidd 10re 3z ere nn0red egt2lt3 simpri 3lt4 lttrd mtbird loglt1b cn 10nn nnledivrp relogbcld logbgt0b rehalfcld nn0ge0i relogbexpd sq2 leidd logblebd 1nn 6nn0 nn0addcli 5p2e7 7re nn0addge1i breqtri declei 7p2e9 eqbrtri 4t4e16 eqcomi leexp1ad divdird 2p1e3 subadd2d lediv1d uzidd - ldiv cuz relogbval df-2 eqnetrd div12d 5cn df-5 reccld addid1d 1e2m1 4cn + ldiv cuz relogbval df-2 eqnetrd div12d 5cn df-5 reccld addridd 1e2m1 4cn ) AEFBUAUBZEUAUBZGHZIJHZFKHZUYNLHZGHZIUYMTJHZLHZUYNIJHZBLHZGHZLHZLHZEUYNE JHZGHZUYMFJHZBGHZLHZKHZTUYMUCJHZLHZUYNTJHZBLHZGHZEFEBUDHZIJHZFKHZUYNLHZGH ZIVURTJHZLHZFBUYNLHZGHZLHZMKHZLHZLHZVUHUYMEFUEHZJHZBGHZLHZKHTVUOGHVUMBGHL @@ -658157,8 +658157,8 @@ fixed reference functional determined by this vector (corresponding to 2re 6nn ceilm1lt ltsubaddd 3lexlogpow5ineq5 5p1e6 nncnd subadd2d leaddsub 5nn 2exp4 eqtr4i uzidd elrpd rpexpcld logblt eqcomi 9pos 3lexlogpow2ineq2 simpld simprd df-3 1zzd 1le2 eluz elfznn nnnn0d fprodm1 2m1e1 fprod1 9nn0 - 4z elnnz sylanbrc orcd elnn0 8cn 8t2e16 mul02i addcomli 4cn addid2i 2t1e2 - 6cn decma2c mulid1i oveq1i 7p4e11 7t6e42 decmul2c 2lt10 3lt10 3exp3 3m1e2 + 4z elnnz sylanbrc orcd elnn0 8cn 8t2e16 mul02i addcomli 4cn addlidi 2t1e2 + 6cn decma2c mulridi oveq1i 7p4e11 7t6e42 decmul2c 2lt10 3lt10 3exp3 3m1e2 7cn 4lt5 sq3 9m1e8 df-9 eqeltrd expaddd 2exp8 2t2e4 4p1e5 5t2e10 subadd2i 8nn0 1p1e2 mpbir ltm1d nn0zd leexp2d lttrd oveq2i eluzle leloed mpjaod ) AIEJKZBLCMNZJKZIEOZAVUFVUHAVUFUAZBCLLEUBNZUFMNZPUCNUBNZVUKLMNZDVUKUGMNZEV @@ -658255,7 +658255,7 @@ fixed reference functional determined by this vector (corresponding to c7 7pos ltled jca elnn0z sylibr wss cfn cn elfznn adantl nnzd ssrdv fzfid ex lcmfcl syl2anc nn0red cfl cmin cprod cmul elnnz sylanbrc flcld 0le1 cc wne recnd gtned logbid1 syl3anc eqcomd breqtrd 2z leidd letrd logblebd wb - 2lt7 0zd flge mpbid nnexpcld nnnn0d zexpcl 1zzd zsubcld 1cnd addid1d 1nn0 + 2lt7 0zd flge mpbid nnexpcld nnnn0d zexpcl 1zzd zsubcld 1cnd addridd 1nn0 caddc 1lt3 exp1d nnge1d elfzuz leexp2ad eqbrtrd ltaddsub2d nnmulcld nnred fprodnncl aks4d1p2 lcmfdvdsb biimpd syldbl2 wi dvdsle mpd lenltd pm2.21dd nn0zd simpr pm2.61dan rexnal ) AFUBBUCJZFKCUDLZUEZUFZUUTUFFUVAUGAUVBUVCAU @@ -658412,7 +658412,7 @@ fixed reference functional determined by this vector (corresponding to 3expa nn0red cr 2re a1i clt 2pos c5 cceil wceq cz c3 cuz eluzelz zred 3re 0red 3pos eluzle ltletrd 1red 1lt2 ltned necomd relogbcld reexpcld ceilcl 5nn0 eqeltrd c9 9re 3lexlogpow5ineq4 lttrd ceilge breqtrrd flcld ad2antrr - 9pos simplr jca elnnz sylibr caddc 1cnd addid2d wne recnd logbid1 syl3anc + 9pos simplr jca elnnz sylibr caddc 1cnd addlidd wne recnd logbid1 syl3anc cc gtned eqcomd eqtrd 2z leidd 2lt9 ltled letrd logblebd eqbrtrd peano2zd wb flge syl2anc mpbid zltp1led mpbird nnnn0d nnexpcld simp3 eqid aks4d1p6 0zd cmul wi rsp imp pcelnn nnge1 lemulge11d cq nnne0d pcexp simpr nn0ge0d @@ -658523,7 +658523,7 @@ fixed reference functional determined by this vector (corresponding to 13-Nov-2024.) $) aks4d1p8d2 $p |- ( ph -> ( P ^ ( P pCnt R ) ) < R ) $= ( cpc co wcel c1 wbr cdvds wn vp cexp cmul cprime cn prmnn nnred reexpcld - syl pccld remulcld clt recnd mulid1d nnrpd nn0zd rpexpcld prmgt1 ltmul2dd + syl pccld remulcld clt recnd mulridd nnrpd nn0zd rpexpcld prmgt1 ltmul2dd 1red eqbrtrrd cle nnzd zexpcld cgcd gcdcomd wceq cv wral wrex wa cc0 0lt1 a1i 0red ltnled mpbid exp1d eqcomd oveq2d cz 1zzd syl2anc eqtrd adantr wb breq1 adantl bicomd biimpd mpd pm2.65da neqcomd pcelnn mpbird prmdvdsexpb @@ -658581,7 +658581,7 @@ fixed reference functional determined by this vector (corresponding to nnred ex ssrdv sstrd cfn c0 wne fzfid ssfid aks4d1p3 rabn0 sylibr fiminre wrex syl3anc breq1 notbid 1zzd cz clogb c5 cexp cceil cfv c3 zred ltletrd syl 1red ltned necomd relogbcld cn0 ad4antr wb ad2antrr wi dvdsval2 mpbid - nnzd cmul nncnd mulid2d sylbir jca dvdsle syl2anc eqbrtrd nnrpd lemuldivd + nnzd cmul nncnd mullidd sylbir jca dvdsle syl2anc eqbrtrd nnrpd lemuldivd lttrd ltled letrd mpbird elfzd simplr zexpcld eqcomd cfl ad3antrrr simprl c9 recnd cc crp redivcld ltnled breq1d infrefilb 3expa mpd elrab3 con2bid con3d r19.29a 2re 2pos cuz eluzelz 0red 3re 3pos eluzle 1lt2 5nn0 ceilcld @@ -658963,7 +658963,7 @@ fixed reference functional determined by this vector (corresponding to cu2 cz cle wa w3a cdiv cr 2re cn simpl 0red zred 2pos simpr ltletrd elnnz cn0 jca sylibr nnnn0 syl nn0red remulcld 3re readdcld wne nngt0d ltaddrpd nnred crp 2rp lttrd ltned redivcld 1nn0 nn0addcld reexpcld 2nn0 nn0mulcld - necomd bccl syl2anc 0le2 2t1e2 1red nnrp rpaddcld rpcnd mulid1d nnre 1le2 + necomd bccl syl2anc 0le2 2t1e2 1red nnrp rpaddcld rpcnd mulridd nnre 1le2 rpge0d lemulge12d 2lt3 leltaddd eqbrtrd ltmuldiv2d mpbid ltmul2dd expge0d simp2 ltmul12ad expaddd expcld mulcomd exp1d eqidd 3eqtrd eqtrd 2np3bcnp1 2cnd eqcomd mulcld addcld nn0cnd nncnd cc 3cn divcld 2z uzindd ) AEUAUCZF @@ -659434,7 +659434,7 @@ fixed reference functional determined by this vector (corresponding to ( c1 wcel vs vw cc0 wceq caddc co cfz cv cfv cmin cif wne adantr iffalsed wa neneqd eqcomd cn0 wf csu cab w3a eleq1 cz cle wbr nnzd zaddcld cn wral clt feq1 fveq1 anbi12d elab adantl 1zzd nnge1d zred leidd elfzd ffvelcdmd - wi elfznn syl zsubcld nnred recnd addid1d elfzle2 eqbrtrd 0red leaddsub2d + wi elfznn syl zsubcld nnred recnd addridd elfzle2 eqbrtrd 0red leaddsub2d mpbid elnn0z sylibr 3impa wn 1red 1cnd elfzle1 3adant3 simp3 neqne necomd readdcld ltlend mpbird wb zleltp1 syl2anc lesubadd syl3anc fveq2 leaddsub jca cr a1i ifbothda cvv eqidd simpr eqeq1d fveq2d oveq12d oveq1d ifbieq2d @@ -659445,7 +659445,7 @@ fixed reference functional determined by this vector (corresponding to ifcld eleqtrdi zltp1le fsump1 ltp1d lelttrd ltned fzfid ad2antrr resubcld nnuz zltlem1 lem1d fsumsub id fsum1p npcand sumeq1d peano2zd lep1d cbvsum fsumshft eqeltrd breqtrd nncnd telfsum2 eleq1d chash cfn fsumconst nnnn0d - pncan3d hashfz1 mulid1d addid2d 0cnd subsub3d subsub4d addsubassd fsumzcl + pncan3d hashfz1 mulridd addlidd 0cnd subsub3d subsub4d addsubassd fsumzcl subcld nn0cnd addlsub ovex mptex simpl fveq1d eleqtrrd eqeltrrd fmptd ) A MEKUCUDZSLUUAUUBZISKSUEUFZUGUFZIUHZUYEUDZLKUEUFZKMUHZUIZUJUFZUYGSUDZSUYJU IZSUJUFZUYGUYJUIZUYGSUJUFZUYJUIZUJUFZSUJUFZUKZUKZUUCZUKZDJAUYJETZUOZVUCVU @@ -659696,7 +659696,7 @@ fixed reference functional determined by this vector (corresponding to 0le1 iftrued ltp1d lelttrd simp3 ltlend 3expa fsumsub nncnd eqeltrd fsum1 eluz 3eqtrd nnuz eleqtrd 3adantl3 iftrue fsum1p lep1d ffvelcdmda fsumshft npcand mpdan ex eleq2d imbi1d imp telfsum2 pncan3d leloed orcomd mpjaodan - wo chash fsumconst hashfz1 mulid1d addid1d 0cnd addcomd subsub2d subsub4d + wo chash fsumconst hashfz1 mulridd addridd 0cnd addcomd subsub2d subsub4d subidd addsubassd eleq1d fsumzcl addcld addlsub mptex simpl simp1 eluzfz1 ovex ltletrd ltled hashfzp1 fsumcl addsubd nncand addassd cbvmpt wfn ffnd oveq1 dffn5 ralrimiva ) AQUHZLUIZKUIZVVQUJQEAVVQEUKZULZVVSUEUMMUNUOZUEUHZ @@ -660338,7 +660338,7 @@ fixed reference functional determined by this vector (corresponding to vs cv cab a1i fveq2d breq2 anbi2d abbidv oveq1d eqeq12d simprl clt simprr wf oveq1 cfn simpr ffvelcdmda fsumnn0cl syldan nn0ge0d 0red nn0red lenltd wn mpbid jca eqleltd mpbird leidd eqbrtrd impbid cmin 0nn0 sticksstones21 - ex eqid c0 wne cn wb hashnncl syl bicomd biimpd nncnd 1cnd subcld addid2d + ex eqid c0 wne cn wb hashnncl syl bicomd biimpd nncnd 1cnd subcld addlidd mpd nnm1nn0 bcnn eqtrd eqcomd wo ad2antrr cr adantl 1red readdcld syl2anc cz nn0zd sylibr cfz cxp cpw fzfid xpfi pwfi sylib wss fsetsspwxp wral 0zd ssfid simpllr difssd adantlr mpdan addge01d nn0cnd breqtrd pm2.01da elfzd @@ -661220,7 +661220,7 @@ D Fn ( I ... ( M - 1 ) ) /\ cr subsub2d cc0 posdifd mpbid elrpd ltsubrpd eqbrtrrd eqbrtrd ltned simpr adantr neeq1d mpbird neneqd iffalsed cle w3a lenltd 3adant2 breq1d notbid simp2 oveq1d eqtrd simp3 iftrued eqcomd eqtr4di oveq2d ltnled biimprd imp - wn 3expa cc subcld addcld addid1d biimpa syld pm2.61dan 1zzd nnzd zsubcld + wn 3expa cc subcld addcld addridd biimpa syld pm2.61dan 1zzd nnzd zsubcld ex zaddcld 1p0e1 1red 0red subge0d le2addd eqbrtrrid ltled elfzd elfzelzd nnge1d 0zd ifcld eleq1d fvmptd ) AKEUBFUBZGUBKJIUCUDZUEUDZGUBYMHUEUDZAYKY MGABDEFIJKLMNOPQRSUFUGACYMCUHZJUIZIYOIUJUKZYOYOULUEUDZUMZUMZYNULJUNUDZGUQ @@ -661268,7 +661268,7 @@ D Fn ( I ... ( M - 1 ) ) /\ nnre ltsubrpd lelttrd ltned adantr wb neeq1 adantl mpbird neneqd iffalsed w3a wn lenltd biimpa 3adant2 breq1 3ad2ant2 mtbird simp2 oveq1d cc0 simp3 iftrued eqcomd eqtrd oveq2d 3expa simpr ltnled nncn 3ad2ant1 nncnd subcld - cc addid1d pm2.61dan 1zzd nnzd zsubcld 1m1e0 nltled neqned ltlend posdifd + cc addridd pm2.61dan 1zzd nnzd zsubcld 1m1e0 nltled neqned ltlend posdifd jca mpbid eqbrtrid zlem1lt syl2anc ltled elfzd 0zd eleq1d zaddcld fvmptd ifcld ) AKEUBFUBZGUBKIUCUDZGUBYIHUEUDZAYHYIGABDEFIJKLMNOPQRSUFUGACYICUHZJ UIZIYKIUJUKZYKYKULUEUDZUMZUMZYJULJUNUDZGUQGCYQYPUOUIATUPAYKYIUIZURZYPYOYJ @@ -661476,7 +661476,7 @@ D Fn ( I ... ( M - 1 ) ) /\ crp w3a cdiv wi cneg cpnf cico neg1rr elicopnf ax-mp sylib simpld readdcl clt mpan2 cmin neg1cn 2cn addcom mp2an negsub 2m1e1 3eqtri simprd mp3an13 leadd1 mpbid eqbrtrrid 0lt1 jctil 0re 1re ltletr mp3an12 mpd imbi2i mpbir - jca elrp remulcl mpan 3re 1red readdcld recn addid1 addcomi negidi leadd2 + jca elrp remulcl mpan 3re 1red readdcld recn addrid addcomi negidi leadd2 a1i eqtri mp3an1 2timesd oveq1d addass mp3an3 anidms eqtrd breqtrrd leidd eqbrtrrd le2addd recnd addassd 1p2e3 oveq2 eqtrdi breqtrd 3jca divge1 ) A BDEFZUBGZDBUAFZHEFZIGZXLXOJKZUCLXOXLUDFJKAXMXPXQAXMUEAXLIGZMXLUOKZNZUEAXR @@ -661501,7 +661501,7 @@ D Fn ( I ... ( M - 1 ) ) /\ factwoffsmonot $p |- ( ( ( X e. NN0 /\ Y e. NN0 /\ X <_ Y ) /\ N e. NN0 ) -> ( ! ` ( X + N ) ) <_ ( ! ` ( Y + N ) ) ) $= ( cn0 wcel cle wbr caddc co cfa cfv cc0 c1 wceq oveq2 fveq2d syl wa jca - cr vx vy w3a cv breq12d facwordi cc nn0cn addid1 3ad2ant1 3ad2ant2 ax-1cn + cr vx vy w3a cv breq12d facwordi cc nn0cn addrid 3ad2ant1 3ad2ant2 ax-1cn 3brtr4d addass mp3an3 syl2an 3ad2antl1 adantr nn0addcl 3adant2 3adant1 wb cmul nn0re leadd1 syl3an biimpa syl3anc 3an1rs 1re mpbid wi cn faccl nnre 3syl clt nngt0 0re ltle mpan mpd 1nn0 nn0ge0 nn0readdcl readdcld lemul12a @@ -662321,7 +662321,7 @@ D Fn ( I ... ( M - 1 ) ) /\ ( A O ( U ++ V ) ) = ( ( A .xb U ) ++ ( A .x. V ) ) ) $= ( vx cc0 cfzo co csn cxp cconcat cmulr cfv cof wcel wf fconstg ffnd chash syl caddc cword iswrdi 3syl ccatvalfn syl2anc cmul wceq fzofi snfi hashxp - wfn cfn mp2an c1 hashsng oveq2d cn0 hashcl nn0cnd mulid1d hashfzo0 3eqtrd + wfn cfn mp2an c1 hashsng oveq2d cn0 hashcl nn0cnd mulridd hashfzo0 3eqtrd mp1i eqtrid oveq12d eqtrd fneq2d mpbid cv wa clt cmin adantr breq2d ifbid wbr cif cuz cz elfzouz ad2antlr ad2antrr nn0zd elfzo2 syl3anbrc fvconst2g simpr syl2an2r wn cle elfzonn0 nn0red elfzoelz adantl zred lenltd biimpar @@ -664449,8 +664449,8 @@ assignment with finite support (as only a finite amount of variables Towards the start of this section are several proofs regarding the different complex number axioms that could be used to prove some results. - For example, ~ ax-1rid is used in ~ mulid1 related theorems, so one could - trade off the extra axioms in ~ mulid1 for the axioms needed to prove that + For example, ~ ax-1rid is used in ~ mulrid related theorems, so one could + trade off the extra axioms in ~ mulrid for the axioms needed to prove that something is a real number. Another example is avoiding complex number closure laws by using real number closure laws and then using ~ ax-resscn ; in the other direction, real number closure laws can be avoided by using @@ -664474,12 +664474,12 @@ assignment with finite support (as only a finite amount of variables This only requires ~ ax-addass , ~ ax-1cn , and ~ ax-addcl . (And in practice, the expression isn't fully expanded into ones.) - Multiplication by 1 requires either ~ mulid2i or ( ~ ax-1rid and ~ 1re ) as + Multiplication by 1 requires either ~ mullidi or ( ~ ax-1rid and ~ 1re ) as seen in ~ 1t1e1 and ~ 1t1e1ALT . Multiplying with greater natural numbers uses ~ ax-distr . Still, this takes fewer axioms than adding zero, which is often implicit in theorems such as ` ( 9 + 1 ) = ; 1 0 ` . Adding zero uses almost every complex number axiom, though notably not ~ ax-mulcom (see - ~ readdid1 and ~ readdid2 ). + ~ readdrid and ~ readdlid ). $) @@ -664536,16 +664536,16 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $} ${ - readdid1addid2d.a $e |- ( ph -> A e. RR ) $. - readdid1addid2d.b $e |- ( ph -> B e. RR ) $. - readdid1addid2d.1 $e |- ( ph -> ( B + A ) = B ) $. + readdridaddlidd.a $e |- ( ph -> A e. RR ) $. + readdridaddlidd.b $e |- ( ph -> B e. RR ) $. + readdridaddlidd.1 $e |- ( ph -> ( B + A ) = B ) $. $( Given some real number ` B ` where ` A ` acts like a right additive identity, derive that ` A ` is a left additive identity. Note that the hypothesis is weaker than proving that ` A ` is a right additive identity (for all numbers). Although, if there is a right additive identity, then by ~ readdcan , ` A ` is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.) $) - readdid1addid2d $p |- ( ( ph /\ C e. RR ) -> ( A + C ) = C ) $= + readdridaddlidd $p |- ( ( ph /\ C e. RR ) -> ( A + C ) = C ) $= ( cr wcel wa caddc co wceq adantr recnd simpr addassd oveq1d eqtr3d wb readdcld readdcan syl3anc mpbid ) ADHIZJZCBDKLZKLZCDKLZMZUGDMZUFCBKLZDKLU HUIUFCBDUFCACHIZUEFNZOUFBABHIUEENZOUFDAUEPZOQUFULCDKAULCMUEGNRSUFUGHIUEUM @@ -664644,7 +664644,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( Multiplication with 1 is commutative for natural numbers, without ~ ax-mulcom . Since ` ( A x. 1 ) ` is ` A ` by ~ ax-1rid , this is - equivalent to ~ remulid2 for natural numbers, but using fewer axioms + equivalent to ~ remullid for natural numbers, but using fewer axioms (avoiding ~ ax-resscn , ~ ax-addass , ~ ax-mulass , ~ ax-rnegex , ~ ax-pre-lttri , ~ ax-pre-lttrn , ~ ax-pre-ltadd ). (Contributed by SN, 5-Feb-2024.) $) @@ -664791,7 +664791,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.) $) sqn5i $p |- ( ; A 5 x. ; A 5 ) = ; ; ( A x. ( A + 1 ) ) 2 5 $= - ( c5 cdc cmul co cc0 caddc c2 0nn0 deccl nn0cni 5nn0 eqid addid2i decaddi + ( c5 cdc cmul co cc0 caddc c2 0nn0 deccl nn0cni 5nn0 eqid addlidi decaddi 5cn 2nn0 cn0 eqtri 5p5e10 decaddci2 sqmid3api 5t5e25 wcel peano2nn0 ax-mp c1 nn0mulcli decmulnc mul01i deceq2i 2cn mul02i oveq1i decma ) ACDZUQEFAG DZAUHHFZGDZEFCCEFZHFAUSEFZIDZCDURUQUTCURAGBJKLQAGCURCBJMURNZCQOZPACUSUQCB @@ -664833,7 +664833,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove by Steven Nguyen, 10-Dec-2022.) $) decpmul $p |- ( ; A B x. ; C D ) = ; I H $= ( co cdc cmul c1 cc0 caddc decpmulnc dfdec10 cn0 nn0mulcli eqeltrri numcl - deccl 0nn0 dec0u eqid decaddcom eqtri nn0cni addid2i decadd 3eqtri ) ABUA + deccl 0nn0 dec0u eqid decaddcom eqtri nn0cni addlidi decadd 3eqtri ) ABUA CDUAUBTEFUAZGHUAZUAUCUDUAVBUBTZVCUETIHUAABCDEFVCJKLMNOPUFVBVCUGVBUDGHIHVD VCEFACUBTEUHNACJLUIUJZADUBTBCUBTZUETFUHODVFAJMBCKLUIUKUJZULZUMRSVBVHUNVCU OVBGUETEGUAFUETIEFGVEVGRUPQUQHHSURUSUTVA $. @@ -664847,9 +664847,9 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove sqdeccom12 $p |- ( ( ; A B x. ; A B ) - ( ; B A x. ; B A ) ) = ( ; 9 9 x. ( ( A x. A ) - ( B x. B ) ) ) $= ( cmul co caddc cdc cmin c1 cc0 c9 cc wcel 0nn0 deccl nn0cni eqid oveq12i - mulcli wceq nn0mulcli subadd4 mp4an addid2i decaddi eqtr2i addcomi decadd + mulcli wceq nn0mulcli subadd4 mp4an addlidi decaddi eqtr2i addcomi decadd numcl 3eqtr4i addsubeq4com mpbi 10nn0 ax-1cn subcli subdiri subdii mul02i - wb dec0u decmul1 eqtri mulid2i decpmulnc 9p1e10 decsucc addcomli mvlladdi + wb dec0u decmul1 eqtri mullidi decpmulnc 9p1e10 decsucc addcomli mvlladdi 9nn0 oveq1i ) AAEFZABEFZBAEFZGFZHZBBEFZHZVQVOHZVLHZIFZJKHZKHZJIFZVLVQIFZE FZABHZWGEFZBAHZWIEFZIFLLHZWEEFVLKHZVQHZVQKHZVLHZIFZWLKHZWNKHZIFZWEIFZWAWF WTWQVQGFZWRVLGFZIFZWPWQMNWRMNVLMNVQMNWTXCUAWQWLKVLKAACCUBZOPZOPQWRWNKVQKB @@ -664871,7 +664871,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove Steven Nguyen, 3-Jan-2023.) $) sq3deccom12 $p |- ( ( ; ; A B C x. ; ; A B C ) - ( ; D B x. ; D B ) ) = ( ; 9 9 x. ( ( ; A B x. ; A B ) - ( C x. C ) ) ) $= - ( cdc cmul co cmin c9 cc0 caddc 0nn0 eqid nn0cni addcomli addid2i decaddi + ( cdc cmul co cmin c9 cc0 caddc 0nn0 eqid nn0cni addcomli addlidi decaddi decadd deccl eqtr3i oveq12i oveq2i sqdeccom12 eqtri ) ABIZCIZUJJKZDBIZULJ KZLKUKCUIIZUNJKZLKMMIUIUIJKCCJKLKJKUMUOUKLULUNULUNJCNIZUIOKULUNCNABDBUPUI GPEFUPQZUIQACDAERCGRHSBBFRTUBCNUIUPUIGPABEFUCZUQUIUIURRTUAUDZUSUEUFUICURG @@ -664894,7 +664894,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove ( c7 c1 cdc c6 c5 c2 c3 2nn0 3nn0 deccl 5nn0 7nn0 1nn0 eqid c4 caddc co 2cn decaddi cmul cc0 c8 8nn0 nn0cni 3p2e5 addcomli 0nn0 4nn0 6nn0 nn0addcli 7cn 7t2e14 mulcomli 4p2e6 3cn 7t3e21 decmul1c 4cn addcli ax-1cn 6p1e7 eqtri 5cn - oveq1i 7t5e35 3p1e4 5p5e10 decaddci2 decmac mulid1i 5p3e8 decma2c decmul2c + oveq1i 7t5e35 3p1e4 5p5e10 decaddci2 decmac mulridi 5p3e8 decma2c decmul2c ) ABCZBBDCZACZUACZUBCEFGCZECZVRVNBCZVREFGHIJZKJZABLMJMVTNKWAABFGVSVQUBVRVNV RLMHIVNNVRNZWBUCWAVREFEAVPUAOVSFVRPQWAKHKVSNVRFFECVRWAUDRFGEVRFHIHWCUESUFLU GUHVOBAVRATQFOPQZBDMUIJMFOHUHUJFGVOBAFVRLHIWCMHBODFATQFMUHHAFBOCUKRULUMUNSA @@ -664909,8 +664909,8 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove ex-decpmul $p |- ( ; ; 2 3 5 x. ; ; 7 1 1 ) = ; ; ; ; ; 1 6 7 0 8 5 $= ( c2 c3 cdc c5 c7 c1 c6 c8 2nn0 3nn0 deccl 5nn0 7nn0 1nn0 eqid cmul decaddi cc0 co 5cn 6nn0 c4 4nn0 7cn 2cn 7t2e14 mulcomli 4p2e6 7t3e21 decmul1c 1p2e3 - 3cn nn0cni mulid1i decmul2c 7t5e35 mulid2i decmul1 5p2e7 5p3e8 decadd dec0h - addcomli eqtri 0nn0 8nn0 caddc 3p3e6 6p1e7 7p3e10 decaddc2 addid2i decpmul + 3cn nn0cni mulridi decmul2c 7t5e35 mullidi decmul1 5p2e7 5p3e8 decadd dec0h + addcomli eqtri 0nn0 8nn0 caddc 3p3e6 6p1e7 7p3e10 decaddc2 addlidi decpmul 8cn ) ABCZDEFCZFFGCZBCZBCZBECZHCZRDVQECZRCZHCABIJKZLEFMNKZNEFVRBVOAVPWDMNVP OZJIVQFBVOEPSAFGNUAKZNIABVQFEAVOMIJVOONIFUBGAEPSANUCIEAFUBCUDUEUFUGUHQEBAFC UDULUIUGUJUKQVOVOWDUMUNZUOABBDCZDVTHVOFPSDVPPSIJBDJLKZLWHVPDWIDCVPWEUMTEFWI @@ -664936,7 +664936,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove ( cc0 clt wbr cexp co wa cr wcel adantr simpr wn wceq cn c2 cz syl3anc ex nnzd expgt0 wo lttrid notbid notnotr 0re ltnri 0expd breq2d mtbiri eqcomd 0red oveq1d mtbird cneg renegcld cc cdvds recnd cdiv cnumer cfv cdenom c1 - cmul cq wb zq adantl qden1elz syl mpbird oveq2d qmuldeneqnum zcnd mulid1d + cmul cq wb zq adantl qden1elz syl mpbird oveq2d qmuldeneqnum zcnd mulridd 3eqtr3rd nnred 2re a1i nngt0d 2pos divgt0d qgt0numnn syl2anr mtand evend2 eqeltrd biimpd nnnn0d reexpcld pm2.46 syl6bi 3syld lt0neg1d lt0neg2d jaod oexpneg 3imtr4d syl5 sylbid impcon4bid ) AGCHIZGCBJKZHIZAXGXIAXGLCMNZBUAN @@ -665098,7 +665098,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove expgcd $p |- ( ( A e. NN /\ B e. NN /\ N e. NN0 ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) $= ( cn wcel cn0 cgcd co cexp c1 nncnd cdiv wceq cz cdvds wbr nnzd wa syl2anc - cc w3a gcdnncl 3adant3 simp3 nnexpcld mulid1d nnexpcl 3adant2 3adant1 simpl + cc w3a gcdnncl 3adant3 simp3 nnexpcld mulridd nnexpcl 3adant2 3adant1 simpl cmul simpr gcddvds simpld simp1 dvdsexpim syl3anc mpd simprd simp2 syl32anc gcddiv nncn 3ad2ant1 nnne0d expdivd 3ad2ant2 oveq12d syl31anc dividd eqtr3d wi divgcdnn nnnn0d divgcdnnr nn0rppwr 3eqtr2d cc0 wb ax-1cn divmul syl12anc @@ -665386,8 +665386,8 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( Negative zero is a left additive identity. (Contributed by Steven Nguyen, 7-Jan-2023.) $) - reneg0addid2 $p |- ( A e. RR -> ( ( 0 -R 0 ) + A ) = A ) $= - ( cc0 cr wcel cresub co caddc wceq elre0re rernegcl renegid readdid1addid2d + reneg0addlid $p |- ( A e. RR -> ( ( 0 -R 0 ) + A ) = A ) $= + ( cc0 cr wcel cresub co caddc wceq elre0re rernegcl renegid readdridaddlidd mpancom ) BCDZACDBBEFZAGFAHAINOBABJBIBKLM $. $( Lemma for ~ resubeu . A value which when added to zero, results in @@ -665403,7 +665403,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove resubeulem2 $p |- ( ( A e. RR /\ B e. RR ) -> ( A + ( ( 0 -R A ) + ( ( 0 -R ( 0 + 0 ) ) + B ) ) ) = B ) $= ( cr wcel wa cc0 cresub co caddc renegid adantr oveq1d simpl recnd rernegcl - wceq readdcld adantl addassd 3eqtr3d elre0re resubeulem1 recn reneg0addid2 + wceq readdcld adantl addassd 3eqtr3d elre0re resubeulem1 recn reneg0addlid syl id ) ACDZBCDZEZAFAGHZIHZFFFIHZGHZBIHZIHFUNIHZAUJUNIHIHBUIUKFUNIUGUKFPUH AJKLUIAUJUNUIAUGUHMNUIUJUGUJCDUHAOKNUIUNUHUNCDUGUHUMBUHULCDUMCDUHFFBUAZUPQU LOUEZUHUFQRNSUHUOBPUGUHFUMIHZBIHFFGHZBIHUOBUHURUSBIBUBLUHFUMBUHFUPNUHUMUQNB @@ -665569,13 +665569,13 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove CUDUESTUCDEUAABLMSUAUDDETACLNTUAUEDESBCPOQR $. ${ - resubidaddid1lem.a $e |- ( ph -> A e. RR ) $. - resubidaddid1lem.b $e |- ( ph -> B e. RR ) $. - resubidaddid1lem.c $e |- ( ph -> C e. RR ) $. - resubidaddid1lem.1 $e |- ( ph -> ( A -R B ) = ( B -R C ) ) $. - $( Lemma for ~ resubidaddid1 . A special case of ~ npncan . (Contributed + resubidaddridlem.a $e |- ( ph -> A e. RR ) $. + resubidaddridlem.b $e |- ( ph -> B e. RR ) $. + resubidaddridlem.c $e |- ( ph -> C e. RR ) $. + resubidaddridlem.1 $e |- ( ph -> ( A -R B ) = ( B -R C ) ) $. + $( Lemma for ~ resubidaddlid . A special case of ~ npncan . (Contributed by Steven Nguyen, 8-Jan-2023.) $) - resubidaddid1lem $p |- + resubidaddlidlem $p |- ( ph -> ( ( A -R B ) + ( B -R C ) ) = ( A -R C ) ) $= ( cresub co caddc cr wcel rersubcl syl2anc readdcld resubaddd mpbid recnd wceq eqcomd oveq1d addassd 3eqtr3d reladdrsub ) ADBCIJZCDIJZKJZBGAUFUGABL @@ -665586,7 +665586,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( Any real number subtracted from itself forms a left additive identity. (Contributed by Steven Nguyen, 8-Jan-2023.) $) - resubidaddid1 $p |- ( ( A e. RR /\ B e. RR ) -> ( ( A -R A ) + B ) = B ) $= + resubidaddlid $p |- ( ( A e. RR /\ B e. RR ) -> ( ( A -R A ) + B ) = B ) $= ( cr wcel wa caddc co cresub wceq readdsub 3anidm13 repncan2 eqtr3d ) ACDZB CDZEABFGAHGZAAHGBFGZBNOPQIABAJKABLM $. @@ -665600,7 +665600,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove UIACUKOHZGHUJULOHUMUIACUKUIAUNQUICUFUGUHRQUIUKUOQSUIUPBAGUGUHUPBTZUFUHUGUQC BUAUBMUCUDUE $. - $( Equality of two left-additive identities. See ~ resubidaddid1 . Uses + $( Equality of two left-additive identities. See ~ resubidaddlid . Uses ~ ax-i2m1 . (Contributed by SN, 25-Dec-2023.) $) re1m1e0m0 $p |- ( 1 -R 1 ) = ( 0 -R 0 ) $= ( c1 cresub co cc0 wceq wtru 0red cr wcel 1re rersubcl mp2an a1i caddc cmul @@ -665625,7 +665625,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( Lemma for ~ sn-00id . (Contributed by SN, 25-Dec-2023.) $) sn-00idlem3 $p |- ( ( 0 -R 0 ) = 1 -> ( 0 + 0 ) = 0 ) $= ( cc0 cresub co c1 wceq cmul caddc oveq2 oveq1d wcel 0re sn-00idlem1 adantr - cr wa resubidaddid1 eqtrd mp2an a1i ax-1rid mp1i 3eqtr3rd ) AABCZDEZAUCFCZA + cr wa resubidaddlid eqtrd mp2an a1i ax-1rid mp1i 3eqtr3rd ) AABCZDEZAUCFCZA GCZADFCZAGCAAAGCUDUEUGAGUCDAFHIUFAEZUDANJZUIUHKKUIUIOZUFUCAGCAUJUEUCAGUIUEU CEUIALMIAAPQRSUDUGAAGUIUGAEUDKATUAIUB $. @@ -665647,23 +665647,23 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove AAABCZAMDEAAAEFZNAAGCADEHIJKL $. $( $j usage 're0m0e0' avoids 'ax-mulcom'; $) - $( Real number version of ~ addid2 . (Contributed by SN, 23-Jan-2024.) $) - readdid2 $p |- ( A e. RR -> ( 0 + A ) = A ) $= - ( cr wcel cc0 caddc co cresub re0m0e0 oveq1i reneg0addid2 eqtr3id ) ABCDAEF + $( Real number version of ~ addlid . (Contributed by SN, 23-Jan-2024.) $) + readdlid $p |- ( A e. RR -> ( 0 + A ) = A ) $= + ( cr wcel cc0 caddc co cresub re0m0e0 oveq1i reneg0addlid eqtr3id ) ABCDAEF DDGFZAEFALDAEHIAJK $. - $( $j usage 'readdid2' avoids 'ax-mulcom'; $) + $( $j usage 'readdlid' avoids 'ax-mulcom'; $) ${ $d A x y $. - $( ~ addid2 without ~ ax-mulcom . (Contributed by SN, 23-Jan-2024.) $) - sn-addid2 $p |- ( A e. CC -> ( 0 + A ) = A ) $= + $( ~ addlid without ~ ax-mulcom . (Contributed by SN, 23-Jan-2024.) $) + sn-addlid $p |- ( A e. CC -> ( 0 + A ) = A ) $= ( vx vy cc wcel cv ci cmul co caddc wceq cr wrex cc0 cnre w3a 0cnd simp2l - wa recnd ax-icn a1i simp2r mulcld addassd readdid2 adantr 3ad2ant2 oveq1d + wa recnd ax-icn a1i simp2r mulcld addassd readdlid adantr 3ad2ant2 oveq1d eqtr3d simp3 oveq2d 3eqtr4d 3exp rexlimdvv mpd ) ADEZABFZGCFZHIZJIZKZCLMB LMNAJIZAKZBCAOUQVBVDBCLLUQURLEZUSLEZSZVBVDUQVGVBPZNVAJIZVAVCAVHNURJIZUTJI VIVAVHNURUTVHQVHURUQVEVFVBRTVHGUSGDEVHUAUBVHUSUQVEVFVBUCTUDUEVHVJURUTJVGU QVJURKZVBVEVKVFURUFUGUHUIUJVHAVANJUQVGVBUKZULVLUMUNUOUP $. - $( $j usage 'sn-addid2' avoids 'ax-mulcom'; $) + $( $j usage 'sn-addlid' avoids 'ax-mulcom'; $) $} ${ @@ -665688,7 +665688,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( ~ 0ne2 without ~ ax-mulcom . (Contributed by SN, 23-Jan-2024.) $) sn-0ne2 $p |- 0 =/= 2 $= ( cc0 c1 caddc co c2 wne cr wcel 1re ax-mp clt wbr 2re ltadd2i biimpi 1p1e2 - c3 1p2e3 3brtr3g 3re wceq readdid2 wo sn-1ne2 lttri2i mpbi 1red lttri mpdan + c3 1p2e3 3brtr3g 3re wceq readdlid wo sn-1ne2 lttri2i mpbi 1red lttri mpdan ltned a1i mpancom gtned jaoi df-3 neeqtri eqnetri oveq1 necon3i ) ABCDZEBCD ZFAEFUTBVABGHUTBUAIBUBJBQVABEKLZEBKLZUCZBQFZBEFVDUDBEIMUEUFVBVEVCVBBQVBUGVB EQKLBQKLVBBBCDZBECDZEQKVBVFVGKLBEBIMINOPRSBEQIMTUHUIUJVCQBQGHVCTUKQEKLVCQBK @@ -665722,25 +665722,25 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove ( cr wcel cc0 cresub co cmul re0m0e0 oveq2i sn-00idlem1 remul01 3eqtr3a ) A BCADDEFZGFADGFAAEFDMDAGHIAJAKL $. - $( Real number version of ~ addid1 without ~ ax-mulcom . (Contributed by SN, + $( Real number version of ~ addrid without ~ ax-mulcom . (Contributed by SN, 23-Jan-2024.) $) - readdid1 $p |- ( A e. RR -> ( A + 0 ) = A ) $= + readdrid $p |- ( A e. RR -> ( A + 0 ) = A ) $= ( cr wcel cresub co cc0 wceq caddc resubid id elre0re resubaddd mpbid ) ABC ZAADEFGAFHEAGAINAAFNJZOAKLM $. - $( $j usage 'readdid1' avoids 'ax-mulcom'; $) + $( $j usage 'readdrid' avoids 'ax-mulcom'; $) $( Real number version of ~ subid1 without ~ ax-mulcom . (Contributed by SN, 23-Jan-2024.) $) resubid1 $p |- ( A e. RR -> ( A -R 0 ) = A ) $= - ( cr wcel cc0 cresub co wceq caddc readdid2 id elre0re resubaddd mpbird ) A + ( cr wcel cc0 cresub co wceq caddc readdlid id elre0re resubaddd mpbird ) A BCZADEFAGDAHFAGAINADANJZAKOLM $. $( $j usage 'resubid1' avoids 'ax-mulcom'; $) $( A real number is equal to the negative of its negative. Compare ~ negneg . (Contributed by SN, 13-Feb-2024.) $) renegneg $p |- ( A e. RR -> ( 0 -R ( 0 -R A ) ) = A ) $= - ( cr wcel cc0 cresub co caddc wceq rernegcl syl id renegid elre0re readdid1 - eqeltrd repncan3 syl2anc oveq2d 3eqtr4d recnd readdid2 recn oveq1d readdcan + ( cr wcel cc0 cresub co caddc wceq rernegcl syl id renegid elre0re readdrid + eqeltrd repncan3 syl2anc oveq2d 3eqtr4d recnd readdlid recn oveq1d readdcan addassd w3a biimpa syl31anc ) ABCZDDAEFZEFZBCZUIAUJGFZBCZUMUKGFZUMAGFZHZUKA HZUIUJBCZULAIZUJIJZUIKUIUMDBALZAMZOUIAUJUKGFZGFZDAGFZUOUPUIADGFAVEVFANUIVDD AGUIUSDBCVDDHUTVCUJDPQRAUASUIAUJUKAUBUIUJUTTUIUKVATUEUIUMDAGVBUCSULUIUNUFUQ @@ -665750,7 +665750,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove 21-Feb-2024.) $) readdcan2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) = ( B + C ) <-> A = B ) ) $= - ( cr wcel caddc co wceq cc0 oveq1 adantl simpl recnd simpr addassd readdid1 + ( cr wcel caddc co wceq cc0 oveq1 adantl simpl recnd simpr addassd readdrid wa oveq2d adantr 3eqtrd w3a cresub rernegcl renegid 3adant2 3adant1 3eqtr3d ex impbid1 ) ADEZBDEZCDEZUAZACFGZBCFGZHZABHZUMUPUQUMUPQUNICUBGZFGZUOURFGZAB UPUSUTHUMUNUOURFJKUMUSAHZUPUJULVAUKUJULQZUSACURFGZFGZAIFGZAVBACURVBAUJULLMV @@ -665761,8 +665761,8 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( Commuted version of ~ renegid . (Contributed by SN, 4-May-2024.) $) renegid2 $p |- ( A e. RR -> ( ( 0 -R A ) + A ) = 0 ) $= - ( cr wcel cresub co caddc wceq renegid oveq2d rernegcl readdid1 eqtrd recnd - cc0 syl recn addassd readdid2 3eqtr4d wb readdcld elre0re readdcan2 syl3anc + ( cr wcel cresub co caddc wceq renegid oveq2d rernegcl readdrid eqtrd recnd + cc0 syl recn addassd readdlid 3eqtr4d wb readdcld elre0re readdcan2 syl3anc id mpbid ) ABCZNADEZAFEZUHFEZNUHFEZGZUINGZUGUHAUHFEZFEZUHUJUKUGUOUHNFEZUHUG UNNUHFAHIUGUHBCZUPUHGAJZUHKOLUGUHAUHUGUHURMZAPUSQUGUQUKUHGURUHROSUGUIBCNBCU QULUMTUGUHAURUGUEUAAUBURUINUHUCUDUF $. @@ -665786,7 +665786,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove sn-it0e0 $p |- ( _i x. 0 ) = 0 $= ( va vb cc wcel cv ci cmul co caddc wceq cr wrex wa remul01 adantr oveq1d cc0 recn syl eqtr3d 0cn cnre cresub oveq2 ax-icn a1i mulassd oveq2d eqtrd - ad2antlr rernegcl recnd mulcld adantl addassd renegid2 sn-addid2 sylan9eq + ad2antlr rernegcl recnd mulcld adantl addassd renegid2 sn-addlid sylan9eq 0cnd eqeq2d biimpa elre0re readdcld ad2antrr ex syl5 rexlimivv mp2b ) QCD QAEZFBEZGHZIHZJZBKLAKLFQGHZQJZUAABQUBVMVOABKKVMQVIUCHZQIHZVPVLIHZJZVIKDZV JKDZMZVOQVLVPIUDWBVSVOWBVSMZVKQGHZVNQWAWDVNJVTVSWAWDFVJQGHZGHVNWAFVJQFCDW @@ -665807,7 +665807,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove ( vy vx cc wcel ci cc0 co cmul caddc wceq wrex wa cr adantl adantr oveq2d addassd ad2antlr cv cresub ax-icn rernegcl recnd mulcld wb eqeq1 rspcedvd addcld eqidd ralrimivva cnre r19.29d2r oveq1 recn renegid adddid sn-it0e0 - a1i 3eqtr3d readdid1 3eqtrd eqtrd oveq1d simpll renegid2 sn-addid2 eqtrdi + a1i 3eqtr3d readdrid 3eqtrd eqtrd oveq1d simpll renegid2 sn-addlid eqtrdi oveq2 syl 3eqtr2d sylan9eqr jca eqeq1d anbi12d syl5ibrcom reximdv expimpd ancomsd rexlimdvva mpd ) AEFZBUAZGHCUAZUBIZJIZHDUAZUBIZKIZLZBEMZAWHGWEJIZ KIZLZNZCOMDOMAWDKIZHLZWDAKIZHLZNZBEMZWCWLWODCOOWCWLDCOOWCWHOFZWEOFZNZNZWK @@ -665848,7 +665848,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove to ~ addcan . (Contributed by SN, 5-May-2024.) $) sn-addcand $p |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) ) $= ( vx caddc co cc0 wceq cc wcel syl wa oveq2 oveq1d adantr addassd cv wrex - wb sn-negex2 simprr sn-addid2 3eqtr3d eqeq12d imbitrid impbid1 rexlimddv + wb sn-negex2 simprr sn-addlid 3eqtr3d eqeq12d imbitrid impbid1 rexlimddv simprl ) AHUAZBIJZKLZBCIJZBDIJZLZCDLZUCHMABMNZUOHMUBEBHUDOAUMMNZUOPZPZURU SURUMUPIJZUMUQIJZLVCUSUPUQUMIQVCVDCVEDVCUNCIJKCIJZVDCVCUNKCIAVAUOUEZRVCUM BCAVAUOULZAUTVBESZACMNZVBFSZTVCVJVFCLVKCUFOUGVCUNDIJKDIJZVEDVCUNKDIVGRVCU @@ -665858,14 +665858,14 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove ${ $d A x $. - $( ~ addid1 without ~ ax-mulcom . (Contributed by SN, 5-May-2024.) $) - sn-addid1 $p |- ( A e. CC -> ( A + 0 ) = A ) $= + $( ~ addrid without ~ ax-mulcom . (Contributed by SN, 5-May-2024.) $) + sn-addrid $p |- ( A e. CC -> ( A + 0 ) = A ) $= ( vx cc wcel caddc cc0 wceq sn-negex2 simprr oveq1d sn-00id eqtrdi simprl cv co wa simpl 0cnd addassd 3eqtr2rd addcld sn-addcand mpbid rexlimddv ) ACDZBNZAEOZFGZAFEOZAGZBCABHUEUFCDZUHPZPZUFUIEOZUGGUJUMUGFUGFEOZUNUEUKUHIZ UMUOFFEOFUMUGFFEUPJKLUMUFAFUEUKUHMZUEULQZUMRZSTUMUFUIAUQUMAFURUSUAURUBUCU D $. - $( $j usage 'sn-addid1' avoids 'ax-mulcom'; $) + $( $j usage 'sn-addrid' avoids 'ax-mulcom'; $) $} ${ @@ -665876,7 +665876,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( ~ addcan2d without ~ ax-mulcom . (Contributed by SN, 5-May-2024.) $) sn-addcan2d $p |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) ) $= ( vx caddc co cc0 wceq cc wcel syl wa oveq1 adantr addassd oveq2d cv wrex - wb sn-negex simprl simprr sn-addid1 eqeq12d imbitrid impbid1 rexlimddv + wb sn-negex simprl simprr sn-addrid eqeq12d imbitrid impbid1 rexlimddv 3eqtrd ) ADHUAZIJZKLZBDIJZCDIJZLZBCLZUCHMADMNZUOHMUBGDHUDOAUMMNZUOPZPZURU SURUPUMIJZUQUMIJZLVCUSUPUQUMIQVCVDBVECVCVDBUNIJBKIJZBVCBDUMABMNZVBERZAUTV BGRZAVAUOUEZSVCUNKBIAVAUOUFZTVCVGVFBLVHBUGOULVCVECUNIJCKIJZCVCCDUMACMNZVB @@ -665904,9 +665904,9 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove sn-addid0.a $e |- ( ph -> A e. CC ) $. sn-addid0.1 $e |- ( ph -> ( A + A ) = A ) $. $( A number that sums to itself is zero. Compare ~ addid0 , - ~ readdid1addid2d . (Contributed by SN, 5-May-2024.) $) + ~ readdridaddlidd . (Contributed by SN, 5-May-2024.) $) sn-addid0 $p |- ( ph -> A = 0 ) $= - ( caddc co cc0 wceq cc wcel sn-addid1 syl eqtr4d 0cnd sn-addcand mpbid ) + ( caddc co cc0 wceq cc wcel sn-addrid syl eqtr4d 0cnd sn-addcand mpbid ) ABBEFZBGEFZHBGHAQBRDABIJRBHCBKLMABBGCCANOP $. $( $j usage 'sn-addid0' avoids 'ax-mulcom'; $) $} @@ -665924,7 +665924,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove sn-subeu $p |- ( ( A e. CC /\ B e. CC ) -> E! x e. CC ( A + x ) = B ) $= ( vy cc wcel wa cv caddc co wceq wreu wrex sn-negex adantr wb wral simprl cc0 addcld simplr simplrr oveq1d simplll simplrl simpllr addassd 3eqtr3rd - sn-addid2 eqeq2d simpr sn-addcand bitrd ralrimiva reu6i syl2anc rexlimddv + sn-addlid eqeq2d simpr sn-addcand bitrd ralrimiva reu6i syl2anc rexlimddv syl ) BEFZCEFZGZBDHZIJZSKZBAHZIJZCKZAELZDEUSVDDEMUTBDNOVAVBEFZVDGZGZVBCIJ ZEFVGVEVLKZPZAEQVHVKVBCVAVIVDRUSUTVJUATVKVNAEVKVEEFZGZVGVFBVLIJZKVMVPCVQV FVPVCCIJSCIJZVQCVPVCSCIVAVIVDVOUBUCVPBVBCUSUTVJVOUDZVAVIVDVOUEZUSUTVJVOUF @@ -666010,7 +666010,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $d A x $. $( Commuted version of ~ ax-1rid without ~ ax-mulcom . (Contributed by SN, 5-Feb-2024.) $) - remulid2 $p |- ( A e. RR -> ( 1 x. A ) = A ) $= + remullid $p |- ( A e. RR -> ( 1 x. A ) = A ) $= ( vx cr wcel cc0 wceq c1 co wn wne df-ne wa ax-rrecex simpll recnd simprl cmul cv mulassd simprr oveq1d remulinvcom ax-1rid eqtrd 3eqtr3d rexlimddv oveq2d syl ex biimtrrid 1re remul01 mp1i oveq2 id 3eqtr4d pm2.61d2 ) ACDZ @@ -666018,10 +666018,10 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove AVDAQHZQHZUTAVIAVDAVIAURVBVHNZOZVIVDVCVGVFPZOVMSVIVEGAQVCVGVFTZUAVIVKAGQH ZAVIVJGAQVIAVDVLVNVOUBUGVIURVPAFVLAUCUHUDUEUFUIUJUSGEQHZEUTAGCDVQEFUSUKGU LUMAEGQUNUSUOUPUQ $. - $( $j usage 'remulid2' avoids 'ax-mulcom'; $) + $( $j usage 'remullid' avoids 'ax-mulcom'; $) $} - $( Lemma for ~ sn-mulid2 and ~ it1ei . (Contributed by SN, 27-May-2024.) $) + $( Lemma for ~ sn-mullid and ~ it1ei . (Contributed by SN, 27-May-2024.) $) sn-1ticom $p |- ( 1 x. _i ) = ( _i x. 1 ) $= ( ci cmul co ax-icn mulcli mulassi oveq2i 3eqtr4i eqtri rei4 oveq1i 3eqtr3i c1 ) AABCZNBCZABCZAOBCZMABCAMBCPNNABCZBCZQNNAAADDEZTDFNANBCZBCAAUABCZBCSQAA @@ -666029,9 +666029,9 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove ${ $d A x y $. - $( ~ mulid2 without ~ ax-mulcom . (Contributed by SN, 27-May-2024.) $) - sn-mulid2 $p |- ( A e. CC -> ( 1 x. A ) = A ) $= - ( vx vy cc wcel cv ci cmul co caddc wceq cr wrex recn adantr a1i remulid2 + $( ~ mullid without ~ ax-mulcom . (Contributed by SN, 27-May-2024.) $) + sn-mullid $p |- ( A e. CC -> ( 1 x. A ) = A ) $= + ( vx vy cc wcel cv ci cmul co caddc wceq cr wrex recn adantr a1i remullid c1 adantl mulassd cnre 1cnd ax-icn mulcld adddid sn-1ticom oveq1i oveq12d wa oveq2d 3eqtrd eqtr3d eqtrd oveq2 id eqeq12d syl5ibrcom rexlimivv syl ) ADEABFZGCFZHIZJIZKZCLMBLMRAHIZAKZBCAUAVDVFBCLLUTLEZVALEZUIZVFVDRVCHIZVCKV @@ -666039,13 +666039,13 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove EVIVKUTVLVBJVGVKUTKVHUTQOVIRGHIZVAHIZVLVBVIRGVAVMVNVOTVIVQGRHIZVAHIZGRVAH IZHIVBVQVSKVIVPVRVAHUFUGPVIGRVAVNVMVOTVIVTVAGHVHVTVAKVGVAQSUJUKULUHUMVDVE VJAVCAVCRHUNVDUOUPUQURUS $. - $( $j usage 'sn-mulid2' avoids 'ax-mulcom'; $) + $( $j usage 'sn-mullid' avoids 'ax-mulcom'; $) $} - $( ` 1 ` is a multiplicative identity for ` _i ` (see ~ sn-mulid2 for + $( ` 1 ` is a multiplicative identity for ` _i ` (see ~ sn-mullid for commuted version). (Contributed by SN, 1-Jun-2024.) $) it1ei $p |- ( _i x. 1 ) = _i $= - ( c1 ci cmul co sn-1ticom cc wcel wceq ax-icn sn-mulid2 ax-mp eqtr3i ) ABCD + ( c1 ci cmul co sn-1ticom cc wcel wceq ax-icn sn-mullid ax-mp eqtr3i ) ABCD ZBACDBEBFGMBHIBJKL $. $( The multiplicative inverse of ` _i ` (per ~ i4 ) is also its additive @@ -666067,7 +666067,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove remulcand $p |- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) ) $= ( vx cmul co wceq c1 cr wcel wa adantr recnd syl 3eqtr3d cv cc0 ax-rrecex wi wne wrex syl2anc simplr simpr remulinvcom ex w3a oveq2 3ad2ant3 oveq1d - simp2 simp1r 3ad2ant1 simp1l mulassd remulid2 3exp syld rexlimddv impbid1 + simp2 simp1r 3ad2ant1 simp1l mulassd remullid 3exp syld rexlimddv impbid1 impr ) ADBJKZDCJKZLZBCLZADIUAZJKMLZVIVJUDZINADNOZDUBUEVLINUFGHIDUCUGAVKNO ZVLVMAVOPZVLVKDJKZMLZVMVPVLVRVPVLPDVKVPVNVLAVNVOGQZQAVOVLUHVPVLUIUJUKVPVR VIVJVPVRVIULZVKVGJKZVKVHJKZBCVIVPWAWBLVRVGVHVKJUMUNVTVQBJKMBJKZWABVTVQMBJ @@ -666085,13 +666085,13 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove ( cc0 ci cmul co wcel caddc wceq cr ax-icn wa syl a1i recnd eqtrdi oveq2d c1 eqtrd oveq1d mulassd ax-1cn va vb vx vy vz wrex 0cn mulcli cnre simplr cc cv wn cresub wne neqne adantl simplll rernegcl 1red readdcld ax-rrecex - sylan 1cnd adddid oveq2i 0cnd mulcld renegid2 ad3antrrr simpllr sn-addid2 + sylan 1cnd adddid oveq2i 0cnd mulcld renegid2 ad3antrrr simpllr sn-addlid it1ei addassd 3eqtr3d sn-mul01 eqtr3d oveq12d reixi 1re remulcld eqeltrrd ax-mp eqeltri remul02 ad2antrr addcld simprl simprr rexlimddv necon1d mpd - renegid readdid2 readdid1 3eqtr3rd 0re oveq1i mulassi eqtr3i 3eqtr4d mp1i + renegid readdlid readdrid 3eqtr3rd 0re oveq1i mulassi eqtr3i 3eqtr4d mp1i ex ax-1rid eqeq2i oveq2 adddii rei4 oveq12i 3eqtr3g readdcli df-2 sn-0ne2 eqtri c2 necomi eqnetrri mp2an addcli addassi ipiiie0 3eqtr2i simpl simpr - eqtrid remulid2 adantr 1t1e1ALT rexlimdvaa mpdan sylbi pm2.18da rexlimivv + eqtrid remullid adantr 1t1e1ALT rexlimdvaa mpdan sylbi pm2.18da rexlimivv mpi mp2b ) ABCDZUKEZYPUAULZBUBULZCDZFDZGZUBHUFUAHUFYPAGZABUGIUHZUAUBYPUIU UBUUCUAUBHHYRHEZYSHEZJZUUBUUCUUGUUBJZUUCUUHUUCUMZJZYPPBPCDZFDZGZUUCUUJYPU UAUULUUGUUBUUIUJZUUJYRPYTUUKFUUJYRAYRUNDZPFDZFDZYRAFDZPYRUUJUUPAYRFUUJYPA @@ -666163,7 +666163,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( ~ ltaddpos without ~ ax-mulcom . (Contributed by SN, 13-Feb-2024.) $) sn-ltaddpos $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( B + A ) ) ) $= - ( cr wcel wa cc0 clt wbr caddc co wb 0re ltadd2 mp3an1 wceq readdid1 adantl + ( cr wcel wa cc0 clt wbr caddc co wb 0re ltadd2 mp3an1 wceq readdrid adantl breq1d bitrd ) ACDZBCDZEZFAGHZBFIJZBAIJZGHZBUEGHFCDTUAUCUFKLFABMNUBUDBUEGUA UDBOTBPQRS $. $( $j usage 'sn-ltaddpos' avoids 'ax-mulcom'; $) @@ -666171,7 +666171,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( ~ ltaddneg without ~ ax-mulcom . (Contributed by SN, 25-Jan-2025.) $) sn-ltaddneg $p |- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < B ) ) $= - ( cr wcel wa cc0 clt wbr caddc co wb 0re ltadd2 mp3an2 wceq readdid1 adantl + ( cr wcel wa cc0 clt wbr caddc co wb 0re ltadd2 mp3an2 wceq readdrid adantl breq2d bitrd ) ACDZBCDZEZAFGHZBAIJZBFIJZGHZUDBGHTFCDUAUCUFKLAFBMNUBUEBUDGUA UEBOTBPQRS $. $( $j usage 'sn-ltaddneg' avoids 'ax-mulcom'; $) @@ -666246,7 +666246,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove $( Addition is commutative for nonnegative integers. Proven without ~ ax-mulcom . (Contributed by SN, 1-Feb-2025.) $) nn0addcom $p |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A + B ) = ( B + A ) ) $= - ( cn0 wcel cn cc0 wceq wo caddc co elnn0 readdid2 readdid1 eqtr4d syl oveq1 + ( cn0 wcel cn cc0 wceq wo caddc co elnn0 readdlid readdrid eqtr4d syl oveq1 cr oveq2 eqeq12d syl5ibrcom nnaddcom nnre impcom jaoian sylanb nn0re jaodan imp sylan2b ) BCDACDZBEDZBFGZHABIJZBAIJZGZBKUJUKUOULUJAEDZAFGZHUKUOAKUPUKUO UQABUAUKUQUOUKUOUQFBIJZBFIJZGZUKBQDZUTBUBVAURBUSBLBMNOUQUMURUNUSAFBIPAFBIRS @@ -666259,7 +666259,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove ( A + B ) = ( B + A ) ) $= ( cr wcel cc0 cresub co cn wa cn0 caddc simpr nn0cnd ad2antrr recnd addassd wceq syl oveq1d addcld rernegcl simpll renegid2 oveq2d nn0re adantl 3eqtrrd - readdid1 readdid2 sylan9eq nnnn0 nn0addcom sylan adantll 3eqtr4d sn-addcand + readdrid readdlid sylan9eq nnnn0 nn0addcom sylan adantll 3eqtr4d sn-addcand adantr 3eqtr3d mpbid ) ACDZEAFGZHDZIZBJDZIZVAABKGZKGZVABAKGZKGZQVFVHQVEVAAK GZBKGZVABKGZAKGZVGVIVEBBVAKGZAKGZVKVMVEVOBVJKGBEKGZBVEBVAAVEBVCVDLMZVEVAUTV ACDVBVDAUANOZVEAUTVBVDUBOZPVEVJEBKUTVJEQVBVDAUCZNUDVDVPBQZVCVDBCDZWABUEZBUH @@ -666272,7 +666272,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove zaddcom $p |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) = ( B + A ) ) $= ( cz wcel cn0 cr cresub co cn wa wo caddc wceq reelznn0nn zaddcomlem oveq1d cc0 nncnd addassd 3eqtr4d nn0addcom eqcomd renegid2 ad2antrl ad2antrr recnd - ancoms simplr simpll simprl readdid2 oveq2d simprr addcld nnaddcom ad2ant2l + ancoms simplr simpll simprl readdlid oveq2d simprr addcld nnaddcom ad2ant2l 3eqtr3d nnaddcld sn-addcand mpbid ccase syl2anb ) ACDAEDZAFDZQAGHZIDZJZKBED ZBFDZQBGHZIDZJZKABLHZBALHZMZBCDANBNVCVHVGVLVOABUAABOVLVCVOVLVCJVNVMBAOUBUGV GVLJZVEVJLHZVMLHZVQVNLHZMVOVPVJVELHZVMLHZVEVJVNLHZLHZVRVSVPVJVEVMLHZLHZVEAL @@ -666292,7 +666292,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove renegmulnnass $p |- ( ph -> ( ( 0 -R A ) x. N ) = ( 0 -R ( A x. N ) ) ) $= ( vx wcel cc0 cresub co cmul wceq caddc oveq2 oveq2d eqeq12d syl ad2antrr c1 cr vy cn cv weq rernegcl ax-1rid eqtr4d wa 0red nnre ad2antlr remulcld - simpr readdsub syl3anc readdid2 oveq1d eqtr3d resubsub4 3eqtr4d nnadd1com + simpr readdsub syl3anc readdlid oveq1d eqtr3d resubsub4 3eqtr4d nnadd1com 3eqtrd cc recnd 1cnd nncn adddid eqtrd nnindd mpdan ) ACUBGHBIJZCKJZHBCKJ ZIJZLZEAVKFUCZKJZHBVPKJZIJZLVKSKJZHBSKJZIJZLVKUAUCZKJZHBWCKJZIJZLZVKWCSMJ ZKJZHBWHKJZIJZLVOFUACVPSLZVQVTVSWBVPSVKKNWLVRWAHIVPSBKNOPFUAUDZVQWDVSWFVP @@ -666467,7 +666467,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove cmul ax-icn a1i w3o lttri4 mpan2 reneg1lt0 rernegcl ltnsymi relt0neg1 sylan mulgt0 wb remulneg2d remulcld ipiiie0 renegadd mpbiri oveq1d mulassi oveq1i id mpdan mulcli eqtr3i rei4 3eqtrd oveq2d eqtrd breq2d adantr mpbid ex mtoi - sylbid 0ne1 neii oveq12 ax-i2m1 remul02 readdid2 eqtri 3eqtr3g mto breqtrdi + sylbid 0ne1 neii oveq12 ax-i2m1 remul02 readdlid eqtri 3eqtr3g mto breqtrdi reixi 3ioran syl3anbrc pm2.65i ) ABCZADEFZADGZDAEFZUAZWGDBCZWKHADUBUCWGWHIW IIZWJIWKIWGWHDDJKLZEFZWNDEFWOIUDWNDJBCZWNBCMJUENHUFNZWGWHDDAKLZEFZWOAUGWGWS WOWGWSODWRWRRLZEFZWOWGWRBCZWSXAAUEZXBWSOXAWRWRUIPUHWGXAWOUJWSWGWTWNDEWGWTDW @@ -666513,9 +666513,9 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove cnreeu $p |- ( ph -> ( ( r + ( _i x. s ) ) = ( t + ( _i x. u ) ) <-> ( r = t /\ s = u ) ) ) $= ( ci cmul co caddc wceq cc0 oveq2d wcel cr syl adantr cv weq cresub oveq1 - recnd ax-icn a1i mulcld rernegcl addassd renegid adddid sn-it0e0 readdid1 - wa 3eqtr3d 3eqtrd oveq1d sn-addid2 renegid2 3eqtr4d addcld eqeq12d biimpa - cc 3eqtr3rd simpr readdcld eqeltrrd itrere syl2anc adantl readdid2 syldan + recnd ax-icn a1i mulcld rernegcl addassd renegid adddid sn-it0e0 readdrid + wa 3eqtr3d 3eqtrd oveq1d sn-addlid renegid2 3eqtr4d addcld eqeq12d biimpa + cc 3eqtr3rd simpr readdcld eqeltrrd itrere syl2anc adantl readdlid syldan oveq2 sylan9req eqtr2d jca ex syl5 id oveqan12d impbid1 ) AEUAZJDUAZKLZML ZCUAZJBUAZKLZMLZNZECUBZDBUBZUOZWLOWHUCLZWGJOWEUCLZKLZMLZMLZWPWKWRMLZMLZNZ AWOWLWSXAWPMWGWKWRMUDPAXCWOAXCWPWDMLZJWIWQMLZKLZNZWOAXCXGAWTXDXBXFAWSWDWP @@ -667343,7 +667343,7 @@ evaluates to zero (the "zero set"). (In other words, scalar multiples 2nn nne bicomi rexbii rexnal bitri wss ax-mp neeq1d addgt0d readdcld 0red breqtrd expeq0 eluzelz expne0d lt2addd 00id breqtrdi ltnsymd cmul simp-5l breq2d 2nn0 nncn 2cnd 2ne0 divcan2d cdvds nndivdvds nnnn0 mvrraddd subcld - mvlraddd pncan3d addcld negidd addassd addid2d 3eqtr3d negnegd 3rspcedvdw + mvlraddd pncan3d addcld negidd addassd addlidd 3eqtr3d negnegd 3rspcedvdw pncand eqtr3d rexlimdva2 reximia 3imtr3i con4i wi dfn2 ssdif eqsstri ssel nn0ssz ss2ralv imim12d ralimdv2 neeq2d cbvral3vw sylib ralimi impbii ) AU AZDUAZHIZBUAZUWOHIZJIZCUAZUWOHIZUCZCKUBZBKUBZAKUBZDUDUEUFZUBZEUAZUWOHIZFU @@ -668159,8 +668159,8 @@ evaluates to zero (the "zero set"). (In other words, scalar multiples adddirp1d expaddd chash fsumconst hashfzo0 eqtrd 3eqtr3d breqtrrd expge0d nnrpd rpge0d leexp1a syl32anc lemul1ad ltexp1dd ltmul1dd leltaddd lelttrd ltled fsumle nncand exp1d 0zd peano2zd 0cn ax-1cn addassi addcli eluzp1m1 - cz addid2i mulcld oveq2 oveq12d nn0zd fzosumm1 1p1e2 3eqtri fveq2d fsumcl - eleqtrrd ltadd2dd addcomd sub32d nnncand subidd exp0d recnd mulid1d sylib + cz addlidi mulcld oveq2 oveq12d nn0zd fzosumm1 1p1e2 3eqtri fveq2d fsumcl + eleqtrrd ltadd2dd addcomd sub32d nnncand subidd exp0d recnd mulridd sylib elnn0uz ltmul2dd pwdif syl3anc mvlraddd ) ADCLMZDENLMZOMZUVACUVAOMZPMZQMZ PMZDEOMZCEOMZLMZBEOMZUAAUVFUUTREUBMZDKUCZOMZCEUVLLMZNLMZOMZPMZKUDZPMZUVIU AAUVEUVRUUTAUVBUVDADUVAADHUEZAEUGUFUHSZEURSZUVAUISZIEUJZEUKULZUMZAUVAUVCA @@ -668212,7 +668212,7 @@ evaluates to zero (the "zero set"). (In other words, scalar multiples fltnlta $p |- ( ph -> N < A ) $= ( co c1 cexp cmul caddc cdiv wcel nnred remulcld cmin c3 cuz cn eluzge3nn cfv resubcld c2 uzuzle23 uz2m1nn 3syl nnnn0d reexpcld readdcld nnrpd nnzd - syl rpexpcld rerpdivcld clt cr nncnd recnd adddird pncan3d oveq1d mulid2d + syl rpexpcld rerpdivcld clt cr nncnd recnd adddird pncan3d oveq1d mullidd 1cnd 3eqtr3rd oveq2d eqeltrrd nn0ge0d 1red wbr fltltc wb nnltp1le syl2anc cle mpbid leidd lesub3d lemulge12d mulassd mulcld nnne0d expne0d divcan4d rpred difrp ltexp1dd ltmul2dd ltdiv1dd eqbrtrrd lelttrd breqtrrd ltadd1dd @@ -668348,10 +668348,10 @@ evaluates to zero (the "zero set"). (In other words, scalar multiples 3cubeslem2 $p |- ( ph -> -. ( ( ( ( 3 ^ 3 ) x. ( A ^ 2 ) ) + ( ( 3 ^ 2 ) x. A ) ) + 3 ) = 0 ) $= ( c3 cexp co c2 cmul caddc cc0 c1 cmin wcel a1i recnd oveq2d oveq1d eqtrd - cr 3eqtr4rd 3re mulid2d sqcld qre syl resqcld mulcld 1cnd adddird mulassd - cq addcld mulcomd wceq df-3 expp1d sqvald 3eqtr4d mulid1d binom2d 2timesd + cr 3eqtr4rd 3re mullidd sqcld qre syl resqcld mulcld 1cnd adddird mulassd + cq addcld mulcomd wceq df-3 expp1d sqvald 3eqtr4d mulridd binom2d 2timesd cn0 2nn0 sq1 addcomd cc sqmuld eqeltrd addassd addsubassd subidd remulcld - addid1d eqtr2d peano2re resubcld 3nn nnq ax-mp sylancr 3cubeslem1 gt0ne0d + addridd eqtr2d peano2re resubcld 3nn nnq ax-mp sylancr 3cubeslem1 gt0ne0d cn qmulcl wne 3ne0 mulne0d eqnetrd neneqd ) ADDEFZBGEFZHFZDGEFZBHFZIFZDIF ZJAWPDBHFZKIFZGEFZWQLFZDHFZJAWPWMWKHFZWQIFZKIFZDHFZXAAXCDHFZKDHFZIFXFDIFZ XEWPAXGDXFIADADDSMAUANZOZUBPAXCKDAXBWQAWMWKADXJUCZAWKABABUKMZBSMCBUDUEZUF @@ -668643,10 +668643,10 @@ evaluates to zero (the "zero set"). (In other words, scalar multiples c2 addcld mulassd 3eqtrd cmin cneg c7 c9 c6 c5 c8 c4 cc0 eqidd cc 3cn cn0 wcel 3nn0 expcld cq qcn ax-1cn negsubd eqtr4d wa wceq negcld binom3 sqcld syl jca cu3addd 2nn0 nn0mulcld nn0cnd addcomd eqcomd cn nnexpcld nnred cr - 3nn qre reexpcld remulcld cdvds wn n2dvds3 negexpidd addid2d 1red addid1d + 3nn qre reexpcld remulcld cdvds wn n2dvds3 negexpidd addlidd 1red addridd wbr sqnegd sqmuld expmuld 3t2e6 6p1e7 3eqtr2rd 1nn0 nn0addcld expaddd 7cn - expp1d df-2 7p1e8 8p1e9 6nn0 adddid mulid1d mulexpd 2cn mulcomli 3eqtr4rd - eqtr2d 3t3e9 3eqtr4d 7nn0 9nn0 mulm1d negidd 8nn0 mulid2d negeqd mulneg1d + expp1d df-2 7p1e8 8p1e9 6nn0 adddid mulridd mulexpd 2cn mulcomli 3eqtr4rd + eqtr2d 3t3e9 3eqtr4d 7nn0 9nn0 mulm1d negidd 8nn0 mullidd negeqd mulneg1d 4p4e8 4p1e5 oveq12d 3p2e5 3p1e4 2t2e4 5nn0 4nn0 exp1d 1p2e3 adddird 2p2e4 eqtr3d 2timesd 4p2e6 df-8 df-3 df-7 sq1 mulneg2d 3eqtr2d 2p1e3 ) ADDEFZBD EFZGFZHUAFZDEFZUUGUBZDQEFZBGFZIFHIFDEFZIFZUUEBQEFZGFZUULIFZDEFZIFZBUCEFZD @@ -671884,7 +671884,7 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, ovexd nnm1nn0 nn0uz eleqtrdi nnz nnre ltm1d fzsdom2 syl21anc rpre elfzelz ad2antrr zred remulcld 1rp modcl flcld wn recnd mul01d modge0 0red lemul2 wb nngt0 syl112anc mpbid eqbrtrrd lenltd fllt mtbid mpbird sylanbrc caddc - 0z elnn0z flle syl modlt 1red ltmul2 mulid1d breqtrd lelttrd wceq cc nncn + 0z elnn0z flle syl modlt 1red ltmul2 mulridd breqtrd lelttrd wceq cc nncn ax-1cn npcan breqtrrd 1z zsubcl zleltp1 syl2anc elfz2nn0 syl3anbrc oveq1d oveq2 oveq2d fveq2d fphpdo ) CUBEZDUCEZUDZABUAFDUEGZFDHUFGZUEGZDCUAUGZIGZ HJGZIGZKLZDCAUGZIGZHJGZIGZKLDCBUGZIGZHJGZIGZKLYAMUHXTYAFUILZMFDUJYQTMFUKU @@ -671971,7 +671971,7 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, cneg rpred nnred remulcld nn0re resubcld recnd rpreccld rprege0d flge0nn0 abscld nn0p1nn 3syl simpr ifcld nnrecred 0red rprecred flcld peano2re syl cz zred max2 syl2anc wb nngt0d lerec syl22anc mpbid fllep1 nnne0d recrecd - nncnd breqtrrd recgt0d rpgt0d mpbird mulid1d nnge1d 1red lemul2d eqbrtrrd + nncnd breqtrrd recgt0d rpgt0d mpbird mulridd nnge1d 1red lemul2d eqbrtrrd letrd subid1d ltletrd absltd simprd ltsub2d sylanbrc elfzle2 max1 syl3anc elnnz maxle mpbir2and weq oveq2 fvoveq1d breq1 id ifbieq2d oveq2d breq12d fveq2d breq1d rspc2ev irrapxlem3 r19.29vva ) CUAGZDHGZIZCEJZUDKZFJZLKZMUB @@ -672009,7 +672009,7 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, oveq1d nncnd qre rpre ad3antrrr resubcld recnd absmuld eqtr4d cc qcn rpcn subdid divcan2d mulcomd oveq12d eqtrd fveq2d abssubd 3eqtrd abscld rpge0d remulcld simpllr rprecred syl2anc ifcld rpred fllep1 letrd lerecd recrecd - max2 mpbid rpne0d mulid2d nnge1d lemul1d eqbrtrd ltletrd wb nngt0d ltmul2 + max2 mpbid rpne0d mullidd nnge1d lemul1d eqbrtrd ltletrd wb nngt0d ltmul2 1red syl112anc mpbird msqgt0d gt0ne0d rereccld qdencl max1 dividd divrecd divdiv1d 3eqtr3rd 3brtr4d cz nnzd divdenle le2msq syl22anc lerec wceq mpd 2nn0 expneg sylancl sqvald oveq2d breqtrrd breq2 fvoveq1 breq1d 3anbi123d @@ -672128,7 +672128,7 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, subsq addcld mulcomd eqtrd 3eqtrd fveq2d 3eqtr3d absmuld remulcld cz 2nn0 nn0negzi a1i reexpclzd 1red 2re readdcld simpr wb divgt0d sqrtgt0 addgt0d nngt0d gt0ne0d absgt0 biimpa ltmul1 syl112anc mpbid sqgt0d ltmul2 expclzd - mulass syl3anc expneg sylancl recidd oveq1d mulid2d addcomd ppncan 2times + mulass syl3anc expneg sylancl recidd oveq1d mullidd addcomd ppncan 2times cn0 syl abstrid 0le2 sqrtge0d mulge0d nnsqcld 0lt1 lerec syl22anc 1div1e1 nnge1d breqtrdi eqbrtrd ltletrd ltled leadd1dd letrd ) CUADZAUADZBUADZUBZ ABEFZCUCGZUDFZHGZBIUEZUFFZJKZVDZAIUFFZCBIUFFZLFZUDFZHGZUUPUUIUUGUUHMFZLFZ @@ -672329,8 +672329,8 @@ trivial solution (1,0). We are not explicitly defining the mpbid nn0ge0d wa absresq sqdivd cc sqne0 3eqtr2d oveq12d mulsubd addcld subdid adddid mulcomd mulassd sqmuld eqtr4d subdird eqtr3d subdi negeqd clt w3a syl3anc 3eqtr3d adantr simpr neqned divne0d nnne0d oveq1 adantl - nnabscl divcan1d csqrt ad2antrr ex mulid2d zcnd crp npcand eqtr2d recnd - absrpcld 0red absne0d 1zzd zred 0mod addid2d zmulcld wn 0lt1 0re ltnlei + nnabscl divcan1d csqrt ad2antrr ex mullidd zcnd crp npcand eqtr2d recnd + absrpcld 0red absne0d 1zzd zred 0mod addlidd zmulcld wn 0lt1 0re ltnlei 1re mpbi mulge0d suble0d breq1 syl5ibrcom mtoi divassd divsubdird mul4d sqge0d nnncan2d addsub4d mulneg2d mulneg1d fvoveq1d div0d abs00bd sq0id negsubdi2 mtand negsub divmuleqd divcan4d nngt0d syl22anc sqrtsqd fveq2 @@ -672989,8 +672989,8 @@ trivial solution (1,0). We are not explicitly defining the simplrl sq0 eqtrdi rpcnd mul01d eqtrd simplrr recnd sqcld subid1d 3eqtr3d cc eqtr2id wb nn0ge0 0le1 sq11 syl22anc mpbid simpr oveq12d 1p0e1 breqtrd ltnri pm2.24 mpisyl wo elnn0 sylib mpjaodan le2sqd suble0d mpbird lemul2d - eqbrtrrd sqsqrtd simprr eqcomd mulcld subdid mulid1d oveq1d eqtr2d 3eqtrd - leadd2dd addsub12d addid1d 3brtr4d rpge0d le2addd ) CUADZAUBDZBUBDZEZEZFA + eqbrtrrd sqsqrtd simprr eqcomd mulcld subdid mulridd oveq1d eqtr2d 3eqtrd + leadd2dd addsub12d addridd 3brtr4d rpge0d le2addd ) CUADZAUBDZBUBDZEZEZFA CUCUDZBGHZIHZUEJZASKHZCBSKHZGHZLHZFMZEZEZCFIHZUCUDZXTAYAYJYLYJYKYJCFXOCUF DXRYICUGUHZFUFDYJULNUIZUJZUKZYJXTYJCYMUJZUKZXRAUMDZXOYIXPYSXQAUNOPZYJXTBY RXRBUMDZXOYIXQUUAXPBUNUOPZUPYJYLAQJYLSKHZYDQJYJYDCFYELHZGHZIHZYDCRGHZIHZU @@ -673230,7 +673230,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by recnd breqtrrd mpd3an23 wo cpell14qr pell1qrss14 sselda pell14qrre syldan wceq wi leloed simp-4l simp-4r simplr simprr ad3antrrr ad4antr wss sseldd ad2antrr simprl wb 2pos lemul2 syl112anc mpbid ltletrd w3a simp1 3ad2ant1 - cdiv simp2l simp2r pell14qrdivcl syl3anc mulid2d simp3l eqbrtrd ltdivmul2 + cdiv simp2l simp2r pell14qrdivcl syl3anc mullidd simp3l eqbrtrd ltdivmul2 pell14qrgt0 ltmuldiv simp3r mpbird wn simpll pell14qrgapw ltnsym pm2.21dd mpd syl22anc syl122anc r19.29a exp32 simp1r eqeltrd 3exp jaod sylbid impd simp2 rexlimdva ) AUAUBUCDZAUDEZBUEZFGZYDHYCIJZKGZLZBAUFEZUGZYCYIDZYBYFMD @@ -673363,7 +673363,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by reglogexpbas $p |- ( ( N e. ZZ /\ ( C e. RR+ /\ C =/= 1 ) ) -> ( ( log ` ( C ^ N ) ) / ( log ` C ) ) = N ) $= ( cz wcel crp c1 wne wa cexp clog cfv cdiv cmul wceq simprl simpl reglogexp - co simpr syl3anc reglogbas adantl oveq2d cc zcn adantr mulid1d 3eqtrd ) BCD + co simpr syl3anc reglogbas adantl oveq2d cc zcn adantr mulridd 3eqtrd ) BCD ZAEDZAFGZHZHZABIRJKAJKZLRZBUNUNLRZMRZBFMRBUMUJUIULUOUQNUIUJUKOUIULPUIULSAAB QTUMUPFBMULUPFNUIAUAUBUCUMBUIBUDDULBUEUFUGUH $. @@ -673546,7 +673546,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by ( c2 cfv wcel cexp co c1 cmin csqrt caddc wceq cle wbr cn clt cmul cz recnd syl a1i cuz cpellfund csquarenn cdif cpell14qr rmspecnonsq eluzelz resubcld zsqcl zred 1red cc0 eluz2b2 simprbi eluzelre 0le1 eluzge2nn0 nn0ge0d lt2sqd - sq1 mpbid eqbrtrrd posdifd elrpd rpsqrtcld rpred mulid1d oveq2d pell1qrss14 + sq1 mpbid eqbrtrrd posdifd elrpd rpsqrtcld rpred mulridd oveq2d pell1qrss14 cpell1qr wss cn0 1nn0 oveq2i eqtrid 1cnd nncand eqtrd pellqrexplicit sseldd syl31anc eqeltrrd readdcld ltaddrpd ltadd1dd lttrd pellfundlb npcand fveq2d syl3anc sqrtsqd oveq1d pellfundge cr pellfundre letri3d mpbir2and ) ABUACDZ @@ -673807,7 +673807,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by rmxy1 $p |- ( A e. ( ZZ>= ` 2 ) -> ( ( A rmX 1 ) = A /\ ( A rmY 1 ) = 1 ) ) $= ( c2 cfv wcel c1 crmx co cexp crmy cmul caddc wceq cz 1z mpan2 rpcnd cq cn0 - fovcl sselid cmin csqrt wa rmxyval rmbaserp exp1d rmspecpos sqrtcld mulid1d + fovcl sselid cmin csqrt wa rmxyval rmbaserp exp1d rmspecpos sqrtcld mulridd cuz eqcomd oveq2d 3eqtrd cc cdif rmspecsqrtnq nn0ssq frmx zssq frmy eluzelz wb zq syl sselii a1i qirropth syl122anc mpbid ) ABUJCZDZAEFGZABHGEUAGZUBCZA EIGZJGKGZAVNEJGZKGZLZVLALVOELUCZVKVPAVNKGZEHGZWAVRVKEMDZVPWBLNAEUDOVKWAVKWA @@ -673894,7 +673894,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by ( A rmX ( N + 1 ) ) = ( ( ( A rmX N ) x. A ) + ( ( ( A ^ 2 ) - 1 ) x. ( A rmY N ) ) ) ) $= ( c2 cuz wcel cz wa c1 caddc co crmx cmul cexp cmin crmy wceq adantr oveq2d - cfv eqtrd 1z rmxadd mp3an3 rmx1 rmy1 frmy fovcl zcnd mulid1d oveq12d ) ACDS + cfv eqtrd 1z rmxadd mp3an3 rmx1 rmy1 frmy fovcl zcnd mulridd oveq12d ) ACDS ZEZBFEZGZABHIJKJZABKJZAHKJZLJZACMJHNJZABOJZAHOJZLJZLJZIJZUPALJZUSUTLJZIJULU MHFEUOVDPUAABHUBUCUNURVEVCVFIUNUQAUPLULUQAPUMAUDQRUNVBUTUSLUNVBUTHLJZUTULVB VGPUMULVAHUTLAUERQUNUTUNUTABFUKFOUFUGUHUITRUJT $. @@ -673905,7 +673905,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by rmyp1 $p |- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A rmY ( N + 1 ) ) = ( ( ( A rmY N ) x. A ) + ( A rmX N ) ) ) $= ( c2 cuz wcel cz wa c1 caddc co crmy crmx cmul wceq 1z rmyadd oveq2d adantr - cfv eqtrd mp3an3 rmx1 rmy1 cn0 frmx fovcl nn0cnd mulid1d oveq12d ) ACDSZEZB + cfv eqtrd mp3an3 rmx1 rmy1 cn0 frmx fovcl nn0cnd mulridd oveq12d ) ACDSZEZB FEZGZABHIJKJZABKJZAHLJZMJZABLJZAHKJZMJZIJZUOAMJZURIJUKULHFEUNVANOABHPUAUMUQ VBUTURIUKUQVBNULUKUPAUOMAUBQRUMUTURHMJZURUKUTVCNULUKUSHURMAUCQRUMURUMURABUD UJFLUEUFUGUHTUIT $. @@ -673918,7 +673918,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by ( ( ( A ^ 2 ) - 1 ) x. ( A rmY N ) ) ) ) $= ( c2 wcel cz c1 cneg caddc co crmx cmul cmin crmy mpan2 eqtrd adantr oveq2d wceq 1z cc cuz cfv wa cexp neg1z rmxadd mp3an3 rmxneg rmx1 cn0 fovcl nn0cnd - frmx mulcomd rmyneg rmy1 negeqd frmy zcnd ax-1cn mulneg2 sylancl mulid1d cn + frmx mulcomd rmyneg rmy1 negeqd frmy zcnd ax-1cn mulneg2 sylancl mulridd cn eluzelcn csquarenn rmspecnonsq eldifad nncnd mulneg2d oveq12d adantl negsub zcn mulcld negsubd 3eqtr3d ) ACUAUBZDZBEDZUCZABFGZHIZJIZAABJIZKIZACUDIFLIZA BMIZKIZGZHIZABFLIZJIWFWILIWAWDWEAWBJIZKIZWGWHAWBMIZKIZKIZHIZWKVSVTWBEDWDWRR @@ -674482,7 +674482,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by wa 3adant3 simpr nn0z peano2zd frmy fovcl syl2anc remulcld 3ad2ant2 simp1 zred expp1d simpl cn 2nn eluz2nn nnmulcl nnm1nn0 nn0ge0 3syl 3jca lemul1a jca stoic3 eqbrtrd nn0cn pncan eqeltrd nn0re lep1d wb lermy syl3anc mpbid - recnd mulid1d lesub2dd subdid mulcomd oveq1d eqtrd rmyluc2 letrd 3exp a2d + recnd mulridd lesub2dd subdid mulcomd oveq1d eqtrd rmyluc2 letrd 3exp a2d nn0ind impcom ) BUCCADUDUEZCZDAEFZGHFZBIFZABGJFZKFZLMZYFYHUAUFZIFZAYMGJFZ KFZLMZNYFYHTIFZATGJFZKFZLMZNYFYHUBUFZIFZAUUBGJFZKFZLMZNYFYHUUDIFZAUUDGJFZ KFZLMZNYFYLNUAUBBYMTOZYQUUAYFUUKYNYRYPYTLYMTYHIUGUUKYOYSAKYMTGJUHPUIUJUAU @@ -675007,14 +675007,14 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by ( wcel cmul co cexp cmin c1 crmx crmy cdvds wbr cz adantr oveq12d syl2anc cc0 wceq oveq2 va vb c2 cuz cfv cn0 wa cv wi caddc eluzelz zmulcl sylancr 2z nn0z adantl zmulcld zsqcl zsubcld peano2zm dvds0 rmx0 rmy0 oveq2d zcnd - mul01d eqtrd 1m0e1 eqtrdi cc nn0cn exp0d 1m1e0 breqtrrd rmx1 rmy1 mulid1d + mul01d eqtrd 1m0e1 eqtrdi cc nn0cn exp0d 1m1e0 breqtrrd rmx1 rmy1 mulridd syl nncand exp1d subidd pm3.43 simpll nnz frmx fovcl nn0zd frmy jca nnnn0 cn zexpcl nnm1nn0 zaddcl w3a 3jca ad2antrr congid simpr congmul syl112anc adantrl simprl congsub zaddcld 0zd iddvds subid1d congadd syl322anc sqcld 0z 1cnd addsubd npcand oveq1d eqtr3d ad2antlr expcld subdid mul12d expm1t eqtr4d 3eqtrrd congtr rmxluc rmyluc subcld 2cn mulcld 2cnd mulcomd 3eqtrd mulcl nn0cnd sub4d eqcomd expp1d breq2d imbi2d nncn npcan sylancl mulassd - ax-1cn addid2d sqvald eqtr2d ex expcom a2d syl5 weq 2nn0ind impcom 3impa + ax-1cn addlidd sqvald eqtr2d ex expcom a2d syl5 weq 2nn0ind impcom 3impa ) AUCUDUEZDZBUFDZCUFDZUCAEFZBEFZBUCGFZHFZIHFZACJFZABHFZACKFZEFZHFZBCGFZHF ZLMZUUTUURUUSUGZUVMUVNUVEAUAUHZJFZUVGAUVOKFZEFZHFZBUVOGFZHFZLMZUIUVNUVEAR JFZUVGARKFZEFZHFZBRGFZHFZLMZUIUVNUVEAIJFZUVGAIKFZEFZHFZBIGFZHFZLMZUIUVNUV @@ -675115,8 +675115,8 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by ( I x. M ) ) ) ) ) $= ( wcel cz crmy co cdvds wbr cmul caddc wb cc0 oveq2d breq2d bibi2d imbi2d wi oveq1 va vb c2 cuz cfv wa cn0 cv wceq weq zcn ad2antrl mul02d ad2antll - c1 cc addid1d eqtr2d w3a simp3 simprl simprrl simprrr nn0z adantr zmulcld - zaddcld jm2.19lem2 syl3anc zcnd addassd nn0cn adddird mulid2d eqtrd bitrd + c1 cc addridd eqtr2d w3a simp3 simprl simprrl simprrr nn0z adantr zmulcld + zaddcld jm2.19lem2 syl3anc zcnd addassd nn0cn adddird mullidd eqtrd bitrd 1cnd 3adant3 3exp a2d nn0ind com12 3impia ) AUCUDUEEZCFEZDFEZUFZBUGEZACGH ZADGHZIJZWIADBCKHZLHZGHZIJZMZWHWDWGUFZWPWQWKWIADUAUHZCKHZLHZGHZIJZMZSWQWK WIADNCKHZLHZGHZIJZMZSWQWKWIADUBUHZCKHZLHZGHZIJZMZSWQWKWIADXIUOLHZCKHZLHZG @@ -675298,7 +675298,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by 0zd elun bitri 3bitr4g eqrdv rmspecpos rpcnd wi con3dimp sylbi orel2 sylc fsumsplit fsummulc1 mulcomd expaddd 3cn npcan eqtr3d sumeq2dv 1nn0 oveq1i 1nn 1m1e0 div0i eqtri 0nn0 eqeltri oveq1 sumsn sylancr eqcomd exp1d exp0d - 2cn bcn1 mulid1d fsumcl pncand breqtrrd ) AEUBUCZFZCGFZBUDFZUEZACUFHZIJHZ + 2cn bcn1 mulridd fsumcl pncand breqtrrd ) AEUBUCZFZCGFZBUDFZUEZACUFHZIJHZ EUAUGZKLZUHZUAIBUIHZUJZBDUGZUKHZACULHZBVURUMHZJHZAEJHMUMHZVURMUMHZEUNHZJH ZVUKVURIUMHZJHZNHZNHZNHZDUOZVULNHZACBNHUFHZBVUTBMUMHZJHZVUKNHZNHZUMHZKVUJ VVLGFVULGFZVULVVMKLVUJVUQVVKDVUQUPFZVUJVUPUPFVUQVUPUQVWAIBURVUOUAVUPUSZVU @@ -675431,11 +675431,11 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by rmY M ) ) ) ) $= ( c2 wcel cz co cmul caddc crmy cmin cdvds wbr syl2anc wceq oveq2d oveq1d c1 zcnd va vb cuz cfv wa crmx cneg wo cc0 simprl simprrr frmx fovcl nn0zd - wi cn0 simprrl frmy congid 2cnd mulcld mul02d adantl addid1d adantr eqtrd + wi cn0 simprrl frmy congid 2cnd mulcld mul02d adantl addridd adantr eqtrd ad2antll breqtrrd orcd ex cv wb peano2zd eluzel2 ad2antrl zmulcld zaddcld zcn simpl znegcld zsubcld dvdsmul2 cexp rmxdbl nn0cnd sqcld npcand sqvald - cc w3a mulass eqcomd syl3anc 3eqtrd dvdsmultr2 mpd mulid1d adddid subnegd - 1cnd 3eqtr4d breqtrd rmydbl mul32d dvds2addd adddird 1zzd addassd mulid2d + cc w3a mulass eqcomd syl3anc 3eqtrd dvdsmultr2 mpd mulridd adddid subnegd + 1cnd 3eqtr4d breqtrd rmydbl mul32d dvds2addd adddird 1zzd addassd mullidd 3eqtr2d rmyadd addsubd jm2.25lem1 syl221anc pm5.74da oveq1 breq2d orbi12d olcd weq imbi2d zindbi mpbid impcom 3impa ) AEUCUDZFZCGFZDGFZUEZBGFZADUFH ZACBEDIHZIHZJHZKHZACKHZLHZMNZYLYPYQUGZLHZMNZUHZYKYGYJUEZUUCYKUUDYLACUIYMI @@ -675646,7 +675646,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by cc0 wa id va vb cn0 cuz cfv cv caddc eluzelz peano2zm syl 0z sylancl rmy0 congid oveq1d breqtrrd 1z rmy1 pm3.43 w3a adantl eluzel2 simpr nnz adantr cn frmy fovcl syl2anc zmulcld zmulcl 3jca jca jctir simp3r iddvds congmul - syl112anc simp3l congsub rmyluc nncn mulid1d oveq2d 2timesd eqtrd pnncand + syl112anc simp3l congsub rmyluc nncn mulridd oveq2d 2timesd eqtrd pnncand 1cnd eqtr2d 3exp a2d syl5 breq2d imbi2d weq 2nn0ind impcom ) BUCCADUDUEZC ZAEFGZABHGZBFGZIJZWSWTAUAUFZHGZXDFGZIJZKWSWTARHGZRFGZIJZKWSWTAEHGZEFGZIJZ KWSWTAUBUFZEFGZHGZXOFGZIJZKZWSWTAXNHGZXNFGZIJZKZWSWTAXNEUGGZHGZYDFGZIJZKZ @@ -675828,7 +675828,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by lermxnn0 jm2.20nn mpbird dvdscmul rmydbl 2cnd mul32d nngt0d ltrmy elnnz eqcomd nnsqcld nnm1nn0 nnne0d divcld npcan pncan2d 3eqtrd zsubcl congid nndivdvds divcan1d eqbrtrd muldvds1 dvdsmultr2 subsub23 congsub congmul - subcl mulcld congadd mulid2d pncan3 jm2.15nn0 jm2.16nn0 rmygeid ) AEUEU + subcl mulcld congadd mullidd pncan3 jm2.15nn0 jm2.16nn0 rmygeid ) AEUEU FZGUEUFZHUEUFZUGIUEUFZJUEUFZKUEUFZUGLUEUFZEUHUIUJZBUHUIUJZUKULUJZDUHUIU JZUMUJZULUJZUKUNZHUHUIUJZUWNGUHUIUJZUMUJZULUJZUKUNZIUHUOUPZUFZUGZKUHUIU JZIUHUIUJUKULUJZJUHUIUJZUMUJZULUJZUKUNZGLUKUQUJZUHUWOUMUJZUMUJZUNZHIBUL @@ -676221,7 +676221,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by eluz2nn syl3anc mpbid nnrpd ltaddrpd lttrd cle peano2re exp1d nnge1d nnuz uz2m1nn eleqtrdi leexp2ad eqbrtrrd lelttrd eluzelz zltp1le syl2anc lemul1 syl112anc leadd1dd 1cnd addsub12d adddird sqvald oveq12d mulcld cc ax-1cn - mulcl pncan2d mulid2d 3eqtrd oveq1d oveq2d subadd23d 3eqtr3d 2cnd mulassd + mulcl pncan2d mullidd 3eqtrd oveq1d oveq2d subadd23d 3eqtr3d 2cnd mulassd 2timesd eqtrd sub32d addsubassd 3brtr4d ltletrd ) ACDUAIZBJBKIZCKIZCJUAIZ LIZMLIZACDACJUDUEZNZCONZFJCUBPZADGUCUFZABXRNZBONZEJBUBPZAXPONMONZXQONAXNX OAXMCAJONYDXMONUGYEJBUHUIYAUJACYAUKZUOULXPMUMUNZABCDEFGHUPZABBCKIZCMLIZQI @@ -679351,7 +679351,7 @@ of ideals (the usual "pure ring theory" definition). (Contributed by flcidc $p |- ( ph -> sum_ i e. S ( ( F ` i ) x. B ) = [_ K / i ]_ B ) $= ( cmul co wcel wa c1 wceq cc0 eqtrd cc csn cv cfv csu csb cif cmpt fveq1d adantr snssd sselda eqeq1 ifbid eqid 1ex c0ex ifex fvmpt syl elsni adantl - iftrued oveq1d syldan mulid2d sumeq2dv ax-1cn eqeltrdi mulcld cdif eldifi + iftrued oveq1d syldan mullidd sumeq2dv ax-1cn eqeltrdi mulcld cdif eldifi 0cn ifcli eldifn velsn sylnib iffalsed mul02d fsumss anbi2d csbeq1 eleq1d eleq1 imbi12d nfv nfcsb1v nfel1 nfim csbeq1a chvarfv vtoclg anabsi7 mpdan wi sumsns syl2anc 3eqtr3d ) AGUAZDUBZFUCZBLMZDUDWRBDUDZCXADUDDGBUEZAWRXAB @@ -679615,9 +679615,9 @@ is in the span of P(i)(X), so there is an R-linear combination of Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) $) algmulr $p |- ( .X. e. V -> .X. = ( .r ` A ) ) $= - ( cmulr c1 c6 cop algstr mulrid cnx cfv csn cbs cplusg ctp csca cvsca cpr - snsstp3 cun ssun1 sseqtrri sstri strfv ) FAIGJKLABCDEFHMNOIPFLZQORPBLZOSP - CLZUJTZAUKULUJUDUMUMOUAPDLOUBPELUCZUEAUMUNUFHUGUHUI $. + ( cmulr c1 c6 cop algstr mulridx cnx cfv csn cbs cplusg ctp snsstp3 cvsca + csca cpr cun ssun1 sseqtrri sstri strfv ) FAIGJKLABCDEFHMNOIPFLZQORPBLZOS + PCLZUJTZAUKULUJUAUMUMOUCPDLOUBPELUDZUEAUMUNUFHUGUHUI $. $( The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) $) @@ -679717,13 +679717,13 @@ is in the span of P(i)(X), so there is an R-linear combination of mendmulrfval $p |- ( .r ` A ) = ( x e. B , y e. B |-> ( x o. y ) ) $= ( cvv cmulr cfv cmpo wceq cnx cbs cop co eqid eqtrid fveq2d c0 cplusg cof wcel cv ccom ctp cvsca csn cxp cpr cun cmend clmhm mendbas eqtr4i mendval - csca fvexi mpoex algmulr mp1i eqtr4d wn fvprc mulrid str0 eqtr4di wo olcd - base0 0mpo0 syl pm2.61i ) EHUCZCIJZABDDAUDZBUDZUEZKZLVNVOMNJDOMUAJABDDVPV - QEUAJUBPKZOMIJZVSOUFMUQJEUQJZOMUGJABWBNJDENJVPUHUIVQEUGJUBPKZOUJUKZIJZVSV - NCWDIVNCEULJZWDFABDVTWBWCVSEHDCNJZEEUMPGCEFUNUOVTQVSQWBQWCQUPRSVSHUCVSWEL - VNABDDVRDCNGURZWHUSWDDVTWBWCVSHWDQUTVAVBVNVCZVOTVSWIVOTIJTWICTIWICWFTFEUL - VDRZSIWAVEVFVGWIDTLZWKVHVSTLWIWKWKWIDTNJZTWIDWGWLGWICTNWJSRVJVGVIABDDVRVK - VLVBVM $. + csca fvexi mpoex algmulr mp1i wn fvprc mulridx str0 eqtr4di wo base0 olcd + eqtr4d 0mpo0 syl pm2.61i ) EHUCZCIJZABDDAUDZBUDZUEZKZLVNVOMNJDOMUAJABDDVP + VQEUAJUBPKZOMIJZVSOUFMUQJEUQJZOMUGJABWBNJDENJVPUHUIVQEUGJUBPKZOUJUKZIJZVS + VNCWDIVNCEULJZWDFABDVTWBWCVSEHDCNJZEEUMPGCEFUNUOVTQVSQWBQWCQUPRSVSHUCVSWE + LVNABDDVRDCNGURZWHUSWDDVTWBWCVSHWDQUTVAVJVNVBZVOTVSWIVOTIJTWICTIWICWFTFEU + LVCRZSIWAVDVEVFWIDTLZWKVGVSTLWIWKWKWIDTNJZTWIDWGWLGWICTNWJSRVHVFVIABDDVRV + KVLVJVM $. mendmulr.q $e |- .x. = ( .r ` A ) $. $( A specific multiplication in the module endormoprhism algebra. @@ -680708,7 +680708,7 @@ is in the span of P(i)(X), so there is an R-linear combination of ( ( ( ( F + E ) / 2 ) - ( ( D + C ) / 2 ) ) x. ( B - A ) ) $= ( carea cfv cr cv cima cvol citg caddc co c2 cdiv cmin cmul cdm wcel wceq wss cmpt cibl cicc wa copab iccssre mp2an sseli c1 cc a1i cc0 wne subne0d - recni mulcld addsub12d subdid oveq2d eqtrid subdird mulid2i oveq1i eqtrdi + recni mulcld addsub12d subdid oveq2d eqtrid subdird mullidi oveq1i eqtrdi ax-mp recnd 3eqtr4d 1red resubcld addcomd mulcomd oveq12d 3eqtrd remulcld readdcld eqeltrd syl2anc syl covol nfcv mblvol cle wbr lesub1dd eqtr4d c0 eqtri eqeltrid adantl wtru cncfmptc mp3an divcli cncfmpt2f mptru cniccibl @@ -683338,7 +683338,7 @@ B C_ ( A ^o B ) ) $= nadd2rabex $p |- ( ( Ord A /\ B e. On /\ C e. On ) -> { x e. A | ( B +no x ) e. C } e. _V ) $= ( word con0 wcel w3a cv cnadd co crab simp3 wa wss c0 wceq 0elon ordelon - wi 3ad2antl1 naddcom sylancr naddid1 syl simpl2 naddssim mp3an2i eqsstrrd + wi 3ad2antl1 naddcom sylancr naddrid syl simpl2 naddssim mp3an2i eqsstrrd eqtrd 0ss mpi simpl3 ontr2 syl2anc mpand 3impia rabssdv ssexd ) BEZCFGZDF GZHZCAIZJKZDGZABLDFUTVAVBMVCVFABDVCVDBGZVFVDDGZVCVGNZVDVEOZVFVHVIVDPVDJKZ VEVIVKVDPJKZVDVIPFGZVDFGZVKVLQRUTVAVGVNVBBVDSUAZPVDUBUCVIVNVLVDQVOVDUDUEU @@ -683407,7 +683407,7 @@ B C_ ( A ^o B ) ) $= 31-Dec-2024.) $) nadd1suc $p |- ( A e. On -> ( A +no 1o ) = suc A ) $= ( va vb vy vx cv c1o cnadd co csuc wceq oveq1 con0 wcel wral wa c0 eleq1d - wb adantr weq suceq eqeq12d crab wel naddid1 anbi1d ad2antrr df1o2 raleqi + wb adantr weq suceq eqeq12d crab wel naddrid anbi1d ad2antrr df1o2 raleqi cint csn 0ex oveq2 ralsn bitri a1i cbvralvw nfv nfra1 nfan simpr r19.21bi ralbida bitrid anbi12d wss onelon ad4ant13 syl simpllr jca word ad3antrrr onsuc eloni simplr ordsucss sylc ontr2 ralrimdva pm4.71d adantlr rabbidva @@ -683475,7 +683475,7 @@ B C_ ( A ^o B ) ) $= syl c0 va weq oveq2 wlim w3a wi ciun simplll simpllr simplr oalim syl2anc jca simpl simp3 wel onelss imp onelon simpll naddss2 syl3anc mpbid adantr wb sstrd ralimdva iunss sylibr syl2an eqsstrd exp31 csuc wrex word dflim3 - ex wn notbii iman bitr4i eloni pm5.5 3syl bitrid ssidd oveq2d oa0 naddid1 + ex wn notbii iman bitr4i eloni pm5.5 3syl bitrid ssidd oveq2d oa0 naddrid wo eqtrd 3sstr4d a1d vex sucid eleqtrrid a1i reximdv2 r19.29r simprr oacl naddcl ordsucsssuc oasuc ad4ant23 naddsuc2 rexlimdva2 syl5 expd syl7 syld simprl jaod sylbid pm2.61d on2ind ) UAFZCFZGHZXQXRIHZJZDFZXRGHZYBXRIHZJZY @@ -683507,7 +683507,7 @@ B C_ ( A ^o B ) ) $= equal to that of ordinal addition. (Contributed by RP, 1-Jan-2025.) $) naddonnn $p |- ( ( A e. On /\ B e. _om ) -> ( A +o B ) = ( A +no B ) ) $= ( vx vy com wcel con0 co cnadd wceq cv wi c0 csuc oveq2 eqeq12d imbi2d wa - coa adantr weq naddid1 eqtr4d nnon suceq adantl oasuc naddsuc2 3eqtr4d ex + coa adantr weq naddrid eqtr4d nnon suceq adantl oasuc naddsuc2 3eqtr4d ex oa0 expcom syl a2d finds impcom ) BEFAGFZABSHZABIHZJZUQACKZSHZAVAIHZJZLUQ AMSHZAMIHZJZLUQADKZSHZAVHIHZJZLUQAVHNZSHZAVLIHZJZLUQUTLCDBVAMJZVDVGUQVPVB VEVCVFVAMASOVAMAIOPQCDUAZVDVKUQVQVBVIVCVJVAVHASOVAVHAIOPQVAVLJZVDVOUQVRVB @@ -686408,7 +686408,7 @@ of all sets ( ~ inex1g ). reabssgn $p |- ( A e. RR -> ( abs ` A ) = ( ( sgn ` A ) x. A ) ) $= ( cr wcel csgn cfv cmul cc0 wceq clt wbr cneg cif cabs cxr rexr ovif adantr co c1 eqtr4d sgnval syl oveq1d ifeq2 ax-mp eqtri wa mul02lem2 simpr abs00bd - wn recn mulm1d mulid2d ifeq12d reabsifneg ifeqda eqtrid eqtr2d ) ABCZADEZAF + wn recn mulm1d mullidd ifeq12d reabsifneg ifeqda eqtrid eqtr2d ) ABCZADEZAF RAGHZGAGIJZSKZSLZLZAFRZAMEZUTVAVFAFUTANCVAVFHAOAUAUBUCUTVGVBGAFRZVCVDAFRZSA FRZLZLZVHVGVBVIVEAFRZLZVMVBGVEAFPVNVLHVOVMHVCVDSAFPVBVNVLVIUDUEUFUTVBVIVLVH UTVBUGZVIGVHUTVIGHVBAUHQVPAUTVBUIUJTUTVLVHHVBUKUTVLVCAKZALVHUTVCVJVQVKAUTAA @@ -686475,7 +686475,7 @@ of all sets ( ~ inex1g ). wb eqtrd cabs cre cim clt wbr cneg cmin sqrtcvallem5 ax-icn cr neg1rr ifcli cif 1re sqrtcvallem3 remulcld mulcld addcld id binom2 syl2anc recl readdcld abscl rehalfcld sqsqrtd sqmuld ovif neg1sqe1 sq1 ifeq12 mp2an ifid resubcld - i2 3eqtri oveq12d mulid2d 3eqtrd mulm1d negsubd pnncand 2timesd eqtr4d 2cnd + i2 3eqtri oveq12d mullidd 3eqtrd mulm1d negsubd pnncand 2timesd eqtr4d 2cnd oveq1d wne 2ne0 divsubdird divcan3d 3eqtr3d mul12d mulassd sqrtcvallem4 syl halfnneg2 mpbird crp 2rp sqrtdivd sqrtcvallem2 resqrtcld 2re 0le2 necon3bii sqrt00 mpbir divmuldivd resqcld imcl absvalsq2 mvrladdd subsq eqtr3d fveq2d @@ -686484,7 +686484,7 @@ of all sets ( ~ inex1g ). replim eqeq1d 3bitrd 3ad2ant1 eqtrdi mpbid crred breqtrrd wi cnsqrt00 half0 3eqtr4d wa addcomd addeq0 wo olc eqcom sqeqor addid0 sqeq0 3bitr3d imbitrid reim ancld sylbid w3a simp2 negidd div0i sqrt0 simp3 0red eqnbrtrd iffalsed - 2cn ltnrd subnegd absge0 addid2d rered le0neg2d eqbrtrd 3expib sqrtcvallem1 + 2cn ltnrd subnegd absge0 addlidd rered le0neg2d eqbrtrd 3expib sqrtcvallem1 ixi syld eqsqrtd eqcomd ) ABCZAUADZAUBDZEFZGHFZIDZJAUCDZKUDUEZLUFZLUMZUWEUW FUGFZGHFZIDZMFZMFZEFZAIDUWDUWSAUWDUWIUWRUWDUWIAUHZNZUWDJUWQJBCUWDUIOZUWDUWQ UWDUWMUWPUWMUJCUWDUWKUWLLUJUKUNULOZAUOZUPZNZUQZURZUWDUSUWDUWSGPFZUWIGPFZGUW @@ -686546,7 +686546,7 @@ of all sets ( ~ inex1g ). ( if ( ( Im ` A ) < 0 , -u _i , _i ) x. ( sqrt ` ( ( ( abs ` A ) - ( Re ` A ) ) / 2 ) ) ) ) ) $= ( cc wcel csqrt cfv cabs caddc co c2 cdiv ci c1 cneg cif cmul ax-icn a1i cr - wceq recnd cre cim cc0 clt wbr cmin sqrtcval neg1cn mulm1i mulcomli mulid1i + wceq recnd cre cim cc0 clt wbr cmin sqrtcval neg1cn mulm1i mulcomli mulridi ovif2 ifeq12 mp2an eqtr2i oveq1d neg1rr 1re ifcli sqrtcvallem3 eqtrd oveq2d mulassd eqtr4d ) ABCZADEAFEZAUAEZGHIJHDEZKAUBEUCUDUEZLMZLNZVFVGUFHIJHDEZOHO HZGHVHVIKMZKNZVLOHZGHAUGVEVPVMVHGVEVPKVKOHZVLOHVMVEVOVQVLOVOVQSVEVQVIKVJOHZ @@ -686577,7 +686577,7 @@ of all sets ( ~ inex1g ). ( c1 c5 cdc c8 cmul co caddc csqrt c2 c4 wceq 1nn0 5nn0 nn0cni c6 7nn0 eqid cfv c7 2nn0 ci cre cabs cdiv cexp cc wcel deccl ax-icn 8cn mulcli resqrtval addcli ax-mp c3 cr nn0rei 8re absreim mp2an c9 sqvali 1p1e2 7p5e12 addcomli - mulid2i decaddci mulid1i oveq1i 5p2e7 eqtri 5t5e25 decmul2c decmul1c 8t8e64 + mullidi decaddci mulridi oveq1i 5p2e7 eqtri 5t5e25 decmul2c decmul1c 8t8e64 oveq12i 6nn0 4nn0 2cn 6p2e8 decaddi 5p4e9 decadd 7p1e8 7p4e11 7t7e49 eqtr2i 9nn0 3eqtri fveq2i cc0 cle wbr nn0ge0i sqrtsqi crrei 2p1e3 decaddc mulcomli 6t2e12 3nn0 2ne0 divmuli mpbir 4t4e16 ) ABCZUADEFZGFZHRUBRZXHUCRZXHUBRZGFZI @@ -686598,8 +686598,8 @@ of all sets ( ~ inex1g ). ( c1 c5 cdc c8 cmul co caddc csqrt cfv cc0 c2 1nn0 5nn0 nn0cni c7 eqid 7nn0 c4 2nn0 eqtri ci cim clt wbr cneg cif cabs cre cmin cdiv cc wcel wceq deccl ax-icn 8cn mulcli addcli imsqrtval ax-mp 8pos 0re 8re ltnsymi nn0rei breq1i - wn crimi sylnibr iffalsei cexp cr absreim mp2an c6 c9 sqvali mulid2i 7p5e12 - 1p1e2 addcomli decaddci mulid1i oveq1i 5p2e7 5t5e25 decmul2c 8t8e64 oveq12i + wn crimi sylnibr iffalsei cexp cr absreim mp2an c6 c9 sqvali mullidi 7p5e12 + 1p1e2 addcomli decaddci mulridi oveq1i 5p2e7 5t5e25 decmul2c 8t8e64 oveq12i 6nn0 4nn0 6p2e8 decaddi 5p4e9 decadd 9nn0 7p1e8 7p4e11 7t7e49 eqtr2i 3eqtri decmul1c fveq2i cle nn0ge0i sqrtsqi crrei subaddrii 2div2e1 sqrt1 1t1e1 ) A BCZUADEFZGFZHIUBIZXNUBIZJUCUDZAUEZAUFZXNUGIZXNUHIZUIFZKUJFZHIZEFZAAEFAXNUKU @@ -687611,7 +687611,7 @@ over the natural numbers (including zero) is equivalent to the ( vx wcel cmul co wceq cn crelexp wi c1 caddc oveq2 oveq2d eqeq12d imbi2d eqtrd vy wa w3a cv weq cvv ovexd relexp1d cr simp1 nnre ax-1rid 3syl ccom eqcomd ovex relexpsucnnr sylancr simp3 coeq1d simp21 nnmulcld relexpaddnn - simp22 syl3anc nncnd 1cnd adddid mulid1d eqtr2d 3exp a2d nnind 3expd impd + simp22 syl3anc nncnd 1cnd adddid mulridd eqtr2d 3exp a2d nnind 3expd impd impcom simplr ) AEGZBCDHIZJZUBZCKGZDKGZUBZUBZACLIZDLIZAVSLIZABLIWDWAWGWHJ ZWDVRVTWIWCWBVRVTWIMMWCWBVRVTWIWBVRVTUCZWFFUDZLIZACWKHIZLIZJZMWJWFNLIZACN HIZLIZJZMWJWFUAUDZLIZACWTHIZLIZJZMWJWFWTNOIZLIZACXEHIZLIZJZMWJWIMFUADWKNJ @@ -687813,10 +687813,10 @@ over the natural numbers (including zero) is equivalent to the c1 eqtrd cn0 cn wo wi elnn0 biimpi relexpaddnn eqimss 3exp cuz cfv elnn1uz2 c2 cid cdm crn wrel relco dfrel2 ax-mp cnvco cnvresid coeq2i coires1 3eqtri cun cnvss resss sstri eqsstrri cnvcnvss a1i simp1 relexp0g relexp1g coeq12d - simp2 oveq12d addid2d 3sstr4d cnvexg relexpuzrel syl2anc eluz2nn relexpnndm + simp2 oveq12d addlidd 3sstr4d cnvexg relexpuzrel syl2anc eluz2nn relexpnndm 1cnd df-rn ssun2 sstrdi relssres eqtrid simp3 eluzge2nn0 relexpcnvd 3eqtr4d cvv 3adant1 cnveqb sylancr mpbird coeq1d oveq1d eluzelcn jaod biimtrid jaoi - wb cc eqsstri addid1d 3adant2 ssun1 coeq2d cin resres inidm reseq2i 3eqtr4a + wb cc eqsstri addridd 3adant2 ssun1 coeq2d cin resres inidm reseq2i 3eqtr4a 00id 3imp ) CUAEZBUAEZADEZACFGZABFGZHZACBIGZFGZJZYBBUBEZBKLZUCYAYCYIUDZBUEY AYJYLYKYACUBEZCKLZUCZYJYLUDZYAYOCUEUFZYMYPYNYMYJYCYIYMYJYCMYFYHLZYIABCDUGYF YHUHZNUIYJBSLZBUMUJUKZEZUCYNYLBULYNYTYLUUBYNYTYCYIYNYTYCMZUNAUOZAUPZVFZOZAH @@ -694696,7 +694696,7 @@ base set if and only if the neighborhoods (convergents) of every point mp2an c9 3t3e9 numexp1 oveq1i 4cn 5p4e9 addcomli eqtri eqtr4i dvds0lem cn 4nn0 wa 4nn nnnn0 nnexpcld nnzd adantr zaddcld simpr dvdsmultr1d dvdsmul1 zmulcld dvds2subd cdc cc adddird 3cn 5t3e15 mulcomli oveq2d expp1d ax-1cn - nncnd 3p1e4 eqcomi pncan3oi oveq2i subdii mulid1i 3eqtr3ri oveq12d mulcld + nncnd 3p1e4 eqcomi pncan3oi oveq2i subdii mulridi 3eqtr3ri oveq12d mulcld 5nn0 deccl nn0cni addsubassd eqtr4d 3eqtr4rd breqtrrd ex nnind ) BCUAUCZD EZFGEZHIBCQDEZFGEZHIZBCUBUCZDEZFGEZHIZBCXMQGEZDEZFGEZHIZBCADEZFGEZHIUAUBA XGQJZXIXKBHYCXHXJFGXGQCDKLMXGXMJZXIXOBHYDXHXNFGXGXMCDKLMXGXQJZXIXSBHYEXHX @@ -695132,7 +695132,7 @@ base set if and only if the neighborhoods (convergents) of every point $( First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) $) int-mul11d $p |- ( ph -> ( A x. 1 ) = B ) $= - ( c1 cmul co recnd mulid1d eqtrd ) ABFGHBCABABDIJEK $. + ( c1 cmul co recnd mulridd eqtrd ) ABFGHBCABABDIJEK $. $} ${ @@ -695141,7 +695141,7 @@ base set if and only if the neighborhoods (convergents) of every point $( Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) $) int-mul12d $p |- ( ph -> ( 1 x. A ) = B ) $= - ( c1 cmul co recnd mulid2d eqtrd ) AFBGHBCABABDIJEK $. + ( c1 cmul co recnd mullidd eqtrd ) AFBGHBCABABDIJEK $. $} ${ @@ -695150,7 +695150,7 @@ base set if and only if the neighborhoods (convergents) of every point $( First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) $) int-add01d $p |- ( ph -> ( A + 0 ) = B ) $= - ( cc0 caddc co recnd addid1d eqtrd ) ABFGHBCABABDIJEK $. + ( cc0 caddc co recnd addridd eqtrd ) ABFGHBCABABDIJEK $. $} ${ @@ -695159,7 +695159,7 @@ base set if and only if the neighborhoods (convergents) of every point $( Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) $) int-add02d $p |- ( ph -> ( 0 + A ) = B ) $= - ( cc0 caddc co recnd addid2d eqtrd ) AFBGHBCABABDIJEK $. + ( cc0 caddc co recnd addlidd eqtrd ) AFBGHBCABABDIJEK $. $} ${ @@ -695271,7 +695271,7 @@ base set if and only if the neighborhoods (convergents) of every point This section formalizes theorems used in an n-digit addition proof generator. - Other theorems required: ~ deccl ~ addcomli ~ 00id ~ addid1i ~ addid2i ~ eqid + Other theorems required: ~ deccl ~ addcomli ~ 00id ~ addridi ~ addlidi ~ eqid ~ dec0h ~ decadd ~ decaddc . $) @@ -695830,13 +695830,13 @@ base set if and only if the neighborhoods (convergents) of every point ) ) = ( .r ` F ) ) $= ( cv co wceq cfv cif cmpt cmpo cgsu cfrlm cbs cmulr eqid mnringbaserd cvv wcel fvexi mpoex a1i cmnring ovexi ovex eqtr3id cplusg mnringaddgd eqcomd - gsumpropd mpoeq123dv cnx csts fvex mulrid setsid mp2an mnringvald eqtr4id - cop fveq2d eqtrd ) ABCEEKOPDDJDJUEOUEZPUEZFUFUGWCBUEUHWDCUEUHHUFNUIUJZUKZ - ULUFZUKBCGDUMUFZUNUHZWIWHWFULUFZUKZKUOUHZABCEEWGWIWIWJADEGIKLWHMQRUAWHUPZ - UCUDUQZWNAWFKWHURURURWFURUSAOPDDWEDLUNUAUTZWOVAVBKURUSAKGLVCQVDVBWHURUSZA - GDUMVEZVBAKUNUHEWIRWNVFAWHVGUHKVGUHADGIKLWHMQUAWMUCUDVHVIVJVKAWKWHVLUOUHW - KVTVMUFZUOUHZWLWPWKURUSWKWSUGWQBCWIWIWJWHUNVNZWTVAURWKUOURWHVOVPVQAKWRUOA - BCDWIFGHIJKLWHMNOPQSTUAUBWMWIUPUCUDVRWAVSWB $. + gsumpropd mpoeq123dv cnx csts fvex mulridx setsid mp2an mnringvald fveq2d + cop eqtr4id eqtrd ) ABCEEKOPDDJDJUEOUEZPUEZFUFUGWCBUEUHWDCUEUHHUFNUIUJZUK + ZULUFZUKBCGDUMUFZUNUHZWIWHWFULUFZUKZKUOUHZABCEEWGWIWIWJADEGIKLWHMQRUAWHUP + ZUCUDUQZWNAWFKWHURURURWFURUSAOPDDWEDLUNUAUTZWOVAVBKURUSAKGLVCQVDVBWHURUSZ + AGDUMVEZVBAKUNUHEWIRWNVFAWHVGUHKVGUHADGIKLWHMQUAWMUCUDVHVIVJVKAWKWHVLUOUH + WKVTVMUFZUOUHZWLWPWKURUSWKWSUGWQBCWIWIWJWHUNVNZWTVAURWKUOURWHVOVPVQAKWRUO + ABCDWIFGHIJKLWHMNOPQSTUAUBWMWIUPUCUDVRVSWAWB $. $} ${ @@ -697409,7 +697409,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing cneg mpbird absrpcld rpcnd recnd breqtrd ralimdva reximdva mpd r19.29a wn ltled 1red elioo2 simp3d ad2antrr fvoveq1 oveq2d cvgrat ad4antlr remulcld simp2d syl2an simplbda ltdivmuld mulcomd ralrimiva ioon0 biimpar r19.3rzv - c0 iserex wo lttri2d orcanai neeq1d dvgrat 1re sylancr mulid2d negsubdi2d + c0 iserex wo lttri2d orcanai neeq1d dvgrat 1re sylancr mullidd negsubdi2d wnel 1cnd simprbda ltmuldivd df-nel sylib mtbird impcon4bid ) AETUDUEZUFD FUGUHUIZUJZAUVQUVSAUVQUKZUVSUFDGUGZUVRUJZUVTUWBUWBUAETULUMZUNZAUWDUVQAUWB UAUWCAUAUOZUWCUJZUKZCUOZTUFUMZDUPZUQUPZUWEUWHDUPZUQUPZURUMZUSUEZCUBUOZUTU @@ -697686,7 +697686,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing ( ( || " { M } ) \ ( || " { ( M x. N ) } ) ) ) $= ( cdvds csn cima cdif co cmul cprime wcel cz prmz syl difeq2d syl2anc c1 clcm cin difin nzin eqtr3id cgcd cabs cfv wceq lcmgcd wne wb prmrp mpbird - oveq2d cn0 lcmcl nn0cnd mulid1d eqtrd zred remulcld prmnn nn0ge0d mulge0d + oveq2d cn0 lcmcl nn0cnd mulridd eqtrd zred remulcld prmnn nn0ge0d mulge0d cn nnnn0d absidd 3eqtr3d sneqd imaeq2d ) AGBHIZGCHIZJZVLGBCUAKZHZIZJZVLGB CLKZHZIZJAVNVLVLVMUBZJVRVLVMUCAWBVQVLABCABMNZBONZDBPQZACMNZCONZECPQZUDRUE AVQWAVLAVPVTGAVOVSAVOBCUFKZLKZVSUGUHZVOVSAWDWGWJWKUIWEWHBCUJSAWJVOTLKVOAW @@ -698017,7 +698017,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing ( vy cdv co cmul ce cmpt wceq wcel cc cr vx cfv csn cxp cof fveq2d oveq2d cv oveq2 cbvmptv eqtri oveq2i cvv cpr wo wi elpri recn syl6bi biimpd jaoi eleq2 3syl imp wa mulcl sylan efcl syl syldan ovexd cnelprrecn a1i adantr - adantl c1 1cnd dvmptid dvmptcmul mulid1d mpteq2dv eqtrd dvef wfn wf ax-mp + adantl c1 1cnd dvmptid dvmptcmul mulridd mpteq2dv eqtrd dvef wfn wf ax-mp eff ffn dffn5 mpbi 3eqtr3i fveq2 dvmptco mulcom syl2anr anabss5 mpteq2dva w3a 3anim123i 3anidm12 mul12 eqtrid fconstmpt offval2 eqtr4d ) ADFLMZKDEC EKUHZNMZOUBZNMZNMZPZDEUCUDZFNUEMAXFDKDXJPZLMZXLFXNDLFBDCEBUHZNMZOUBZNMZPX @@ -698124,7 +698124,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing e. CC Y = ( t e. S |-> ( c x. ( exp ` ( K x. t ) ) ) ) ) ) $= ( vu vx vy co cmul wceq ce cc wcel adantr vz cdv csn cxp cof cv cmpt wrex cfv wa cneg cc0 caddc cr cpr cnelprrecn a1i wss recnprss syl sseld syl6an - mulcl imp negcld efcl adantl c1 ax-1cn dvmptid dvmptcmul mulid1d mpteq2dv + mulcl imp negcld efcl adantl c1 ax-1cn dvmptid dvmptcmul mulridd mpteq2dv eqtrd dvmptneg dvef wfn wf eff ffn ax-mp dffn5 mpbi 3eqtr3i fveq2 dvmptco oveq2i oveq2d mulcld fmpttd feq1d mpbird mulcom eqtr3d fconst6g fconstmpt caofcom eqidd offval2 cdm dmeqd eqid dmmptd dvmulf 3eqtr4rd ofmul12 oveq1 @@ -698279,7 +698279,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing Steve Rodriguez, 22-Apr-2020.) $) bccn1 $p |- ( ph -> ( C _Cc 1 ) = C ) $= ( c1 cbcc co cmul cc0 caddc cmin cdiv cn0 wcel 0nn0 a1i bccp1k wceq 0p1e1 - oveq12d eqtrd oveq2i bccn0 subid1d div1d 3eqtr3d mulid2d ) ABDEFZDBGFZBAB + oveq12d eqtrd oveq2i bccn0 subid1d div1d 3eqtr3d mullidd ) ABDEFZDBGFZBAB HDIFZEFZBHEFZBHJFZUIKFZGFUGUHABHCHLMANOPUJUGQAUIDBERUAOAUKDUMBGABCUBAUMBD KFBAULBUIDKABCUCUIDQAROSABCUDTSUEABCUFT $. $} @@ -698421,7 +698421,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing ( cn0 wcel cc0 co cmul caddc cc wceq ad2antrr c1 cr vj wa cfz cbcc cmin cv ccxp cexp csu cbc rpcnd recnd binom 3expia syl2anc imp adantr addcld simpr cxpexp elfznn0 simplr sylan2 cle wbr elfzle2 adantl nn0sub ancoms - wi bccbc adantll mpbid oveq1d oveq12d sumeq2dv 3eqtr4d eqeltrrd addid1d + wi bccbc adantll mpbid oveq1d oveq12d sumeq2dv 3eqtr4d eqeltrrd addridd wb cxpcld cuz cfv cmpt nn0uz eqid 1nn0 a1i nn0addcld eqidd oveq2d bcccl nn0cnd subcld expcld mulcld fvmptd csn cxp peano2nn0 wf c0ex 0red snssd fconst fssd ffvelcdmda eqeltrd cseq cli cdm climrel xpeq1i seqeq3 ax-mp @@ -698580,7 +698580,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing c0ex dvmptid dvmptadd 0p1e1 mpteq2i crest fvex ctps cnfldtps cnfldbas eqtrdi cuni tpsuni restid eqcomi ctop cnfldtop ccom cnbl0 eqtri cxmet cnt cbl cnxmet cnfldtopn blopn mp3an isopn3i dvmptres2 eqidd dvcncxp1 - 0cn 3eqtr3g syl dvmptco subcld mulcld mulid1d mpteq2dva 3eqtrd eqtrid + 0cn 3eqtr3g syl dvmptco subcld mulcld mulridd mpteq2dva 3eqtrd eqtrid ) ADUEUFUGZUHZUIMEUJMUKZULUMZDUNZUOUMZUPZUQUMUIUCEUJUCUKZULUMZUXGUOUM ZUPZUQUMZMEUXGUXFUXGUJURUMZUOUMZUSUMZUPZUXIUXMUIUQMUCEUXHUXLMEUTVAZVB FVCUMZVDZUBMUXSUXTMUXSVEMVBFVCMVBVEZMVCVEMFULLUKZGVFZVBVGZVHVIZUFZLVJ @@ -698737,7 +698737,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing adantllr 1cnd negcld cxpcld cdv eqtrdi fmpt3d subcld nnuz sylan2 cshi cn 3eqtr3d 3eqtrd mpteq2dva oveq12d cuz eqtr4d offval2f ccnv cico cdm vy wn cima crab csup nfmpt1 nfseq nfrabw nfsup nfov nfima nfsum simpl - rpcnd 0red absge0d lelttrd gt0ne0d abs00ad necon3bid mulid1d breqtrrd + rpcnd 0red absge0d lelttrd gt0ne0d abs00ad necon3bid mulridd breqtrrd divcld 1red elrpd ltdivmuld absdivd binomcxplemradcnv 3brtr4d w3a 0re mpbird ssrab2 ressxr supxrcl eqeltri elico2 syl3anbrc eleq2i elpreima sstri wfn ffn mp2b bitri sylanbrc cof ccom cbl eqid cnbl0 mulcl nfcri @@ -698750,7 +698750,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing nnnn0 nnne0 ovex 1pneg1e0 fveq2i eqtr4i znegcld uzmptshftfval cbvmptv oveq1 subnegd pncand nncn mul12d expp1d eqtr3id climrel simprd climdm nn0cn wrel sylib cz 0z neg1z fvex seqshft subnegi 0p1e1 seqeq1 oveq1i - 0cn breq1i climshft releldm sylancr isermulc2 isumadd adddird mulid2d + 0cn breq1i climshft releldm sylancr isermulc2 isumadd adddird mullidd seqex sylibr isumshft cbvsumv pncan2d isum1p 0nn0 subid1d bccn0 exp0d eqcomi sumeq1i eqeltrrd addassd 3eqtr4rd binomcxplemwb mulassd eqtrid isummulc2 3eqtrrd simprbi eleq2s simp3bi syl absnegd eqcomd abssubne0 @@ -713013,7 +713013,7 @@ not even needed (it can be any class). (Contributed by Glauco number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) sub2times $p |- ( A e. CC -> ( A - ( 2 x. A ) ) = -u A ) $= ( cc wcel c2 cmul co cmin caddc cneg 2times oveq2d id addcld negsubd negdid - negcl addassd cc0 negid 3eqtr2d oveq1d addid2d eqtrd ) ABCZADAEFZGFAAAHFZGF + negcl addassd cc0 negid 3eqtr2d oveq1d addlidd eqtrd ) ABCZADAEFZGFAAAHFZGF AUFIZHFZAIZUDUEUFAGAJKUDAUFUDLZUDAAUJUJMNUDUHAUIUIHFZHFAUIHFZUIHFZUIUDUGUKA HUDAAUJUJOKUDAUIUIUJAPZUNQUDUMRUIHFUIUDULRUIHASUAUDUIUNUBUCTT $. @@ -713335,7 +713335,7 @@ not even needed (it can be any class). (Contributed by Glauco ( vg vh cfz co clt wex c1 wcel wceq c0 caddc wiso wmo weu ccnv ccom chash cv cfv wa wor cfn cr wss cz fzssz zssre sstri ltso soss fzfi fz1iso mp2an mp2 wb cc0 cmin fveq2 eqtrdi oveq1d eqtrid oveq2d adantl zcnd 1cnd subcld - hash0 addid2d wbr ltm1d peano2zm syl fzn syl2anc mpbid eqtrd adantr eqcom + hash0 addlidd wbr ltm1d peano2zm syl fzn syl2anc mpbid eqtrd adantr eqcom zred biimpi 3eqtrd fveq2d wn cneg cuz cle pncan3d eqcomd 1red cn hashnncl wne neqne mpbird nnred nnge1d leadd1dd breqtrrdi eqbrtrd hashcl nn0z 3syl cn0 zaddcld eqeltrid eluz hashfz oveq1i nn0cnd addsubassd negcld mvrladdd @@ -713409,8 +713409,8 @@ not even needed (it can be any class). (Contributed by Glauco multiple of ` T ` . (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) fperiodmullem $p |- ( ph -> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) ) $= ( wcel cmul co caddc cfv wceq wi oveq2d cr vn vm cn0 cv c1 oveq1 fveqeq2d - cc0 imbi2d recnd mul02d addid1d eqtrd fveq2d w3a simp3 simp1 wa simpr mpd - simpl 3adant1 cc nn0cn adantl 1cnd adantr adddird addassd mulid2d 3eqtr2d + cc0 imbi2d recnd mul02d addridd eqtrd fveq2d w3a simp3 simp1 wa simpr mpd + simpl 3adant1 cc nn0cn adantl 1cnd adantr adddird addassd mullidd 3eqtr2d mulcld 3adant3 nn0re remulcld readdcld ex imdistani eleq1 fvoveq1 eqeq12d anbi2d fveq2 imbi12d vtoclg sylc 3eqtrd syl3anc 3exp nn0ind mpcom ) EUCLA FECMNZONZDPFDPZQZIAFUAUDZCMNZONZDPWNQZRAFUHCMNZONZDPWNQZRAFUBUDZCMNZONZDP @@ -713660,11 +713660,11 @@ not even needed (it can be any class). (Contributed by Glauco ABEZFAAGHZPABGHZOQPAIJOPKPQRABAGLMN $. ${ - xaddid2d.1 $e |- ( ph -> A e. RR* ) $. + xaddlidd.1 $e |- ( ph -> A e. RR* ) $. $( ` 0 ` is a left identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.) $) - xaddid2d $p |- ( ph -> ( 0 +e A ) = A ) $= - ( cxr wcel cc0 cxad co wceq xaddid2 syl ) ABDEFBGHBICBJK $. + xaddlidd $p |- ( ph -> ( 0 +e A ) = A ) $= + ( cxr wcel cc0 cxad co wceq xaddlid syl ) ABDEFBGHBICBJK $. $} ${ @@ -713673,7 +713673,7 @@ not even needed (it can be any class). (Contributed by Glauco $( A number is less than or equal to itself plus a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) $) xadd0ge $p |- ( ph -> A <_ ( A +e B ) ) $= - ( cc0 cxad co cle cxr wcel wceq xaddid1 syl eqcomd wa wbr a1i jca cpnf + ( cc0 cxad co cle cxr wcel wceq xaddrid syl eqcomd wa wbr a1i jca cpnf cicc iccssxr sselid xrleidd pnfxr iccgelb syl3anc xle2add sylc eqbrtrd 0xr ) ABBFGHZBCGHZIAULBABJKZULBLDBMNOAUNFJKZPZUNCJKZPZPBBIQZFCIQZPULUMIQA UPURAUNUODUOAUKRZSAUNUQDAFTUAHZJCFTUBEUCSSAUSUTABDUDAUOTJKZCVBKUTVAVCAUER @@ -714290,9 +714290,9 @@ not even needed (it can be any class). (Contributed by Glauco cr remulcld rexrd adantlr simplr 0xr pnfxr icogelb syl3anc ltaddrpd ltled a1i id syl lemulge12d xrletrd ex ralrimi ad3antrrr eqeltrd adantll c2 1rp oveq2 oveq1d breq2d rspcva syl2anc 1p1e2 breqtrd simpr simpl mul01d eqtrd - 0red 2cnd ad4ant24 clt rpgt0 oveq1 recnd addid2d eqtr2d xrlelttrd xrltled + 0red 2cnd ad4ant24 clt rpgt0 oveq1 recnd addlidd eqtr2d xrlelttrd xrltled cc wne necon3bi leneltd elrpd ad4ant14 cdiv rpdivcld simpll adantlll 1cnd - rpcn rpne0 divcld adddird mulid2d divcan1d oveq12d eqidd 3eqtrd pm2.61dan + rpcn rpne0 divcld adddird mullidd divcan1d oveq12d eqidd 3eqtrd pm2.61dan wn ralrimiva wb xralrple mpbird impbid ) ACDHIZCJBUDZKLZDMLZHIZBNUBZAYHYM AYHOZYLBNAYHBEYHBUCUEYNYINPZYLYNYOOZCDYKACUFPZYHYOFUGYPQUHUILZUFDQUHUJADY RPZYHYOGUGUKAYOYKUFPYHAYOOZYKYTYJDYTJYIYTULYOYIUPPAYIUMZRUNADUPPZYOAYRUPD @@ -714650,7 +714650,7 @@ not even needed (it can be any class). (Contributed by Glauco ( cle wbr cmul co caddc crp wa wcel cc0 wceq c1 vy cv wral ad2antrr rexrd cxr cr rpre adantl remulcld readdcld simplr adantr rpge0 mulge0d addge01d mpbid adantlr xrletrd ralrimiva ex 1rp oveq2d breq2d rspcva mpan ad2antlr - oveq2 oveq1 0cn mulid1i a1i eqtrd cc recnd addid1d breqtrd wne neqne 0red + oveq2 oveq1 0cn mulridi a1i eqtrd cc recnd addridd breqtrd wne neqne 0red wn simpr leneltd elrpd syldan cdiv simpl rpdivcld adantll simpll adantlll syl2anc rpcnd rpne0d divcan2d wb xralrple mpbird pm2.61dan impbid ) ACDJK ZCDEBUBZLMZNMZJKZBOUCZAXAXFAXAPZXEBOXGXBOQZPZCDXDACUFQZXAXHFUDADUFQXAXHAD @@ -717145,7 +717145,7 @@ similar theorems would not hold (see ~ rexanre and ~ rexanuz ). sqrlearg $p |- ( ph -> ( ( A ^ 2 ) <_ A <-> A e. ( 0 [,] 1 ) ) ) $= ( co cle wbr cc0 c1 wcel wa cr 0re a1i clt simpr 1red adantr cmul mpbid wn c2 cexp cicc ltnled mpbird 0lt1 lttrd elrpd ltmul2dd wceq recnd sqvald - mulid1d eqcomd breq12d syldan adantlr resqcld lenltd condan sqge0d eliccd + mulridd eqcomd breq12d syldan adantlr resqcld lenltd condan sqge0d eliccd letrd ex unitssre sseli rexrd id iccgelbd iccleubd lemul2ad adantl impbid cxr 0xr ) ABUAUBDZBEFZBGHUCDZIZAVQVSAVQJZGHBGKIZVTLMZAVQBHEFZHKIVTWCBVPNF ZAWCTZWDVQAWEHBNFZWDAWEJZWFWEAWEOWGHBWGPABKIZWECQUDUEAWFJZBHRDZBBRDZNFWDW @@ -717659,7 +717659,7 @@ similar theorems would not hold (see ~ rexanre and ~ rexanuz ). wb cmul cseq fveq1i seq1 eqtrid breqtrrd eqbrtrd jca a1i cfzo w3a elfzouz cr 3ad2ant1 simpl3 wss elfzouz2 fzss2 sselda nfel1 eleq1d chvarfv remulcl adantl seqcl simp3 fzofzp1 anabsi7 pm3.35 ancoms 3adant1 breqtrdi mulge0d - simpl simp1 seqp1 breqtrrdi remulcld lemul2ad recnd mulid1d breqtrd simp2 + simpl simp1 seqp1 breqtrrdi remulcld lemul2ad recnd mulridd breqtrd simp2 1red simprd eqbrtrrid letrd eqbrtrid 3exp fzind2 mpcom ) EFGUDUEZQAREBUFZ SUGZYMTSUGZUHZMARUAUIZBUFZSUGZYRTSUGZUHZUNARFBUFZSUGZUUBTSUGZUHZUNZARUBUI ZBUFZSUGZUUHTSUGZUHZUNZARUUGTUJUEZBUFZSUGZUUNTSUGZUHZUNAYPUNUAUBEFGYQFUOZ @@ -717880,7 +717880,7 @@ closed under the multiplication ( ' X ' ) of a finite number of chvarfv adantlr seqsplit eluzfz1 ancli sylc eqeltrd elfzelz lep1d elfzle1 vtoclg1f peano2re letrd elfzle2 elfzd syldan seqcl remulcld 1red cc0 nfbr rpred breq2d breqtrrd peano2zd gtned orel1 zltp1le syl2anc mpbid ad2antrr - eqid leidd simpll fmul01 simprd lemul2ad mulid1d breqtrd lelttrd eqbrtrid + eqid leidd simpll fmul01 simprd lemul2ad mulridd breqtrd lelttrd eqbrtrid recnd pm2.61dan ) AGFUDZGBUEZEUFUGZAUUKUHZUULFCUEZEUFUUNUULFBUEFUICFUJZUE ZUUOUUNGFBAUUKUNUKUUNFBUUPBUUPUDUUNJULUMAUUQUUOUDZUUKAFUORZUURKUICFUPUQZS URAUUOEUFUGUUKQSUSAUUKUTZFGUFUGZUUMAUVAUHZUVBGFVAZUVCGFAUVAUNVBUVCFTRZGTR @@ -717943,7 +717943,7 @@ closed under the multiplication ( ' X ' ) of a finite number of zcnd 1cnd eleqtrrd 1zzd zsubcld wo eqcom sylnib leloed mpbid sylc zltlem1 orel2 syl2anc eluz2 syl3anbrc seqsplit seqeq1d fveq1d eqtrd 1red resubcld wb oveq2d lem1d eqid eluzfz2 fmul01 simpld 3jca jca elfz2 sylanbrc simprd - letr lemul1ad eqbrtrd mulid2d breqtrd lelttrd eqbrtrid pm2.61dan ) AFGUEZ + letr lemul1ad eqbrtrd mullidd breqtrd lelttrd eqbrtrid pm2.61dan ) AFGUEZ HBUFZEUGUHAUUHUIZBCDEGHIAUUHDJUUHDUJUKKAGULTZUUHLUMAHGUNUFZTZUUHMUMADUQZG HUOURZTZUUNCUFZUSTZUUHNUPAUUPUTUUQVAUHZUUHOUPAUUPUUQVJVAUHZUUHPUPAEVBTUUH QUMUUJFCUFZGCUFEUGUUJFGCAUUHVCVDAUVAEUGUHUUHSUMVEVFAUUHVKZUIZUUIHVGCGVHZU @@ -718194,7 +718194,7 @@ closed under the multiplication ( ' X ' ) of a finite number of ( cfv cfa cdiv co cmul wcel syl cn0 csn cun cv csu cprod cn cdif nfv nfcv c1 cfn wss ssfi syl2anc wn eldifn wa elmapi adantr elun1 adantl ffvelcdmd wf cmap faccld nncnd 2fveq3 snidg elun2 fprodsplitsn oveq2d eldifad snssi - unssd ffvelcdmda fsumnn0cl fprodclf mulcld nnne0d fprodn0 mulne0d mulid2d + unssd ffvelcdmda fsumnn0cl fprodclf mulcld nnne0d fprodn0 mulne0d mullidd divcld eqcomd cc0 wne nnne0 dividd mulcomd eqtrd oveq12d divmul13d 3eqtrd divdiv1d cbc cmin csb caddc nfcsb1v nn0cnd csbeq1a wceq csbfv a1i eqeltrd fsumsplitsn oveq1d pncan2d 3eqtrrd fveq2d cfz 0zd nn0ge0d cle wbr breqtrd @@ -719326,7 +719326,7 @@ closed under the multiplication ( ' X ' ) of a finite number of 3eltr4d crab cdif incom inss2 ssrin eqsstri disjdif sseqtri uncom inundif ss0 fsumsplit ssrab2 simprbi vtoclga w3a 3anbi23d fveqeq2 imbi12d chvarvv sselid eleq1w syl3anc sumeq2dv wo inss1 ssfi olcd sumz oveq2d difss mp2an - fzfi sylan2 addid1d fveq2 cbvsumv wf1o wf1 wfo 1re remulcli elfzle2 elfzd + fzfi sylan2 addridd fveq2 cbvsumv wf1o wf1 wfo 1re remulcli elfzle2 elfzd fsumcl wne divsubdird divcan3d rereccld halflt1 ltsub2dd crp 2rp ltsubrpd rpreccl breqtrrd btwnnz eqneltrd intnand sylnibr eldifd subcan2d mulcanad nsyl fmpttd 3eqtrd syldan adantll ex ralrimivva dff13 elfzelzd zeo eldifn @@ -720058,7 +720058,7 @@ closed under the multiplication ( ' X ' ) of a finite number of simp3r simp113 mp2and simp13 min2 eleq1d eqeltrd mpd3an23 simp-7l simp-4r cdif eldifsni divcld nfov nfeq eqeq12d simp-6l simplr nfne neeq1d eqnetrd nfel1 absne0d redivcld rpre ad2antlr remulcld simp-4l ltdiv1dd recnd 1red - divassd 1rp div1d ltdiv23d adantllr ltmul2dd mulid1d breqtrd 3exp ralrimi + divassd 1rp div1d ltdiv23d adantllr ltmul2dd mulridd breqtrd 3exp ralrimi lttrd brimralrspcev ralrimiva mpbir2and ) AUFJGUGUHSUFUISZUAUJZGUKZUYEGUL UHZUMUNZUBUJZUOUPUQUYEJUNZUFULUHZUMUNZUCUJZUOUPZURUACUSUBTUTZUCTUSAVAAUYO UCTAUYMTSZUQZUYFUYHUDUJZUOUPZUQZFUMUNZVBVCUHZUYEIUNZUMUNZUOUPZURZUACUSZUD @@ -720303,7 +720303,7 @@ closed under the multiplication ( ' X ' ) of a finite number of cmul wa eldifad subcld mulcld wceq cdif wne eldifsni mulne0d neneqd elsng syl wb mtbird eldifd caddc negcld limcmptdm cdm wf wss w3a limcrcl simp3d constlimc neglimc addlimc negidd negsubd mpteq2dva oveq1d 3eltr3d mullimc - 0ellimcdiv divsubdivd mulid2d oveq12d eqtr2d eleqtrd reccld ellimcabssub0 + 0ellimcdiv divsubdivd mullidd oveq12d eqtr2d eleqtrd reccld ellimcabssub0 cneg 1cnd mpbird ) AUAEUBNZHFONPQBCUADUBNZWNUCNZRZFONZPAQBCEDUCNZDEUINZUB NZRZFONWRABCWSWTEEUINFBCWSRZBCWTRZXBXCSXDSZXBSABUDCPZUJZEDAETPXFAGFONZTEF GUELUFZUGZXGDTQUHZKUKZULXGWTTXKXGDEXLXJUMZXGWTXKPZWTQUNZXGWTQXGDEXLXJXGDT @@ -724962,7 +724962,7 @@ distinct definitions for the same symbol (limit of a sequence). ( vn cz wcel c2 cpi cmul co ccos cfv c1 wceq wa wi picn a1i adantr adantl eqtrd caddc cdvds wbr cneg cif cv wrex wb 2z divides mpan biimpa w3a 2cnd zcn cc mulassd eqcomd oveq1 eqtr2d fveq2d cos2kpi 3adant1 iftrue 3ad2ant1 - 3exp rexlimdv mpd odd2np1 mulcld 1cnd adddird mulcomd mulid2i oveq12d 2cn + 3exp rexlimdv mpd odd2np1 mulcld 1cnd adddird mulcomd mullidi oveq12d 2cn wn oveq1d mulcli addcomd 3eqtrrd cosper cospi 3eqtrd iffalse pm2.61dan ) ACDZEAUAUBZAFGHZIJZWGKKUCZUDZLZWFWGMZBUEZEGHZALZBCUFZWLWFWGWQECDWFWGWQUGU HBEAUIUJUKWMWPWLBCWGWNCDZWPWLNNWFWGWRWPWLWGWRWPULWIKWKWRWPWIKLWGWRWPMZWIW @@ -725006,10 +725006,10 @@ distinct definitions for the same symbol (limit of a sequence). -u 1 ) ) $= ( vn cz wcel c2 cpi cneg cmul co ccos cfv c1 wceq wa cc0 caddc 3eqtrd syl cc adantr cdvds wbr cif cv simpr wb 2z simpl divides sylancr mpbid wi w3a - wrex zcn negcl 2cn picn mulcli a1i mulcld addid2d mulassd eqcomd mulneg1d + wrex zcn negcl 2cn picn mulcli a1i mulcld addlidd mulassd eqcomd mulneg1d 2cnd id oveq1d mulneg12 syl2anc adantl 3eqtrrd fveq2d 3adant1 0cnd znegcl cosper 3ad2ant2 cos0 iftrue eqtr4id 3ad2ant1 3exp rexlimdv mpd wn odd2np1 - oveq1 biimpa 1cnd negpicn adddird mulneg2d mulcomd eqtr3d mulid2d oveq12d + oveq1 biimpa 1cnd negpicn adddird mulneg2d mulcomd eqtr3d mullidd oveq12d addcomd eqtrd 3adant1r cosnegpi iffalse rexlimdv3a pm2.61dan ) ACDZEAUAUB ZAFGZHIZJKZXFLLGZUCZMZXEXFNZBUDZEHIZAMZBCUNZXLXMXFXQXEXFUEXMECDXEXFXQUFUG XEXFUHBEAUIUJUKXMXPXLBCXFXNCDZXPXLULULXEXFXRXPXLXFXRXPUMXIOXNGZEFHIZHIZPI @@ -726186,7 +726186,7 @@ distinct definitions for the same symbol (limit of a sequence). c1 cv co crp 1rp nnrp rpdivcld rpcnd adantl fmptd cmpt 1cnd divcnv breq1d a1i syl mpbird 0cnd climcncf wf caddc wa nfv adantr adantlr simplr addcld nfan cfn fprodclf fcompt syl2anc fvmpt2 fveq2d cvv oveq2 prodeq2ad prodex - id fvmptd3 eqtr2d mpteq2dva eqtrd eqtr4d ad2antlr addid1d prodeq2d fvmptd + id fvmptd3 eqtr2d mpteq2dva eqtrd eqtr4d ad2antlr addridd prodeq2d fvmptd ex ralrimi breq12d mpbid ) AHIUAZPHUBZUCUDECDFUEZUCUDAQQPHIUJRUFAUGABCDFH JKLNUHAGRUJGUKZUIULZQIXERSZXFQSZAXGXFXGUJXEUJUMSXGUNVDXEUOUPUQZURZOUSZAIP UCUDGRXFUTZPUCUDZAUJQSXMAVAUJGVBVEAIXLPUCIXLTAOVDVCVFAVGZVHAXBEXCXDUCAXBG @@ -726267,7 +726267,7 @@ distinct definitions for the same symbol (limit of a sequence). wtru wral ax-1cn rgenw fmpt mpbi dvcmulf dmeqd ovexd fveq1i mpan2 eqeltrd adantr dmmptd 3eqtrd dvcof ccncf coscn crn wss frnd sseqtrrd dmcosseq syl coexg ovex dmmpti coscld simpl mulcomd coeq1i fveq1d wfun ffund eleqtrrdi - fvco cc0 caddc 0cnd dvmptc dvmptmul mul02d mulid2d addid2d 3eqtr4d ) BDEZ + fvco cc0 caddc 0cnd dvmptc dvmptmul mul02d mullidd addlidd 3eqtr4d ) BDEZ DADBAUAZFRZGHZIZJRDGADYOIZUBZJRDGJRZYRUBZDYRJRZFUCZRZADBYOKHZFRZIZYMYQYSD JYMYSCDCUAZYRHZGHZIZCDBUUHFRZGHZIZYQYMDDGUDZDDYRUDYSUUKLUUOYMUMMZYMADYODB YNUEUFZCGYRDDDUGNYMCDUUJUUMYMUUHDEZUHZUUIUULGUUSAUUHYOUULDYRDUUSYROYNUUHL @@ -726483,7 +726483,7 @@ distinct definitions for the same symbol (limit of a sequence). ( cc wcel wa cmul co cfv cmpt cdv cc0 caddc cnelprrecn a1i adantr adantll wceq mulcld eqtrd cv csin ccos cr cpr simpll id dvmptc mulcl sincld simpl coscld dvsinax adantl dvmptmul mul02d mul32d simpr mulcomd oveq1d oveq12d - 0cnd addid2d mpteq2dva ) BDEZCDEZFZDADBCAUAZGHZUBIZGHJKHADLVJGHZCVIUCIZGH + 0cnd addlidd mpteq2dva ) BDEZCDEZFZDADBCAUAZGHZUBIZGHJKHADLVJGHZCVIUCIZGH ZBGHZMHZJADBCGHZVLGHZJVGABLVJVMDDDDDUDDUEEZVGNOVEVFVHDEZUFZVGVSFZVBVEDADB JKHADLJRVFVEABDVRVENOVEUGUHPVFVSVJDEVEVFVSFZVICVHUIZUJQZVFVSVMDEVEWBCVLVF VSUKZWBVIWCULZSQVFDADVJJKHADVMJRVEACUMUNUOVGADVOVQWAVOLVQMHVQWAVKLVNVQMWA @@ -726620,7 +726620,7 @@ distinct definitions for the same symbol (limit of a sequence). wf fveq2 cpr fmpttd dvcos dmeqi dmmptg sincl negcld mprg eqtri simpl 0red c1 wa id dvmptc simpr 1red dvmptid dvmptmul dmeqd eqtrdi dvcof cin negeqd ovex oveq1d cnex mptex offval3 mp2an wral sincld ralrimiva ineq12d adantr - syl inidm oveq2 fveq2d adantl negex fvmptd mul02 mulid2 oveqan12rd addid2 + syl inidm oveq2 fveq2d adantl negex fvmptd mul02 mullid oveqan12rd addlid eleqtrd eqtrd sylan9eqr oveq12d mulcomd syldan mpteq12dva 3eqtrd cbvmptv ) BDEZDADBAUAZFGZHIZJZKGZCDBBCUAZFGZLIZMZFGZJZADBXKLIZMZFGZJXIXNDHADXKJZU BZKGDHKGZYDUBZDYDKGZFUCZGZXTXIXMYEDKXIYEXMXIACDDXKXOHIXLYDHBXJUDZXIYDUEZX @@ -726951,7 +726951,7 @@ distinct definitions for the same symbol (limit of a sequence). 2halvesd iffalse eleq1d anbi12d crab cinf wb ax-resscn fssd dvcn syl31anc ccncf cncfcdm wrel climrel fvex fvmptd climconst eqeltri cn 1cnd elnnnn0b mptex sylanbrc divcnvg oveq2 reccld oveq2d fvmptd3 oveq12d eqtr4d climadd - addid1d releldm raleqbidv cbvrabv rgenw rabbi mpbi eqtri ioodvbdlimc1lem1 + addridd releldm raleqbidv cbvrabv rgenw rabbi mpbi eqtri ioodvbdlimc1lem1 infeq1i mpteq2dva eqtrid clim anbi2d rexralbidv r19.29a ioosscn mpbir2and breq2 ellimc3 ) AIUOUPZKFUUBUQURVXMUSURZEUTZFVAZVXOFVBUQZVCUPZDUTVDVEVFVX OKUPZVXMVBUQZVCUPZCUTZVDVEZVGEFGUUCUQZVHDVIVJZCVIVHAVXMALVKUPZJUGIUHVYFVL @@ -727441,7 +727441,7 @@ distinct definitions for the same symbol (limit of a sequence). ( vy co cexp cmpt c1 cmul cc cr wcel adantr caddc cdv cmin cc0 cnelprrecn cv cpr a1i wss dvdmsscn simpr sseldd addcld 1red 0red readdcld cn0 nnnn0d wa expcld nn0cnd cn nnm1nn0 syl mulcld dvmptidg dvmptconst dvmptadd dvexp - wceq oveq1 oveq2d dvmptco 1p0e1 oveq2i nncnd mulid1d eqtrd mpteq2dva ) AD + wceq oveq1 oveq2d dvmptco 1p0e1 oveq2i nncnd mulridd eqtrd mpteq2dva ) AD BFBUFZCUALZEMLZNUBLBFEWAEOUCLZMLZPLZOUDUALZPLZNBFWENABKWAWFKUFZEMLZEWHWCM LZPLZDQWBWERQFQGQRQUGSAUEUHAVTFSZUSZVTCWMFQVTAFQUIWLADFGHUJTAWLUKULZACQSW LITZUMZWMOUDWMUNZWMUOZUPAWHQSZUSZWHEAWSUKZAEUQSWSAEJURTZUTWTEWJWTEXBVAWTW @@ -727494,13 +727494,13 @@ distinct definitions for the same symbol (limit of a sequence). c1 oveq12d eqeq12d cc wss cpm cr cpr recnprss syl cvv wf a1i ccnfld ctopn cnex wa crest cpw restsspw id sselid elpwi sstrd sseldd addcld cn0 expcld simpr fmptd elpm2r syl22anc dvn0 syl2anc cle wn nn0ge0d 0red nn0red mpbid - lenltd iffalsed nn0cnd subid1d cn faccl nncnd nnne0d dividd eqtrd mulid2d + lenltd iffalsed nn0cnd subid1d cn faccl nncnd nnne0d dividd eqtrd mullidd 3eqtrrd mpteq2dva 3eqtrd cdv dvnp1 adantl iftrue 0cnd dvmptconst ad2antrr syl3anc nn0re ad2antlr ltled wb nn0zd zleltp1 iftrued eqcomd 3syl adantlr cz ad3antrrr wne mulcld oveq1d nnne0 pm2.61dan mpbird lttri3d subidd fac0 - simpl div1d exp0d mulid1d ltp1d oveq1 breqtrd simpll simplr neqne leneltd + simpl div1d exp0d mulridd ltp1d oveq1 breqtrd simpll simplr neqne leneltd necomd nn0sub divcld posdifd elnnz sylibr nnm1nn0 dvxpaek dvmptmul mul02d - jca addid2d zltp1le peano2re mulassd div32d facnn2 divcan8d 1cnd subsub4d + jca addlidd zltp1le peano2re mulassd div32d facnn2 divcan8d 1cnd subsub4d divrecd eqtr2d mulcomd nn0indd ) AUAUCZDEUDNZUEZBHFUVTUFUGZOFUJUEZFUVTUHN ZUJUEZUINZBUCZCUKNZUWEULNZPNZUMZUNZQOUWAUEZBHFOUFUGZOUWDFOUHNZUJUEZUINZUW IUWPULNZPNZUMZUNZQUBUCZUWAUEZBHFUXCUFUGZOUWDFUXCUHNZUJUEZUINZUWIUXFULNZPN @@ -727592,7 +727592,7 @@ distinct definitions for the same symbol (limit of a sequence). wi eleq1 fveq2 oveq2 sumeq1d fveq1d oveq2d oveq12d eqeq12d imbi2d imbi12d eqtrd c1 caddc wa simpl simpll oveq1d fvoveq1d sumeq12rdv mpteq2dva cc cr cvv mulcld a1i syl2anc cz adantl fvexd adantr adantlr simpr fveq2d 3eqtrd - nfv mulid2d 0re eqtr2d cdv fzfid elfzelz bccld nn0cnd adantll cle elfzle1 + nfv mullidd 0re eqtr2d cdv fzfid elfzelz bccld nn0cnd adantll cle elfzle1 wf wbr zred elfzle2 ltled elfzd mpbird ffvelcdmd zsubcld subge0d resubcld feq1d ltletrd recnd ovex anbi2d chvarfv vtocl 3expa 1red letrd ffvelcdmda peano2zd nfcv nffv nff nfim eqcomd nfov 1zzd ad2antlr vtoclf cdvn eluzfz2 @@ -727605,7 +727605,7 @@ distinct definitions for the same symbol (limit of a sequence). jca lesub2dd breqtrd peano2re leadd1dd elfzop1le2 lesub1dd mulcomd nfmpt1 ltp1d nfcxfr addcld eqeltrrd dvmptconst feqmptd elfznn0 fznn0sub dvmptmul 3impa 1cnd addsubd mul02d dvmptfsum an32s anass ancom anbi2i bicomi bitri - addid2d imbi1i mpbi adddid sumeq2dv cfn fsumadd cbvsum nfel nfan fsumshft + addlidd imbi1i mpbi adddid sumeq2dv cfn fsumadd cbvsum nfel nfan fsumshft eleq1d 0p1e1 oveq1i cdif zcnd npcand subsub3d readdcld simplr leidd elrpd 0lt1 crp ltsubrpd eqcomi nn0ge0d 3jca eluz2 sylibr fsumsplit1 pncand bcnn eqbrtrd subidd fzofzp1 1m1e0 fveq2i eqtr2i eleqtrd fzdifsuc2 eleqtrrd wal @@ -728741,7 +728741,7 @@ distinct definitions for the same symbol (limit of a sequence). 0re ccncf cibl eqid fvmpt2 mpteq2dva nfmpt1 nfcv sincn cncfmptss cniccibl expcnfg syl3anc iblss itgcl adddirp1d clt wbr eluz2b2 sylib simpld expm1t syl2anr ccos cneg itgsinexplem1 subsub4d 1p1e2 eqtrd sincossq sincl sqcld - oveq2d coscl subaddd mpbird nnm1nn0 subdird mulid2d expaddd eqtr3d itgsub + oveq2d coscl subaddd mpbird nnm1nn0 subdird mullidd expaddd eqtr3d itgsub cr pncan3d oveq12d 3eqtrd fvmptd3 mulcld subdid 3eqtr2d divcan3d 3eqtr3d mpbid nnne0d div23d ) AEEDUAZHIZEUBIEJKIZELKIZDUAZHIZEUBIUUHUUJEUBIUULHIA UUIUUMEUBAUUIUUJJUCIZUUHHIUUJUUHHIZUUHUCIZUUMAEUUNUUHHAUUNEAEJAELUEUAMZEU @@ -729060,7 +729060,7 @@ distinct definitions for the same symbol (limit of a sequence). ioc0 eqcom biimpi adantl recn adantr eqeltrd subeq0bd 3eqtr4a 3ad2antl1 clt wn simpl1 simpl2 simpl3 wne necon3bi leneltd csn cun 3ad2ant1 simp3 ioounsn cioo cdm cin ioombl a1i snmbl ubioo mpbir ioovolcl 3adant3 volsn 0red volun - disjsn syl32anc simp1 simp2 ltled volioo oveq12d recnd subcld addid1d eqtrd + disjsn syl32anc simp1 simp2 ltled volioo oveq12d recnd subcld addridd eqtrd 3eqtrd pm2.61dan ) ACDZBCDZABEFZGZABHZABIJZKLZBAUAJZHZXAXBXEXIXCXAXEMZNKLOX GXHUBXJXFNKXEXAXFAAIJZNXEXKXFABAIUCPXAXKNHZAAEFZAUDXAASDZXNXLXMUEAUFZXOAAUL UGUHUIUJXJBAXJBAUKXEBAHZXAXEXPABUMUNUOZXAAUKDZXEAUPZUQURXQUSUTVAXDXEVCZMZXA @@ -729617,7 +729617,7 @@ distinct definitions for the same symbol (limit of a sequence). cres cxr rexrd ctg cnt oveq2d addcld fmpttd ccnfld tgioo2 syl22anc iccntr ctopn reseq2d cpr reelprrecn dvmptid 0cnd dvmptc dvmptadd reseq1d ioossre dvres 1p0e1 mpteq2i fveq2 itgsubsticc mulcld cbvitgv iccgelb eqtri fveq2d - iccleub mulid1d sylan9eqr fvmptd itgeq2dv ) ABCEUCOZDEUCOZUDOZBUEZGUFZUGZ + iccleub mulridd sylan9eqr fvmptd itgeq2dv ) ABCEUCOZDEUCOZUDOZBUEZGUFZUGZ BUVOUVPUVSUHZNCDNUEZEUCOZGUFZPUIOZUHZBCDUDOZUVRFUFZUGZAUWABUVOUVPUKOUVSUG UVTABUVOUVPUVSACDEHIAELUJZJULUMABUVOUVPUVSACEHUWJUNZADEIUWJUNZAUVQQUVRGAB UVQUVREUOOZFUFZQGAUVRUVQRZUPZUWGQUWMFAUWGQFUQZUWOAFUWGQURORUWQKUWGQFUSUTZ @@ -729687,7 +729687,7 @@ distinct definitions for the same symbol (limit of a sequence). iccshift constcncfg cxp fconstmpt ioombl ioovolcl eqeltrrid elind crn ctg iblconst cnt oveq2d addcld fmpttd ccnfld ctopn tgioo2 syl22anc iccntr cpr dvres reelprrecn dvmptid 0cnd dvmptc dvmptadd reseq1d ioossre 1p0e1 fveq2 - mpteq2i itgsubsticc mulcld fvoveq1 cbvitgv mulid1d itgeq2dv eqtrid ) ABCE + mpteq2i itgsubsticc mulcld fvoveq1 cbvitgv mulridd itgeq2dv eqtrid ) ABCE UBPZDEUBPZUCPZBUDZFUEZUFZBUWJUWKUWNUGZNCDNUDZEUBPZFUEZUHUIPZUGZBCDUCPZUWN UFZAUWPBUWJUWKUKPUWNUFUWOABUWJUWKUWNACDEGHAEJUJZIULUMABUWJUWKUWNACEGUXDUN ZADEHUXDUNZAUWMUWLQZUOZRSUWMFARSFVLZUXGKTUXHUWJRQZUWKRQZUXGUWMRQZAUXJUXGU @@ -729752,7 +729752,7 @@ distinct definitions for the same symbol (limit of a sequence). volioo 1cnd iblconst eqeltrrid elind cdv cc0 crn ctg ccnfld tgioo2 iccntr ctopn cnt dvmptntr cpr reelprrecn ioossre sseli dvmptid iooretop dvmptres adantl 0cnd fveq2d oveq1 eqtrd dvmptc dvmptsub subid1d 3eqtrd itgsubsticc - oveq2 pncan3d oveq1d cncff ioossicc ffvelcdmd mulid1d ditgeq3d ) AGCFUAMZ + oveq2 pncan3d oveq1d cncff ioossicc ffvelcdmd mulridd ditgeq3d ) AGCFUAMZ DFUAMZFGUBZUCMZENZUDBCDFBUBZFUAMZUCMZENZOUEMZUDBCDUUSENZUDABGUUTOUURUVBUU NUUOCDHIJABCDUHMZUUTPZBUVEUUSFUIUCMZPZUVEUUNUUOUHMZUFMZABUVEUUTUVGAUUSUVE QZUJZUVGUUTUVLUUSFUVLUUSAUVERUUSACDHIUGZUKZULAFSQZUVKAFKULZTUMUNUOZAUVHUV @@ -729790,7 +729790,7 @@ distinct definitions for the same symbol (limit of a sequence). ( cr wcel wa wbr co cvol cfv cc0 wceq caddc rexr 3ad2ant1 syl3anc fveq2d c0 cxr a1i wn clt cico cmin cif w3a csn cun 3ad2ant2 simp3 snunioo1 eqcomd cdm cioo cin ioombl snmbl lbioo mpbir ioovolcl 3adant3 volsn 0red eqeltrd volun - disjsn syl32anc cle simp1 simp2 ltled volioo oveq12d cc recn subcld addid1d + disjsn syl32anc cle simp1 simp2 ltled volioo oveq12d cc recn subcld addridd recnd eqtrd 3eqtrd 3expa iftrue adantl eqtr4d simpl simprd simpld lenltd wb simpr mpbird ad2antrr ad2antlr ico0 syl2anc vol0 iffalse pm2.61dan ) ACDZBC DZEZABUAFZABUBGZHIZXABAUCGZJUDZKWTXAEXCXDXEWRWSXAXCXDKWRWSXAUEZXCABUMGZAUFZ @@ -730088,7 +730088,7 @@ distinct definitions for the same symbol (limit of a sequence). nnnn0d resubcld nn0expcld cn0 nn0mulcld nnred remulcld nncnd rpcnd nnne0d 2nn0 wne rpne0d mulne0d cc cn mulcld expne0 syl2anc mpbird redivcld caddc readdcld exple1 syl31anc subge0d expge0d expcld dividd 0red 0le1 leadd1dd - wb addid2d eqbrtrrd clt cz nngt0d rpgt0d mulgt0d expgt0 syl3anc syl112anc + wb addlidd eqbrtrrd clt cz nngt0d rpgt0d mulgt0d expgt0 syl3anc syl112anc nnzd lediv1 mpbid mulexpd oveq2d oveq1d breqtrd lemulge11d subcld divassd 1cnd addcld breqtrrd mulcomd renegcld le0neg2 ax-mp letrd bernneq eqbrtrd cneg mpbi lemul2ad wceq subsq sq1 expmuld 2cnd eqtr3d oveq12d 3eqtr3d jca @@ -730400,9 +730400,9 @@ distinct definitions for the same symbol (limit of a sequence). recnd fsumsplit syl112anc chash fsumconst syl2anc hashfz breqtrd ltadd2dd cfn eqbrtrd subcld 3eqtrd mulcomd leadd2dd lemul2d wf 0zd 0red w3a elfzuz 0le1 sylib simp3d eluzfz2 ne0i 4syl cn rpgt0d ltmul2 fsumlt nnne0d divcld - fzss1 lemul2 mulid1d fsumle 1e0p1 fveq2i eluzp1m1 sylancr subid1d ltletrd + fzss1 lemul2 mulridd fsumle 1e0p1 fveq2i eluzp1m1 sylancr subid1d ltletrd 0z leadd1dd addcld mul12d div12d elfzle1 suble0d mpbird addsub12d addcomd - addid1d 3brtr3d nngt0d lediv1 dividd eqbrtrrd adddird ltmul1dd lelttrd + addridd 3brtr3d nngt0d lediv1 dividd eqbrtrrd adddird ltmul1dd lelttrd eqtr4d lttrd ) ABUAZBCQGUBRZFUWIDUAZHUCZUCZUDRZDUEZUFZUCZUWOEUAZSUGUHRZUI RZFUDRZUJAUWICTZUWOUKTUWQUWOULJUWJUWNDUNBCUWOUKUWPUWPUMUOUPAUWOFUWRUDRZGU WRVERZSUIRZFFGUHRZUDRZUDRZUIRZUXAAUWJUWNDAQGUQZAUWKUWJTZURZFUWMAFUSTZUXKA @@ -730480,11 +730480,11 @@ distinct definitions for the same symbol (limit of a sequence). wbr 2re remulcl sylancr recnd negsubdi2d 1red remulcld cdiv 3ne0 rereccli cv 3re a1i cc0 wne w3a 4re 3pm3.2i redivcl mp1i lesub1dd ltsub2dd lelttrd subcld subdird sub4d subsub2d oveq1d eqtr3d breqtrd subidd 4cn 3cn divcli - eqtrd ax-1cn 1div1e1 oveq2i ax-1ne0 divadddivi addcomi df-4 1t1e1 mulid2i + eqtrd ax-1cn 1div1e1 oveq2i ax-1ne0 divadddivi addcomi df-4 1t1e1 mullidi eqtr3i oveq12i 3eqtr4ri oveq1i 3t1e3 3eqtri subaddrii 1e0p1 eqtr4i eqtrdi 1lt2 ltmul1d mpbii lttrd eqbrtrd ltnegcon1d c5 redivcld renegcld readdcld ltnegd mpbid lt2addd negsubd adddird eqcomd negcld mulneg1d oveq2d 3eqtrd - negeqd mulcld negdid negnegd oveq12d add4d negidd addcld addid2d divcan2i + negeqd mulcld negdid negnegd oveq12d add4d negidd addcld addlidd divcan2i 5re adddid df-5 3eqtr4i mvllmuld 3eqtr2d 3brtr3d 5lt6 3t2e6 breqtrri 3pos c6 wa wb pm3.2i ltdivmul mp3an mpbir ltmul1dd absltd mpbir2and ) AEDUANZU BUCUDCONZPUJUUQUEUUPPUJUUPUUQPUJAUUPUUQAEDHGUFZAUDUGUHZCUGUHUUQUGUHUKACFU @@ -730531,7 +730531,7 @@ distinct definitions for the same symbol (limit of a sequence). biimpri jca reximi2 rexn0 4syl nnwo syl2anc df-rex sylib simplbi ad2antrl elrab2 simpl simprl simprr nfcv nfrab1 nfcxfr nfv cbvralfw simprbi wb cc0 1red adantl rpregt0d adantr ltdivmul2 syl3anc syldan mpbid syl12anc oveq1 - nnre wceq cc rpcnd mulid2d eqtrd oveq1d rpred rehalfcld halfre 2re ltdiv1 + nnre wceq cc rpcnd mullidd eqtrd oveq1d rpred rehalfcld halfre 2re ltdiv1 2pos syl112anc halflt1 lttrd eqbrtrd adantlr cuz cfv simpll simplrl neqne cmin syl eluz2b3 sylanbrc peano2rem ad2antrr rpne0d rereccld cz w3a caddc 1zzd df-2 fveq2i eleq2i eluzsub sylibr mpd 3adant3 3ad2ant1 resubcld 1cnd @@ -731113,15 +731113,15 @@ functions with (possibly) negative values. (Contributed by Glauco vtoclgaf fsumle ltletrd zre nndivred wss elfzelzd nnzd nnred 0le2 lesub2d 2z zcnd eluz2 fzss2 sselda ltadd2dd divdiri 3p1e4 3eqtr3ri breqtrdi rpcnd subsub4d mulcomd subdid eqtr3d 1zzd zlem1lt nngt0d ltdiv1 syl112anc nncnd - nnne0d dividd mulgt0d ltmul2 mulid1d divcld subdird mulid2d div32d eqtr4d + nnne0d dividd mulgt0d ltmul2 mulridd divcld subdird mullidd div32d eqtr4d ltmul1dd mulassd cn elnnuz elfzp12 orcanai 1p1e2 elfzle1 mpbird subadd23d subge0d hashfz cneg negsubdi2i 2m1e1 negeqi eqtr3i negsubd c0 fzn0 simpll - ad2antrr jca ffvelcdm leadd1dd lemul1 addid1 eqcomd mp1i subidd addsubass + ad2antrr jca ffvelcdm leadd1dd lemul1 addrid eqcomd mp1i subidd addsubass wf subsubd divcli addsubassd df-3 subaddrii pm3.2i 3pm3.2i peano2zd lep1d elfzd npcand 0p1e1 3eltr4d fzaddel neleqtrd sylnib ianor olc anim1i orcom 1lt2 anbi2i pm4.43 ltnled ltled sylanbrc eleqtrrd 3ad2ant1 simp2 3ad2ant3 sseldd elex nfrab1 nfmpt nfcxfr 3anbi3d imbi12d vtoclg1f syld3an2 mpd3an3 - nfel2 nf3an fsumlt fsumcl addid1d 1m1e0 lesub1dd eqbrtrrid lemul2 fsumge0 + nfel2 nf3an fsumlt fsumcl addridd 1m1e0 lesub1dd eqbrtrrid lemul2 fsumge0 simpr leadd2dd fsummulc2 cin wral ltsub2d lelttrd intnanrd mtbird ralrimi ex cun mpbir2and fzsplit uneq2d fsumsplit 3brtr4d ltletr mp2and pm2.61dan disj sumex sumeq2sdv eqid fvmptg ) AKUIUJUKULZUMULZIUNULZUOLUPULZIEGUQZMU @@ -732005,7 +732005,7 @@ the final h is a normalized version of G ( divided by its norm, see the ( cle c1 wcel wa cc0 cfv wbr cdiv co cfz cv cmul cr nnrecred adantr fzfid csu stoweidlem15 simp1d an32s fsumrecl clt 1red a1i nnred nngt0d syl22anc 0le1 divge0 simp2d fsumge0 stoweidlem30 breqtrrd simp3d fsumle wceq chash - mulge0d cfn cc ax-1cn fsumconst sylancl cn0 nnnn0d hashfz1 oveq1d mulid1d + mulge0d cfn cc ax-1cn fsumconst sylancl cn0 nnnn0d hashfz1 oveq1d mulridd syl nncnd 3eqtrd breqtrd wb 0lt1 divgt0 syl21anc lemul2 syl112anc eqbrtrd jca mpbid wne w3a nnne0d 3jca divcan1 ) AFGUAZUBZUCFDUDZSUEXGTSUEXFUCTLUF UGZTLUHUGZFJUIZKUDUDZJUOZUJUGZXGSXFXHXLAXHUKUAZXEALPULUMZXFXIXKJXFTLUNZAX @@ -732217,7 +732217,7 @@ the final h is a normalized version of G ( divided by its norm, see the divcan1d 1cnd divcld mulcld mulneg1d 3eqtr2d renegcld nnred 3ne0 rereccld 1lt3 0lt1 ltdiv2 syl222anc 1div1e1 breqtrdi lttrd nnge1d ltletrd rpregt0d 3pos ltled nngt0d syl121anc cc rpcnne0d syl lenegd bernneq syl3anc oveq1d - divid fmptdf feq1d mpbird r19.21bi an32s crp addid2d syl221anc div1d 0red + divid fmptdf feq1d mpbird r19.21bi an32s crp addlidd syl221anc div1d 0red ltaddsubd elrpd stoweidlem3 fmuldfeq ex ralrimi ) AUNJUOUPZBUQZMURZUSUTZB CQAUWKCVAZUWMAUWNVBZUWJUWKOURZUWLUSUWOUWJLVCUWKKURZUNVDZURZUWPUSUWOUWJUNJ LVEUPZUOUPZLVFUPZUWSAUWJVGVAUWNAUNJAVIZAJUFVJZVHVKUWOUXALAUXAVGVAUWNAUNUW @@ -733802,7 +733802,7 @@ approximated is nonnegative (this assumption is removed in a later ( I ` N ) = ( ( ( N - 1 ) / N ) x. ( I ` ( N - 2 ) ) ) ) ) $= ( cc0 cfv cpi wceq c1 c2 wcel cmin co cn0 cexp citg cr pire eqtri cuz cdiv wi cmul 0nn0 cioo cv csin wa oveq2 adantr cc ioosscn sseli sincld exp0d itgeq2dv - adantl eqtrd cvol cdm ioombl 0re ioovolcl mp2an ax-1cn itgconst mp3an mulid2i + adantl eqtrd cvol cdm ioombl 0re ioovolcl mp2an ax-1cn itgconst mp3an mullidi recni cle wbr pipos ltleii volioo subid1i eqtrdi elexi fvmpt ax-mp 1nn0 simpl oveq2d exp1d itgex itgsin0pi id itgsinexp 3pm3.2i ) FCGHIZJCGZKIDKUAGLZDCGDJM NDUBNDKMNCGUDNIUCFOLWJUEBFAFHUFNZAUGZUHGZBUGZPNZQZHOCWPFIZWRAWMJQZHWSAWMWQJWS @@ -733893,13 +733893,13 @@ approximated is nonnegative (this assumption is removed in a later eluz1i itgsinexp remulcld nnge1 3pos mpbii addcld divcld cn0 wallispilem3 mulne0d rpcnd syl clt rpcnne0d adantr resubcld 1lt2 nnrp redivcld eqeltrd crp fvmptd3 2rp rpmulcld rpdivcld mulge0d ge0p1rpd eqcomd rpne0d mpbir3an - cseq uzid subidi simp1i 2cnne0 div32 mp3an mulid2i eluz2 eqcomi subaddrii + cseq uzid subidi simp1i 2cnne0 div32 mp3an mullidi eluz2 eqcomi subaddrii 2lt3 simp2i mulcomi 3eqtr2i 1z seq1 1nn div1i fvmpt oveq2i mulcli divne0i - ovex mulne0i eqnetrri divreci 2cnd adddid mulid1d peano2zd nnnn0 addge02d + ovex mulne0i eqnetrri divreci 2cnd adddid mulridd peano2zd nnnn0 addge02d nnz nn0ge0d pncand addassd zaddcld 0re addge01d peano2nn nnne0d nn0mulcld letrd 2nn0 0red nngt0 mulgt0d jctir sylibr ltaddrpd lttrd gt0ne0d divne0d 2pos zsubcld subge0d mpbird elnn0z jca31 divmuldiv syl21anc elnnuz biimpi - elrp seqp1 ltaddrp2d mulgt1d gtned subne0d eqnetrd elfznn mulid2d ltmul1d + elrp seqp1 ltaddrp2d mulgt1d gtned subne0d eqnetrd elfznn mullidd ltmul1d id eqbrtrrd lelttrd posdifd elrpd rpge0d rpmulcl rpaddcld divdiv1 syl3anc cfz seqcl divassd eqnetrrd divmuldivd divdiv2d divdivdivd 3eqtr2d halfcld reccld mulcomd ex nnind mpteq2ia 3eqtr4i ) CUDLCUEZMNZGOZVUBPQNGOZRNZUFCU @@ -734005,7 +734005,7 @@ approximated is nonnegative (this assumption is removed in a later divcld rpcnd rpne0d divcan4d cuz 2re nnre remulcld readdcld nn0ge0d nnge1 1red lemulge11d ltp1d lelttrd ltled cz nn0zd eluz mpbird itgsinexp pncand syl2anc oveq1d wceq 1e2m1 oveq2d subsub3d eqtr2d oveq12d peano2nnd nnne0d - eqtrd mulassd divcan6d mulid2d 3eqtr2d eqtr3d breqtrrd rpdivcld nfcv cioo + eqtrd mulassd divcan6d mullidd 3eqtr2d eqtr3d breqtrrd rpdivcld nfcv cioo wb cpi nffv adantl csin cexp citg nfmpt1 nfcxfr nfov oveq2 fvoveq1d mpdan fvmptf cc wa simpr fvmptd 3brtr4d dividd climsqz2 mptru eqbrtrri ) EFNUAA BCDEFGIJKLUBENUAUCUDNBHENUIOUEUDUFUDPNCHMUDUGPUJUKZUDUHQUDULUMEUIRUDECOPC @@ -734056,7 +734056,7 @@ approximated is nonnegative (this assumption is removed in a later elrpd nnred rpge0d mulge0d ge0p1rpd eqeltrd adantl wa rpmulcl seqcl rpcnd rpne0d reccld fmpti ffvelcdmda wceq fveq2 eqidd oveq2d fvmptd 2cn 2ne0 id eleq1d vtoclga divrecd divcan6d eqcomd mulassd rpreccld 3eqtr4d climmulc2 - 3eqtrd divcli mulid1i breqtrdi divne0i csn cdif wn recne0d eqnetrd eldifd + 3eqtrd divcli mulridi breqtrdi divne0i csn cdif wn recne0d eqnetrd eldifd nelsn recrecd fvmpt3 3eqtr4rd eqeltri climrec mptru recdiv mp4an breqtri ) DGHIJKZJKZIHJKZUDDUVMUDUELUVLUABMGBUFZNCGUGZOZJKZUHZDGUIMUJLUKZLUVSUVLG NKUVLUDLGUVLUABMUVNUVRNKZUHZUVSGUIMUJUVTUWBGUDUELUBABCBMHUVONKZBULUBPIUMK @@ -734104,7 +734104,7 @@ approximated is nonnegative (this assumption is removed in a later ( cmul cn c2 co c1 cmin cdiv caddc c4 cexp wceq oveq1d oveq2d oveq12d wcel c3 cfv a1i vx vy vw cv cmpt cseq fveq2 oveq2 eqeq12d cz seq1 ax-mp 1nn eqid ovex 1z fvmpt 2t1e2 oveq1i 2m1e1 eqtri oveq12i 2p1e3 2cn ax-1cn ax-1ne0 divmuldivi - 3cn 3ne0 2t2e4 mulid2i 3eqtri cc cc0 wne divrec2 mp3an eqcomi oveq2i c6 2exp4 + 3cn 3ne0 2t2e4 mullidi 3eqtri cc cc0 wne divrec2 mp3an eqcomi oveq2i c6 2exp4 4cn cdc sq2 4t4e16 4ne0 divcan3i cuz elnnuz biimpi adantr seqp1 syl simpr crp wa eqidd adantl peano2nn 2rp nnre nnnn0 nn0ge0d ge0p1rpd rpmulcld cr 2re 1red readdcld remulcld resubcld clt wbr 1lt2 rpdivcld cle fvmptd rpne0d divmuldivd @@ -734112,7 +734112,7 @@ approximated is nonnegative (this assumption is removed in a later nnmulcld nnne0d reccld mulne0d 2z expne0d eqtrd 3eqtrd nnrp ltaddrp2d mulgt1d posdifd mpbid elrpd rpge0d addge0d mulge0d gt0ne0d 3eqtr2d divrec2d peano2nnd 0le1 rpcnd cfz elfznn cn0 4nn0 nnexpcld nncnd subcld nnne0 breqtrrid lemul2ad - 0le2 nnge1 ltletrd gtned subne0d eqeltrd mulcl mul12d mulcomd mulassd mulid2d + 0le2 nnge1 ltletrd gtned subne0d eqeltrd mulcl mul12d mulcomd mulassd mullidd seqcl adddid addsubassd eqtr4d 2p2e4 mvlladdi 4z expsubd expclzd sqmuld nnind ex ) UAUDZCADEAUDZCFZUWKGHFZIFZUWKUWKGJFZIFZCFZUEZGUFZSZGEUWICFZGJFZIFZUWICAD UWKKLFZUWKUWLCFZELFZIFZUEZGUFZSZCFZMGUWRSZGEGCFZGJFZIFZGUXHSZCFZMUBUDZUWRSZGE @@ -734185,7 +734185,7 @@ approximated is nonnegative (this assumption is removed in a later ( cmul c2 co c4 cexp cmin cdiv cfv cfa wceq caddc oveq2 oveq2d oveq1d oveq12d c1 wcel a1i vx vy cv cn cmpt cseq fveq2 fveq2d eqeq12d cz seq1 ax-mp 1nn eqid 1z ovex fvmpt 2t1e2 oveq1i cdc 2exp4 cc0 1nn0 6nn0 0nn0 1t1e1 1p0e1 eqtri 6cn - c6 mulid1i dec0h decmul1c eqtr4i 2t2e4 sq1 numexp2x eqcomi oveq2i 4cn oveq12i + c6 mulridi dec0h decmul1c eqtr4i 2t2e4 sq1 numexp2x eqcomi oveq2i 4cn oveq12i 2nn0 fac1 3eqtri 2m1e1 fveq2i fac2 wa cuz elnnuz biimpi adantr seqp1 simpr cc syl eqidd adantl peano2nn 2cnd nncn 1cnd addcld mulcld cn0 4nn0 expcld subcld sqcld clt wbr 2pos gt0ne0d nnne0d mulne0d 1red cr 2re nnre readdcld 1lt2 nnrp @@ -734255,7 +734255,7 @@ approximated is nonnegative (this assumption is removed in a later sqcld wne 2ne0 nnne0d mulne0d 1red cr 2re remulcld clt wbr 1lt2 breqtrrdi nnred 2t1e2 cle 0le2 elfzle1 lemul2ad ltletrd gtned subne0d cz 2z expne0d divcld fvmptd eqeltrd adantl mulcl seqcl 2nn id nnmulcld peano2nnd div32d - mulid2d wallispi2lem2 3eqtrd mpteq2ia wallispi2lem1 3eqtr4ri wallispi ) D + mullidd wallispi2lem2 3eqtrd mpteq2ia wallispi2lem1 3eqtr4ri wallispi ) D ADFGDHZIJZXMKUAJZLJXMXMKUCJLJIJMZBXOUDAFKGAHZIJZKUCJZLJXPIDFXMNOJZXMXNIJZ GOJZLJZMZKUEPZIJZMAFGNXPIJOJXPUFPNOJIJXQUFPGOJLJZXRLJZMAFXPIXOKUEPZMBAFYE YGXPFQZYEKYDXRLJZIJYJYGYIKXRYDYIUGZYIXQKYIGXPYIUHXPUIUJYKUKZYIEUBIRYCKXPY @@ -734292,13 +734292,13 @@ approximated is nonnegative (this assumption is removed in a later cv cmpt wcel ax-1cn divcnv ax-mp eqbrtri caddc nnex mptex eqeltri cfv crp cc cr wceq wa simpr id nnrp rpreccld fvmptd nnrecre eqeltrd adantl oveq1d 2re nnre remulcld cle 0le2 rpge0d mulge0d ge0p1rpd 1red 0le1 readdcld clt - rpred nncn mulid2d 1lt2 ltmul1dd eqbrtrrd ltp1d lediv2ad 3brtr4d breqtrrd + rpred nncn mullidd 1lt2 ltmul1dd eqbrtrrd ltp1d lediv2ad 3brtr4d breqtrrd lttrd ltled climsqz2 1cnd recnd mulcld addcld rpne0d reccld subcld eqcomd - eqtrd climsubc2 1m0e1 breqtrdi halfcld sqcld adddid mul12d sqvald mulid1d + eqtrd climsubc2 1m0e1 breqtrdi halfcld sqcld adddid mul12d sqvald mulridd 2cnd cexp oveq12d wne 2ne0 divcan2d eqtr4d 3eqtrd cz 2z rpexpcld rpaddcld rphalfcld divmuldivd pncand dividd nnne0 divcld divne0d divcan5rd adddird divsubdird divcan6d div12d 1e2m1 oveq2i exp1d expm1d 3eqtr3a eqtr3d eqtri - 3eqtr2d mpteq2ia climmulc2 mptru halfcn mulid1i breqtri ) DJKLMZJNMZUUNUB + 3eqtr2d mpteq2ia climmulc2 mptru halfcn mulridi breqtri ) DJKLMZJNMZUUNUB DUUOUBUCOJUUNUABDJUDPUEOUFZOBJQRMJUBOQJUACBJUDPUEUUPOQUAECJUDPUEUUPEQUBUC OEAPJAUGZLMZUHZQUBIJUTUIUUSQUBUCUJJAUKULUMSCUDUIOCAPJKUUQNMZJUNMZLMZUHZUD HAPUVBUOUPUQSUAUGZPUIZUVDEURZVAUIOUVEUVFJUVDLMZVAUVEAUVDUURUVGPEUSEUUSVBU @@ -734381,8 +734381,8 @@ approximated is nonnegative (this assumption is removed in a later mpteq2ia mulassd addcld 0red nnred 2re nnre remulcld 1red mulexpd 3eqtr4d cr sqmuld cle wbr wa rprege0d resqrtth syl 2t2e4 expmuld 3eqtr3d divdiv1d 4ne0 readdcld nngt0d ltp1d lttrd div12d eqcomi sq2 dividi sqvald divcan4d - divassd 4cn nn0cnd mulid2d mul32d eqeltrrd divdiv2d dividd 3eqtr2d reccld - mul12d mulid1d recidd 3eqtri ) ECJKLCUBZMNZONZUVEUCUDZLONZMNZKUVEMNZUCUDZ + divassd 4cn nn0cnd mullidd mul32d eqeltrrd divdiv2d dividd 3eqtr2d reccld + mul12d mulridd recidd 3eqtri ) ECJKLCUBZMNZONZUVEUCUDZLONZMNZKUVEMNZUCUDZ KONZPNZUVKVAUENZPNZUGCJUVGUVEAUDZLONZUVEDUDZLONZMNZMNZUVEBUDZKONZUVKDUDZK ONZMNZPNZUVOPNZUGCJUVRUWDPNZUVEKONZUVEUVOMNPNZMNZUGICJUVPUWIUVEJQZUVNUWHU VOPUWNUVJUWBUVMUWGPUWNUVIUWAUVGMUWNUWAUVIUWNUWAUVHUVKUFUDZUVERPNZUVEONZMN @@ -734472,7 +734472,7 @@ approximated is nonnegative (this assumption is removed in a later cfa cmpt nffv 2fveq3 fvmptf mpdan cbvmpt eqtri wa simpr fveq2d fvmptd cn0 faccl 3syl cc simpl rpcnd adantr sqrtcld ere recni cc0 wne nnnn0d expne0d expcld mulne0d divassd divdiv32d divdiv1d eqcomd dividd div23d divmuldivd - sqrtmuld 3eqtr2d mulid2d expdivd expp1d div32d 3eqtr4d gtneii fveq2 oveq2 + sqrtmuld 3eqtr2d mullidd expdivd expp1d div32d 3eqtr4d gtneii fveq2 oveq2 0re epos nnzd oveq1 id peano2zd facp1 recdivd rpred rpge0d nnred sqrtdivd divdivdivd reccld addcld subcld ) EIJZEKUALZEMLZUBNZUVBEUCLZOLZPMLZQNZKRE OLZUALZRMLZUVBQNZOLZKUDLZEBNZUVABNZUDLZEDNZUUTUVGUVEQNZPQNZUDLUVMUUTUVEPU @@ -734578,9 +734578,9 @@ approximated is nonnegative (this assumption is removed in a later 3brtr4d cdif eldifi eldifn num0h eqeq2d rspcev mp2an ax-1cn ax-mp mpbir eleq1 crn 0nn0 nn1m1nn ord mpd nfcv nfmpt1 nfcxfr nfdif nfcri mtbid sylibr r19.21bi wo nfrn necomd adantlr simplr zred mulcomd elnn0z sylnib nan anabss1 rpregt0d - mpbi mulltgt0 addid2d nnge1 adantll mtand pm2.61dan neneqd ralrimi sylib nnzd + mpbi mulltgt0 addlidd nnge1 adantll mtand pm2.61dan neneqd ralrimi sylib nnzd ex odd2np1 mtbird notnotrd npcand nn0zd oddp1even oexpneg syl3anc 1exp negeqd - mulm1d addcomd negidd pncand 0le2 nn0ge0d mulge0d 0lt1 addgegt0d nn0z mulid2d + mulm1d addcomd negidd pncand 0le2 nn0ge0d mulge0d 0lt1 addgegt0d nn0z mullidd m1expeven 2timesd divrec2d 3eqtr2d eqtr2d 3eqtr4d isercoll2 resubcld ltsub1dd reccld subidd elrpd relogdivd ) APHUCUDZQCPRZUEUFZQCUGRZUEUFZUGRZVVOVVQUHRUEU FUIAVVNVVSUIUJPFQUDZVVSUIUJAVVTVVPVVRUKZPRVVSUIAVVPVWAUAPBQUDZPEQUDZVVTQULUOU @@ -734670,7 +734670,7 @@ approximated is nonnegative (this assumption is removed in a later crp 1rp 1cnd div1d 2rp rpmulcld ltaddrp2d eqbrtrd ltrec1d absltd stirlinglem5 nnrp mpbir2and 2cnd nncn mulcld addcld readdcld mulgt0d ltp1d gt0ne0d oveq12d 2pos dividd divdird divsubdird addassd wceq 1p1e2 oveq2d adddid eqtr4d 3eqtrd - mulid1d pncand eqtrd divcan7d divmuldivd divcld mulid2d fveq2d breqtrd ) CEFZ + mulridd pncand eqtrd divcan7d divmuldivd divcld mullidd fveq2d breqtrd ) CEFZ GBHUBIIJCKLZIGLZMLZGLZIXNNLZMLZUCUDCIGLZCMLZUCUDUEXKAEIUFZAUAZINLUGLXNYAUGLYA MLZKLZOZXNAAEYBOZAEYCYBGLOZAUHJYAKLIGLOZBYDUIYEUIYFUIDYGUIXKXMXKXLXKJCJUJFXKU KPZCULZUMZXKJCYHYIHJUNQXKUOPXKHCXKUPZYICUQZURUSUTZVAZXKXNVBUDIRQXTXNRQXNIRQXK @@ -734706,7 +734706,7 @@ approximated is nonnegative (this assumption is removed in a later ltned reccld nncn adantr nnre nngt0 gt0ne0d 2nn0 nn0mulcld 1nn0 nn0addcld ltled expcld fvmptd3 eqeltrd stirlinglem6 clim2ser eqbrtrd seq1 mp1i cmpt cr cz 0z mul01d eqcomd eqtrd breqtrd eqeltrrdi fvmptd eqtr4di div1d exp1d - 0cnd mulid2d divassd mulid1d 3eqtr2d 3eqtrd halfcld seqex elnnuz readdcld + 0cnd mullidd divassd mulridd 3eqtr2d 3eqtrd halfcld seqex elnnuz readdcld cuz lttrd syl ad2antrr 2ne0 nn0zd expne0d addcomd eqtr4d divcld id biimpi cfz elfzuz biimpri nnnn0 3syl nn0cnd elfznn nnrp rpmulcld ltaddrp2d addcl crp 2rp seqcl simprl simprr adddid div32d divcan3d mul12d exprecd divrecd @@ -734789,7 +734789,7 @@ approximated is nonnegative (this assumption is removed in a later wceq simpr ffvelcdmda rpcnd expcld fvmpt2 syl2anc climexp cmul adantr 2nn id nnmulcld adantl ffvelcdmd fmptdf fex 1nn 2cnd 1cnd mulcld oveq2 fvmptg eqid sylancr eqeltrd caddc cuz cle wbr nnred nnge1d leadd2dd mpdan oveq1d - 1red cbvmptv oveq2d peano2nn fvmptd nncn adddid mulid1d 3eqtrd 3brtr4d cz + 1red cbvmptv oveq2d peano2nn fvmptd nncn adddid mulridd 3eqtrd 3brtr4d cz wb nnzd peano2zd eluz mpbird eqcomd fveq2d eqtrd climsuse sqcld rpne0d 2z 2nn0 expne0d cc0 csn eqeltrrd rpexpcld neneqd 0cn elsn2g nn0zd 2cn eldifd ax-mp sylnibr rpdivcld oveq12d eqtr4d climdivf mvlladdi expsubd breqtrrd @@ -734864,8 +734864,8 @@ approximated is nonnegative (this assumption is removed in a later 2rp ltaddrp2d lttrd gt0ne0d cz expne0d reccld cabs cneg renegcld rereccld 2z resqcld wb 1re lt0neg2 ax-mp sylib sqgt0d recgt0d expgt1 syl3anc elrpd 2nn recgt1d mpbid cn0 wa cc weq oveq2d adantl adantr nnnn0d expcld eqcomd - wceq oveq1d eqtrd mulid1d 3eqtr2d oveq12d adddid eqtr4d 3eqtrd divmuldivd - wne mulne0d mulid2d breqtrd oveq2 syl fvmptd3 adantlr reexpcld seqcl 1exp + wceq oveq1d eqtrd mulridd 3eqtr2d oveq12d adddid eqtr4d 3eqtrd divmuldivd + wne mulne0d mullidd breqtrd oveq2 syl fvmptd3 adantlr reexpcld seqcl 1exp eqeltrd cle expdivd nn0ge0d mulge0d ge0p1rpd ltled rpexpcld mpbir2and cuz absltd 1nn0 simpr elnnuz biimpri fvmptd geolim2 dividd rpcnne0d divsubdir exp1d ax-1cn binom2 sylancl sqmuld sq2 mulassd 4cn sqvald sq1 pncand 4pos @@ -734946,7 +734946,7 @@ approximated is nonnegative (this assumption is removed in a later 0le1 ge0p1rpd rpreccld rpge0d cz 2z zmulcld rpexpcld rpgt0d cseq cuz eqid 1z peano2zd oveq2 adantl adantr nnnn0 stirlinglem9 clim2ser divge0d recnd nnuz wss uznnssnn mp2b sseld imdistani sseli 1p1e2 eluzle eqbrtrrid letrd - eluzelre expge0d breqtrrd iserge0 seq1 mp1i breqtrd addid2d subcld npcand + eluzelre expge0d breqtrrd iserge0 seq1 mp1i breqtrd addlidd subcld npcand leadd1dd 3brtr3d ltletrd posdifd mpbird ) FJKZFLMNZBOZFBOZUBUCPUVIUVHUDNZ UBUCUVFPLEOZUVJUVFUEZUVFUVKLQLRNZLMNZUFNZLQFRNZLMNZUFNZUVMUHNZRNZSUVFCLLQ CUGZRNZLMNZUFNZUVRUWBUHNZRNZUVTJEUIECJUWFUJUKZUVFITUVFUWALUKZVAZUWDUVOUWE @@ -735009,8 +735009,8 @@ approximated is nonnegative (this assumption is removed in a later cli climrel releldmi mp1i adantr eqcomd nnuz eqbrtrd reccld wtru nnre cdm nnne0 fvmptd seqeq1 trireciplem wn simpl wo elnn1uz2 sylib ord mpd npcand uz2m1nn seqeq1d clim2ser pm2.61dan isumrecl nnrpd rpge0d ge0p1rpd isumge0 - rpmulcld leadd2dd addid1d mulcld isumsplit 3brtr4d 1zzd isumclim breqtrdi - mptru lemul2ad 4cn gt0ne0d fsummulc2 mulid1d 3brtr3d letrd subled ) EJKZL + rpmulcld leadd2dd addridd mulcld isumsplit 3brtr4d 1zzd isumclim breqtrdi + mptru lemul2ad 4cn gt0ne0d fsummulc2 mulridd 3brtr3d letrd subled ) EJKZL BMZEBMZLUGUBNZUWCOKUWBUWCLAMZPMZOLJKZUWGOKZUWCUWGUCUDUWHUWFUEKUWIUDACLFUH UWFUIUJZCLCUKZAMPMZUWGJBOCLULZCUWFPCPULZCLACACJUWKUMMUNUWKQNUOMUWKUPUBNUW KUQNQNUBNZURFCJUWOUSUTZUWMVAVAUWKLPAVBGVCVDUWJVEVFUWBUWDEAMZPMZOUWBUWROKU @@ -735398,7 +735398,7 @@ approximated is nonnegative (this assumption is removed in a later dirkerper $p |- ( ( N e. NN /\ x e. RR ) -> ( ( D ` N ) ` ( x + T ) ) = ( ( D ` N ) ` x ) ) $= ( wcel caddc co c2 cpi cmul wceq cdiv csin cfv a1i cc cn cv cr wa cmo cc0 - c1 cif eqcomi oveq2i 2re pire remulcli eqeltri recni mulid2i eqtri oveq1i + c1 cif eqcomi oveq2i 2re pire remulcli eqeltri recni mullidi eqtri oveq1i crp cz id ad2antlr 2rp pirp rpmulcl mp2an 1z modcyc syl3anc simpr iftrued 3eqtrd iftrue adantl eqtr4d cneg iffalse nncn halfcn addcld adantr mulcld wn recn sincld halfcld dirkerdenne0 adantll div2negd mp3an23 eqtrd neqned @@ -735454,14 +735454,14 @@ approximated is nonnegative (this assumption is removed in a later ( c1 c2 cmul co cfz cpi ccos cfv csu cc0 wceq caddc wcel cz a1i cc mulcld picn vx vy oveq2 oveq2d sumeq1d eqeq1d ax-1cn 2timesi oveq2i sumeq1i cneg cv wtru cuz 1z uzid ax-mp wa elfzelz zcnd adantl coscld id 1p1e2 fvoveq1d - eqtrdi fsump1 mptru coscl oveq1 mulid2i fveq2d fsum1 mp2an cos2pi oveq12i + eqtrdi fsump1 mptru coscl oveq1 mullidi fveq2d fsum1 mp2an cos2pi oveq12i cospi neg1cn 1pneg1e0 addcomli 3eqtri cn 2cnd nncn adddid addassd 3eqtr4a eqtri adantr cle wbr 1red cr 2re nnre remulcld readdcld crp nnrp rpmulcld 2rp ltaddrp2d ltled wb 2z nnz zmulcld peano2zd sylancr mpbird fvoveq1 clt eluz 1lt2 2t1e2 nnge1 lemul2d mpbid eqbrtrrid ltletrd eqtr4i 3eqtr4d cdiv addcld mulassd oveq1d wne 0re pipos gtneii rpne0d divcan5d 3eqtrd eqeltrd divcan4d peano2cn syl coseq1 eqtrd oveq12d simpr adddird eqeltrid addcomd - mulcomd cosper addid2i oveq1i ex nnind ) CDUAULZEFZGFZAULZHEFZIJZAKZLMCDC + mulcomd cosper addlidi oveq1i ex nnind ) CDUAULZEFZGFZAULZHEFZIJZAKZLMCDC EFZGFZUUPAKZLMCDUBULZEFZGFZUUPAKZLMZCDUVACNFZEFZGFZUUPAKZLMZCDBEFZGFZUUPA KZLMUAUBBUUKCMZUUQUUTLUVNUUMUUSUUPAUVNUULUURCGUUKCDEUCUDUEUFUUKUVAMZUUQUV DLUVOUUMUVCUUPAUVOUULUVBCGUUKUVADEUCUDUEUFUUKUVFMZUUQUVILUVPUUMUVHUUPAUVP @@ -735508,7 +735508,7 @@ approximated is nonnegative (this assumption is removed in a later ( c1 c2 cdiv co cmul cfv caddc cpi csin cmin wcel oveq1d cc0 cfz ccos csu vj cv 1cnd halfcld fzfid wa cc elfzelz adantl adantr mulcld coscld fsumcl zcnd recnd addcld sincld divcan4d eqcomd fsummulc1 mulcomd wceq sinmulcos - syl2anc cneg adddird addcomd mulid2d 3eqtrrd fveq2d negsubdi2d subcld syl + syl2anc cneg adddird addcomd mullidd 3eqtrrd fveq2d negsubdi2d subcld syl sinneg eqtrd oveq12d eqeltrrd mulsubfacd oveq2d 3eqtrd sumeq2dv peano2cnm negsubd 2cnd wne 2ne0 a1i fsumdivc divrec2d eqtr3d adddid 3eqtr4d npncand fsumadd fvoveq1 nnzd cuz eleqtrdi peano2uz telfsum2 pncand fvoveq1d oveq1 @@ -735576,16 +735576,16 @@ approximated is nonnegative (this assumption is removed in a later 2 x. _pi ) x. ( sin ` ( A / 2 ) ) ) ) ) $= ( c2 co cc0 wceq c1 cdiv cmul caddc cpi wcel a1i adantr cmo cfz ccos csin cv cfv csu wa oveq2d cc elfzelz zcnd adantl 2cnd mulcld 1cnd picn mulcomd - addcld mulassd adddird addcomd mulid2d mul4d eqtrd oveq12d 3eqtrd zmulcld + addcld mulassd adddird addcomd mullidd mul4d eqtrd oveq12d 3eqtrd zmulcld fveq2d cz cosper syl2anc sumeq2dv oveq1d wne 2ne0 divcan2d eqcomd sumeq1d nncnd wral rgen sumeq2d cn clt wbr simpr cr crp wb nnred 2rp mod0 sylancl - mpbid 2re nngt0d 2pos divgt0d elnnz sylanbrc dirkertrigeqlem1 syl addid1i + mpbid 2re nngt0d 2pos divgt0d elnnz sylanbrc dirkertrigeqlem1 syl addridi halfcn eqtrdi ax-1cn 2cnne0 pipos gt0ne0ii pm3.2i divdiv1 oveq2i peano2cn - pire mp3an muladdd mul12d mulid1d eqtrid addassd sinper halfcld eqtr3d wn - 2cn rehalfcld neqned divdiv1d 3eqtrrd cneg cfl addid2i 1z oveq2 cospi cle + pire mp3an muladdd mul12d mulridd eqtrid addassd sinper halfcld eqtr3d wn + 2cn rehalfcld neqned divdiv1d 3eqtrrd cneg cfl addlidi 1z oveq2 cospi cle neg1cn pm2.61dan cmin recidi 3eqtr2d nnzd div23d divcan3d sincld divcan4d divdird rpreccld ltaddrpd halflt1 ltadd2dd btwnnz syl3anc eqneltrd sineq0 - zred mtbird mulne0d dividd oddfl fvoveq1 halffl 2t0e0 coscl ax-mp mulid1i + zred mtbird mulne0d dividd oddfl fvoveq1 halffl 2t0e0 coscl ax-mp mulridi 1red fsum1 mp2an cuz 2nn flcld 2div2e1 nnne1ge2 syl2an lediv1dd eqbrtrrid neqne elnnz1 nnmulcld eleqtrdi coscld fsump1 sumeq2i adddid mul13d eqtr4d flge nnuz sinppi divnegd negeqd negcld negsubdi2i 1mhlfehlf negeqi divneg @@ -735682,7 +735682,7 @@ approximated is nonnegative (this assumption is removed in a later wral elfzelz zcnd recn 2cn mulcli mulne0i divassd simpr crp wb simpl pirp 2rp rpmulcl mp2an mod0 sylancl mpbid adantr zmulcld eqeltrd coseq1 mpbird adantlr ralrimiva adantll sumeq2d cfn fzfid 1cnd fsumconst syl2anc nnnn0d - cn0 hashfz1 oveq1d mulid1d eqtrd 3eqtrd oveq2d ax-1ne0 divadddivd addcomd + cn0 hashfz1 oveq1d mulridd eqtrd 3eqtrd oveq2d ax-1ne0 divadddivd addcomd div1d mulcomd oveq12d 3eqtr4rd iffalse cfl divcan1d rpreccl ax-mp mp3an13 wn moddi divrec2d reccld mulassd eqtr2d recnd adantlll fvoveq1d pm2.61dan fveq2d recidi oveq2i eqtrid id modcld divne0i neqne mulne0d eqnetrd oddfl @@ -735831,13 +735831,13 @@ approximated is nonnegative (this assumption is removed in a later pire ltsubpos syl2anc eqbrtrid ltaddpos breqtrrdi eliood wceq cmul eldifi mpbii cz elioored adantl recnd 2cnd cc picn 2ne0 gt0ne0ii divdiv1d wn cmo crp wb 2rp pirp rpmulcld mod0 mpbid peano2zm syl ad2antrr zred rerpdivcld - rpred rpne0d redivcld dividd eqcomd oveq2d divsubdird eqtr4d mulid2i 1lt2 + rpred rpne0d redivcld dividd eqcomd oveq2d divsubdird eqtr4d mullidi 1lt2 eqcomi 1re 2re ltmul1ii mpbi eqbrtri ltsub2dd ltdiv1dd eqbrtrd w3a elioo2 simp2d lttrd ad2antlr simpr divcld 1cnd npcand breqtrd btwnnz syl3anc cle eldifsni necomd 1red oveq1i divdird eqneltrd halfcld mtbird neqned oveq1d eqtrd 3eqtrd ltadd1dd jca leneltd stoic1a ltnled mpbird simp3d 2cn oveq2i mulcomi pm3.2i 2cnne0 divdiv1 mp3an dividi 3eqtr2i ltadd2dd sineq0 eqtr2d - halflt1 pm2.61dan mulid1d mulcomd divcan5d rehalfcld 2halvesd ralrimiva + halflt1 pm2.61dan mulridd mulcomd divcan5d rehalfcld 2halvesd ralrimiva oveq12d addassd coseq0 ) AECDUAJZKBUBZLMJZUCUDZUEUGZUVKUHUDZUEUGZUFZBUVIE UIZUJZUKACDEACENULJZUMFAUVSAENHNUNKZAUTOZUOZUPUQZADENPJZUMGAUWDAENHUWAURZ UPUQZHACUVSEQFAUVTEUNKZUVSEQRZUWAHUVTUWGUFZUENQRZUWHUSNEVAVJVBVCAEUWDDQAU @@ -735919,7 +735919,7 @@ approximated is nonnegative (this assumption is removed in a later pire crn ctg iooretop cnt nncnd pm3.2i ccnfld ctopn tgioo2 dvres syl21anc retop ccld cha rehaus elioored uniretop difopn isopn3i reseq2d reelprrecn sncld dvsinax sseqtrrid dvres3 syl22anc eqtrdi eqimss oveq2i mulcli sseli - cpr 2cn 1cnd div13d halfcl mulid1d dvasinbx mul32d recidd mulid2d resmptd + cpr 2cn 1cnd div13d halfcl mulridd dvasinbx mul32d recidd mullidd resmptd ccom wfn ralrimiva fnmpt eldifi eleq1w anbi2d neeq1d imbi12d elioore 3syl oveq1 0re pipos gtneii sineq0 mtbid eqneltrd crp 2rp rpmulcl mp2an neqned pirp mod0 chvarvv simpll ad2antlr 0red 1red nngt0d rpreccld lttrd gt0ne0d @@ -736350,7 +736350,7 @@ approximated is nonnegative (this assumption is removed in a later fourierdlem4 $p |- ( ph -> E : RR --> ( A (,] B ) ) $= ( cr cmin co caddc wcel clt wbr adantr cc0 cv cdiv cfl cmul cioc wa simpr cfv cle resubcld eqeltrid wne wceq a1i recnd gtned subne0d redivcld flcld - eqnetrd zred remulcld readdcld addid1d eqcomd subcld subidd oveq2d nncand + eqnetrd zred remulcld readdcld addridd eqcomd subcld subidd oveq2d nncand c1 addsub12d addcomd eqtrd 3eqtrd oveq1d cc addsubd divdird dividd fveq2d eqeltrrd peano2re syl reflcl crp posdifd mpbid breqtrrd elrpd flltp1 1zzd cz fladdz syl2anc ltmul1dd ltadd2dd divcan1d pncan3d 3brtr3d flle lemul1d @@ -737210,7 +737210,7 @@ approximated is nonnegative (this assumption is removed in a later 1cnd 3eqtr3d 3eqtrd elrpd addcomd eleqtrd simp3d lesubaddd mpbird subge0d subsub2d breqtrd divge0d divdird dividd simp2d sublt0d divlt0gt0d 1red cz ltaddneg zcnd mulcld pncan2d divcan4d id oveq2 adantl reflcl syl eqeltrrd - zsubcld flbi2 mpbir2and subdird addcld addsubassd mulid2d eqeltrd addsubd + zsubcld flbi2 mpbir2and subdird addcld addsubassd mullidd eqeltrd addsubd 1zzd add32d nncand ) AHFUAHDHQRZEUBRZUCUAZEUDRZSRZHDGQRZEUBRZUCUAZUEQRZEU DRZSRZCHGQRZSRZABHBUFZDUULQRZEUBRZUCUAZEUDRZSRZUUCUGFUGFBUGUUQUHUIAMUJZAU ULHUIZUKZUULHUUPUUBSAUUSUMZUUTUUOUUAEUDUUTUUNYTUCUUTUUMYSEUBUUTUULHDQUVAU @@ -737283,7 +737283,7 @@ approximated is nonnegative (this assumption is removed in a later syl3anc iooltub eliood 1red wf ffvelcdmd recnd ffvelcdmda cc0 0red ccnfld ctopn crest crn iooretop eqid tgioo2 eleqtri dvmptconst dvmptidg dvmptadd ctg wceq 0p1e1 mpteq2dva eqtrd feqmptd reseq1d wss ioossre resmptd eqtr2d - cres oveq2d eqcomi 3eqtrd fveq2 dvmptco mulid1d ) AOGBCUAPZFGUIZUBPZEUCZQ + cres oveq2d eqcomi 3eqtrd fveq2 dvmptco mulridd ) AOGBCUAPZFGUIZUBPZEUCZQ UDPGXAXCDUCZRUEPZQGXAXEQAGNXCRNUIZEUCZXGDUCZOOXDXEOOXAFBUBPZFCUBPZUAPZOOU JUFSAUGUHZXMAXBXASZUKZXJXKXCAXJULSXNAXJAFBIJUMUNTAXKULSXNAXKAFCIKUMUNTXOF XBAFOSXNITZXNXBOSAXBBCUOUPZUMXOBXBFABOSXNJTZXQXPXOBULSZCULSZXNBXBUQURXOBX @@ -737336,7 +737336,7 @@ approximated is nonnegative (this assumption is removed in a later wcel rpne0d redivcld cle absge0d breqtrrdi addge0d divge0d addge02d mpbid ltletrd gt0ne0d divnegd oveq2d divassd eqtr4d oveq12d divsubdird itgmulc2 letrd reccld divrec2d adantr wne 3eqtr2d itgeq2dv 3eqtr4rd subcld absdivd - fveq2d ltled absidd 3eqtrd elrpd absmuld absnegd eqbrtrd lemul2ad mulid1d + fveq2d ltled absidd 3eqtrd elrpd absmuld absnegd eqbrtrd lemul2ad mulridd abs2dif2d breqtrd le2addd leadd1dd lediv1dd ltp1d lelttrd ge0p1rpd eqcomi oveq12i lediv2ad oveq1i wceq oveq1 adantl addcld rpcnd div0d eqtrd oveq1d 0p1e1 eqtrdi crp ax-1ne0 eqnetrd neqne ne0gt0d rpdivcld rpaddcld ltdiv23d @@ -737755,7 +737755,7 @@ approximated is nonnegative (this assumption is removed in a later wbr wrex nfra1 rspa ex ralrimi reximdva mpd simpr elioore adantl remulcld resincld mulcld ralrimiva dmmptg eleqtrd ad4ant14 simplr adantlr c1 eqidd syl2anc fveq2 oveq2 fveq2d oveq12d sselid sincld fvmptd absmuld ad3antrrr - eqtrd abscld 1red simpllr absge0d eqbrtrd mulid1d cnbdibl redivcld coscld + eqtrd abscld 1red simpllr absge0d eqbrtrd mulridd cnbdibl redivcld coscld ffvelcdmda divcld negcld oveq1d negeqd oveq2d 3eqtrd renegcld mpteq2dva cdv wne rpcnne0d eldifsn sylibr difssd sincn ioosscn ioombl cmin resubcld volioo ioossicc sselda cncfmptssg cniccbdd sseli sylan2 nfv nfan adantllr @@ -737938,7 +737938,7 @@ approximated is nonnegative (this assumption is removed in a later leidd sseld mulneg1d negsubd simpl1 ltsubaddd iocleub eliood sseldd negex znegcld 3anbi3d 3anbi2d ralrimiva dfss3 sylibr anbi12d syl12anc nnzd 0le1 breq1 nnge1d oveq2i 3eqtr4rd nnncan2d nngt0d fzolb 0re vtoclg ax-mp mpdan - pncan3d fveq2i negeqd mulid2d adddird negsubdid 3eqtr4d eqsstrd sylan9req + pncan3d fveq2i negeqd mullidd adddird negsubdid 3eqtr4d eqsstrd sylan9req 0p1e1 negcld addsubassd subsub23d ltaddsubd peano2zd breq2 cico 3ad2antl3 subaddd 3eqtrd ad2antlr pm2.65da neqned leneltd elfzfzo sylanbrc 3adant1r simp2 neneq simp1l 3ad2ant3 chvarvv elicod ioossico ex icoltub icogelb cc @@ -738125,7 +738125,7 @@ approximated is nonnegative (this assumption is removed in a later wor neneqd ideqg mtbird eldifd fveq2d 3eqtrrd fnfvelrn resss rneqi rncoss rnss sstri eqsstri fiinfcl syl13anc lptre2pt simpll 3jca rexbidv cbvrexvw bitrdi elrab reeanv simplll simpl1 simpl2 2rexbidv anbi12d simprbi biimpi - 3anbi13d iccsuble breqtrrdi ltnled remulcld simp2d simp1d mulid2d zltlem1 + 3anbi13d iccsuble breqtrrdi ltnled remulcld simp2d simp1d mullidd zltlem1 zre readdcld peano2rem resubcl zcn pncan3d npncand 3brtr3d syldan posdifd leadd1dd 1cnd lemul1d simp3d ltaddrp2d addsub4d ad2antll ad2antrl subdird eqtr2d lelttrd 3adantl3 condan sselid jca simp2l 0re fvelrnb rpge0d simp3 @@ -738133,10 +738133,10 @@ approximated is nonnegative (this assumption is removed in a later simp1 fveq2i syld3an3 ltled abssubge0d eqeq1d elex bibi12d imbi2d infrelb eqeq2 eqbrtrid pncan2d addsubd addcomd divdird 3eqtr2d sublt0d divlt0gt0d divcan4d subidd redivcld ltaddsub2d zltp1le leaddsub2d 3adantll3 3adantl2 - cvv letrd mulcld addid2d ad3antrrr 3ad2antl1 simpl3 wo lenltd eqcom ioran + cvv letrd mulcld addlidd ad3antrrr 3ad2antl1 simpl3 wo lenltd eqcom ioran notbii leloed 3adantll2 adantllr simp2r subge0d mulge0d pm2.61dan addassd addge01d subadd4b adddird oveq12d 00id 3eqtr3d 3adant3l 3adant2l 3adant3r - mul02d addid1d 3adant2r ltsub1dd ltdiv1dd lttrd 3ad2antl2 1zzd lemulge12d + mul02d addridd 3adant2r ltsub1dd ltdiv1dd lttrd 3ad2antl2 1zzd lemulge12d zsubcld lesubd sub32d lesub2dd subsub2d simpll2 letri3d mpbir2and simpl2l simpl3l 3ad2antl3 elicc2 pncand oveq2i eqtr2id mulsubfacd sylan2b leneltd nne subsub3d sylan syl121anc adantlrr npcand mpdan simpl3r neqne lesub1dd @@ -738625,7 +738625,7 @@ approximated is nonnegative (this assumption is removed in a later sylancl nn0cnd recnd addcomd eqtr2d breqtrd elnnz sylanbrc peano2nnd cmpt ltletrd cibl elioore sylan2 adantlr simpll simpr ad2antlr syl21anc adantr crp adantl eqcomi oveq12i oveq1i negcld mulcld itgcl eqeltrrid wne itgabs - eqid eqbrtrd cxr syldan absmuld recn absnegd lemul2ad mulid1d itgle letrd + eqid eqbrtrd cxr syldan absmuld recn absnegd lemul2ad mulridd itgle letrd leadd2dd lediv1dd flltp1 breqtrrd lelttrd rexrd pnfxr ioogtlb lttrd ltled ltadd1dd syl3anc fourierdlem30 ralrimiva oveq1 raleqdv rspcev ) ALUMUNCDP UOZUPUQURUSUQEFUVEUPUQURUSUQUTUQBKIJUVEUPUQURUSUQVAUTUQVBVCHVDVIZPLVEVFUQ @@ -739354,7 +739354,7 @@ approximated is nonnegative (this assumption is removed in a later cfn cioc wss wf wi wral fvres simprbi nfv simpl cxr iocssre adantr elrabi wa rexrd simpr wne ad2antrr syl3anc eqeltrd eqcomd adantlr simpllr mpbird ad2antlr ad3antrrr eleq1 anbi2d anbi1d anbi12d imbi12d simplr syl21anc cc - cle 3eqtrd wn sselda ralrimiva crn cpr 0red addid2d eqbrtrrd eliccd lttrd + cle 3eqtrd wn sselda ralrimiva crn cpr 0red addlidd eqbrtrrd eliccd lttrd posdifd eqcomi gt0ne0d oveq2 fveq2d oveq12d fvmptd rspcev chash wiso prfi cfz wfo csn snfi cres wf1 nfre1 nfcv nfrabw nfan sseldd w3a fvmpt2 lbicc2 nfcri ad4ant14 jca31 iocssicc fourierdlem4 ffvelcdmda iocgtlb 3brtr3d zre @@ -739664,7 +739664,7 @@ approximated is nonnegative (this assumption is removed in a later cpr 2z expne0d redivcld 1cnd crn ctg ccnfld recn dvmptid wss ioossre eqid ctopn tgioo2 iooretop dvmptres wn elsni necon3ai syl eldifd coscld mulcld cnelprrecn sinf ffvelcdmda cosf dvmptdivc wfn ffn ax-mp dffn5 mpbi eqcomi - oveq2i dvsin 3eqtri fveq2 dvmptco dvmptdiv mulid2d divrecd eqcomd oveq12d + oveq2i dvsin 3eqtri fveq2 dvmptco dvmptdiv mullidd divrecd eqcomd oveq12d wf oveq1d mpteq2ia ) AIEBCUBJZEUCZDKZUDZUEJIEYSYTYTLMJZNKZMJZLMJZUDZUEJEY SOUUDUFJZUUCPKZOLMJZUFJZYTUFJZUGJZUUDLUHJZMJZLMJZUDZEYSUUDUUILMJZYTUFJZUG JZUUNMJZLMJZUDZAUUBUUGIUEAEYSUUAUUFAYTYSQZVAZUUAYTRUIZOYTLUUDUFJZMJZUJZUV @@ -739718,7 +739718,7 @@ approximated is nonnegative (this assumption is removed in a later wss mp4an cdm 2cn sseqtrri reseq1i 3eqtri iooretop tgioo2 eleqtri subid1d dvmptconst dvmptsub mpteq2dva csn eldifsn sylanbrc divrec2d eqcomd coscld cdif recn id 3syl eqeltrrd cnt eqcomi dvres eqtr2i ioontr reseq12i dmmptg - ioossre mulcli dvasinbx mp2an dmeqi dvres3 resabs1 ioosscn recidi mulid2d + ioossre mulcli dvasinbx mp2an dmeqi dvres3 resabs1 ioosscn recidi mullidd mprg oveq1i 3eqtrd eqtri dvmptdiv feq1d mpbird jca pm3.2i ) ABCUARZSSFUBR ZUCZUWFHUWEGHUDZUERZSEGBUERZGCUERZUARZUFUBRZUGZUHUWHUHUIRZUJUGZUKRZUKRZUW OULUGZUWIEUGZDUMRZUKRZUMRZUWQUHUNRZUIRZUOZUPZUQURSHUWEUWQUOZUBRZHUWEUWSUO @@ -739775,7 +739775,7 @@ approximated is nonnegative (this assumption is removed in a later dvmptidg cres cnt resmptd eqcomd recn fmpti dvres syl22anc retop uniretop ctop isopn3 mpbid reseq2d resmpt ax-mp fveq2d mpteq2ia halfcn eqtrd eqtri id 2cn reseq1i 3eqtrd divrec2d eqtr2i oveq2i dvasinbx mp2an recidi halfcl - oveq12d coscld mulid2d dmeqi dmmptg sseqtrri dvres3 mp4an 3eqtri resabs1d + oveq12d coscld mullidd dmeqi dmmptg sseqtrri dvres3 mp4an 3eqtri resabs1d mprg coscn idcncfg eldifsn sylanbrc divcncf cncfmpt1f dvdivcncf cncff fdm difssd 3syl feq2d cncfcdm mpbird ) AICJKZBIUAKLZBIUVMUFZAUVMUBZIUVMUFZUVO ABICUFBIUCZUVQADBDUGZMUVSMUDKZUENZOKZUDKZICAUVSBLZUHZUVSUWBUWEUIUJZUIUKKZ @@ -739874,14 +739874,14 @@ approximated is nonnegative (this assumption is removed in a later fourierdlem60 $p |- ( ph -> E e. ( H limCC 0 ) ) $= ( vx cmin co cc0 cioo cfv cdiv cmpt climc resubcld rexrd 0red clt sublt0d cv wbr mpbird caddc cr wcel wa wf adantr cxr elioore adantl readdcld wceq - recnd pncan3d eqcomd 0xr simpr ioogtlbd ltadd2dd eqbrtrd iooltubd addid1d + recnd pncan3d eqcomd 0xr simpr ioogtlbd ltadd2dd eqbrtrd iooltubd addridd a1i breqtrd eliood ffvelcdmd cc wss ioossre ax-resscn sstrdi ccnfld ctopn eqid lptioo2cn limcrecl fmptd cdv cdm oveq2i dmeqd cpr reelprrecn syl2anc dvfre feq1d eqtr2d eleqtrd c1 1red ffvelcdmda crn dvmptc dvmptres feqmptd tgioo4 oveq2d 3eqtrd fveq2 mpteq2dva eqtrd wral ralrimiva syl wne adantrr dmmptg constlimc idlimc oveq1d wn simplrr condan limcco rneqd neleqtrd wb eqtr4di fveq1d fvmpt4 oveq12d fvmpt2 cmul feq2d ctg iooretop recn dvmptid - dvmptadd mpteq2i eqtrdi dvmptco mulid1d dvmptsub subid1d addlimc eqeltrrd + dvmptadd mpteq2i eqtrdi dvmptco mulridd dvmptsub subid1d addlimc eqeltrrd 0p1e1 limccl sselid ltned neneqd sublimc subidd eqcomi oveq1i 3eltr3d cid ubioo cres mptresid rnresi eqtr2di 0ne1 neii elsng mtbiri c0 ioon0 rnmptc csn div1d eqtrid eqtr3d lhop2 ) AEKBCUDUEZUFUGUEZKUQZIUHZUWFDUHZUIUEZUJZU @@ -739953,14 +739953,14 @@ approximated is nonnegative (this assumption is removed in a later fourierdlem61 $p |- ( ph -> E e. ( H limCC 0 ) ) $= ( vx cc0 cmin co cioo cfv cdiv cmpt climc 0red resubcld rexrd clt posdifd cv wbr mpbid caddc cr wcel wa wf adantr cxr elioore adantl readdcld recnd - addid1d eqcomd 0xr a1i ioogtlbd ltadd2dd eqbrtrd iooltubd pncan3d breqtrd + addridd eqcomd 0xr a1i ioogtlbd ltadd2dd eqbrtrd iooltubd pncan3d breqtrd wceq simpr eliood ffvelcdmd cc ioossre ax-resscn sstrdi ccnfld ctopn eqid wss lptioo1cn limcrecl fmptd cdv cdm oveq2i dmeqd reelprrecn dvfre mpbird cpr syl2anc eqtr2d eleqtrd c1 1red ffvelcdmda crn dvmptc dvmptres feqmptd tgioo4 oveq2d 3eqtrd fveq2 mpteq2dva eqtrd wral ralrimiva syl wne adantrr dmmptg constlimc idlimc oveq1d wn simplrr condan limcco rneqd neleqtrd wb eqtr4di fveq1d fvmpt4 oveq12d fvmpt2 feq1d cmul ctg iooretop recn dvmptid - feq2d dvmptadd 0p1e1 mpteq2i eqtrdi dvmptco mulid1d limccl sselid subid1d + feq2d dvmptadd 0p1e1 mpteq2i eqtrdi dvmptco mulridd limccl sselid subid1d dvmptsub addlimc eqeltrd gtned neneqd sublimc subidd eqcomi 3eltr3d lbioo oveq1i cid cres mptresid rnresi eqtr2di csn 0ne1 neii elsng mtbiri rnmptc c0 ioon0 div1d eqtrid eqtr3d lhop1 ) AEKUDCBUEUFZUGUFZKUQZIUHZUWGDUHZUIUF @@ -740030,7 +740030,7 @@ approximated is nonnegative (this assumption is removed in a later cicc ctopn ccn fss csn elioore iooretop negpilt0 pipos cxr renegcli rexri w3a elioo2 mpbir3an 1ex dmmpti cpr reelprrecn 1red dvmptid tgioo2 sncldre ccld toponunii difopn dvmptres eqtr3i eqimssi fvex divrec2d mulcli dmmptd - dvcnre reseq1i recidi eqcomd mulid2d eqtrd idcncfg cnlimc eleq12d rspccva + dvcnre reseq1i recidi eqcomd mullidd eqtrd idcncfg cnlimc eleq12d rspccva sseqtrri dvasinbx sincn divccncf cncfmpt1f div0i sin0 2t0e0 wrex ioossicc 3eltr3i eldifsni fourierdlem44 eqnetrd nrex fnmpt mprg fvelimab cre rered picn divneg crp 2rp ioogtlb eqbrtrid iooltub eliood cosne0 fnmpti imaeq1i @@ -740332,7 +740332,7 @@ approximated is nonnegative (this assumption is removed in a later reximdva rabn0 sselda simprbi leaddsub2d lemuldivd sylanl2 syl21anc breq2 ralbidv suprzcl sylib ne0i nnred elfzoelz fzofzp1 elioore ioogtlb lelttrd elfzolt2 zssre peano2zd 1cnd npcand sylan9eqr fveq2d eqcomi addcomd eqtrd - cc addassd mulid2d adddird 3eqtr4d eqtr2d eqbrtrd syl31anc ltp1d peano2re + cc addassd mullidd adddird 3eqtr4d eqtr2d eqbrtrd syl31anc ltp1d peano2re suprub peano2rem elfzoel2 zltlem1 necomd leneltd ltadd1dd breqtrd elfzfzo neqne 1red anim1i pm2.61dan lbicc2 ubicc2 jca prssg letrd simpll ad4ant14 iccleub iccgelb eliccd cneg znegcld mulneg1d negsubd wfn fnfvelrn rexbidv @@ -740488,7 +740488,7 @@ approximated is nonnegative (this assumption is removed in a later ioossre sselid ioogtlb eqcomi cbvrabv uneq2i fourierdlem54 elmapi elfzofz pnfxr fzofzp1 elfzoelz ltp1d wfo isof1o f1ofo elicc2 crp posdifd ltaddrpd wf1o elrpd lelttrd ltnled eliccd oveq2 remulcld pncan2d divcan4d peano2zm - flcld rpne0d 1cnd subdird mulid2d mulcld ppncand flid 3eqtrrd wfn ffn cuz + flcld rpne0d 1cnd subdird mullidd mulcld ppncand flid 3eqtrrd wfn ffn cuz nnnn0d nn0uz eleqtrdi eluzfz2 fnfvelrn rspcev elrab sylanbrc elun2 foelrn cn0 eqcom rexbii sylib biimpri eqbrtrrd simplll elfzelz ad2antlr ad3antlr adantrr adantrl btwnnz pm2.65da nrexdv condan elioc2 fourierdlem26 subcld @@ -741169,7 +741169,7 @@ approximated is nonnegative (this assumption is removed in a later ltled sselda rsp sylc ralrimi reximdva cnbdibl coscn ioosscn ssid idcncfg constcncfg cncfmpt1f negcncfg fveq1d sylan9eq reximdv eqidd eleq1w anbi2d eqeq12d imbi12d chvarvv sylan9eqr oveq2 oveq12d ffvelcdmda eqeltrrd simpl - negeqd negcld fvmptd simpllr 1red absge0d absnegd absmuld mulid1d 3brtr4d + negeqd negcld fvmptd simpllr 1red absge0d absnegd absmuld mulridd 3brtr4d abscosbd lemul2ad ralimdaa breq1d cbvralvw r19.21bi dmmptg raleqdv ancoms sylib biimpri breqtrd iftrue lbicc2 rpdivcld cbvitgv oveq2i oveq1i fveq2i eqcom leidd nnrpd fourierdlem47 simplll 0red nnre nngt0 pnfxr lttrd elrpd @@ -742689,7 +742689,7 @@ approximated is nonnegative (this assumption is removed in a later sumeq1d fveq2 fourierdlem22 imp vtocl rehalfcld fzfid simpl fourierdlem16 0nn0 cres chvarvv wne 0re pipos gtneii elfznn nnnn0d nn0red fourierdlem21 fmptd resincld readdcld fsumrecl cneg cioo ioosscn eleqtrdi mul02d eqtrdi - cos0 ioossre mulid1d feqmptd reseq1d cicc renegcld ioossicc eqsstri sseli + cos0 ioossre mulridd feqmptd reseq1d cicc renegcld ioossicc eqsstri sseli id eliccre ssriv resmpt mp1i 2cnd divdiv32d divrecd reccld fvmpt2 divassd 2ne0 nn0re ad2antlr cof cvol ioombl eqeltri coscn sstri idcncfg cncfmpt1f ssid mulcncf 1re abscosbd mul12d eqtr3d nnre adantll sincn abssinbd sylan @@ -743134,7 +743134,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u recnd abscld ralrimiva mpd 3rp rpdivcld eqeltrid ad2antrr sylbi cmpt cibl c3 itgrecl rpred cxr cmnf cc0 syl3anc redivcld lelttrd wb itgeq2dv breq1d cpnf adantlr fourierdlem5 ad2antlr halfre readdcld renegcli iccssre sseli - nnre resincld absmuld rpre ad4antlr 1red absge0d abssinbd mulid1d breqtrd + nnre resincld absmuld rpre ad4antlr 1red absge0d abssinbd mulridd breqtrd letrd reximdva w3a id 3adant3 nf3an simpl1l fourierdlem67 simplrl feqmptd sselda simprbi iblss itgcl iblabs simpl1r rehalfcld itgabs simpl2 iccssxr eqeltrrd cc volf sselid iccvolcl mnfxr 0xr mnflt0 volge0 xrltletrd iccmbl @@ -743250,7 +743250,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u cmpt climc rexri fourierdlem8 feqresmpt eleqtrd cnlimci mullimc mpteq2dva limcresi wral cn wb fourierdlem2 mpbid simpld elmapi fourierdlem14 ccnfld frn 3syl ctopn pnfxr ltpnfd lptioo1cn mnfxr mnfltd lptioo2cn fourierdlem5 - fourierdlem55 cncfss mp2an elfzofz fzofzp1 fourierdlem12 addid2d iccssred + fourierdlem55 cncfss mp2an elfzofz fzofzp1 fourierdlem12 addlidd iccssred ssid fveq2 cbvmptv eqtri fveq2d fvmptd oveq12d eleq12d eliooshift reseq2d 0red 3eltr4d fourierdlem78 renegcli simplr ioossicc fourierdlem9 ad2antrr sseli fourierdlem43 fourierdlem18 fssd fourierdlem75 sstrid fourierdlem62 @@ -743776,7 +743776,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u S. ( A [,] B ) ( F ` x ) _d x ) $= ( vj vy vw vz cc0 clt wbr caddc co cicc cv cfv citg wceq wa cfz cmpt cmin cif cr wcel adantr cn simpr elrpd adantlr fveq2 oveq1d cbvmptv cc wf cfzo - cioo cres ccncf climc eqeq1 ifbieq2d eqid fourierdlem81 wn oveq2d addid1d + cioo cres ccncf climc eqeq1 ifbieq2d eqid fourierdlem81 wn oveq2d addridd recnd eqtrd oveq12d itgeq1d simpll simplr syl mpbird mpbid negsubd pncand readdcld eqcomd wral cmap crab renegcld wb fourierdlem2 simpld cvv adantl a1i cle ffvelcdmd fvmptd eleqtrdi jca syl2anc syl3anc eqtr2d cxr leadd1dd @@ -744215,7 +744215,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u difss2d fourierdlem67 ffvelcdmda cvol cdm wral cmap cbvmptv fourierdlem88 crab fveq2 itgrecl pipos gt0ne0ii redivcld fvmpt2 2cnd 2ne0 divrecd eqtrd crn cdif fourierdlem66 difss renegcli iccssre mp2an sstri sselid readdcld - ffvelcdmd ifcld resubcld dirkerre div23d divrec2d 3eqtr3rd dividi mulid2d + ffvelcdmd ifcld resubcld dirkerre div23d divrec2d 3eqtr3rd dividi mullidd 3eqtrrd mpteq2dva reccld eqeltrd divcan3d sylan2 adantll ax-resscn cncfss itgcl sseli sylancl dirkerf dirkercncf cncfmptssg sseldd cniccibl syl3anc adantl ad2antrr itgadd subdird npcand eqtr3d ) ALVPZVQVRZVSZUXOMVTZUAWAWB @@ -745080,14 +745080,14 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u fourierdlem74 fourierdlem70 ralbii 3anbi3i fourierdlem87 halfpos lt0neg2d anbi1i iftrue negeqd ltnegd rpgt0 iffalse 3pm3.2i ltnegi divgt0ii lt0neg2 pm3.2i elioo3g mpbir2an pm2.61dan 3ad2ant2 lenegd rpge0 iftrued breqtrrid - 2pos breqtrrd ifcld le0neg2d volioo 0cn subnegd addid2d min1 sseq1 itgeq1 + 2pos breqtrrd ifcld le0neg2d volioo 0cn subnegd addlidd min1 sseq1 itgeq1 anbi12d rspcva 3adant1 itgeq1d rspcev rexlimdv3a nnex mptex syl2an clim0c r19.29a eqeltri climconst gtneii fourierdlem67 feqmptd csn eldifsn divcld picn cdif climdivf div0i limccl eleq2i velsn sylnibr eldifd ssriv lensymd dirkeritg ubicc2 mp3an elfzelz zcnd mul01d sin0 elfzle1 div0d sumeq2dv wo olci sumz 00id fvmpt lbicc2 sumeq2sdv cz mulneg12 negcld divcan4d znegcld - negpicn sineq0 sumeq2i eqtri oveq2i divcli addid1i divneg recni divdiv32i - csu dividi negeqi 3eqtri subnegi addid2i fourierdlem95 dirkerf dirkercncf + negpicn sineq0 sumeq2i eqtri oveq2i divcli addridi divneg recni divdiv32i + csu dividi negeqi 3eqtri subnegi addlidi fourierdlem95 dirkerf dirkercncf halfcn fourierdlem84 itgrecl climadd ) AUPWMUMWNWOWPZWQWPYWTUUIAWMYWTUAUB TWRYWTWSZUPWTXAWTUUJXBZYXBXCZAUUKZAUBWMXDWOWPZWMUUIAWMXDUAUAWRUQXDUULZWMX EWPZUQXIZUDXBZUUMZWSZUAWRXDWSZUBWTXAWRAUAUUPUAWRYXJUUQUAWRXDUUQUAUBUAWRYX @@ -745620,9 +745620,9 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u nnex clim0c eqeltri recni climconst gtneii fourierdlem67 feqmptd csn cdif mpan2 eldifsn divcld climdivf div0i limccl eleq2i velsn sylnibr ssriv csu eldifd dirkeritg ubicc2 mp3an cz elfzelz divcan4d sineq0 elfzle1 sumeq2dv - zcnd div0d wo olci sumz fvmpt lbicc2 2cn sin0 00id c0ex addid1i divdiv32i + zcnd div0d wo olci sumz fvmpt lbicc2 2cn sin0 00id c0ex addridi divdiv32i mul01d dividi 3eqtri subid1i eqtri fourierdlem95 dirkercncf fourierdlem84 - halfcn dirkerf itgrecl climadd addid2d ) AUPWMUOWNWOWPZWQWPYWKUUKAWMYWKUA + halfcn dirkerf itgrecl climadd addlidd ) AUPWMUOWNWOWPZWQWPYWKUUKAWMYWKUA UBTWRYWKWSZUPWTXAWTUULXBZYWMXCZAUUMZAUBWMXDWOWPZWMUUKAWMXDUAUAWRUQWMXDXEW PZUQXIZUDXBZUUOZWSZUAWRXDWSZUBWTXAWRAUAUUPUAWRYWTUUNUAWRXDUUNUAUBUAWRYWTX DWOWPZWSZVSUAWRYXCUUNUUQUURYWOAYXAWMUUKXJUQYWQYWROXBZSXIZWTWNWOWPZWQWPZYW @@ -746136,11 +746136,11 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u fourierdlem107 $p |- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x ) $= ( vj cmin co cicc cv cfv citg clt wbr wa cc0 caddc cc oveq2i recnd eqtrid - wceq subidd oveq1d resubcld addid2d 3eqtrd eqcomd cfzo cmpt cif cfz cn c1 + wceq subidd oveq1d resubcld addlidd 3eqtrd eqcomd cfzo cmpt cif cfz cn c1 wral cr crab wcel wb fveq2 a1i ltsubrpd simpld simpr eliccre syl3anc eqid adantr cioo cres wf adantlr rexrd cxr pnfxr ltpnfd eliood climc ffvelcdmd cpnf eqtrd fourierdlem105 itgcl eqeltrrd cle ltled mpbid cibl itgspliticc - eliccd addsubassd oveq2d subsub4d cneg df-neg 3eqtr3d addid1d eleqtrd wss + eliccd addsubassd oveq2d subsub4d cneg df-neg 3eqtr3d addridd eleqtrd wss addsub12d leidd rpred subadd4b pncan3d oveq12d itgeq1d cmap oveq1 breq12d fveq2d cbvralvw anbi2d rabbidv mpteq2ia eqtri wiso fourierdlem54 eqeltrid simprd syldan cbvmptv fourierdlem90 fourierdlem89 fourierdlem91 eqbrtrrid @@ -747632,16 +747632,16 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u ( cpi co cmul cc0 caddc wceq cr wcel a1i wbr c1 cc cneg cioo cfv ccos cif citg cdiv pire cicc cle 0re ltleii pipos wa cmo clt adantl adantr elioore remulcld recoscld recnd cmpt cibl syl2anc2 crp c2 pirp mp2an eqeltri picn - eqtri oveq2i recni oveq1i cxr rexri 0red rexrd id ioogtlb syl3anc mulid2i - ltadd1dd lttrd ltled iooltub addid2d modid syl22anc iffalsed eqtrd oveq1d + eqtri oveq2i recni oveq1i cxr rexri 0red rexrd id ioogtlb syl3anc mullidi + ltadd1dd lttrd ltled iooltub addlidd modid syl22anc iffalsed eqtrd oveq1d breqtrd cz mpteq2dva 1cnd negcld ioossicc cvol ioombl iccssre sseli ccncf wss ax-resscn sstri constcncfg idcncfg mulcncf cncfmpt1f cniccibl eqeltrd iblmulc2 itgeq2dv itgmulc2 mul02d fveq2d eqtrdi adantll ioovolcl itgconst iblss cos0 cmin volioo mp3an oveq2d 3eqtrd iftrue mul01d csin itgcoscmulx - mulid2d syl oveq12d 3eqtr4d syldan pm2.61dan 3eqtr2d cv renegcli negpilt0 + mullidd syl oveq12d 3eqtr4d syldan pm2.61dan 3eqtr2d cv renegcli negpilt0 elicc2i mpbir3an wf 1red renegcld ifcld fmptd ffvelcdmd nn0red fvmpt2 2rp rpmulcl modcld 2timesi addassi negidi addcomli 3eqtr2ri remulcli eqbrtrid - addid2i 2re eqcomi readdcld 1zzd modcyc 3eqtr3a ltnsymd cdm ssid 2timesgt + addlidi 2re eqcomi readdcld 1zzd modcyc 3eqtr3a ltnsymd cdm ssid 2timesgt coscn ax-mp breqtrri eqbrtrd iftrued itgsplitioo ioosscn sylan9eq subnegi oveq1 0cn 3eqtri mulm1i eqcomd wn nn0ge0d wne neqne ne0gt0d simpr gt0ne0d sin0 mulneg2d mulcld sinneg 0cnd sincld subnegd nn0zd sinkpi div0d neneqd @@ -747725,10 +747725,10 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u ioombl sstri iblss eliood sylan2 itgeq2dv ccos itgsincmulx eqtrdi oveq12d cmin syl 1m1e0 iftrue 3eqtr4a iffalse ax-1cn 1p1e2 eqtr4id 3eqtrd pm2.61i negeqd 0cn 2cn 3eqtr2d oveq2d eqcomd cdvds elicc2i mpbir3an fvmpt2 sincld - nncnd 2rp rpmulcl modcld 2timesi addassi negidi addcomli addid2i 3eqtr2ri - cv remulcli mulid2i readdcld addid2d 1zzd modcyc 3eqtr3a ltnsymd iffalsed + nncnd 2rp rpmulcl modcld 2timesi addassi negidi addcomli addlidi 3eqtr2ri + cv remulcli mullidi readdcld addlidd 1zzd modcyc 3eqtr3a ltnsymd iffalsed 2re neg1cn nnred cvol sincn ssid iblmulc2 2timesgt ax-mp breqtrri iftrued - cdm mulid2d itgsplitioo mulm1d itgneg nnne0d nnzd cosknegpi mul01d fveq2d + cdm mullidd itgsplitioo mulm1d itgneg nnne0d nnzd cosknegpi mul01d fveq2d cos0 wn negdi2 negeqi negcli divnegd neg0 negnegi coskpi2 subnegi divdird 00id div0d 2p2e4 eqtr4d pm2.61dan gtneii div0i 4cn wne divdiv1d ) ABIUAZI UBJZBUUPZDUCZEUXJUDJZUEUCZUDJZUFZIKJBUXHLUBJZUXNUFZBLIUBJZUXNUFZMJZIKJNEU @@ -747810,7 +747810,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u 0re pm2.61dan cxr eliocd iftrued necon2bi iffalsed eqtrid 1pneg1e0 eqtrdi 0xr rexri 2cn iftrue eqtr2id ax-1cn mulcomi oveq1i oveq2i leneltd 2div2e1 eqtri mulcli eqtr2i eqbrtrdi mp3an 3eqtr2i nsyl ltnled negeqi 2pos elrpii - mulgt0ii odd2np1 biimpa 1cnd adddird eqcomi oveq2d mulid2i addcomd modcyc + mulgt0ii odd2np1 biimpa 1cnd adddird eqcomi oveq2d mullidi addcomd modcyc peano2cn ltleii 2timesgt modid syl22anc iocgtlb gt0ne0d neneqd pm2.53 imp olcd modcl breqtrrid eqlei2 oveq1 breq1d ifbid 1ex negex ifex fvmpt ltned syl2anr adantll div0i ad2antlr mpdan cfl cmin eqtr3i gtneii redivcli flcl @@ -747909,13 +747909,13 @@ its Fourier expansion has only sine terms (coefficients for cosine terms nnre 0le2 nnge1 lemul2ad lesub1dd ltletrd gtned 4ne0 cioc cneg 0cnd mulcl nncn nnne0 gtneii mulne0d ifcld fmpti breq2 ifbieq2d c0ex simpr nndivdvds 4cn ifex 2nn mpbird 3adant1 cn0 1re renegcli ifcli ifbid cbvmptv remulcli - breq1d eqeltri mulid2i 2pos mulgt0ii elrpii modcyc readdcld fvmpt2 eqtr4d + breq1d eqeltri mullidi 2pos mulgt0ii elrpii modcyc readdcld fvmpt2 eqtr4d crp mpan2 csn cdif snfi eldifi negpilt0 ltleii iooss1 reseq1i resmpt pirp cfn 2timesgt modid eqbrtrd mpteq2ia 3eqtrri cpr reelprrecn ccnfld crn ctg ctopn iooretop tgioo2 1cnd dvmptconst ssid ax-resscn fss dvresioo 3eqtr3i dmeqi dmmpti eqtr3i ssdmres mpbir elind dmres eleqtrrdi ad2antrr ad2antlr adantlr lttri5d iooss2 2timesi negpicn addassi addcomli ltadd1dd readdcli - modcld negidi addid2i breqtrdi addcomd ltaddneg jca modid2 eqtr3d lensymd + modcld negidi addlidi breqtrdi addcomd ltaddneg jca modid2 eqtr3d lensymd neqne negcld sylan eldifn condan velsn sylibr ssriv ccncf eqsstri ioosscn ssfi inss1 sstri dvf fresin ffdm simpli cuni cun sselid elun1 nltled ccnp clp mnfltd lptioo2 incom df-ss fveq2i ltpnfd lptioo1 mnfle sstrdi resabs1 @@ -747929,16 +747929,16 @@ its Fourier expansion has only sine terms (coefficients for cosine terms mp4an reseq2i limcresiooub negpitopissre iocgtlb iocleub necon3bi xrltnle 3syl mnflt iffalsei ioossioc modcl resubcli elrpd ltsubrpd ioossico ltmod feqresmpt mpteq2dva sselda addcli subadd23 pncan3oi 3eqtri pncan2 modabs2 - addid1d modaddabs 3eqtrd npcand 3eqtrrd 3brtr4d ltsubrp eqtr2di wo pm4.56 + addridd modaddabs 3eqtrd npcand 3eqtrrd 3brtr4d ltsubrp eqtr2di wo pm4.56 lbioc rexrd biimpi olc modge0 orcd pm2.61dane modlt elicod ubioc1 eleqtrd nsyl eqtr2id neg1cn gtnelioc nnncan2d sub31 posdifd eqbrtri eqeltrd nncan ltsub2dd eqbrtrrd modsubmodmod nncand ltaddsublt pm2.61i subge02 ltadd2dd - subidi addid1i lelttrd pncan3i pncan3d ltaddrpd ltnled ibir fvmpt readdcl + subidi addridi lelttrd pncan3i pncan3d ltaddrpd ltnled ibir fvmpt readdcl 1ex sylancr addgegt0d eqeltrrd ccos citg sqwvfoura sqwvfourb nnnn0 coscld - mul02d addid2d iftrue sylan9eqr iffalse fourierclim div0i divcli sumnnodd + mul02d addlidd iftrue sylan9eqr iffalse fourierclim div0i divcli sumnnodd 0nn0 subid1i elnnz sylanbrc dvdsmul1 mulcomd divdiv1d div32d fourierswlem oddp1even seqeq3 3brtr3i subcld 1lt2 breqtrrdi subne0d ffvelcdmda divcan6 - eqeltrid mulid2d mulassd 3eqtr3a climrel releldmi climuni mulassi 3eqtr2i + eqeltrid mullidd mulassd 3eqtr3a climrel releldmi climuni mulassi 3eqtr2i eqeltrrdi isermulc2 isumclim2 mulcomli ) UANUBOZUCUDDUUBZUEOZUFUGOZGUEOZU HUIZXXHUBOZDUUCZUEOZHPULBUFUUDZNUAUBOZHUEOZUJUKXXMXXEXXPUEOXXEXXOUEOZHUEO ZHXXLXXPXXEUEULEUCUDEUUBZUEOZUFUGOZGUEOZUHUIZXYAUBOZUMZUFUUDZXXLUJUKZXYFX @@ -748370,7 +748370,7 @@ those for the more general case of a piecewise smooth function (see 3eqtr4d fvmptd 3eqtrd expcld mulcld oveq12d npcand sumeq2dv lenltd condan zred expne0d fzfid syl2anc2 fmptd elplyr syl3anc iftrued iffalse resubcld cif wo nn0re ltnled ltsubaddd olc dgrlt pm2.61dan fssd fvmpt2d sumeq12rdv - simpl coeeq2 addid2d 0nn0 sylib eqnetrd caa aasscn fvoveq1 oveq2 fsumshft + simpl coeeq2 addlidd 0nn0 sylib eqnetrd caa aasscn fvoveq1 oveq2 fsumshft sumeq1d elfzelz expsubd oveq2d 0red nn0ge0d elfzle1 divassd eqcomd eqtr2d letrd eleqtrdi fzss1 divcld cdif eldifi elfzle2 elfzd eldifn neqne mul02d elfzelzd adantll div0d fsumss syldan fvmpt2 fsumcl coeid2 fsumdivc eqeq1d @@ -748747,7 +748747,7 @@ those for the more general case of a piecewise smooth function (see ( cfa cfv cc0 co c1 cmul cfz cv cprod cdiv cmin clt wbr cexp cneg wn wceq cif wa cr fzssre cuz wcel cn0 nn0uz eleqtrdi eluzfz1 syl ffvelcdmd sselid eqeltrrd lttri3d mpbid simprd iffalsed eqcomd subeq0bd fveq2d fac0 eqtrdi - recnd oveq2d cn nnm1nn0 faccld nncnd div1d oveq12d 0exp0e1 mulid1d 3eqtrd + recnd oveq2d cn nnm1nn0 faccld nncnd div1d oveq12d 0exp0e1 mulridd 3eqtrd eqtrd oveq1d df-neg eqtr4di ifeq2d prodeq2ad eqtrid ) ADHOPQGUARZEUBZBPZO PEUCUDRZCSUERZQBPZUFUGZQWQOPZWQWRUERZOPZUDRZFXAUHRZTRZULZSGUARZCWOUFUGZQC OPCWOUERZOPUDRZFWNUERZXIUHRZTRZULZEUCZTRZTRWPWTXGXHQXJWNUIZXIUHRZTRZULZEU @@ -749033,7 +749033,7 @@ those for the more general case of a piecewise smooth function (see nfv negsubdi2d nn0red lenegcon1d elfzel2 iooltub ltletrd absled mpbir2and leexp1a syl32anc nnge1d cuz iftrue lem1d wn iffalse leidd pm2.61dan eluz2 syl3anbrc leexp2ad fprodle chash cfn fprodconst syl2anc hashfz0 lemul12ad - mp1i mulid2d itgle itgconst lediv1dd lelttrd ) AGUDUEZQIUFRZEUGZCUEZUHVUP + mp1i mullidd itgle itgconst lediv1dd lelttrd ) AGUDUEZQIUFRZEUGZCUEZUHVUP UIRZUJRZBQVUPUKRZUHBUGZULZUIRZVVAFUEZUJRZUOZUJRZEUMZUDUEZDUNUPRZUQUEZURRZ UNUSAVUNVVHVVKURRZUDUEZVVIVVKUDUEZURRVVLVUNVVNUTAGVVMUDGHVVKURRZVVMLHVVHV VKURKVAVBVCVDAVVHVVKAVUOVVGEAQIVEZAVUPVUOSZVFZVUSVVFVVSVUQVURVVSVUQVVSVJV @@ -749138,12 +749138,12 @@ those for the more general case of a piecewise smooth function (see pm2.61dan oveq2 iffalsed mul01d ralnex eleqtrdi fzsscn wf ssrab2 eqsstrdi wral nn0uz sseldd elmapi ffvelcdmda ad4ant14 fsum1p simplr oveq1i sumeq1i 0p1e1 eleq1d notbid rspccva adantll fzssnn0 fz1ssfz0 elnn0 orel1 sumeq2dv - sylan2 fzfi olci sumz mp1i nnm1nn0 nn0red addid1d eqnetrd neneqd sylan2br + sylan2 fzfi olci sumz mp1i nnm1nn0 nn0red addridd eqnetrd neneqd sylan2br recnd condan nfcv nn0ex mapss mccl csn cdif difss ssfi sstri dvds0 iftrue fzssre zred subge0d mpbird elfzle1 subge02d elnn0z zexpcl nn0zd 1red nnre nnred nnge1 lesub2dd 3adant2 dvdsmul1 npcand facp1 mulcomd eqtr2d iffalse 1cnd simp2 breq2d ifbieq2d fprodsplit1 dvdsmultr2 3adant1r eluzfz1 subidd - lttri3d sylan9eqr fac0 eqtrdi div1d exp0d mulid1d divcan3d rexlimdv3a mpd + lttri3d sylan9eqr fac0 eqtrdi div1d exp0d mulridd divcan3d rexlimdv3a mpd mulcld iftrued simpll ad2antlr leneltd posdifd elnnz nnnn0d divcld mul02d id 0expd div0d pm2.61dane ) ADGUCUDZRIUESZEUFZCUDZUCUDEUGZUHSZDUIUJSZRCUD ZUKUOZRVUKUCUDZVUKVULUJSZUCUDZUHSZHVUOULSZUMSZUPZUIIUESZDVUHUKUOZRDUCUDZD @@ -749238,7 +749238,7 @@ those for the more general case of a piecewise smooth function (see mccl ax-mp id 1e0p1 oveq1i eleqtrdi adantl etransclem3 fprodzcl 3jca cdif csn zcnd subidd oveq2d ifeq2d fzfi diffi mp1i eldifi dvds0 iftrue breqtrd cfn wceq wn iddvds ad2antrr iffalse ad2antlr oveq1 ffvelcdmd eqtrd eqtrdi - cc fac0 nncnd div1d 0cnd exp0d oveq12d mulid1d eqtr2d simpr nnne0d divcld + cc fac0 nncnd div1d 0cnd exp0d oveq12d mulridd eqtr2d simpr nnne0d divcld adantlr cr pm2.61dan fprodcl iffalsed simpll nnred cle nltled wne leneltd zred neqne zsubcld posdifd mpbid elnnz sylanbrc 0expd syl2anc dvdsmultr1d fveq2 oveq2 sylan9eqr ifbieq2d fprodsplit1 breqtrrd dvdsmultr2 ffvelcdmda @@ -749703,15 +749703,15 @@ those for the more general case of a piecewise smooth function (see wa ad2antrr sseli adantl fprodzcl zmulcld zcnd cuz nn0uz eleqtrdi eluzfz1 wf ffvelcdmda fveq2 fsum1p sumeq1i fvmpt2 syl2anr elfzelz elfzle1 ltletrd ovex gtned neneqd iffalsed eqtrd sumeq2dv cfn wo sumz mp1i 3eqtrd oveq12d - nncnd addid1d sylanbrc prodeq2ad oveq2d breq2d ifbieq2d fsumsplit1 faccld + nncnd addridd sylanbrc prodeq2ad oveq2d breq2d ifbieq2d fsumsplit1 faccld fveq1 fac0 eqtrdi prodeq2dv nnne0d dividd wn nn0red ffvelcdmd cle lensymd - mpbid nnnn0d mulid2d cz adantllr cdvn cneg cpr reelprrecn ctopn crest crn + mpbid nnnn0d mullidd cz adantllr cdvn cneg cpr reelprrecn ctopn crest crn ccnfld cioo ctg eqid tgioo2 eleqtri etransclem5 etransclem31 etransclem16 reopn nnm1nn0 etransclem12 eleqtrd rabid sylib nn0ex fzssnn0 mapss simpld crab mp2an mccl nnzd elmapi etransclem10 fz1ssfz0 etransclem3 ifcld fmptd eluzfz2 elmap sylibr fzsscn iftrued fvmptd 0p1e1 1red zred 0lt1 fzfi olci oveq1i 1cnd subcld sumeq2sdv eqeq1d elrab eleqtrrd fprod1p prodeq1i prod1 - mulid1d lttri3d div1d 0cnd exp0d fzssre nnred nngt0d ltled eqbrtrd df-neg + mulridd lttri3d div1d 0cnd exp0d fzssre nnred nngt0d ltled eqbrtrd df-neg subeq0bd subid1d eqcomi znegcld expcld zexpcl mulcld eldifi elfzle2 nn0zd sylan2 elfzelzd zsubcld ffnd id ad2antlr eqtr2d adantlr ad4antr fsumnn0cl wfn ad5antr ad3antrrr simp-4l sylancom 1zzd elfzel2 elfznn0 neqne elnnne0 @@ -750199,7 +750199,7 @@ those for the more general case of a piecewise smooth function (see c1 a1i wa cdv cmpt oveq2i adantr ere negcld cxpcld adantl simpr fzfid cn0 cc wf elfznn0 syl eqeltrid sylan2 adantlr simplr ffvelcdmd fsumcl syl2anc cz fvmpt2 eqeltrd mulcld caddc cle wbr 0re epos neg1rr dvmptneg epr ax-mp - wtru mulid2d mpteq2ia eqtri oveq2 recnd mulcomd eqtrd peano2nn0 wi anbi2d + wtru mullidd mpteq2ia eqtri oveq2 recnd mulcomd eqtrd peano2nn0 wi anbi2d eleq1 fveq2 feq1d imbi12d eqcomd oveq2d addcomd negsubd fsumsub cexp eqid subdid fveq1d wss cvv oveq12d mpteq2dva 0red ccncf adantll cmnf cxr cprod 3eqtrd ad2antrr oveq1d sumeq2dv cxp c1st c2nd cdvn cfa cdiv crn ctg ctopn @@ -752243,7 +752243,7 @@ the same as the infinite group sum (that's always convergent, in this 3exp csu cico feqresmpt adantlr fveq2i oveq1i icossicc pm3.2i ressabs 0xr sselda adantll ffvelcdmd cle wbr iccgelb wne cdm ffund sylan fdmd eleqtrd wn fvelrn eqeltrd adantl3r ad3antrrr pm2.65da neqned ge0xrre ltpnfd fmptd - 0e0icopnf eliccxr xaddid2 xaddid1 jca gsumress csubmnd gsumsubm eqidd vex + 0e0icopnf eliccxr xaddlid xaddrid jca gsumress csubmnd gsumsubm eqidd vex elicod mptex ax-resscn cnfldbas rge0srg simpl srgacl rexadd eqtr3i eqtr4i csrg 3eqtr4d funmpt gsumpropd2 3eqtrd recnd gsumfsum eqtr3d rneqd supeq1d mpteq2dva pm2.61dan xrge0tsms eleq12d ) ABUAKZCBUBLZMIEUCZUDUEZCBIUHZUIZU @@ -752818,7 +752818,7 @@ the supremum (in the real numbers) of finite subsums. Similar to $= ( wceq cle adantr cc0 wcel cpr cmpt csumge0 cfv co wbr wa csn preq1 dfsn2 cxad eqcomi a1i eqtrd mpteq1d fveq2d adantl sge0snmpt cpnf iccssxr sselid - cicc cxr xaddid2d eqcomd 0xr pnfxr iccgelb syl3anc xleadd1d eqbrtrd neqne + cicc cxr xaddlidd eqcomd 0xr pnfxr iccgelb syl3anc xleadd1d eqbrtrd neqne wn wne sge0pr xaddcld xrleidd pm2.61dan ) ABCPZFBCUAZDUBZUCUDZEGUKUEZQUFA VSUGZWBGWCQWDWBFCUHZDUBZUCUDZGVSWBWGPAVSWAWFUCVSFVTWEDVSVTCCUAZWEBCCUIWHW EPVSWEWHCUJULUMUNUOUPUQAWGGPVSACDGFIKMOURRUNAGWCQUFVSAGSGUKUEZWCQAWIGAGAS @@ -753180,7 +753180,7 @@ the supremum (in the real numbers) of finite subsums. Similar to ) ) ) $= ( cmpt csumge0 cfv cxad co cvv wcel wceq cc0 cun wss ssexg syl2anc difexd cdif cin c0 disjdif cv wa cpnf cicc 0e0iccpnf eqeltrd sge0splitmpt eqcomd - a1i mpteq2da fveq2d sge0z eqtrd oveq2d eqid fmptdf sge0xrcl xaddid1 eqidd + a1i mpteq2da fveq2d sge0z eqtrd oveq2d eqid fmptdf sge0xrcl xaddrid eqidd cxr syl 3eqtrrd undif sylib mpteq1d 3eqtr4d ) AEBDLZMNZECBUFZDLZMNZOPZEBV RUAZDLZMNZVQECDLZMNAWDWAAEBVRDQQGABCUBZCFRBQRIHBCFUCUDZACBFHUEZBVRUGUHSAB CUIURJAEUJVRRUKZDTTULUMPZKTWJRWIUNURUOUPUQAWAVQTOPZVQVQAVTTVQOAVTEVRTLZMN @@ -754667,7 +754667,7 @@ the supremum (in the real numbers) of finite subsums. Similar to Siliprandi, 17-Aug-2020.) $) meadjun $p |- ( ph -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) ) $= ( vx c0 wceq cfv wa cc0 wcel adantr a1i adantl cun cxad cpnf cicc iccssxr - cxr meaf ffvelcdmd sselid xaddid2 syl eqcomd uneq1 0un eqtrd fveq2d fveq2 + cxr meaf ffvelcdmd sselid xaddlid syl eqcomd uneq1 0un eqtrd fveq2d fveq2 co mea0 oveq1d 3eqtr4d wn wne simpl cin ad2antrr inidm eqcomi ineq2 neqne eqtr2id eqnetrd neneqd adantll pm2.65da neqned cpr csumge0 uniprg syl2anc cuni cres prssd com cdom wbr cfn prfi isfinite biimpi sdomdom ax-mp wdisj @@ -754832,7 +754832,7 @@ the supremum (in the real numbers) of finite subsums. Similar to adantllr reseq1d resmptd ssequn2 sylib eqcomd mpteq1d 3eqtrd cxad cvv nfv simplr cin disjsn biimpri ad2antrr sselda ffvelcdmd ad4ant14 sge0splitmpt p0ex elsni fveq2 sylan2 mpteq2dva sge0z oveq2d fssresd feq1dd pm2.61dan - ex sge0xrcl xaddid1d 3eqtrrd nfdisj1 nfan eqidd sylan sge0fodjrn syl21anc + ex sge0xrcl xaddridd 3eqtrrd nfdisj1 nfan eqidd sylan sge0fodjrn syl21anc nnex eqtr2d exlimdv sylc ralrimiva jca31 ismea sylibr ) AEUCZKUDUEUFZEUGZ UURUHLZMZNEOZKPZMUAUIZUJUKULZJUVEJUIZUMZMZUVEUNZEOZEUVEUOZQOZPZUPZUAUURUQ ZURZMEUSLAUVBUVDUVQAUUTUVAAUUTBUUSEUGZGAUURBUUSEABUUSEGVBZUTVAAUURBUHUVSF @@ -755720,7 +755720,7 @@ nondecreasing measurable sets (with bounded measure) then the measure of caragen0 $p |- ( ph -> (/) e. S ) $= ( va c0 cdm cuni eqid wcel a1i cfv cxad co cc0 wceq fveq2i adantr cpnf cv cpw 0elpw cin cdif in0 dif0 oveq12i ome0 oveq1d cicc cxr iccssxr come wss - wa elpwi adantl omecl sselid xaddid2d 3eqtrd carageneld ) ABGCCHIZFDVDJZE + wa elpwi adantl omecl sselid xaddlidd 3eqtrd carageneld ) ABGCCHIZFDVDJZE GVDUBZKAVDUCLAFUAZVFKZUPZVGGUDZCMZVGGUEZCMZNOZGCMZVGCMZNOZPVPNOVPVNVQQVIV KVOVMVPNVJGCVGUFRVLVGCVGUGRUHLVIVOPVPNAVOPQVHACDUISUJVIVPVIPTUKOULVPPTUMV IVGCVDACUNKVHDSVEVHVGVDUOAVGVDUQURUSUTVAVBVC $. @@ -755748,7 +755748,7 @@ nondecreasing measurable sets (with bounded measure) then the measure of caragenunidm $p |- ( ph -> X e. S ) $= ( va cvv wcel come syl cfv cxad co cc0 wceq fveq2d adantl c0 cpw cdm cuni dmexg uniexg 3syl eqeltrid pwidg cv wa cin cdif elpwi df-ss biimpi ssdif0 - wss sylib ome0 adantr eqtrd oveq12d cpnf cicc cxr iccssxr sselid xaddid1d + wss sylib ome0 adantr eqtrd oveq12d cpnf cicc cxr iccssxr sselid xaddridd omecl eqidd 3eqtrd carageneld ) ABDCDHEFGADIJDDUAZJADCUBZUCZIFACKJZVNIJVO IJECKUDVNIUEUFUGDIUHLAHUIZVMJZUJZVQDUKZCMZVQDULZCMZNOVQCMZPNOWDWDVSWAWDWC PNVRWAWDQAVRVTVQCVRVQDUQZVTVQQZVQDUMZWEWFVQDUNUOLRSVSWCTCMZPVRWCWHQAVRWBT @@ -756474,7 +756474,7 @@ This is the proof of the statement in the middle of Step (e) in the necon3bi df-pss pssnel ad2antll ad2antlr ex exlimimdd simplll elsni con3i wpss elunnel2 foelcdmi sylancl simp3 eqtr2d cdif uncom undif difexg ssexd mpteq1d cin disjdifr cpnf cicc eldifi syldan sge0splitmpt sge0xrcl eldifn - xaddid2d iffalsed sge0z oveq1d feqresmpt fveq2 sge0f1o 3eqtrd 3eqtr4d + xaddlidd iffalsed sge0z oveq1d feqresmpt fveq2 sge0f1o 3eqtrd 3eqtr4d eqidd ) AHUBZFUCZFHUDZUEUCZUFUGDQDUHZBUCZUIZFUCZDQUWPFUCZUJZUEUCZUFUGZAAQ GUKZBUNZUXBAVGAQHRULZUMZUXCBADQUWOCUOZUWOEUCZRUPZUXFBAUWOQUOZSZUXGUXIUXFU OZAUXGUXLUXJAUXGSZUXIUXHUXFUXGUXIUXHTZAUXGUXHRUQZURZACUXFUWOEACHUXFEACHEU @@ -756618,7 +756618,7 @@ This is the proof of the statement in the middle of Step (e) in the caragencmpl $p |- ( ph -> E e. S ) $= ( va wcel wss cvv come cfv cxad cc0 adantr adantl cpw unidmex ssexd elpwg wb syl mpbird cv wa cin cdif co cle wceq inss2 omess0 oveq1d difssd elpwi - a1i cxr sstrd omexrcl xaddid2 eqtrd omessle eqbrtrd caragenel2d ) ABCDEKF + a1i cxr sstrd omexrcl xaddlid eqtrd omessle eqbrtrd caragenel2d ) ABCDEKF GJACEUAZLZCEMZHACNLVJVKUEACENADOEFGUBHUCCENUDUFUGAKUHZVILZUIZVLCUJZDPZVLC UKZDPZQULZVRVLDPUMVNVSRVRQULZVRVNVPRVRQVNCVODEADOLVMFSZGAVKVMHSACDPRUNVMI SVOCMVNVLCUOUTUPUQVNVRVALVTVRUNVNVQDEWAGVMVQEMAVMVQVLEVMVLCURZVLEUSZVBTVC @@ -758643,7 +758643,7 @@ This is the proof of the statement in the middle of Step (e) in the eqtr2d mpteq1d fvexd cin incom nnuzdisj eqtri icossicc peano2nnd uznnssnn ssid breq1d ifbieq1d ifbieq12d cbvmptv mpteq2i hsphoif sge0ssrempt rexadd sge0splitmpt nnex oveq1d addassd sge0fsummpt ax-resscn sstri fsumadd 0le1 - syldan rpge0d addge0d hsphoidmvle2 sge0lempt oveq2 mul01d addid1d eqbrtrd + syldan rpge0d addge0d hsphoidmvle2 sge0lempt oveq2 mul01d addridd eqbrtrd syl21anc neqne cprod hoiprodp1 oveqd feq2d neneq ad2antlr pm2.65da neqned hoidmv0val hoidmvn0val hoidmvval0b condan iftrued fvres prodeq2dv 3eqtrrd hsphoival iffalsed ltnled volico anabss5 icogelb eqleltd resubcld adddird @@ -760389,8 +760389,8 @@ This is Step (a) of Lemma 115F of [Fremlin1] p. 31. (Contributed by wf fveq2 fveq2d fprodsplitsn cle hsphoival iftrue adantl oveq2d prodeq2dv wbr cif oveq1d 3eqtrd eqeltrrd hoidifhspval3 wne wn eldifsni syl iffalsed neneq fvoveq1d cc eqid iftruei c0 ad2antrr simplr simpr letrd cxr wb ico0 - rexrd mpbird iffalse pm2.61dan vol0 addid1d clt simpl ltnled adantlr cmin - volico resubcld 3eqtrrd addid2d iftrued addcomd npncand 3eqtr4d fprodcl + rexrd mpbird iffalse pm2.61dan vol0 addridd clt simpl ltnled adantlr cmin + volico resubcld 3eqtrrd addlidd iftrued addcomd npncand 3eqtr4d fprodcl lttrd eqtr2d adddid ) ADEMGUEUEZLKUEZUFZDMFUEUEZEUUKUFZUGUFUULUUNUHUFZLIU IZDUEZUUPEUEZUJUFZUKUEZIULZDEUUKUFZAUULUUNAUMUNUJUFZWBUULUOABDUUJIKLNOUBQ TACMEHGUPLLJUSZUQZPUCSQUAURZUTVAAUVCWBUUNUOABUUMEIKLNOUBQABDFHJUPLMPUDSQT @@ -765078,7 +765078,7 @@ is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] 1red letrd elrpd rpdivcld eqeltrd ifcld resqcl resubcld resqcld cc leabsd rpred absmuld cioo rexrd ioogtlb syl3anc lttrd iooltub absdifltd ioogtlbd mpbird iooltubd ltmul12ad eqbrtrd lelttrd sqvald ltled lemul1ad leadd12dd - cxr breqtrd ltleaddd cicc min1 adddid eqbrtrid sqrlearg 1cnd mulid1d min2 + cxr breqtrd ltleaddd cicc min1 adddid eqbrtrid sqrlearg 1cnd mulridd min2 eliccd leadd1dd addcomd gtned divcan1d ltsub1d ) AGHUEUFZBUGUHUULFIUEUFZU IUFZBUUMUIUFZUGUHAUUNGFUIUFZHIUIUFZUEUFZUUPIUEUFZFUUQUEUFZUJUFZUJUFZUUOUG AUVBUULIFUEUFZUJUFZGIUEUFZHFUEUFZUJUFZUIUFZUVGUUMUUMUJUFZUIUFZUJUFZUULUUM @@ -767418,7 +767418,7 @@ that every function in the sequence can have a different (partial) ( wceq cmin co cmul cc0 cc wcel subcld adantr wa cr simp3d simp1d sigarim simpld syl2anc recnd mul01d simp2d simpr subeq0bd oveq2d ccj cfv sigarval cim eqcomd cjcld eqtrd fveq2d 0red reim0d 3eqtrd oveq1d mul02d 3eqtr4d wn - cdiv npncand sigaraf syl3anc eqtr3d simprd sigarperm addid1d 3eqtr2d cneg + cdiv npncand sigaraf syl3anc eqtr3d simprd sigarperm addridd 3eqtr2d cneg c1 negsubdi2d neqned subne0d divnegd dividd negeqd mulm1d div32d sigardiv caddc 3jca sigarls divcld mulassd divcan1d pm2.61dan ) AEGLZFDMNZGDMNZHNZ EGMNZONZFEMNZGEMNZHNZDGMNZONZLAWPUAZWSPONPXAXFXGWSXGWSXGWQQRZWRQRZWSUBRAX @@ -773156,7 +773156,7 @@ Negated membership (alternative) cnambpcma $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) + C ) - A ) = ( C - B ) ) $= ( cc wcel w3a cmin co caddc subcl 3adant3 simp3 simp1 addsubd wa oveq1d cc0 - wceq 3ad2ant1 3eqtrd simpl simpr sub32 anidms simp2 subadd23d subid addid2d + wceq 3ad2ant1 3eqtrd simpl simpr sub32 anidms simp2 subadd23d subid addlidd 3jca syl ancoms 3adant1 ) ADEZBDEZCDEZFZABGHZCIHAGHUQAGHZCIHAAGHZBGHZCIHZCB GHZUPUQCAUMUNUQDEUOABJKUMUNUOLZUMUNUOMNUPURUTCIUPUMUNUMFZURUTRUMUNVDUOUMUNO UMUNUMUMUNUAZUMUNUBVEUIKABAUCUJPUPVAUSVBIHZQVBIHZVBUPUSBCUMUNUSDEZUOUMVHAAJ @@ -773267,7 +773267,7 @@ Ordering on reals (cont.) - extension (Contributed by AV, 23-Jan-2023.) $) readdcnnred $p |- ( ph -> ( A + B ) e/ RR ) $= ( caddc co cr wcel wn cc cim cfv cc0 wceq wb recnd reim0b syl eqeq1d wnel - eldifbd df-nel eldifad addcld reim0d oveq1d addid2d imaddd 3bitr4d notbid + eldifbd df-nel eldifad addcld reim0d oveq1d addlidd imaddd 3bitr4d notbid imcld eqtrd bitrd bitrid mpbird ) ABCFGZHUAZCHIZJZACKHEUBURUQHIZJAUTUQHUC AVAUSAVAUQLMZNOZUSAUQKIVAVCPABCABDQZACKHEUDZUEUQRSABLMZCLMZFGZNOVGNOZVCUS AVHVGNAVHNVGFGVGAVFNVGFABDUFUGAVGAVGACVEULQUHUMTAVBVHNABCVDVEUITACKIUSVIP @@ -773367,7 +773367,7 @@ Nonnegative integers (as a subset of complex numbers) - extension 10-Sep-2021.) $) deccarry $p |- ( A e. NN -> ( ; A 9 + 1 ) = ; ( A + 1 ) 0 ) $= ( cn wcel c1 caddc co cc0 cdc c9 cmul df-dec 9nn peano2nn nnmulcld nncnd cc - a1i nncni eqtr2id eqtrd ax-mp addid1d nncn adddid mulid1d oveq2d id addassd + a1i nncni eqtr2id eqtrd ax-mp addridd nncn adddid mulridd oveq2d id addassd 1cnd oveq1i ) ABCZADEFZGHIDEFZULJFZGEFZAIHZDEFZULGKUKUOUNUQUKUNUKUNUKUMULUM BCZUKIBCURLIMUAZQZAMNOUBUKUNUMAJFZUMDJFZEFZUQUKUMADUMPCUKUMUSRQZAUCUKUIZUDU KVCVAUMEFZUQUKVBUMVAEUKUMVDUEUFUKUQVAIEFZDEFVFUPVGDEAIKUJUKVAIDUKVAUKUMAUTU @@ -776627,7 +776627,7 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . cexp oveq1d adantl cv cfz cprod wa peano2nn0 3syl 2cnd expp1d oveq2d 2nn0 fmtno a1i nn0expcld nn0cnd sqvald expmuld fmtnom1nn adantr oveq1 fzfid cc 3eqtr4d elfznn0 fmtnonn nncnd fprodcl 1cnd addsubassd 2m1e1 oveq2i eqtrdi - subcld muls1d mulid2d joinlmuladdmuld eqcom subadd2d bitr4id oveq2 eqcoms + subcld muls1d mullidd joinlmuladdmuld eqcom subadd2d bitr4id oveq2 eqcoms cn mulcld addassd cuz elnn0uz biimpi fveq2 eqcomd npcan1 subadd23d nncand fprodp1 3eqtrd sylan9eqr ex sylbid imp ) AUAZCDZWREFGZHIZJWRUBGZBUAZHIZBU CZKFGZLZWTEFGZHIZJWTUBGZXDBUCZKFGZLWSXGUDZXIXAEMGZXNNGZEFGZXFEMGZXNNGZEFG @@ -776748,7 +776748,7 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . cc wceq oveq1d cuz cc0 cfz cv cprod fzfid cn elfznn0 fmtnonn nncnd adantl fprodcl uznn0sub fmtnorec2 mvlraddd 2nn0 eluz2nn nnm1nn0 nn0expcld nn0cnd 2cn peano2nn0 subdid ax-1cn w3a subsub syl3anc 2m1e1 oveq2i eqtrdi fveq2d - eluzelcn eqtrd mulid2d adddird addcomd fmtno eqtr4d sqvald 3eqtr2d mulcld + eluzelcn eqtrd mullidd adddird addcomd fmtno eqtr4d sqvald 3eqtr2d mulcld addsubassd npcan1 binom2sub1 nnsqcld subcld addassd 2timesi eqcomi mulcli fmtnorec1 subadd23d mulneg2d negsubdi2d fmtnom1nn subnegd mulcomd 3eqtr3d cneg eqtr3d 3eqtrd 3eqtrrd ) BCUADEZBFGHZIDZCCXDJHZJHZUBBCGHZUCHZAUDZIDZA @@ -776835,7 +776835,7 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . $( Lemma 1 for ~ fmtno5 . (Contributed by AV, 22-Jul-2021.) $) fmtno5lem1 $p |- ( ; ; ; ; 6 5 5 3 6 x. 6 ) = ; ; ; ; ; 3 9 3 2 1 6 $= ( c6 c5 cdc c3 c9 c2 c1 6nn0 5nn0 deccl 3nn0 eqid c8 cmul co 1nn0 8nn0 0nn0 - cc0 decmul1c 0p1e1 6t6e36 6p3e9 decaddi 6cn 5cn 6t5e30 mulcomli 3cn addid2i + cc0 decmul1c 0p1e1 6t6e36 6p3e9 decaddi 6cn 5cn 6t5e30 mulcomli 3cn addlidi 9nn0 decsuc 6t3e18 1p1e2 8p3e11 decaddci ) ABCZBCZDCZADECZDCZFCZGCAADUSACZH URDUQBABHIJZIJZKJHVCLHKVAGCZMGVBUSANODVAGUTDDEKUKJZKJZPJQKURDVFMAGUSHVEKUSL QPVASGURANOVHRUAUQBVASADURHVDIURLRKUTSDUQANODVGRKABUTSADUQHHIUQLRKDAEAANODK @@ -776845,7 +776845,7 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . $( Lemma 2 for ~ fmtno5 . (Contributed by AV, 22-Jul-2021.) $) fmtno5lem2 $p |- ( ; ; ; ; 6 5 5 3 6 x. 5 ) = ; ; ; ; ; 3 2 7 6 8 0 $= ( c6 c5 cdc c3 c2 c7 c8 5nn0 6nn0 deccl 3nn0 eqid 0nn0 cmul co 2nn0 decaddi - cc0 c1 decmul1c 7nn0 1nn0 5p1e6 6t5e30 2cn addid2i 5t5e25 decsuc 5cn 5t3e15 + cc0 c1 decmul1c 7nn0 1nn0 5p1e6 6t5e30 2cn addlidi 5t5e25 decsuc 5cn 5t3e15 5p2e7 3cn mulcomli 5p3e8 ) ABCZBCZDCZADECZFCZACZGCRBDUQACZHUPDUOBABIHJZHJZK JIVALMKUTBGUQBNODUSAURFDEKPJZUAJZIJHKUPDUTBBSUQHVCKUQLHUBUSBAUPBNOVEHUCUOBU SBBEUPHVBHUPLHPURBFUOBNOEVDHPABURBBEUOHIHUOLHPDREABNOEKMPUDEUEUFQUGTUKQUGTU @@ -776866,7 +776866,7 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . c4 eqid cexp cmul 5nn0 nn0cni sqvali fmtno5lem1 fmtno5lem2 decmul10add 4nn0 eqcomi fmtno5lem3 2nn0 9nn0 7nn0 1nn0 7p1e8 3p1e4 9p3e12 decaddci 3p2e5 7cn 2cn 7p2e9 addcomli 6cn ax-1cn 6p1e7 decsuc 8cn 8p6e14 decaddc deceq1i 5p1e6 - 00id 1p1e2 8p1e9 5p3e8 decaddi 9p2e11 8p7e15 6p4e10 addid2i decaddm10 2p1e3 + 00id 1p1e2 8p1e9 5p3e8 decaddi 9p2e11 8p7e15 6p4e10 addlidi decaddm10 2p1e3 4cn 9cn 9p6e15 eqtri 9p7e16 4p3e7 3eqtri ) ABCZBCZDCZACZEUAFWOWOUBFDGCZDCZE CZHCZACZICZDECZJCZACZKCZICZLFZICZXFLFZICZHGCZACZACZICZKCZLFZICZWTLFSECZGCZS CZGCZACZJCZECZGCZACWOWOWNAWMDWLBABMUCNZUCNZONZMNZUDUEWNAXPWTWOYHMYIWOWNUBFX @@ -776919,11 +776919,11 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . 257prm $p |- ; ; 2 5 7 e. Prime $= ( c2 c5 cdc c7 2nn0 5nn0 deccl decnncl c4 c1 4nn0 7nn0 1nn0 c3 3nn0 cmul co caddc eqid c9 7nn c8 8nn0 2lt8 5lt10 7lt10 3decltc 5nn 1lt10 declti c6 df-7 - 3t2e6 dec2dvds cdvds wbr 3nn 2nn 3cn mulid1i oveq1i 3p2e5 eqtri 2lt3 ndvdsi + 3t2e6 dec2dvds cdvds wbr 3nn 2nn 3cn mulridi oveq1i 3p2e5 eqtri 2lt3 ndvdsi 3dvds2dec 5cn 2cn 5p2e7 addcomli 7p7e14 breq2i 3dvdsdec ax-1cn 4p1e5 3bitri 4cn bitri mtbir 2lt5 dec5dvds2 6nn0 7t3e21 decaddi 7t6e42 decmul2c 5lt7 1nn - 4nn nn0cni mulcomi 11multnc mulcomli 4p3e7 4lt10 cc0 9nn0 10nn 0nn0 mulid2i - 1p1e2 decsuc decadd 9cn 9p2e11 9t3e27 addid1i decmac declt 8cn 7cn decaddci + 4nn nn0cni mulcomi 11multnc mulcomli 4p3e7 4lt10 cc0 9nn0 10nn 0nn0 mullidi + 1p1e2 decsuc decadd 9cn 9p2e11 9t3e27 addridi decmac declt 8cn 7cn decaddci 3pos 8p7e15 5p3e8 7t5e35 decmul1c 2lt10 9nn 9pos prmlem2 ) ABCZDCZYBDABEFGZ UAHAUBBIDJEUCFKLMUDUEUFUGYBDJABEUHHLMUIUJYBNUKDYDOUMULUNNYCUOUPZNBUOUPZNBJA UQMURNJPQZARQNARQBYGNARNUSUTVAVBVCVDVEYENABRQZDRQZUOUPNJICZUOUPZYFABDEFLVFY @@ -777211,7 +777211,7 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) $= ( vn wcel c2 c1 caddc co cexp cmul wceq cn0 c5 c4 a1i wa oveq1 oveq1d cc0 cn cfmtno cfv cdvds wbr cv wrex wo wi elnn1uz2 cprime 5prm dvdsprime mpan - cuz wb 1nn0 simpl adantl eqeq12d 4cn mulid2i eqcomi oveq1i eqtri rspcedvd + cuz wb 1nn0 simpl adantl eqeq12d 4cn mullidi eqcomi oveq1i eqtri rspcedvd df-5 0nn0 mul02i 0p1e1 syl6bi fveq2 fmtno1 eqtrdi breq2d 1p1e2 oveq2d sq2 jaoi eqeq2d rexbidv imbi12d syl5ibr w3a fmtnofac2 id 2nn0 nn0mulcld simpr adantr eqeqan12d cc eluzge2nn0 nn0cnd add1p1 eqcomd 2cnd peano2nn0 expp1d @@ -777253,8 +777253,8 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . wrex 4re 2lt4 ltleii eluz2 mpbir3an fmtnoprmfac2 mp3an1 wo cun elnnuz 4nn cfzo nnuz eleqtri fzouzsplit ax-mp eleq2i elun ctp fzo1to4tp bitri orbi1i vex eltp 3bitri 4p2e6 oveq2i 2exp6 eqtri oveq1i eqeq2i simpl oveq1 nn0cni - deccl mulid2i eqtrdi oveq1d 4p1e5 eqid decsuc adantl eqtrd ex cc0 6cn 2cn - 2nn0 6t2e12 mulcomli eqcomi 4cn 4t2e8 decmul10add 1nn0 8nn0 8p1e9 addid2i + deccl mullidi eqtrdi oveq1d 4p1e5 eqid decsuc adantl eqtrd ex cc0 6cn 2cn + 2nn0 6t2e12 mulcomli eqcomi 4cn 4t2e8 decmul10add 1nn0 8nn0 8p1e9 addlidi 0nn0 8cn decaddi 3nn0 6t3e18 mulcomi eqtr3i 4t3e12 2p1e3 decadd 3orim123d 3cn 9nn0 a1i com13 wb clt cr nn0rei nn0zi adantr mpbid sylbi biimtrid w3a fmtno4sqrt breq2i breq1 wn 6t4e24 4t4e16 zre decnncl nngt0i pm3.2i lemul1 @@ -777295,7 +777295,7 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . -> P = ; ; 1 9 3 ) $= ( cprime wcel c4 cfv wbr c6 c5 cdc wceq c1 c9 c3 cmul co 1nn0 3nn 3nn0 eqid caddc cfmtno cdvds csqrt cfl cle w3a c2 w3o fmtno4prmfac wi 5nn decnncl 1nn - 1lt5 1lt10 declti nprmi 5nn0 5cn mulid1i oveq1i 5p1e6 eqtri 5t3e15 decmul2c + 1lt5 1lt10 declti nprmi 5nn0 5cn mulridi oveq1i 5p1e6 eqtri 5t3e15 decmul2c eqtr4di eleq1d mtbiri pm2.21d 4nn0 4nn 1lt3 4t3e12 3t3e9 decmul1 ax-1 3jaoi id com12 3ad2ant1 mpd ) ABCZADUAEZUBFZAWCUCEUDEUEFZUFAGHIZJZAKUGIZLIZJZAKLI MIJZUHZWKAUIWBWDWLWKUJWEWLWBWKWGWBWKUJWJWKWGWBWKWGWBHKMIZNOZBCHWMWNUKKMPQUL @@ -777308,10 +777308,10 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . fmtno4nprmfac193 $p |- -. ; ; 1 9 3 || ( FermatNo ` 4 ) $= ( c1 c9 c3 c4 c6 c5 c7 1nn0 9nn0 deccl 3nn0 c2 co 5nn0 2nn0 7nn0 eqid caddc cdc c8 cfmtno cfv wbr cc0 3nn decnncl 1nn decnncl2 cmul 6nn0 4nn0 0nn0 8nn0 - cdvds mulid2i oveq1i 3p2e5 eqtri 9t3e27 decmul1c 3t3e9 decmul1 ax-1cn 5p1e6 + cdvds mullidi oveq1i 3p2e5 eqtri 9t3e27 decmul1c 3t3e9 decmul1 ax-1cn 5p1e6 3cn 5cn addcomli 6p1e7 8cn 7cn 8p7e15 decaddc 7p7e14 decaddci decsuc 9p5e14 4p1e5 7p1e8 9p3e12 decma2c 9cn 9p8e17 1p2e3 decaddi mulcomli decmul2c 2p1e3 - 9t9e81 decadd addid1i 10pos 1lt9 declt decltc ndvdsi fmtno4 breq2i mtbir + 9t9e81 decadd addridi 10pos 1lt9 declt decltc ndvdsi fmtno4 breq2i mtbir 9nn ) ABSZCSZDUAUBZUNUCXAEFSZFSZCSZGSZUNUCXAXFCCSZBSZAASZUDSZWTCABHIJZUEUFX GBCCKKJZIJXIAAHUGUFUHXCDSZLSZGXIUDXEGXAXHUIMXJXMLXCDEFUJNJZUKJZOJPAAHHJZULX GBXNGXAAGSZCSZXHWTCXKKJZXLIXHQPXRCAGHPJZKJCCXRCXAXMLFTSZXGXSKKYAKXGQXSQXTOF @@ -777364,7 +777364,7 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . fmtno5faclem1 $p |- ( ; ; ; ; ; ; 6 7 0 0 4 1 7 x. 4 ) = ; ; ; ; ; ; ; 2 6 8 0 1 6 6 8 $= ( c6 c7 cdc cc0 c4 c1 c2 4nn0 6nn0 7nn0 deccl 0nn0 1nn0 eqid 8nn0 2nn0 cmul - c8 co decaddi 6t4e24 4p2e6 7t4e28 decmul1c 4cn mul02i decmul1 0p1e1 mulid2i + c8 co decaddi 6t4e24 4p2e6 7t4e28 decmul1c 4cn mul02i decmul1 0p1e1 mullidi 4t4e16 ) ABCZDCZDCZECZFCZBGACZRCZDCZFCZACZACREGUOBCZHUNFUMEULDUKDABIJKZLKZL KZHKZMKJVANOPUTEAUOEQSGUSAURFUQDUPRGAPIKOKLKZMKIKHPUNFUTEEUOHVEMUONUMEUSAEF UNHVDHUNNIMURDFUMEQSFVFLMULDURDEUMHVCLUMNUKDUQDEULHVBLULNABUPREGUKHIJUKNOPG @@ -777374,8 +777374,8 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . fmtno5faclem2 $p |- ( ; ; ; ; ; ; 6 7 0 0 4 1 7 x. 6 ) = ; ; ; ; ; ; ; 4 0 2 0 2 5 0 2 $= ( c6 c7 cdc c4 c1 c2 6nn0 7nn0 deccl 0nn0 4nn0 1nn0 eqid 2nn0 cmul decmul1c - cc0 co 6cn decmul1 c5 c3 3nn0 6t6e36 6p4e10 decaddci2 7t6e42 mul02i addid2i - 3p1e4 2cn decaddi 4cn 6t4e24 mulcomli mulid2i 4p1e5 ) ABCZQCZQCZDCZECZBDQCZ + cc0 co 6cn decmul1 c5 c3 3nn0 6t6e36 6p4e10 decaddci2 7t6e42 mul02i addlidi + 3p1e4 2cn decaddi 4cn 6t4e24 mulcomli mullidi 4p1e5 ) ABCZQCZQCZDCZECZBDQCZ FCZQCZFCZUACZQCFADVBBCZGVAEUTDUSQURQABGHIZJIZJIZKIZLIHVHMNKVFDCZAVGVBAORDVF DVEFVDQVCFDQKJINIJIZNIZKIGKVAEVMAAVBGVLLVBMUTDVFDAFVAGVKKVAMKNVEQFUTAORFVNJ NUSQVEQAUTGVJJUTMURQVDQAUSGVIJUSMABVCFADURGGHURMNKUBADAAORDUCGKUDUJUEUFUGPA @@ -777385,8 +777385,8 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . fmtno5faclem3 $p |- ( ; ; ; ; ; ; ; ; 4 0 2 0 2 5 0 2 0 + ; ; ; ; ; ; ; 2 6 8 0 1 6 6 8 ) = ; ; ; ; ; ; ; ; 4 2 8 8 2 6 6 8 8 $= - ( c4 cc0 cdc c2 c5 c6 c8 c1 4nn0 0nn0 2nn0 5nn0 6nn0 8nn0 1nn0 eqid addid2i - deccl 2cn decadd decaddi 6cn 6p2e8 addcomli 8cn addid1i 5p1e6 ) ABCZDCZBCZD + ( c4 cc0 cdc c2 c5 c6 c8 c1 4nn0 0nn0 2nn0 5nn0 6nn0 8nn0 1nn0 eqid addlidi + deccl 2cn decadd decaddi 6cn 6p2e8 addcomli 8cn addridi 5p1e6 ) ABCZDCZBCZD CZECZBCZDCZBDFCZGCZBCZHCZFCZFCZGADCZGCZGCZDCZFCZFCZGCGUNBCZUTGCZUMDULBUKEUJ DUIBUHDABIJRZKRZJRZKRZLRZJRZKRJUSFURFUQHUPBUOGDFKMRZNRZJRZORZMRZMRNVGPVHPUM DUSFVFGUNUTVNKVSMUNPUTPULBURFVEFUMUSVMJVRMUMPUSPUKEUQHVDFULURVLLVQOULPURPUJ @@ -777399,8 +777399,8 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . = ( ; ; ; ; ; ; 6 7 0 0 4 1 7 x. ; ; 6 4 1 ) $= ( c4 cc0 c2 c6 c8 c1 co c7 c9 cmul 4nn0 2nn0 deccl 8nn0 6nn0 0nn0 7nn0 eqid cdc decadd c5 caddc cfmtno cfv 1nn0 fmtno5faclem3 deceq1i 9nn0 6p1e7 decsuc - 8p1e9 8p6e14 decaddci 7cn 2cn 7p2e9 addcomli addid1i 8p4e12 decaddc addid2i - 6cn fmtno5faclem2 eqcomi fmtno5faclem1 decmul10add nn0cni mulid1i 3eqtr4ri + 8p1e9 8p6e14 decaddci 7cn 2cn 7p2e9 addcomli addridi 8p4e12 decaddc addlidi + 6cn fmtno5faclem2 eqcomi fmtno5faclem1 decmul10add nn0cni mulridi 3eqtr4ri fmtno5 ) ABSCSBSCSUASBSCSZBSCDSESBSFSDSDSESZUBGZBSZDHSZBSZBSZASZFSZHSZUBGAC SZISZASZISZDSZHSZCSZISZHSVTDASZFSJGUAUCUDWAESZESZCSZDSZDSZESZESZBVSHWHHVNVT WOEWNEWMDWLDWKCWJEWAEACKLMZNMZNMZLMZOMZOMZNMZNMPVRFVQAVPBVOBDHOQMZPMZPMZKMZ @@ -777669,9 +777669,9 @@ be a Fermat number (i.e., a Fermat prime), see ~ 2pwp1prmfmtno . cc0 c2 deccl 9nn decnncl 8nn0 4nn0 1lt8 3lt10 9lt10 3decltc 3nn 1lt10 4t2e8 declti dec2dvds cdvds wbr 3ndvds4 3dvdsdec 3cn ax-1cn 3p1e4 addcomli breq2i df-9 bitri mtbir 3dvds2dec oveq1i 9cn 4cn 9p4e13 eqtri 4lt5 5p4e9 dec5dvds2 - 4nn 7nn 6nn eqid dec0h 7cn mulid1i 6cn addid2i 7p6e13 9t7e63 mulcomli 6p3e9 - oveq12i decaddi decma2c 6lt7 ndvdsi 1nn 2cn oveq2i nncni 1p2e3 mulid2i 00id - addid1i 7p2e9 3eqtri decmac 7lt10 nn0cni mul01i c5 5nn0 8p5e13 8t7e56 6lt10 + 4nn 7nn 6nn eqid dec0h 7cn mulridi 6cn addlidi 7p6e13 9t7e63 mulcomli 6p3e9 + oveq12i decaddi decma2c 6lt7 ndvdsi 1nn 2cn oveq2i nncni 1p2e3 mullidi 00id + addridi 7p2e9 3eqtri decmac 7lt10 nn0cni mul01i c5 5nn0 8p5e13 8t7e56 6lt10 8cn 5cn 2p1e3 6t2e12 decsuc 8p1e9 6t3e18 decrmac 2nn prmlem2 ) ABCZDCZYCDAB EFUAZUBUCAGBHDAEUDFUEIEUFUGUHUIYCDAABEUJUCZIEUKUMYCHGDYEUEULVDUNBYDUOUPZBYC UOUPZYHBHUOUPZUQYHBABJKZUOUPYIABEFURYJHBUOBAHUSUTVAVBZVCVEVFYGBYJDJKZUOUPYH @@ -777735,9 +777735,9 @@ proof is much longer (regarding size) than the proof of ~ 37prm (1810 c5 5nn0 deccl 7nn decnncl c4 8nn0 4nn0 2lt10 7lt10 3decltc 2nn 1lt10 declti 1lt8 3t2e6 df-7 dec2dvds cdvds wbr 3nn 3t3e9 oveq1i 9p1e10 ndvdsi 3dvds2dec 1nn 1lt3 1p2e3 7p3e10 addcomli breq2i bitri mtbir 2lt5 5p2e7 dec5dvds2 0nn0 - 7cn 3cn dec0h mulid1i 5cn addid2i oveq12i 7p5e12 6nn0 8t7e56 mulcomli 6p1e7 - 8cn decaddi decma2c 1lt7 6nn nn0cni ax-1cn 1p1e2 6cn 6lt10 9nn0 9cn mulid2i - 10nn 9p3e12 9t3e27 addid1i decmac 3pos declt 7t7e49 9p8e17 decaddci decrmac + 7cn 3cn dec0h mulridi 5cn addlidi oveq12i 7p5e12 6nn0 8t7e56 mulcomli 6p1e7 + 8cn decaddi decma2c 1lt7 6nn nn0cni ax-1cn 1p1e2 6cn 6lt10 9nn0 9cn mullidi + 10nn 9p3e12 9t3e27 addridi decmac 3pos declt 7t7e49 9p8e17 decaddci decrmac 8nn 4p1e5 8lt10 9nn 5p1e6 6p6e12 9t6e54 4p3e7 3lt9 2cn 5t2e10 dec10p 5t3e15 1lt2 decltc prmlem2 ) ABCZDCZYIDABEFUAZUBUCAGBUDDAEUEFUFHEUMUGUHUIYIDAABEUJ UCZHEUKULYIIJDYKKUNUOUPIYJUQURZIALCZUQURZIYNIAUSKVEIIMNZAONPAONYNYPPAOUTVAV @@ -777768,7 +777768,7 @@ proof is much longer (regarding size) than the proof of ~ 37prm (1810 m11nprm $p |- ( ( 2 ^ ; 1 1 ) - 1 ) = ( ; 8 9 x. ; 2 3 ) $= ( c2 c1 cdc co cc0 c4 c7 c8 c9 c3 cmul 2nn0 deccl 4nn0 8nn0 1nn0 eqid caddc c5 3nn0 cexp cmin 0nn0 2exp11 4p1e5 decsuc 8m1e7 decsubi 9nn0 7nn0 c6 2p2e4 - 8t2e16 oveq12i 6nn0 1p1e2 6p4e10 eqtri 8t3e24 oveq1i nn0cni addid1i decma2c + 8t2e16 oveq12i 6nn0 1p1e2 6p4e10 eqtri 8t3e24 oveq1i nn0cni addridi decma2c decaddci2 9t2e18 8p2e10 9t3e27 decmul2c decmul1c eqtr4i ) ABBCUADZBUBDAECZF CZGCHICZAJCZKDVMHGVLSCVKBVLFAELUCMZNMOPUDVLFSVMVPNUEVMQUFUGUHHIVMGVOVLVNAJL TMOUIVNQUJVPAJAEHVLFAVOVLLTLUCVOQZVLQONLHAKDZAARDZRDBUKCZFRDVLVRVTVSFRUMULU @@ -778627,7 +778627,7 @@ become clearer and sometimes also shorter (see, for example, ~ divgcdoddALTV m1expoddALTV $p |- ( N e. Odd -> ( -u 1 ^ N ) = -u 1 ) $= ( codd wcel c1 cneg cexp co cmin caddc cmul cc wceq oddz zcnd npcan1 eqcomd syl oveq2d a1i cz cc0 wne neg1ne0 peano2zm expp1zd oddm1eveni m1expevenALTV - neg1cn ceven oveq1d mulid2d eqtrd 3eqtrd ) ABCZDEZAFGUOADHGZDIGZFGUOUPFGZUO + neg1cn ceven oveq1d mullidd eqtrd 3eqtrd ) ABCZDEZAFGUOADHGZDIGZFGUOUPFGZUO JGZUOUNAUQUOFUNAKCZAUQLUNAAMZNUTUQAAOPQRUNUOUPUOKCUNUHSZUOUAUBUNUCSUNATCUPT CVAAUDQUEUNUSDUOJGUOUNURDUOJUNUPUICURDLAUFUPUGQUJUNUOVBUKULUM $. @@ -779381,7 +779381,7 @@ Perfect Number Theorem (revised) simp1d 1rp cc0 peano2rem expgt1 posdif elrp ltdiv2d div1d lelttrd eluz2b2 nnrp cpr crab cfz fzfid dvdsssfz1 ssfi ssrab2 prssi snssd simp2d dvdsmul2 simplrl nnne0d divcan2d iddvds simplrr incom disjsn2 eqtr3id prfi divdird - df-pr subdird mulid2d pncan3d divassd 3eqtr4d ccxp nnexpcl mulcom mulassd + df-pr subdird mullidd pncan3d divassd 3eqtr4d ccxp nnexpcl mulcom mulassd expp1 2nn codd isodd7 sylbi rpexp1i sgmmul syl13anc pncan 1sgm2ppw eqtr3d 2z 3eqtrd sgmnncl sgmval sselid cxp1d sumeq2dv remulcld ltaddrpd readdcld 1nn0 3eqtrrd condan elpri ralrimiva 1dvds orbi12d imbi12d rspcv syl3c ord @@ -779620,7 +779620,7 @@ pseudoprimes shall be composite (positive) integers, they must be $( 341 is the product of 11 and 31. (Contributed by AV, 3-Jun-2023.) $) 11t31e341 $p |- ( ; 1 1 x. ; 3 1 ) = ; ; 3 4 1 $= - ( c1 c3 c4 cdc 3nn0 1nn0 deccl eqid cmul co nn0cni mulid2i 3cn ax-1cn 3p1e4 + ( c1 c3 c4 cdc 3nn0 1nn0 deccl eqid cmul co nn0cni mullidi 3cn ax-1cn 3p1e4 addcomli decaddi decmul1c ) AABCDABADZBAADZBAEFGZFFTHFEBACASIJBEFESSUAKLZBA CMNOPQUBR $. @@ -779630,8 +779630,8 @@ pseudoprimes shall be composite (positive) integers, they must be ( c2 c3 c4 cdc cc0 co c1 3nn0 4nn0 deccl 2nn 1nn0 c8 2nn0 oveq1i caddc cmul c6 2cn eqtri cexp cmo c7 1nn decnncl 7nn0 0nn0 0z c5 8nn0 5nn0 3z 2exp5 5cn 6nn0 5t2e10 mulcomli 3p1e4 eqid decsuc decmulnc 3t3e9 9p1e10 4cn 3cn 4t3e12 - decmul2c mulid1i deceq12i 9nn0 3t2e6 6p6e12 decaddci 2t2e4 decmul1c 3eqtr4i - mod2xi 2t0e0 0p1e1 nncni mul02i 1t1e1 addid2i mulid2i modxp1i nn0cni 4t4e16 + decmul2c mulridi deceq12i 9nn0 3t2e6 6p6e12 decaddci 2t2e4 decmul1c 3eqtr4i + mod2xi 2t0e0 0p1e1 nncni mul02i 1t1e1 addlidi mullidi modxp1i nn0cni 4t4e16 c9 4t2e8 4p1e5 2p1e3 6t2e12 6p1e7 8cn 8t2e16 7cn 7t2e14 cr wcel crp cle wbr clt wceq 1re cn nnrp ax-mp 0le1 4nn 9re 1lt9 ltleii decltdi modid mp4an ) A BCDZEDZUAFXQGDZUBFGXSUBFZGAGUCDZEDZEXRGGXSXQGBCHIJZUDUEZKYAEGUCLUFJZUGJUHLL @@ -779721,7 +779721,7 @@ pseudoprimes shall be composite (positive) integers, they must be ( cn wcel cfppr cfv cexp co cmo wceq fpprbasnn c1 cmul syl2an adantr oveq1d wi wa cz sylbid cuz cprime wnel cmin w3a fpprel cn0 nnz eluz4nn nnm1nn0 syl c4 zexpcl zred crp nnrpd adantl modcld recnd 1cnd cc nncn cc0 nnne0 mulcand - oveq1 cr modmulmodr syl3anc eqeq1d zcnd mulcomd expm1t eqcomd eqtrd mulid1d + oveq1 cr modmulmodr syl3anc eqeq1d zcnd mulcomd expm1t eqcomd eqtrd mulridd wne eqeq12d biimpd syl5 sylbird a1d ex 3impd mpcom ) ACDZBAEFDZABGHZBIHZABI HZJZABKWFWGBULUAFDZBUBUCZABLUDHZGHZBIHZLJZUEWKABUFWFWLWMWQWKWFWLWMWQWKQZQWF WLRZWRWMWSWQAWPMHZALMHZJZWKWSWPLAWSWPWSWOBWSWOWFASDZWNUGDZWOSDWLAUHZWLBCDZX @@ -779742,7 +779742,7 @@ pseudoprimes shall be composite (positive) integers, they must be cexp fpprwppr jca simprll simprlr eluz4nn cn0 nnnn0d zexpcl moddvds syl3anc mpcom nnz cc nncn expm1t oveq1d nnm1nn0 syl mulsubfacd eqtrd breq2d zsubcld zcnd 1zzd dvdsmulgcd syl2anc eluzelz gcdcom syl2an eqeq1d biimpd imp oveq2d - mulid1d com23 expimpd impcom eluz4eluz2 modm1div mpbird mpbir3and impbid2 + mulridd com23 expimpd impcom eluz4eluz2 modm1div mpbird mpbir3and impbid2 ex c2 ) BAUACZDEZBAUBFGZBUCUDFGZBUEUFZHZAUGGZABUMCZBICABICEZHZHZWTXCXGXDWTX CABUHZXDWTXAXBABDJCZUMCZBICDEZUIZXCABUJZXMXCKXDXAXBXLUKULLVDWTXDXFXIABUNUOU OWSXHWTWSXHHZWTXAXBXLWSXAXBXGUPWSXAXBXGUQXOXLBXKDJCZMNZXHWSXQXCXGWSXQKZXAXG @@ -779786,8 +779786,8 @@ pseudoprimes shall be composite (positive) integers, they must be ( c8 cexp co cmo wceq cprime wcel wi wn c3 wa c9 cn 9nn eleq1 wbr cc0 caddc c1 cv cuz cfv wrex elexi oveq2 id oveq12d eqeq12d imbi12d notbid anbi12d cz wex cle 3z nnzi 3re 9re 3lt9 ltleii eluz2 mpbir3an 8nn 8nn0 0z 8exp8mod9 cr - 1nn0 crp clt 1re nnrp ax-mp 0le1 1lt9 modid mp4an eqtr4i 8p1e9 cmul addid2i - 8cn mul02i oveq1i mulid2i 3eqtr4i modxp1i 9nprm pm3.2i annim mpbi ceqsexv2d + 1nn0 crp clt 1re nnrp ax-mp 0le1 1lt9 modid mp4an eqtr4i 8p1e9 cmul addlidi + 8cn mul02i oveq1i mullidi 3eqtr4i modxp1i 9nprm pm3.2i annim mpbi ceqsexv2d 9cn df-rex mpbir ) BAUAZCDZWQEDZBWQEDZFZWQGHZIZJZAKUBUCZUDWQXEHZXDLZAUNXGMX EHZBMCDZMEDZBMEDZFZMGHZIZJZLAMMNOUEWQMFZXFXHXDXOWQMXEPXPXCXNXPXAXLXBXMXPWSX JWTXKXPWRXIWQMEWQMBCUFXPUGUHWQMBEUFUIWQMGPUJUKULXHXOXHKUMHMUMHKMUOQUPMOUQKM @@ -781490,7 +781490,7 @@ This is proven by Helfgott (see section 7.4 in [Helfgott] p. 70) for ( vm vo c7 clt wbr wcel wi codd c1 cc0 co cle cr c8 wa pm3.2i a1i com23 cn cv cgbo cdc c2 cexp oddz zred cn0 10re 2nn0 7nn decnncl nnnn0i reexpcl wo mp2an lelttric sylancl c3 cmul wral wrex tgoldbachlt wceq breq2 eleq1w - breq1 anbi12d imbi12d rspcv recni mulid2i 1re 8re 0le1 3nn decnncl2 10nn0 + breq1 anbi12d imbi12d rspcv recni mullidi 1re 8re 0le1 3nn decnncl2 10nn0 1lt8 nn0expcli nn0ge0i w3a nnzi 3pm3.2i 1lt10 3nn0 7nn0 0nn0 7lt10 decltc cz 2lt3 ltexp2a ltmul12a syl22anc eqbrtrrid remulcli adantl syl3anc mpand nnre lttr imp adantr 3jca lelttr syl mpan2d anim1i ancomd pm2.27 ex exp41 @@ -783271,7 +783271,7 @@ their group (addition) operations are equal for all pairs of elements of nn0mnd $p |- M e. Mnd $= ( vx vy vz ve wcel cv caddc co cn0 wceq wral wa nn0cn jca cc0 eqeq1d cvv cc cmnd wrex nn0addcl w3a 3anim123i 3expa addass syl ralrimiva rgen2 c0ex - wex eleq1 oveq1 oveq2 anbi12d ralbidv 0nn0 addid2d addid1d rgen ceqsexv2d + wex eleq1 oveq1 oveq2 anbi12d ralbidv 0nn0 addlidd addridd rgen ceqsexv2d pm3.2i df-rex mpbir cbs nn0ex grpbase ax-mp cplusg addex grpplusg ismnd cfv ) AUAGCHZDHZIJZKGZVQEHZIJVOVPVSIJIJLZEKMZNZDKMCKMZFHZVOIJZVOLZVOWDIJZ VOLZNZCKMZFKUBZNWCWKWBCDKKVOKGZVPKGZNZVRWAVOVPUCWNVTEKWNVSKGZNVOTGZVPTGZV @@ -784832,7 +784832,7 @@ Ring homomorphisms (extension) $( 2 is an even integer. (Contributed by AV, 12-Feb-2020.) $) 2even $p |- 2 e. E $= ( c2 cv cmul co wceq cz wrex crab wcel 2z cc 2cn c1 1zzd wb oveq2 rexbidv - eqeq2d adantl mulid1 eqcomd rspcedvd ax-mp eqeq1 elrab mpbir2an eleqtrri + eqeq2d adantl mulrid eqcomd rspcedvd ax-mp eqeq1 elrab mpbir2an eleqtrri ) EBFZEAFZGHZIZAJKZBJLZCEUQMEJMEUNIZAJKZNEOMZUSPUTUREEQGHZIZAQJUTRUMQIZUR VBSUTVCUNVAEUMQEGTUBUCUTVAEEUDUEUFUGUPUSBEJULEIUOURAJULEUNUHUAUIUJDUK $. @@ -784932,8 +784932,8 @@ Ring homomorphisms (extension) $( R is an (additive) monoid. (Contributed by AV, 11-Feb-2020.) $) 2zrngamnd $p |- R e. Mnd $= ( vy cmnd wcel csgrp cv caddc co wceq wa wral wrex cc0 adantl cz 0even id - 2zrngasgrp wb oveq1 eqeq1d ovanraleqv cmul crab elrabi eleq2s zcnd addid2 - cc c2 addid1 ralrimiva rspcedvd ax-mp 2zrngbas 2zrngadd ismnddef mpbir2an + 2zrngasgrp wb oveq1 eqeq1d ovanraleqv cmul crab elrabi eleq2s zcnd addlid + cc c2 addrid ralrimiva rspcedvd ax-mp 2zrngbas 2zrngadd ismnddef mpbir2an jca syl ) CHICJIAKZGKZLMZVGNZVGVFLMVGNOGDPZADQZABCDEFUCRDIZVKABDEUAVLVJRV GLMZVGNZVGRLMVGNZOZGDPZARDVLUBVFRNZVJVQUDVLVIVNGVGVFVGLDRVRVHVMVGVFRVGLUE UFUGSVLVPGDVGDIZVPVLVSVGUNIZVPVSVGVGTIVGBKUOVFUHMNATQZBTUIDWABVGTUJEUKULV @@ -785111,26 +785111,26 @@ Ring homomorphisms (extension) ( vb wcel wceq wa cfv cplusg co syl va vc cn cabl cmgp csgrp cv cmpo wral w3a crng ccrg wi cn0 nnnn0 zncrng crg crngring ring0cl eleq1a imp cznabel adantlr eqid cznrnglem mgpbas cnx cmulr cop csts fveq2i cvv czn fvexi cbs - mpoex mulrid setsid mp2an mgpplusg eqcomi wne ne0i adantl simpr copissgrp - c0 oveq1 ad3antlr cmnd adantr anim1i mndlid eqtrd eqidd weq simpr1 simpr2 - ringmnd ovmpod oveq12d ad3antrrr ringacl syl3anc 3eqtr4rd jca ralrimivvva - simpr3 mpdan plusgid plusgndxnmulrndx setsnid eqtr4i eqtri isrng sylibr - 3jca ) EUCNZDHOZPZFUDNZFUEQZUFNZUAUGZMUGZUBUGZGRQZSZABCCDUHZSZYDYEYISZYDY - FYISZYGSZOZYDYEYGSZYFYISZYLYEYFYISZYGSZOZPZUBCUIMCUIUACUIZUJZFUKNXTDCNZUU - BXRXSUUCXRGULNZXSUUCUMZXREUNNUUDEUOEGIUPTZUUDGUQNZUUEGURZUUGHCNUUECGHJLUS - HCDUTTTTVAXTUUCPZYAYCUUAXRUUCYAXSABCDEFGIJKVBVCUUIABCDYBCFYBYBVDZABCDEFGI - JKVEZVFYIYBRQGVGVHQZYIVIVJSZYIYBFUUMUEKVKGVLNYIVLNYIUUMVHQZOGEVMIVNABCCDC - GVOJVNZUUOVPVLYIVHVLGVQVRVSZVTWAUUCCWGWBXTCDWCWDXTUUCWEZWFUUIYTUAMUBCCCUU - IYDCNZYECNZYFCNZUJZPZYNYSUVBDDYGSZDYMYJUVBUVCHDYGSZDXSUVCUVDOXRUUCUVADHDY - GWHWIUVBGWJNZUUCPZUVDDOUUIUVFUVAXTUVEUUCXRUVEXSXRUUGUVEXRUUDUUGUUFUUHTZGW - STWKWLWKCYGGDHJYGVDZLWMTWNZUVBYKDYLDYGUVBABYDYECCDDYICUVBYIWOZUVBAUAWPZBM - WPPPDWOUUIUURUUSUUTWQZUUIUURUUSUUTWRZUUIUUCUVAUUQWKZWTUVBABYDYFCCDDYICUVJ - UVBUVKBUBWPZPPDWOUVLUUIUURUUSUUTXHZUVNWTZXAUVBABYDYHCCDDYICUVJUVBUVKBUGYH - OPPDWOUVLUVBUUGUUSUUTYHCNXRUUGXSUUCUVAUVGXBZUVMUVPCYGGYEYFJUVHXCXDUVNWTXE - UVBUVCDYRYPUVIUVBYLDYQDYGUVQUVBABYEYFCCDDYICUVJUVBAMWPUVOPPDWOUVMUVPUVNWT - XAUVBABYOYFCCDDYICUVJUVBAUGYOOUVOPPDWOUVBUUGUURUUSYOCNUVRUVLUVMCYGGYDYEJU - VHXCXDUVPUVNWTXEXFXGXQXIUAMUBCYGFYIYBUUKUUJYGUUMRQFRQYIUULRGXJXKXLFUUMRKV - KXMYIUUNFVHQUUPUUMFVHFUUMKWAVKXNXOXP $. + mpoex mulridx setsid mp2an mgpplusg eqcomi c0 ne0i adantl simpr copissgrp + wne oveq1 ad3antlr ringmnd adantr anim1i mndlid eqtrd eqidd simpr1 simpr2 + cmnd ovmpod simpr3 oveq12d ad3antrrr ringacl syl3anc 3eqtr4rd ralrimivvva + weq 3jca mpdan plusgid plusgndxnmulrndx setsnid eqtr4i eqtri isrng sylibr + jca ) EUCNZDHOZPZFUDNZFUEQZUFNZUAUGZMUGZUBUGZGRQZSZABCCDUHZSZYDYEYISZYDYF + YISZYGSZOZYDYEYGSZYFYISZYLYEYFYISZYGSZOZPZUBCUIMCUIUACUIZUJZFUKNXTDCNZUUB + XRXSUUCXRGULNZXSUUCUMZXREUNNUUDEUOEGIUPTZUUDGUQNZUUEGURZUUGHCNUUECGHJLUSH + CDUTTTTVAXTUUCPZYAYCUUAXRUUCYAXSABCDEFGIJKVBVCUUIABCDYBCFYBYBVDZABCDEFGIJ + KVEZVFYIYBRQGVGVHQZYIVIVJSZYIYBFUUMUEKVKGVLNYIVLNYIUUMVHQZOGEVMIVNABCCDCG + VOJVNZUUOVPVLYIVHVLGVQVRVSZVTWAUUCCWBWGXTCDWCWDXTUUCWEZWFUUIYTUAMUBCCCUUI + YDCNZYECNZYFCNZUJZPZYNYSUVBDDYGSZDYMYJUVBUVCHDYGSZDXSUVCUVDOXRUUCUVADHDYG + WHWIUVBGWRNZUUCPZUVDDOUUIUVFUVAXTUVEUUCXRUVEXSXRUUGUVEXRUUDUUGUUFUUHTZGWJ + TWKWLWKCYGGDHJYGVDZLWMTWNZUVBYKDYLDYGUVBABYDYECCDDYICUVBYIWOZUVBAUAXGZBMX + GPPDWOUUIUURUUSUUTWPZUUIUURUUSUUTWQZUUIUUCUVAUUQWKZWSUVBABYDYFCCDDYICUVJU + VBUVKBUBXGZPPDWOUVLUUIUURUUSUUTWTZUVNWSZXAUVBABYDYHCCDDYICUVJUVBUVKBUGYHO + PPDWOUVLUVBUUGUUSUUTYHCNXRUUGXSUUCUVAUVGXBZUVMUVPCYGGYEYFJUVHXCXDUVNWSXEU + VBUVCDYRYPUVIUVBYLDYQDYGUVQUVBABYEYFCCDDYICUVJUVBAMXGUVOPPDWOUVMUVPUVNWSX + AUVBABYOYFCCDDYICUVJUVBAUGYOOUVOPPDWOUVBUUGUURUUSYOCNUVRUVLUVMCYGGYDYEJUV + HXCXDUVPUVNWSXEXQXFXHXIUAMUBCYGFYIYBUUKUUJYGUUMRQFRQYIUULRGXJXKXLFUUMRKVK + XMYIUUNFVHQUUPUUMFVHFUUMKWAVKXNXOXP $. $( The ring constructed from a ` Z/nZ ` structure with ` 1 < n ` by replacing the (multiplicative) ring operation by a constant operation is @@ -785138,17 +785138,17 @@ replacing the (multiplicative) ring operation by a constant operation is cznnring $p |- ( ( N e. ( ZZ>= ` 2 ) /\ C e. B ) -> X e/ Ring ) $= ( c2 cfv wcel co cmulr wceq cvv c1 va vb vc cuz wa wn wnel cgrp cmgp cmnd crg cv cplusg cnx cmpo cop csts wral w3a eqid cznrnglem mgpbas fveq2i czn - fvexi cbs mpoex mulrid setsid mp2an mgpplusg eqcomi simpr chash clt eluz2 - wbr cz cle 1lt2 wi 1red 2re a1i zre ltletr syl3anc expcomd 3imp mpi sylbi - cr cn eluz2nn znhash syl breqtrrd adantr copisnmnd df-nel sylib intn3an2d - isring sylnibr sylibr ) EMUDNOZDCOZUEZFUKOZUFFUKUGXHFUHOZFUINZUJOZUAULZUB - ULZUCULZFUMNZPGUNQNABCCDUOZUPUQPZQNZPXMXNXSPXMXOXSPZXPPRXMXNXPPXOXSPXTXNX - OXSPXPPRUEUCCURUBCURUACURZUSXIXHXLXJYAXHXKUJUGXLUFXHABCDXKCFXKXKUTZABCDEF - GIJKVAZVBXQXKUMNXRXQXKFXRUIKVCGSOXQSOXQXSRGEVDIVEABCCDCGVFJVEZYDVGSXQQSGV - HVIVJVKVLXFXGVMXFTCVNNZVOVQXGXFTEYEVOXFMVROZEVROZMEVSVQZUSZTEVOVQZMEVPYIT - MVOVQZYJVTYFYGYHYKYJWAZYGYHYLWAWAYFYGYKYHYJYGTWLOMWLOZEWLOYKYHUEYJWAYGWBY - MYGWCWDEWETMEWFWGWHWDWIWJWKXFEWMOYEEREWNCEGIJWOWPWQWRWSXKUJWTXAXBUAUBUCCX - PFXSXKYCYBXPUTXRFQFXRKVLVCXCXDFUKWTXE $. + fvexi cbs mpoex mulridx setsid mp2an mgpplusg eqcomi simpr clt wbr cz cle + chash eluz2 1lt2 wi cr 1red 2re a1i zre ltletr syl3anc expcomd 3imp sylbi + cn eluz2nn znhash breqtrrd adantr copisnmnd df-nel sylib intn3an2d isring + mpi syl sylnibr sylibr ) EMUDNOZDCOZUEZFUKOZUFFUKUGXHFUHOZFUINZUJOZUAULZU + BULZUCULZFUMNZPGUNQNABCCDUOZUPUQPZQNZPXMXNXSPXMXOXSPZXPPRXMXNXPPXOXSPXTXN + XOXSPXPPRUEUCCURUBCURUACURZUSXIXHXLXJYAXHXKUJUGXLUFXHABCDXKCFXKXKUTZABCDE + FGIJKVAZVBXQXKUMNXRXQXKFXRUIKVCGSOXQSOXQXSRGEVDIVEABCCDCGVFJVEZYDVGSXQQSG + VHVIVJVKVLXFXGVMXFTCVRNZVNVOXGXFTEYEVNXFMVPOZEVPOZMEVQVOZUSZTEVNVOZMEVSYI + TMVNVOZYJVTYFYGYHYKYJWAZYGYHYLWAWAYFYGYKYHYJYGTWBOMWBOZEWBOYKYHUEYJWAYGWC + YMYGWDWEEWFTMEWGWHWIWEWJXBWKXFEWLOYEEREWMCEGIJWNXCWOWPWQXKUJWRWSWTUAUBUCC + XPFXSXKYCYBXPUTXRFQFXRKVLVCXAXDFUKWRXE $. $} @@ -788043,7 +788043,7 @@ Basic algebraic structures (extension) ztprmneprm $p |- ( ( Z e. ZZ /\ A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) $= ( wcel cprime cmul co wceq wi wa wo cc0 adantr eqeq1d adantl ex syl clt wbr - sylbi cz cn0 cr cneg cn elznn0nn elnn0 c1 c2 cuz cfv elnn1uz2 oveq1 mulid2d + sylbi cz cn0 cr cneg cn elznn0nn elnn0 c1 c2 cuz cfv elnn1uz2 oveq1 mullidd prmz zcnd biimpd sylbid prmuz2 nprm sylan2 eleq1 notbid pm2.24 com12 syl6bi wn com3l mpcom jaoi prmnn nnred mul02lem2 wne elnnne0 eqneqall eqcoms com23 elnnz lt0neg1 nngt0d simpr anim12ci orcd simprl mul2lt0bi mpbird wb nn0nlt0 @@ -788130,7 +788130,7 @@ The binomial coefficient operation (extension) sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( N _C k ) ) = 0 ) $= ( cn wcel cc0 cexp co c1 cneg caddc cmul csu cc wceq syl oveq1d cz syl2an cn0 eqtrd cfz cv cbc 1cnd negid eqcomd 0exp negcld nnnn0 binom syl3anc wa - cmin nnz elfzelz zsubcl 1exp neg1cn a1i elfznn0 expcl mulid2d oveq2d bccl + cmin nnz elfzelz zsubcl 1exp neg1cn a1i elfznn0 expcl mullidd oveq2d bccl nn0cnd mulcomd sumeq2dv 3eqtr3rd ) BCDZEBFGHHIZJGZBFGZEEBUAGZVJAUBZFGZBVN UCGZKGZALZVIEVKBFVIHMDZEVKNVIUDZVSVKEHUEUFOPBUGVIVLVMVPHBVNUMGZFGZVOKGZKG ZALZVRVIVSVJMDZBSDZVLWENVTVIHVTUHBUIZHVJABUJUKVIVMWDVQAVIVNVMDZULZWDVPVOK @@ -788153,9 +788153,9 @@ instead of the binomial theorem ( ~ binom ). (Contributed by AV, sumeq2dv neg1cn mulcld 1z zsubcld fsumadd 0zd nnz oveq12d fsumshft oveq1i oveq2 0p1e1 sumeq1d cuz cfv elnnuz biimpi elfznn expcld elfzel1 oveq1 clt fsump1 wbr nncn pncan1 nnnn0 eqeltrd nn0zd nnre ltm1 breqtrrd olcd bcval4 - wo syl3anc mul01d weq cbvsumv oveq1d fsumcl addid1d 3eqtrd elnn0uz fsum1p + wo syl3anc mul01d weq cbvsumv oveq1d fsumcl addridd 3eqtrd elnn0uz fsum1p cr sylib exp0d 0z zsubcl mp2an 0re mp1i orcd npcan1 expp1 mulcomd mulassd - zcnd sumeq12rdv fsummulc2 eqtr4d addid2d mulm1d negidd ) BUADZEBFGZHUBZAU + zcnd sumeq12rdv fsummulc2 eqtr4d addlidd mulm1d negidd ) BUADZEBFGZHUBZAU CZIGZBYMJGZKGZASYKYNBHLGZYMJGZKGZYNYQYMHLGZJGZKGZMGZASZHBFGZYLYTIGZUUAKGZ ASZYLUUHKGZMGZEYJYKYPUUCAYJYMYKDZUDZYPYNYRUUAMGZKGUUCUULYOUUMYNKUULUUMYOU UKYJYMNDZUUMYOOYMEBUEZYMBUFUGUHPUULYNYRUUAUUKYNQDZYJHQDZUUKUUPUIUUQYLQDZY @@ -791678,7 +791678,7 @@ Differences between (left) modules and (left) vector spaces equal to the dividend (the real number). (Contributed by AV, 26-May-2020.) $) divge1b $p |- ( ( A e. RR+ /\ B e. RR ) -> ( A <_ B <-> 1 <_ ( B / A ) ) ) $= - ( crp wcel cr wa cle c1 cmul co cdiv wceq rpcn mulid2d eqcomd adantr breq1d + ( crp wcel cr wa cle c1 cmul co cdiv wceq rpcn mullidd eqcomd adantr breq1d wbr cc0 clt wb 1red simpr rpregt0 lemuldiv syl3anc bitrd ) ACDZBEDZFZABGRHA IJZBGRZHBAKJGRZUJAUKBGUHAUKLUIUHUKAUHAAMNOPQUJHEDUIAEDSATRFZULUMUAUJUBUHUIU CUHUNUIAUDPHBAUEUFUG $. @@ -791687,7 +791687,7 @@ equal to the dividend (the real number). (Contributed by AV, the divisor (the positive real number) is less than the dividend (the real number). (Contributed by AV, 30-May-2020.) $) divgt1b $p |- ( ( A e. RR+ /\ B e. RR ) -> ( A < B <-> 1 < ( B / A ) ) ) $= - ( crp wcel cr wa clt wbr c1 cmul cdiv rpcn adantr mulid2d eqcomd breq1d cc0 + ( crp wcel cr wa clt wbr c1 cmul cdiv rpcn adantr mullidd eqcomd breq1d cc0 co cc wb 1red simpr rpregt0 ltmuldiv syl3anc bitrd ) ACDZBEDZFZABGHIAJRZBGH ZIBAKRGHZUIAUJBGUIUJAUIAUGASDUHALMNOPUIIEDUHAEDQAGHFZUKULTUIUAUGUHUBUGUMUHA UCMIBAUDUEUF $. @@ -792437,7 +792437,7 @@ Logarithm to an arbitrary base (extension) -> ( ( B logb X ) < 1 <-> X < B ) ) $= ( c2 cuz cfv wcel crp wa clogb co c1 clt wbr cr wb relogcl syl adantr bitrd clog cdiv relogbval breq1d cmul adantl 1red eluz2nn eluz2gt1 loggt0b mpbird - cc0 nnrpd ltdivmul syl3anc recnd mulid1d breq2d anim2i ancoms logltb bicomd + cc0 nnrpd ltdivmul syl3anc recnd mulridd breq2d anim2i ancoms logltb bicomd jca wceq ) ACDEFZBGFZHZABIJZKLMBTEZATEZUAJZKLMZBALMZVFVGVJKLABUBUCVFVKVHVIK UDJZLMZVLVFVHNFZKNFVINFZUKVILMZHZVKVNOVEVOVDBPUEVFUFVDVRVEVDVPVQVDAGFZVPVDA AUGULZAPQZVDVQKALMZAUHVDVSVQWBOVTAUIQUJVBRVHKVIUMUNVFVNVHVILMZVLVFVMVIVHLVD @@ -793295,7 +793295,7 @@ last digit of the integer part (for ` b = ` 10 the first digit before 2nn0 mulcld oveq1 oveq12d 2cn exp0 oveq2i eqtrdi fsum1p 0dig2nn0e syl2anr ax-mp cr 1re mul02lem2 1z eqeltri 2cnd elfznn nnnn0d oveq1i eleq2s expcld 0p1e1 fsumshftm ad4antr elfzonn0 dignn0ehalf expp1d elfzoelz 2re reexpcld - recnd w3a mulass eqtrd 0cn fzoval oveq2d fzofi peano2zd peano2nn0 addid2d + recnd w3a mulass eqtrd 0cn fzoval oveq2d fzofi peano2zd peano2nn0 addlidd cfn fsumcl fsummulc1 3eqtr4d 3eqtrd weq cbvsumv ex com25 biimpac wne 2ne0 divcan1d ad3antlr 3eqtrrd imim2i com13 com23 exp31 com14 expdcom mpid mpd sylbid impcom imp ) DUBZUCFZUURGUAHZUCFZIZBUBZUCFZAUBZUDUEUVCJZUVEKUVCUFH @@ -793451,7 +793451,7 @@ last digit of the integer part (for ` b = ` 10 the first digit before ( vy cv wceq cc0 cfzo co c2 csu wi cn0 wral c1 eqeq2 oveq2 sumeq1d wcel cc cblen cfv cdig cexp cmul csn caddc fzo01 eqtrdi eqeq2d imbi12d ralbidv vx weq wa 0cnd cpnf cico 2nn a1i 0zd nn0rp0 digvalnn0 syl3anc nn0cnd 1cnd - cn cz mulcld jca adantr oveq1 2cn exp0 ax-mp oveq12d sumsn syl mulid1d wo + cn cz mulcld jca adantr oveq1 2cn exp0 ax-mp oveq12d sumsn syl mulridd wo cpr blen1b biimpa vex elpr sylibr 0dig2pr01 3eqtrrd nn0sumshdiglem1 nnind ex rgen ) CEZUAUBZUMEZFZWMGWOHIZAEZWMJUCUBZIZJWRUDIZUEIZAKZFZLZCMNWNOFZWM GUFZXBAKZFZLZCMNWNDEZFZWMGXKHIZXBAKZFZLZCMNWNXKOUGIZFZWMGXQHIZXBAKZFZLZCM @@ -794214,7 +794214,7 @@ According to Wikipedia ("Ackermann function", 8-May-2024, itcovalpclem1 $p |- ( C e. NN0 -> ( ( IterComp ` F ) ` 0 ) = ( n e. NN0 |-> ( n + ( C x. 0 ) ) ) ) $= ( cn0 wcel cc0 citco cfv cv cmpt cmul co caddc cvv wral nn0ex ovexd nn0cn - wceq rgen itcoval0mpt mp2an wa mul01d adantr oveq2d addid1d adantl eqtr2d + wceq rgen itcoval0mpt mp2an wa mul01d adantr oveq2d addridd adantl eqtr2d mpteq2dva eqtrid ) AEFZGCHIIZBEBJZKZBEUOAGLMZNMZKEOFUOANMZOFZBEPUNUPTQUTB EUOEFZUOANRUAEUSBCOODUBUCUMBEUOURUMVAUDZURUOGNMZUOVBUQGUONUMUQGTVAUMAASUE UFUGVAVCUOTUMVAUOUOSUHUIUJUKUL $. @@ -794228,7 +794228,7 @@ According to Wikipedia ("Ackermann function", 8-May-2024, = ( n e. NN0 |-> ( n + ( C x. ( y + 1 ) ) ) ) ) ) $= ( vm cv cn0 wcel wa cfv cmul co caddc cmpt wceq c1 cvv simpr nn0cnd citco ccom nn0ex mptex eqeltri simpl itcovalsucov mp3an2ani nn0mulcld nn0addcld - simplr adantr eqidd oveq1 cbvmptv eqtri a1i fmptco addassd mulid1d adantl + simplr adantr eqidd oveq1 cbvmptv eqtri a1i fmptco addassd mulridd adantl nn0cn eqcomd oveq2d 1cnd adddid eqtr4d eqtrd mpteq2dva ex ) AGZHIZBHIZJZV KDUAKZKCHCGZBVKLMZNMZOZPZVKQNMZVOKZCHVPBWALMZNMZOZPVNVTJWBDVSUBZWEDRIVNVL VTVTWBWFPDCHVPBNMZOZRECHWGUCUDUEVLVMUFZVNVTSDVSRVKUGUHVNWFWEPVTVNWFCHVRBN @@ -794287,7 +794287,7 @@ According to Wikipedia ("Ackermann function", 8-May-2024, = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ 0 ) ) - C ) ) ) $= ( cn0 wcel cc0 citco cfv cmpt caddc co c2 cmul cvv wa wceq nn0cnd eqtrd c1 cv cexp cmin wral nn0ex ovexd rgen pm3.2i itcoval0mpt mp1i simpr simpl - 2nn0 numexp0 a1i oveq2d nn0addcld mulid1d mvrraddd eqcomd mpteq2dva ) AEF + 2nn0 numexp0 a1i oveq2d nn0addcld mulridd mvrraddd eqcomd mpteq2dva ) AEF ZGCHIIZBEBUAZJZBEVDAKLZMGUBLZNLZAUCLZJEOFZMVDNLZAKLZOFZBEUDZPVCVEQVBVJVNU EVMBEVDEFZVKAKUFUGUHEVLBCOODUIUJVBBEVDVIVBVOPZVIVDVPVHVDAVPVDVBVOUKZRVPAV BVOULZRVPVHVFTNLVFVPVGTVFNVGTQVPMUMUNUOUPVPVFVPVFVPVDAVQVRUQRURSUSUTVAS @@ -794374,7 +794374,7 @@ According to Wikipedia ("Ackermann function", 8-May-2024, ackval1 $p |- ( Ack ` 1 ) = ( n e. NN0 |-> ( n + 2 ) ) $= ( vi c1 cack cfv cc0 caddc co cn0 cv cmpt c2 wcel wceq 1nn0 nn0cn syl a1i cc 3eqtrd citco 1e0p1 fveq2i 0nn0 ackvalsuc1mpt ax-mp peano2nn0 itcovalpc - cmul ackval0 sylancl mulid2d oveq2d mpteq2dv eqtrd fveq1d cvv eqidd oveq1 + cmul ackval0 sylancl mullidd oveq2d mpteq2dv eqtrd fveq1d cvv eqidd oveq1 adantl ovexd fvmptd peano2cn addcomd addassd 1p1e2 oveq2i mpteq2ia 3eqtri 1cnd ) CDEFCGHZDEZAICAJZCGHZFDEZUAEEZEZKZAIVMLGHZKCVKDUBUCFIMVLVRNUDAFUEU FAIVQVSVMIMZVQCBIBJZVNGHZKZECVNGHZVSVTCVPWCVTVPBIWACVNUIHZGHZKZWCVTVNIMZC @@ -794488,7 +794488,7 @@ According to Wikipedia ("Ackermann function", 8-May-2024, ( ( Ack ` 1 ) ` 1 ) , ( ( Ack ` 1 ) ` 2 ) >. = <. 2 , 3 , 4 >. $= ( vn c1 cack cfv cn0 cv c2 caddc co cmpt wceq cc0 cotp c3 oveq1 eqtrdi wcel - c4 a1i fvmptd3 ackval1 2cn addid2i 0nn0 2nn0 1p2e3 1nn0 3nn0 2p2e4 oteq123d + c4 a1i fvmptd3 ackval1 2cn addlidi 0nn0 2nn0 1p2e3 1nn0 3nn0 2p2e4 oteq123d 4nn0 ax-mp ) BCDZAEAFZGHIZJKZLUMDZBUMDZGUMDZMGNRMKAUAZUPUQGURNUSRUPALUOGEUM EUTUNLKUOLGHIGUNLGHOGUBUCPLEQUPUDSGEQUPUESZTUPABUONEUMEUTUNBKUOBGHINUNBGHOU FPBEQUPUGSNEQUPUHSTUPAGUOREUMEUTUNGKUOGGHIRUNGGHOUIPVAREQUPUKSTUJUL $. @@ -794499,7 +794499,7 @@ According to Wikipedia ("Ackermann function", 8-May-2024, ( ( Ack ` 2 ) ` 1 ) , ( ( Ack ` 2 ) ` 2 ) >. = <. 3 , 5 , 7 >. $= ( vn c2 cfv cn0 cmul co c3 caddc cc0 c1 c5 oveq2 oveq1d oveq1i eqtri eqtrdi - wceq c7 wcel a1i cack cmpt cotp ackval2 2t0e0 3cn addid2i 0nn0 3nn0 fvmptd3 + wceq c7 wcel a1i cack cmpt cotp ackval2 2t0e0 3cn addlidi 0nn0 3nn0 fvmptd3 cv 2t1e2 2cn 3p2e5 addcomli 1nn0 5nn0 2t2e4 4p3e7 2nn0 7nn0 oteq123d ax-mp c4 ) BUACZADBAUKZEFZGHFZUBQZIVECZJVECZBVECZUCGKRUCQAUDZVIVJGVKKVLRVIAIVHGDV EDVMVFIQZVHBIEFZGHFZGVNVGVOGHVFIBELMVPIGHFGVOIGHUENGUFUGOPIDSVIUHTGDSVIUITU @@ -794513,7 +794513,7 @@ According to Wikipedia ("Ackermann function", 8-May-2024, ( ( Ack ` 3 ) ` 1 ) , ( ( Ack ` 3 ) ` 2 ) >. = <. 5 , ; 1 3 , ; 2 9 >. $= ( vn c3 cfv cn0 c2 caddc co cexp cmin wceq cc0 c1 c5 cdc 3cn eqtrdi c8 wcel - a1i c4 cack cv cmpt cotp ackval3 oveq1 addid2i oveq2d oveq1d cu2 oveq1i 5cn + a1i c4 cack cv cmpt cotp ackval3 oveq1 addlidi oveq2d oveq1d cu2 oveq1i 5cn c9 5p3e8 eqcomi mvrraddi eqtri 0nn0 5nn0 fvmptd3 ax-1cn 3p1e4 addcomli 1nn0 2exp4 3nn0 deccl nn0cni eqid 3p3e6 decaddi 2cn 3p2e5 2exp5 2nn0 9nn0 9p3e12 c6 2p1e3 decaddci oteq123d ax-mp ) BUACZADEAUBZBFGZHGZBIGZUCJZKWCCZLWCCZEWC @@ -794688,7 +794688,7 @@ Elementary geometry (extension) affineid.x $e |- ( ph -> T e. CC ) $. $( Identity of an affine combination. (Contributed by AV, 2-Feb-2023.) $) affineid $p |- ( ph -> ( ( ( 1 - T ) x. A ) + ( T x. A ) ) = A ) $= - ( c1 cmin co cmul caddc 1cnd subdird mulid2d oveq1d eqtrd mulcld npcand ) + ( c1 cmin co cmul caddc 1cnd subdird mullidd oveq1d eqtrd mulcld npcand ) AFCGHBIHZCBIHZJHBSGHZSJHBARTSJARFBIHZSGHTAFCBAKEDLAUABSGABDMNONABSDACBEDP QO $. $} @@ -794698,7 +794698,7 @@ Elementary geometry (extension) 1subrec1sub $p |- ( ( A e. CC /\ A =/= 1 ) -> ( 1 - ( 1 / ( 1 - A ) ) ) = ( A / ( A - 1 ) ) ) $= ( cc wcel c1 wne wa cmin co cdiv cmul 1cnd simpl subcld simpr necomd eqcomd - subne0d oveq1d cneg eqtrd divcan4d mulcld divsubdird mulid2d adantr negsubd + subne0d oveq1d cneg eqtrd divcan4d mulcld divsubdird mullidd adantr negsubd negcl caddc mvrladdd divneg2d divnegd negsubdi2d oveq2d 3eqtr3d 3eqtr2d ) A BCZADEZFZDDDAGHZIHZGHDUSJHZUSIHZUTGHVADGHZUSIHZAADGHZIHZURDVBUTGURVBDURDUSU RKZURDAVGUPUQLZMZURDAVGVHURADUPUQNOQZUAPRURVADUSURDUSVGVIUBVGVIVJUCURVDASZU @@ -795281,7 +795281,7 @@ left module (or any extended structure having a base set, an addition, subne0d sylbi ad2antlr renegcld adantl eqcom 3ad2ant2 ad2antrr ffvelcdmda wf elmapi eldifi 3ad2ant3 mulcld subcld subadd2d bitr4id divmuld divrec2d syl divsubdird div23d bitrid divcld mulneg1d eqcomd oveq2d reccld negsubd - eqtrd subnegd cc muldivdir syl112anc mulid1d npcand 3eqtr2d biimpd sylbid + eqtrd subnegd cc muldivdir syl112anc mulridd npcand 3eqtr2d biimpd sylbid 3eqtr3d ralimdva imp rspcedvd rexlimdva2 0xr 1re elioc2 gt0ne0 negsubdi2d 3adant3 rereccld eqtr3d eqeq1d a1i 3bitr3d subaddd addcld elioc1 divmul2d anim1i divdird dividd comraddd 3jaod ) GUAJZAUFZKLGUBMZUCMZJZBUFZUUMUUKUD @@ -795356,7 +795356,7 @@ left module (or any extended structure having a base set, an addition, negsubd negsubdi2d 3eqtrd 3eqtr2d sylibrd ralimdva imp exp31 com23 simplr rspcedvd ex recrecd eqcomd sublt0d negelrpd le0neg1d divge0d div2negd crp breqtrrd elrpd rpreccld ltsubrpd elicod gtned subeq0ad sub32d subidd 0cnd - posdifd addid1d mulid1d recid2d mvllmuld eqtr3d eqeltrrd divne1d 3ad2ant2 + posdifd addridd mulridd recid2d mvllmuld eqtr3d eqeltrrd divne1d 3ad2ant2 subdivcomb2 syl112anc eldifi syl 3ad2ant3 subadd2d divneg2d 3mix2d 3mix1d recgt0d recgt1i simprr mpdan 3jca 1re pm3.2i elioc2 gt0ne0 recne0d dividd mp1i syld divnegd mulneg1d addcomd 3mix3d 3jaod mpid rexlimdva ) GUAJZAUF @@ -795494,7 +795494,7 @@ two different points in a left module (see ~ rrxlines ). (Contributed w3a fveq1 necon3i adantl rrx2line syl3an3 oveq2 oveq2d eqcoms 3ad2ant3 cc rrx2pxel recnd 3ad2ant1 recn affineid eqtrd eqeq2d anbi1d rexbidva wi a1i simpl rexlimdva cdiv rrx2pyel resubcld 3ad2ant2 cc0 simpr necomd redivcld - subne0d oveq1d oveq1 oveq12d anbi2d mulid2d subcld 3adant3 pncan3d eqtr2d + subne0d oveq1d oveq1 oveq12d anbi2d mullidd subcld 3adant3 pncan3d eqtr2d wb divcan1d 1cnd submuladdmuld eqtr4d jca rspcedvd impbid bitrd rabbidva ex ) EALZFALZMEUBZMFUBZNZUCEUBZUCFUBZUDZOZUIZEFDPZMGUEZUBZMUAUEZUFPZXDQPZ XOXEQPZRPZNZUCXMUBZXPXGQPZXOXHQPZRPZNZOZUASUGZGAUHZXNXDNZGAUHXJXBXCEFUDZX @@ -795778,7 +795778,7 @@ between the two points ("point-slope form"), sometimes also written as 0re eqid impbid1 adantl xor3 sylibr simp1 recnd mul01d simp2 oveq12d 00id eqeq1d eqcom bitrdi adantr bibi1d mtbird fveq1 1ex c0ex 1ne2 fvpr1g mp3an oveq2d fvpr2g bibi12d notbid rspcev sylancr expcom notnotb cneg cdiv 1red - 2ex sylbir renegcld simprl redivcld syl2anc ax-1ne0 neii mpbir mulid1d cc + 2ex sylbir renegcld simprl redivcld syl2anc ax-1ne0 neii mpbir mulridd cc 2th negcld divcan2d negidd eqtrd simprr eqeq12d mtbiri ovex ex nne bicomi 1re oveq1 ax-1cn mul02i id eqeqan12d syl2anbr jaoi3 orcoms sylbi biimtrid a1d imp rexnal syl6ib con4d df-3an syl6ibr ) AUAIZBUAIZCUAIZUBZAJFUCZUDZK @@ -795901,7 +795901,7 @@ between the two points ("point-slope form"), sometimes also written as cfv ianor df-ne orbi12i bitr4i wrex cop cpr simp3 adantr simpl 3ad2ant2 wi 0red simp2r redivcld adantl prelrrx2 syl2anc necomd neneqd a1d eqidd id a1i impbid xor3 sylibr fv1prop syl oveq2d recn mul01d 3ad2ant1 eqtrd - cvv ovexd fv2prop cc recnd divcan2d oveq12d addid2d eqeq1d mtbird fveq1 + cvv ovexd fv2prop cc recnd divcan2d oveq12d addlidd eqeq1d mtbird fveq1 bibi12d notbid rspcev nne 1red jca eqneqall com12 pm2.24 eqcoms simprl1 ex addcomd anim12ci 3adant2 addid0 bitrd bibi1d 1ex mpbird sylanb jaoi3 ax-1rid orcoms rexnal syl6ib biimtrid con4d ) AUAUBZBUAUBZBUCUDZUEZCUAU @@ -795946,7 +795946,7 @@ between the two points ("point-slope form"), sometimes also written as 1re olci jctil orcd prneimg 3netr4g 3jca adantl eqid rrx2linest eqeq12d mpsyl fveq1i 3pm3.2i fvpr1g eqtrid oveq12d eqtrdi oveq1d fvpr2g mp3an13 rabbi 1m0e1 subidd eqtrd mp3an12i ax-1rid mul02d subid1d rrx2pyel recnd - wral recn mulid2d rrx2pxel bibi2d ralbidva addid2d ad2antlr cc 3ad2ant2 + wral recn mullidd rrx2pxel bibi2d ralbidva addlidd ad2antlr cc 3ad2ant2 adantr ad3antrrr eqeq1d sylan9bb eqeq2d oveq1 mulcld simp3 simpl simp2r line2xlem divmuld eqcomd 3bitr2d bitrdi ralrimiva impbid 3bitrd bitr3id eqcom ex bitrd ) AUAUBZBUAUBZBUCUDZUEZCUAUBZUFZIUAUBZUEZFJKHUGZUHATLUIZ @@ -796004,7 +796004,7 @@ between the two points ("point-slope form"), sometimes also written as mpan wb elmapg mp1i mpbird 3ad2ant2 fveq1i 3pm3.2i fvpr1g eqtrid eqtr4d 0re simp3 simp1 fvpr2g simp2 3netr4d jca adantl rrx2vlinest syl eqeq12d 3jca ax-mp eqtri eqeq2d rabbidv wral rabbi wi line2ylem adantr ad2antlr - oveq1 oveq2d rrx2pyel recnd mul02d cc ad3antrrr rrx2pxel mulcld addid1d + oveq1 oveq2d rrx2pyel recnd mul02d cc ad3antrrr rrx2pxel mulcld addridd eqtrd eqeq1d wo mul0ord eqneqall com12 idd jaod impbid1 bitrd ralrimiva olc 3bitrd ex impbid bitr3id ) AUAUBZBUAUBZCUAUBZUCZIUAUBZJUAUBZIJUDZUC ZUEZFKLHUFZUGAUHMUIZUJZUKUFZBURUUMUJZUKUFZULUFZCUGZMDUMZUUNUHKUJZUGZMDU @@ -796135,7 +796135,7 @@ between the two points ("point-slope form"), sometimes also written as oveq2 oveq1d negeqd oveq1 oveq12d eqcoms simp12 recnd simp3r sqcld 2cnd cc mulcld negcld add32r syl3anc addcld negsubd mulassd 2timesd subeq0bd mul32d sqvald eqtr4d eqtrd sylan9eqr simp3l mul02d eqeq1d sqmuld simp13 - 3eqtrrd ex addid2d mulneg1d rpcn 3ad2ant2 subid1d 3imtr4d 3exp 3adant1r + 3eqtrrd ex addlidd mulneg1d rpcn 3ad2ant2 subid1d 3imtr4d 3exp 3adant1r 3imp adantld wb anbi2d eqtrid oveq1i a1i imbi12d adantl 3ad2ant1 mpbird sq0i ) AUAMZANOZUBZBUAMZCUAMZUCZEUDMZHUAMZIUAMZUBZUCZHPQRIPQRZSREPQRZOZ AHTRZBITRZSRZCOZUBZDXMTRZFITRZGSRZSRZNOZUEZXONHTRZXQSRZCOZUBZNBPQRZSRZX @@ -796183,7 +796183,7 @@ holds even for degenerate lines ( ` A = B = 0 ` ). (Contributed by w3a simpl2 simpl3 sqneg syl sqmuld sq2 a1i oveq12d 3eqtrd eqtrid simpl1 oveq12i sqcld addcld simpr subdid adddird mul12d oveq2d assraddsubd 4cn addcomd simp1 adantr eqeltrid subcld mulassd subsub4d cc0 subidd oveq1d - subsub2d addid2d eqtr3d adddid eqcomd eqtr4d mulcomd eqtrd 3eqtr2rd + subsub2d addlidd eqtr3d adddid eqcomd eqtr4d mulcomd eqtrd 3eqtr2rd 0cnd ) AMUAZBMUAZCMUAZUHZFMUAZUBZGNOPZQEHRPZRPZSPQBNOPZCNOPZRPZRPZQXAAN OPZWTRPZXCXCFNOPZRPZRPZXCWSXERPZRPZUCPZSPZUCPZRPZSPZQXCRPDRPZWOWPXBWRXM SWOWPNBCRPZRPZUDZNOPZXBGXRNOJUEWOXSXQNOPZNNOPZXPNOPZRPXBWOXQMUAXSXTUFWO @@ -796360,8 +796360,8 @@ holds even for degenerate lines ( ` A = B = 0 ` ). (Contributed by ( wcel cc0 wceq wa c2 co cmul cdiv adantr cc cr w3a wne crp cle wbr caddc cexp csqrt cfv cneg wo cmin animorr anim2i itsclc0yqsol syl3an1 imp oveq1 adantl rpcn sqcld resum2sqcl recnd 3adant3 mulcld simpll3 subcld eqeltrid - wi sqrtcld mul02d eqtrd oveq2d simpll2 subid1d sq0i oveq1d addid2d eqtrid - recn sqvald 3ad2ant2 oveq12d simplrr divcan5d eqeq2d biimpd addid1d simp2 + wi sqrtcld mul02d eqtrd oveq2d simpll2 subid1d sq0i oveq1d addlidd eqtrid + recn sqvald 3ad2ant2 oveq12d simplrr divcan5d eqeq2d biimpd addridd simp2 simpl3 simpr jaod eqeq1d simp1rr divcld simp3l subadd2d wb sqdivd resqcld sqgt0d elrpd subdivcomb1 syl3anc eqtr4d eqcomd oveq1i eqeq1i eqcom sqrtth rpcnne0d mulcomd syl jca sqeqor orcom a1i 3bitrd biimtrrid sylbid sylbird @@ -796411,10 +796411,10 @@ holds even for degenerate lines ( ` A = B = 0 ` ). (Contributed by ( wcel cc0 wceq wa co caddc cmul cdiv adantr eqtrd cr w3a wne crp cle wbr c2 cexp csqrt cfv cneg wo cmin itschlc0xyqsol1 orcom oveq1 ad2antrl recnd simpll3 mul02d oveq1d simpll2 rpre adantl sqcld resum2sqcl 3adant3 mulcld - subcld eqeltrid sqrtcld addid2d sq0i simp2 sqvald eqtrid simplrr divcan5d + subcld eqeltrid sqrtcld addlidd sq0i simp2 sqvald eqtrid simplrr divcan5d oveq2d eqcomd eqeq2d biimpd subid1d oveq12d eqtr2d biimpa jctird mulneg2d cc simp1rr negcld df-neg eqtr4di divnegd 3eqtr4d 3ad2ant1 simp1l3 simp1l2 - 3ad2ant2 addid1d orim12d biimtrid expimpd syld ) AUAKZBUAKZCUAKZUBZALMZBL + 3ad2ant2 addridd orim12d biimtrid expimpd syld ) AUAKZBUAKZCUAKZUBZALMZBL UCZNZNZFUDKZLDUEUFZNZGUAKHUAKNZUBZGUGUHOHUGUHOPOFUGUHOZMAGQOBHQOPOCMNHCBR OZMZGDUIUJZBROZUKZMZGYBMZULZNGACQOZBYAQOZPOZEROZMZHBCQOZAYAQOZUMOZEROZMZN ZGYGYHUMOZEROZMZHYLYMPOZEROZMZNZULZABCDEFGHIJUNXQXTYFUUEYFYEYDULXQXTNZUUE @@ -796921,7 +796921,7 @@ coordinates of the intersection points of a (nondegenerate) line and a cr simpl syl2an ltaddsub2d ltmul2dd ltaddsubd lt2addd lelttrd biimtrrid syl2and imp wn wceq eqcom subeq0ad biimprd biimtrid eqeq1i 0red mulge0d simprl syl2an2r oveq1 adantl mul02d sylan9eqr oveq1d eqtrd oveq2d 2t0e0 - nne eqtrdi addid1d 3brtr4d simprr addcomd eqbrtrd mulcld mul01d addid2d + nne eqtrdi addridd 3brtr4d simprr addcomd eqbrtrd mulcld mul01d addlidd sq0i wo ioran pm2.24d 4casesdan posdifd 2itscplem3 ) AUDIUEUFUGZGUEUFUG ZBUEUFUGZUHUGZUIUGZEUEUFUGZYHCUEUFUGZUHUGZUIUGZUJUGZUEEBUIUGZICUIUGZUIU GZUIUGZUHUGZHUKAYTYPUKULZUDUUAUKULACKUMZBJUMZUUBAUUCUUDUNUUBAUUCCKUHUGZ @@ -803595,7 +803595,7 @@ to Davis and Putnam (1960). (Contributed by David A. Wheeler, clt cress cvsca 3expa clmod qdrng drngring frlmlmod qrngbas frlmsca qrng0 crg csca c0g frlm0 islindf4 mp3an12 simpr feqmptd ffvelcdm cmulr cnfldmul ressmulr fconstmpt qmulcl snex xpex fsuppmptdm frlmgsum cnfldbas cnfldadd - frlmvscafval cnfldex zq addid2 addid1 jca gsumress simplr gsumfsum eqtr3d + frlmvscafval cnfldex zq addlid addrid jca gsumress simplr gsumfsum eqtr3d 3eqtrd qaddcl fsumcllem eqeltrd nfmpt1 nfmpt eqfnfv2f fveq1d fvmpt2 mpan2 sylancl sylan9eq fvconst2 eqeq12d ralbidva bitrd imbi1d cdm cnzr islindf3 drngnzr dmmpti f1eq2 anbi1i bitri wnel con34b df-nel velsn xchbinx imbi1i