diff --git a/changes-set.txt b/changes-set.txt index d0f61c613e..6da7a06e88 100644 --- a/changes-set.txt +++ b/changes-set.txt @@ -76,16 +76,17 @@ proposed sseqtr4d sseqtrrd proposed syl6sseqr sseqtrrdi proposed syl6eqss eqsstrdi compare to eqsstri or eqsstrd proposed syl6eqssr eqsstrrdi compare to eqsstrri or eqsstrrd -proposed syl6eqbr eqbrtrdi compare to eqbrtri or eqbrtrd -proposed syl6eqbrr eqbrtrrdi compare to eqbrtrri or eqbrtrrd (Please send any comments on these proposals to the mailing list or make a github issue.) DONE: Date Old New Notes + 5-Jan-24 syl6eqbr eqbrtrdi compare to eqbrtri or eqbrtrd + 5-Jan-24 syl6eqbrr eqbrtrrdi compare to eqbrtrri or eqbrtrrd 4-Jan-24 fimaproj [same] moved from TA's mathbox to main set.mm 4-Jan-24 fproj [same] moved from TA's mathbox to main set.mm 4-Jan-24 mptima [same] moved from GS's mathbox to main set.mm + 3-Jan-24 unima [same] moved from GS's mathbox to main set.mm 29-Dec-23 uzidd [same] moved from GS's mathbox to main set.mm 28-Dec-23 eqri [same] moved from TA's mathbox to main set.mm 28-Dec-23 domep dmep moved from SF's mathbox to main set.mm diff --git a/iset.mm b/iset.mm index d275f653a4..b8f0d204f6 100644 --- a/iset.mm +++ b/iset.mm @@ -34330,21 +34330,21 @@ same disjoint variable group (meaning ` A ` cannot depend on ` x ` ) and $} ${ - syl6eqbr.1 $e |- ( ph -> A = B ) $. - syl6eqbr.2 $e |- B R C $. + eqbrtrdi.1 $e |- ( ph -> A = B ) $. + eqbrtrdi.2 $e |- B R C $. $( A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.) $) - syl6eqbr $p |- ( ph -> A R C ) $= + eqbrtrdi $p |- ( ph -> A R C ) $= ( wbr breq1d mpbiri ) ABDEHCDEHGABCDEFIJ $. $} ${ - syl6eqbrr.1 $e |- ( ph -> B = A ) $. - syl6eqbrr.2 $e |- B R C $. + eqbrtrrdi.1 $e |- ( ph -> B = A ) $. + eqbrtrrdi.2 $e |- B R C $. $( A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.) $) - syl6eqbrr $p |- ( ph -> A R C ) $= - ( eqcomd syl6eqbr ) ABCDEACBFHGI $. + eqbrtrrdi $p |- ( ph -> A R C ) $= + ( eqcomd eqbrtrdi ) ABCDEACBFHGI $. $} ${ @@ -69364,7 +69364,7 @@ elements or fails to hold (is equal to ` (/) ` ) for some element. ( vx vy c1o cen wbr wa cin c0 wceq cun c2o csn wi 1on ensn1 entr wn ex cv con0 elexi ensymi mpan2 wcel onirri disjsn mpbir unen mpanr2 sylan2 df-2o csuc df-suc eqtri breq2i syl6ibr wex en1 1nen2 unidm sneq uneq2d syl5reqr - wne a1i syl6eqbr ensymd sylan mtand necon2ai disjsn2 uneq12 breq1d ineq12 + wne a1i eqbrtrdi ensymd sylan mtand necon2ai disjsn2 uneq12 breq1d ineq12 vex syl eqeq1d 3imtr4d exlimdv exlimiv imp syl2anb impbid ) AEFGZBEFGZHZA BIZJKZABLZMFGZWHWJWKEENZLZFGZWLWGWFBWMFGZWJWOOWGEWMFGWPWMEEEUBPUCQUDBEWMR UEWFWPHZWJWOWQWJEWMIJKZWOWREEUFSEPUGEEUHUIAEBWMUJUKTULMWNWKFMEUNWNUMEUOUP @@ -95649,7 +95649,7 @@ Positive reals (as a subset of complex numbers) equal to the integer. (Contributed by AV, 19-Jun-2021.) $) nn0ledivnn $p |- ( ( A e. NN0 /\ B e. NN ) -> ( A / B ) <_ A ) $= ( cn0 wcel cn cdiv co cle wbr cc0 wceq wo wi elnn0 wa c1 nnge1 adantl wb ex - crp nnrp nnledivrp sylan2 mpbid cap nncn nnap0 jca div0ap syl 0le0 syl6eqbr + crp nnrp nnledivrp sylan2 mpbid cap nncn nnap0 jca div0ap syl 0le0 eqbrtrdi cc oveq1 id breq12d adantr mpbird jaoi sylbi imp ) ACDZBEDZABFGZAHIZVCAEDZA JKZLVDVFMZANVGVIVHVGVDVFVGVDOPBHIZVFVDVJVGBQRVDVGBUADVJVFSBUBABUCUDUETVHVDV FVHVDOZVFJBFGZJHIZVKVLJJHVKBUNDZBJUFIZOZVLJKVDVPVHVDVNVOBUGBUHUIRBUJUKULUMV @@ -96221,7 +96221,7 @@ Infinity and the extended real number system (cont.) ( cxr wcel clt wbr cxne cr cpnf wceq cmnf w3o wa wi elxr cneg syl adantr wn sylbid rexneg breqan12rd bitr4d biimpd xnegeq xnegpnf syl6eq adantl renegcl ltneg eqeltrd mnflt eqbrtrd a1d simpr breq2d nltmnf pm2.21d 3jaodan sylan2b - rexr expimpd simpl breq1d pnfnlt breq1 anbi2d ltpnf mnfltpnf syl6eqbr breq2 + rexr expimpd simpl breq1d pnfnlt breq1 anbi2d ltpnf mnfltpnf eqbrtrdi breq2 mnfxr ax-mp pm2.21i syl6bi imp 3jaoian sylanb xnegmnf syl5ibr 3jaoi 3impib sylbi ) ACDZBCDZABEFZBGZAGZEFZWDAHDZAIJZAKJZLWEWFMZWINZAOWJWNWKWLWJWEWFWIWE WJBHDZBIJZBKJZLZWFWINZBOZWJWOWSWPWQWJWOMZWFWIXAWFBPZAPZEFWIABUJWOWJWGXBWHXC @@ -106168,7 +106168,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ leexp1a $p |- ( ( ( A e. RR /\ B e. RR /\ N e. NN0 ) /\ ( 0 <_ A /\ A <_ B ) ) -> ( A ^ N ) <_ ( B ^ N ) ) $= ( cr wcel cc0 cle wa cexp co wi c1 wceq oveq2 breq12d imbi2d adantr sylan - wbr adantlr vj vk cn0 cv caddc cc recn exp0 1le1 syl6eqbr adantl breqtrrd + wbr adantlr vj vk cn0 cv caddc cc recn exp0 1le1 eqbrtrdi adantl breqtrrd syl2an simpll reexpcl simplll simpr simplrl expge0 syl3anc simplr anim12i cmul jca31 simpl simpllr jca32 simplrr jca lemul12a expp1 adantll 3brtr4d sylc ex expcom a2d nn0ind exp4c com3l 3imp1 ) ADEZBDEZCUCEZFAGSZABGSZHZAC @@ -107465,7 +107465,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( ! ` M ) <_ ( ! ` N ) ) $= ( cn0 wcel cle wbr cfa cfv wa wi cc0 breq2 anbi2d fveq2 breq2d imbi12d cr wceq adantl nnred vj vk cv caddc nn0le0eq0 biimpa fveq2d fac0 1re eqeltri - c1 co leidi syl6eqbr impexp wo cz wb simpl nn0zd peano2nn0 zleloe syl2anc + c1 co leidi eqbrtrdi impexp wo cz wb simpl nn0zd peano2nn0 zleloe syl2anc clt nn0leltp1 cmul faccl nn0re peano2re syl nnnn0d nn0ge0d nn0p1nn nnge1d lemulge11d facp1 breqtrrd adantr faccld letr syl3anc mpan2d com23 sylbird imim2d leidd syl5ibcom syl5 a1dd jaod sylbid com13 com4l a2d imp4a syl5bi @@ -107549,7 +107549,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( cn0 wcel cn cc0 wceq wo cexp co cfv cle wbr wa c1 cr adantr sylan reexpcl cmul cfa elnn0 caddc nnre nnge1 cuz nn0z adantl uzid peano2uz 3syl leexp2ad cz nnnn0 faclbnd wi nn0re peano2nn0 syl2an mpancom faccl nnred remulcl letr - syl3anc mp2and 0exp 0le1 syl6eqbr oveq2 0exp0e1 1le1 eqbrtri jaoi sylbi 1nn + syl3anc mp2and 0exp 0le1 eqbrtrdi oveq2 0exp0e1 1le1 eqbrtri jaoi sylbi 1nn nnmulcl sylancr nnge1d 0re 1re mp3an2 syl2anc wb oveq1 oveq12 anidms syl6eq mpan oveq1d breq12d mpbird jaoian sylanb ) ACDZAEDZAFGZHBCDZABIJZAAIJZBUAKZ TJZLMZAUBWPWRXCWQWPWRNZWSABOUCJZIJZLMZXFXBLMZXCXDABXEWPAPDZWRAUDQWPOALMWRAU @@ -111257,7 +111257,7 @@ A Cauchy sequence (as defined here, which has a rate of convergence ( 1 / ( 2 ^ N ) ) < ( F ` N ) ) $= ( wcel c1 c2 cexp co cdiv clt wbr wi oveq2d adantr vw vk cn cfv cv wceq caddc oveq2 fveq2 breq12d crp 2cnd exp1d 2rp syl6eqel rprecred readdcld - imbi2d 1red halflt1 syl6eqbr cc0 addge01d ltletrd resqrexlemf1 breqtrrd + imbi2d 1red halflt1 eqbrtrdi cc0 addge01d ltletrd resqrexlemf1 breqtrrd cle mpbid wa cmul a1i nnz ad2antlr rpexpcld rpcnd rpap0d recdivap2d cn0 cz nnnn0 expp1d cr resqrexlemf ffvelrnda rpred rerpdivcld simpr divge0d eqtr4d ltdiv1dd resqrexlemfp1 eqbrtrrd ex expcom a2d nnind impcom ) FUC @@ -121008,7 +121008,7 @@ Infinite sums (cont.) Jim Kingdon, 12-Dec-2022.) $) efap0 $p |- ( A e. CC -> ( exp ` A ) =//= 0 ) $= ( cc wcel ce cfv cneg efcl negcl syl cmul co cc0 cap efcan 1ap0 mulap0bad - c1 syl6eqbr ) ABCZADEZAFZDEZAGSUABCUBBCAHUAGISTUBJKQLMANORP $. + c1 eqbrtrdi ) ABCZADEZAFZDEZAGSUABCUBBCAHUAGISTUBJKQLMANORP $. $} $( The exponential of a complex number is nonzero. Corollary 15-4.3 of @@ -123255,7 +123255,7 @@ Infinite sums (cont.) Carneiro, 2-Jul-2015.) $) dvds1 $p |- ( M e. NN0 -> ( M || 1 <-> M = 1 ) ) $= ( cn0 wcel c1 cdvds wbr wceq wa simpl 1nn0 a1i simpr cz 1dvds adantr dvdseq - nn0z syl syl22anc ex id 1z iddvds ax-mp syl6eqbr impbid1 ) ABCZADEFZADGZUGU + nn0z syl syl22anc ex id 1z iddvds ax-mp eqbrtrdi impbid1 ) ABCZADEFZADGZUGU HUIUGUHHZUGDBCZUHDAEFZUIUGUHIUKUJJKUGUHLUGULUHUGAMCULAQANROADPSTUIADDEUIUAD MCDDEFUBDUCUDUEUF $. @@ -133328,7 +133328,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on $( ` { (/) } ` is the only topology with one element. (Contributed by FL, 18-Aug-2008.) $) en1top $p |- ( J e. Top -> ( J ~~ 1o <-> J = { (/) } ) ) $= - ( ctop wcel c1o cen wbr c0 csn wceq wi en1eqsn ex syl id 0ex ensn1 syl6eqbr + ( ctop wcel c1o cen wbr c0 csn wceq wi en1eqsn ex syl id 0ex ensn1 eqbrtrdi 0opn impbid1 ) ABCZADEFZAGHZIZTGACZUAUCJARUDUAUCGAKLMUCAUBDEUCNGOPQS $. ${ @@ -142651,7 +142651,7 @@ S C_ ( P ( ball ` D ) T ) ) $= wa fconstmpt eqidd offval2 feqmptd 3eqtr4d oveq2d cabs ccom cmopn fconstg syl snssd fssd ssidd fmpttd cop c0ex opelxpi mpan2 dvconst eleqtrrd df-br snid sylibr cofmpt wi simpr simpl subap0d wb eqid oveq1 fvmptd3 id subcld - cap subid breq12d mpbird ex ralrimiva dveflem syl6eqbr 1ex 1cnd dvmptidcn + cap subid breq12d mpbird ex ralrimiva dveflem eqbrtrdi 1ex 1cnd dvmptidcn eqtrd 0cnd dvmptccn dvmptsubcn 1m0e1 mpteq2i eqtr4i syl6eq dvcoapbr 1t1e1 breqtrdi breqdi dvmulxxbr ffvelrnd mul02d mpancom mulid2d oveq12d addid2d fvconst2g breqtrd funbrfv mpsyl mpteq2ia wss ssid dvbsssg mp2an mpd3an23 diff --git a/set.mm b/set.mm index b8794a313f..1dd51386e9 100644 --- a/set.mm +++ b/set.mm @@ -25494,6 +25494,13 @@ the definition of class equality ( ~ df-cleq ). Its forward implication ( wceq eqcom mpbi ) ABDBADCABEF $. $} + ${ + neqcomd.1 $e |- ( ph -> -. A = B ) $. + $( Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + neqcomd $p |- ( ph -> -. B = A ) $= + ( wceq eqcom sylnib ) ABCECBEDBCFG $. + $} + ${ eqeq2d.1 $e |- ( ph -> A = B ) $. $( Deduction from equality to equivalence of equalities. (Contributed by @@ -32600,6 +32607,16 @@ Such interpretation is rarely needed (see also ~ df-ral ). (Contributed ( wrex rspcedv mpd ) ACBDFJIABCDEFGHKL $. $} + ${ + $d x ph $. $d x B $. $d x A $. + rspcime.1 $e |- ( ( ph /\ x = A ) -> ps ) $. + rspcime.2 $e |- ( ph -> A e. B ) $. + $( Prove a restricted existential. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + rspcime $p |- ( ph -> E. x e. B ps ) $= + ( cv wceq wa simpl 2thd id rspcedvd ) ABACDEGACHDIZJBAFAOKLAMN $. + $} + ${ $d x y A $. $d x B $. $d x C $. $d x ps $. $d x ch $. rspceaimv.1 $e |- ( x = A -> ( ph <-> ps ) ) $. @@ -35860,6 +35877,24 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use CAUMULUQCUDJKBCDEFGHICDRCERULDRLUEUHUFUIMUJUHUK $. $} + ${ + $d A x y $. $d B y $. $d C x $. $d D y $. $d E x $. $d ph x $. + $d ch x $. $d ps y $. + rspc2vd.a $e |- ( x = A -> ( th <-> ch ) ) $. + rspc2vd.b $e |- ( y = B -> ( ch <-> ps ) ) $. + rspc2vd.c $e |- ( ph -> A e. C ) $. + rspc2vd.d $e |- ( ( ph /\ x = A ) -> D = E ) $. + rspc2vd.e $e |- ( ph -> B e. E ) $. + $( Deduction version of 2-variable restricted specialization, using + implicit substitution. Notice that the class ` D ` for the second set + variable ` y ` may depend on the first set variable ` x ` . + (Contributed by AV, 29-Mar-2021.) $) + rspc2vd $p |- ( ph -> ( A. x e. C A. y e. D th -> ps ) ) $= + ( csb wcel wral csbied eleqtrrd wi nfcsb1v nfv nfral cv csbeq1a raleqbidv + wceq rspc syl rspcv sylsyld ) AHEGJQZRDFJSZEISZCFUNSZBAHKUNPAEGJKINOTUAAG + IRUPUQUBNUOUQEGICEFUNEGJUCCEUDUEEUFGUIDCFJUNEGJUGLUHUJUKCBFHUNMULUM $. + $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -45977,21 +46012,21 @@ proper class (see for example ~ iprc ). (Contributed by NM, $} ${ - syl6eqbr.1 $e |- ( ph -> A = B ) $. - syl6eqbr.2 $e |- B R C $. + eqbrtrdi.1 $e |- ( ph -> A = B ) $. + eqbrtrdi.2 $e |- B R C $. $( A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.) $) - syl6eqbr $p |- ( ph -> A R C ) $= + eqbrtrdi $p |- ( ph -> A R C ) $= ( wbr breq1d mpbiri ) ABDEHCDEHGABCDEFIJ $. $} ${ - syl6eqbrr.1 $e |- ( ph -> B = A ) $. - syl6eqbrr.2 $e |- B R C $. + eqbrtrrdi.1 $e |- ( ph -> B = A ) $. + eqbrtrrdi.2 $e |- B R C $. $( A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.) $) - syl6eqbrr $p |- ( ph -> A R C ) $= - ( eqcomd syl6eqbr ) ABCDEACBFHGI $. + eqbrtrrdi $p |- ( ph -> A R C ) $= + ( eqcomd eqbrtrdi ) ABCDEACBFHGI $. $} ${ @@ -53416,6 +53451,29 @@ of the relation (see ~ op1stb ). (Contributed by NM, 28-Jul-2004.) $) NRABGOABCDEHFPQ $. $} + ${ + $d D x $. $d x A $. $d x C $. $d x ph $. + elrnmptdv.1 $e |- F = ( x e. A |-> B ) $. + elrnmptdv.2 $e |- ( ph -> C e. A ) $. + elrnmptdv.3 $e |- ( ph -> D e. V ) $. + elrnmptdv.4 $e |- ( ( ph /\ x = C ) -> D = B ) $. + $( Elementhood in the range of a function in maps-to notation, deduction + form. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + elrnmptdv $p |- ( ph -> D e. ran F ) $= + ( crn wcel wceq wrex rspcime wb elrnmpt syl mpbird ) AFGMNZFDOZBCPZAUCBEC + LJQAFHNUBUDRKBCDFGHISTUA $. + $} + + ${ + $d x C $. + elrnmpt2d.1 $e |- F = ( x e. A |-> B ) $. + elrnmpt2d.2 $e |- ( ph -> C e. ran F ) $. + $( Elementhood in the range of a function in maps-to notation, deduction + form. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + elrnmpt2d $p |- ( ph -> E. x e. A C = B ) $= + ( crn wcel wceq wrex elrnmpt ibi syl ) AEFIZJZEDKBCLZHQRBCDEFPGMNO $. + $} + ${ $d y A $. $d y B $. $d x y $. $( Alternate definition of indexed union when ` B ` is a set. (Contributed @@ -61176,6 +61234,19 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). FNEOBDPQGHBCDEFRS $. $} + ${ + $d A y $. $d B x y $. $d C x y $. $d F x y $. + $( Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) + unima $p |- ( ( F Fn A /\ B C_ A /\ C C_ A ) + -> ( F " ( B u. C ) ) = ( ( F " B ) u. ( F " C ) ) ) $= + ( vy vx wfn wss w3a cun cima cv wcel wo cfv wceq wrex simp1 wb fvelimab + simpl simpr unssd 3adant1 fvelimabd syl6bb 3adant3 3adant2 orbi12d bitr4d + wa rexun elun syl6bbr eqrdv ) DAGZBAHZCAHZIZEDBCJZKZDBKZDCKZJZUSELZVAMZVE + VBMZVEVCMZNZVEVDMUSVFFLDOVEPZFBQZVJFCQZNZVIUSVFVJFUTQVMUSFAUTVEDUPUQURRUQ + URUTAHUPUQURUKBCAUQURUAUQURUBUCUDUEVJFBCULUFUSVGVKVHVLUPUQVGVKSURFABVEDTU + GUPURVHVLSUQFACVEDTUHUIUJVEVBVCUMUNUO $. + $} + $( The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) $) fvi $p |- ( A e. V -> ( _I ` A ) = A ) $= @@ -84599,6 +84670,20 @@ the first case of his notation (simple exponentiation) and subscript it VFSUEUFUGUHUIUJUK $. $} + ${ + enpr2d.1 $e |- ( ph -> A e. C ) $. + enpr2d.2 $e |- ( ph -> B e. D ) $. + enpr2d.3 $e |- ( ph -> -. A = B ) $. + $( A pair with distinct elements is equinumerous to ordinal two. + (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + enpr2d $p |- ( ph -> { A , B } ~~ 2o ) $= + ( c1o cen csn cun wbr cin c0 wceq wcel syl con0 1on cpr csuc ensn1g en2sn + c2o sylancl wne neqned disjsn2 wn onirri a1i disjsn sylibr syl22anc df-pr + unen df-suc 3brtr4g df-2o breqtrrdi ) ABCUAZIUBZUEJABKZCKZLZIIKZLZVBVCJAV + DIJMZVEVGJMZVDVENOPZIVGNOPZVFVHJMABDQVIFBDUCRACEQISQVJGTCIESUDUFABCUGVKAB + CHUHBCUIRAIIQUJZVLVMAITUKULIIUMUNVDIVEVGUQUOBCUPIURUSUTVA $. + $} + $( Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.) $) ssct $p |- ( ( A C_ B /\ B ~<_ _om ) -> A ~<_ _om ) $= @@ -85473,7 +85558,7 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily domunsn $p |- ( A ~< B -> ( A u. { C } ) ~<_ B ) $= ( vz csdm wbr cv wcel csn cun cdom c0 wceq wn wex sdom0 sylib wa cvv cin breq2 mtbiri con2i neq0 cdif domdifsn adantr cen en2sn endom snprc biimpi - elvd syl snex 0dom syl6eqbr pm2.61i incom disjdif eqtri undom mpan2 uncom + elvd syl snex 0dom eqbrtrdi pm2.61i incom disjdif eqtri undom mpan2 uncom sylancl wss simpr snssd undif syl5eq breqtrd exlimddv ) ABEFZDGZBHZACIZJZ BKFDVMBLMZNVODOVRVMVRVMALEFAPBLAEUAUBUCDBUDQVMVORZVQBVNIZUEZVTJZBKVSAWAKF ZVPVTKFZVQWBKFZVMWCVOABVNUFUGCSHZWDWFVPVTUHFZWDWFWGDCVNSSUIUMVPVTUJUNWFNZ @@ -85764,7 +85849,7 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily ( vx cdom wbr cvv wcel cmap co wa cv cen wss wex wb reldom adantl syl2anc c0 brrelex2i domeng syl ibi adantr simpl enrefg mapen syl2anr ovex simprr ad2antrr mapss ssdomg mpsyl endomtr exlimddv wn wceq elmapex simprd con3i - eq0rdv 0dom syl6eqbr pm2.61dan ) ABEFZCGHZACIJZBCIJZEFZVGVHKZADLZMFZVMBNZ + eq0rdv 0dom eqbrtrdi pm2.61dan ) ABEFZCGHZACIJZBCIJZEFZVGVHKZADLZMFZVMBNZ KZVKDVGVPDOZVHVGVQVGBGHZVGVQPABEQUAZDABGUBUCUDUEVLVPKZVIVMCIJZMFZWAVJEFZV KVPVNCCMFZWBVLVNVOUFVHWDVGCGUGRAVMCCUHUIVJGHVTWAVJNZWCBCIUJZVTVRVOWEVGVRV HVPVSULVLVNVOUKVMBCGUMSWAVJGUNUOVIWAVJUPSUQVGVHURZKVITVJEWGVITUSVGWGDVIVM @@ -85905,7 +85990,7 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily Carneiro, 30-Apr-2015.) $) mapdom2 $p |- ( ( A ~<_ B /\ -. ( A = (/) /\ C = (/) ) ) -> ( C ^m A ) ~<_ ( C ^m B ) ) $= - ( vx cdom wbr c0 wceq wa wn cvv wcel cmap wne simplr syl syl6eqbr syl2anc + ( vx cdom wbr c0 wceq wa wn cvv wcel cmap wne simplr syl eqbrtrdi syl2anc co cen simpr oveq1d idd jctird mtod neqned map0b eqtrd ovex cv wss simpll 0dom wex wb reldom brrelex2i ad2antrr domeng mpbid ad2antlr simprrl mapen enrefg cdif cxp ovexd simprl difexg map0g simpl syl6bi necon3d mpd xpdom3 @@ -86204,6 +86289,20 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily php5 $p |- ( A e. _om -> -. A ~~ suc A ) $= ( com wcel csuc csdm wbr cen wn php4 sdomnen syl ) ABCAADZEFALGFHAIALJK $. + ${ + phpeqd.1 $e |- ( ph -> A e. Fin ) $. + phpeqd.2 $e |- ( ph -> B C_ A ) $. + phpeqd.3 $e |- ( ph -> A ~~ B ) $. + $( Corollary of the Pigeonhole Principle using equality. Strengthening of + ~ php expressed without negation. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + phpeqd $p |- ( ph -> A = B ) $= + ( cen wbr wceq wn wa csdm cfn wcel wpss wss adantr simpr neqcomd dfpss2 + sylanbrc php3 syl2an2r sdomnen ensym nsyl syl ex mt4d ) ABCGHZBCIZFAUKJZU + JJZAULKZCBLHZUMABMNULCBOZUODUNCBPZCBIJUPAUQULEQUNBCAULRSCBTUABCUBUCUOCBGH + UJCBUDBCUEUFUGUHUI $. + $} + $( A singleton ` { A } ` is never equinumerous with the ordinal number 2. This holds for proper singletons ( ` A e. _V ` ) as well as for singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, 6-Aug-2019.) $) @@ -96975,7 +97074,7 @@ several of their earlier lemmas available (which would otherwise be ( vx vy c1o cen wbr wa cin c0 wceq cun c2o csn wi ensn1 wn ex cv csdm 1on 1oex ensymi entr mpan2 wcel onirri disjsn mpbir unen mpanr2 sylan2 df-suc csuc df-2o eqtri breq2i syl6ibr wex en1 unidm sneq uneq2d syl5reqr 1sdom2 - wne vex ensdomtr mp2an syl6eqbr sdomnen syl necon2ai uneq12 breq1d ineq12 + wne vex ensdomtr mp2an eqbrtrdi sdomnen syl necon2ai uneq12 breq1d ineq12 disjsn2 a1i eqeq1d 3imtr4d exlimdv exlimiv imp syl2anb impbid ) AEFGZBEFG ZHZABIZJKZABLZMFGZWHWJWKEENZLZFGZWLWGWFBWMFGZWJWOOWGEWMFGWPWMEEUBPUCBEWMU DUEWFWPHZWJWOWQWJEWMIJKZWOWREEUFQEUAUGEEUHUIAEBWMUJUKRULMWNWKFMEUNWNUOEUM @@ -100430,7 +100529,7 @@ Because we use a disjoint union for cardinal addition (as explained in the sylib elmapd ad2antrr mpbird wfo f1ofo eqcomd ex eximdv mpd df-rex sylibr forn ss2abdv eqid rnmpt syl6sseqr mpsyl wfn wral rgenw mp1i fodomnum sylc fnmpt dffn4 domtr sbth 3ad2ant1 map0e 1oex enref csn df-sn en0 anbi2i 0ss - df1o2 sseq1 mpbiri pm4.71ri bitr4i 3eqtr4ri breqtrri syl6eqbr pm2.61ne + df1o2 sseq1 mpbiri pm4.71ri bitr4i 3eqtr4ri breqtrri eqbrtrdi pm2.61ne abbii ) UABEFZCBEFZBCGUBZUCUDZHZUEZUUJAUFZBIZUUNCJFZKZALZJFZBMGUBZUUOUUNM JFZKZALZJFCMCMNZUUJUUTUURUVCJCMBGUGUVDUUQUVBAUVDUUPUVAUUOCMUUNJUHUIUJUKUU MCMULZKZUUJUUREFZUURUUJEFZUUSUVFUUJUUNCBUNZIZUUPKZALZEFZUVLUURJFZUVGUVFUV @@ -110293,7 +110392,7 @@ prove that every set is contained in a weak universe in ZF (see cvv wss vw cina cen cxp cv ciun cwina inawina winaon winalim r1lim onelon wlim sylan csuc eleq1 fveq2 breq1d imbi12d weq ne0i 0sdomg syl5ibr breq1i wne r10 syl6ibr 3syl wtr word eloni ordtr trsuc ex adantl ccrd r1suc fvex - cpw cardid ensymi pwen ax-mp syl6eqbr wb winacard eleq2d cardsdom sylancr + cpw cardid ensymi pwen ax-mp eqbrtrdi wb winacard eleq2d cardsdom sylancr bitr3d ccf elina simp3bi pweq rspccv sylbird ensdomtr syl2an expr imim12d imp cun vex mpan nfcv nfiu1 nfbr wel iunex ssiun2 ssdomg endomtr vtoclgaf mpsyl iundom mp2an mp2 domtr com domentr eqbrtrd ad2antlr wrex wfn sylibr @@ -110943,7 +111042,7 @@ values in the universe (see ~ gruiun for a more intuitive version). grudomon inawinalem winainflem syl3anc wb vex sdomtr iscard cardlim sseq2 limeq bibi12d mpbii mpbid cflm syl2anc eleq1 mpbiri abssi syl6eqelr intex fvex sylibr onint eqeltrd eqeq1 anbi1d exbidv elab simp2rr simp1l simp2rl - sylib syl6ss 3ad2ant3 simp2l syl6eqbr gruen syl112anc gruuni 3exp exlimdv + sylib syl6ss 3ad2ant3 simp2l eqbrtrdi gruen syl112anc gruuni 3exp exlimdv w3a mpd wo cfon cfle onsseleq mpan ord elina syl3anbrc ) BUAFZBGHZIZAGHZA UBJZAKZDUCZUDZALMZDANZAUEFUVHUVGUVJUVHUVMBFZDUFUVGUVJUGZDBUHUVQUVRDUVGUVQ UVJUVGUVQIZAGUVSGBOUIZAUVSGBFZGOFGUVTFUVGUVQGUVMPUWAUVMUOUVMGBUJUKULGBOUM @@ -111006,7 +111105,7 @@ values in the universe (see ~ gruiun for a more intuitive version). wo wb cina cwina c0 wne ne0i gruina sylan2 inawina winaon 3syl wfn r1fnon fndm syl6eleqr rankr1ag mtbid w3o rankon word eloni syl2an sylancr 3orass ordtri3or sylib ord mpjaod exlimdv syl5bi cdom simpll fveq2 ad2antll csdm - mpd cpw wral elina simp2bi eqtrd rankcf fvex domtri mp2an mpbir syl6eqbrr + mpd cpw wral elina simp2bi eqtrd rankcf fvex domtri mp2an mpbir eqbrtrrdi grudomon syl112anc elin biimpri ordirr adantl pm2.21dd rexlimdvaa pm2.18d cin syld grur1a eqssd ) BUAFZBGHUBUCZFZIZBAGJZUUCBUUDKZUUCUUELZDUDZMJZANZ DBUEZUUEUUFEUDZBFZUUKUUDFZLZIZEUFUUCUUJEBUUDUGUUCUUOUUJEUUCUUOUUJUUCUUOIZ @@ -131682,7 +131781,7 @@ Useful for working with integer logarithm bases (which is a common case, equal to the integer. (Contributed by AV, 19-Jun-2021.) $) nn0ledivnn $p |- ( ( A e. NN0 /\ B e. NN ) -> ( A / B ) <_ A ) $= ( cn0 wcel cn cdiv co cle wbr cc0 wceq wo wi elnn0 wa c1 nnge1 adantl wb ex - crp nnrp nnledivrp sylan2 mpbid wne nncn nnne0 jca div0 0le0 syl6eqbr oveq1 + crp nnrp nnledivrp sylan2 mpbid wne nncn nnne0 jca div0 0le0 eqbrtrdi oveq1 cc syl id breq12d adantr mpbird jaoi sylbi imp ) ACDZBEDZABFGZAHIZVCAEDZAJK ZLVDVFMZANVGVIVHVGVDVFVGVDOPBHIZVFVDVJVGBQRVDVGBUADVJVFSBUBABUCUDUETVHVDVFV HVDOZVFJBFGZJHIZVKVLJJHVKBUNDZBJUFZOZVLJKVDVPVHVDVNVOBUGBUHUIRBUJUOUKULVHVF @@ -132638,7 +132737,7 @@ Infinity and the extended real number system (cont.) ( cxr wcel clt wbr cxne cr cpnf wceq cmnf w3o wa wi elxr cneg rexneg adantr wn sylbid bitr4d biimpd xnegeq xnegpnf syl6eq adantl renegcl eqeltrd mnfltd ltneg breqan12rd eqbrtrd a1d breq2d rexr nltmnf syl pm2.21d 3jaodan sylan2b - simpr simpl breq1d pnfnlt breq1 anbi2d ltpnfd mnfltpnf syl6eqbr breq2 mnfxr + simpr simpl breq1d pnfnlt breq1 anbi2d ltpnfd mnfltpnf eqbrtrdi breq2 mnfxr expimpd ax-mp pm2.21i syl6bi imp 3jaoian sylanb xnegmnf syl5ibr 3jaoi sylbi 3impib ) ACDZBCDZABEFZBGZAGZEFZWDAHDZAIJZAKJZLWEWFMZWINZAOWJWNWKWLWJWEWFWIW EWJBHDZBIJZBKJZLZWFWINZBOZWJWOWSWPWQWJWOMZWFWIXAWFBPZAPZEFWIABUJWOWJWGXBWHX @@ -138210,15 +138309,11 @@ Finite intervals of nonnegative integers (or "finite sets of sequential $( Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, - 1-Aug-2018.) $) - elfzom1p1elfzo $p |- ( ( N e. NN /\ X e. ( 0 ..^ ( N - 1 ) ) ) - -> ( X + 1 ) e. ( 0 ..^ N ) ) $= - ( cc0 c1 cmin co cfzo wcel cn caddc cn0 clt wbr w3a elfzo0 peano2nn0 adantr - wi wa cr 3ad2ant1 simpr 1red nnre adantl ltaddsubd biimprd impancom 3adant2 - nn0re imp syl3anbrc ex sylbi impcom ) BCADEFZGFHZAIHZBDJFZCAGFHZUQBKHZUPIHZ - BUPLMZNZURUTRBUPOVDURUTVDURSUSKHZURUSALMZUTVDVEURVAVBVEVCBPUAQVDURUBVDURVFV - AVCURVFRVBVAURVCVFVAURSZVFVCVGBDAVABTHURBUJQVGUCURATHVAAUDUEUFUGUHUIUKUSAOU - LUMUNUO $. + 1-Aug-2018.) (Proof shortened by Thierry Arnoux, 14-Dec-2023.) $) + elfzom1p1elfzo $p |- ( ( N e. NN /\ X e. ( 0 ..^ ( N - 1 ) ) ) -> ( X + 1 ) + e. ( 0 ..^ N ) ) $= + ( cn wcel cz cc0 c1 cmin co cfzo caddc nnz elfzom1elp1fzo sylan ) ACDAEDBFA + GHIJIDBGKIFAJIDALBAMN $. ${ $d k N $. @@ -143932,7 +144027,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ leexp1a $p |- ( ( ( A e. RR /\ B e. RR /\ N e. NN0 ) /\ ( 0 <_ A /\ A <_ B ) ) -> ( A ^ N ) <_ ( B ^ N ) ) $= ( vj vk cr wcel cc0 cle wa cexp co wi c1 wceq oveq2 breq12d imbi2d adantr - wbr cn0 caddc recn exp0 1le1 syl6eqbr adantl breqtrrd syl2an cmul reexpcl + wbr cn0 caddc recn exp0 1le1 eqbrtrdi adantl breqtrrd syl2an cmul reexpcl cv cc ad4ant14 simplll simpr simplrl syl3anc ad4ant24 jca31 simpl anim12i simpllr jca32 simplrr anim1ci lemul12a sylc expp1 sylan ad5ant14 ad5ant24 expge0 3brtr4d ex expcom a2d nn0ind exp4c com3l 3imp1 ) AFGZBFGZCUAGZHAIT @@ -144723,7 +144818,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ 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 - mul01i 0m0e0 0le0 eqbrtri syl6eqbr wo eqid discr1 leloe mpjaodan ) AJCUFK + mul01i 0m0e0 0le0 eqbrtri eqbrtrdi wo eqid discr1 leloe mpjaodan ) AJCUFK ZDLUAMZNCEOMZOMZUBMZJPKJCUGZAUUOUCZUUSNCOMZJOMZJPUVAUUSUVBQMZJPKUUSUVCPKU VAUVDUUPUVBQMZEUBMZJPUVAUVDUVEUURUVBQMZUBMUVFUVAUUPUURUVBUVAUUPUVADRSZUUP RSAUVHUUOGUDZDUEUHZUIZUVAUURUVANRSUUQRSUURRSUJUVACEACRSZUUOFUDZAERSZUUOHU @@ -145267,7 +145362,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( ! ` M ) <_ ( ! ` N ) ) $= ( vj vk cn0 wcel cle wbr cfa cfv wa wi cc0 wceq breq2 anbi2d fveq2 breq2d imbi12d cr cv c1 caddc weq nn0le0eq0 biimpa fveq2d fac0 1re eqeltri leidi - co syl6eqbr impexp clt wo nn0re peano2re syl leloe syl2an nn0leltp1 faccl + co eqbrtrdi impexp clt wo nn0re peano2re syl leloe syl2an nn0leltp1 faccl cmul nnred nnnn0d nn0ge0d nn0p1nn nnge1d lemulge11d facp1 breqtrrd adantl wb adantr peano2nn0 faccld letr syl3anc mpan2d imim2d com23 sylbird leidd syl5ibcom syl5 a1dd jaod sylbid ex com13 com4l imp4a syl5bi nn0ind 3impib @@ -145350,7 +145445,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( cn0 wcel cn cc0 wceq wo cexp co cfv cle wbr wa c1 cr adantr sylan reexpcl cmul cfa elnn0 caddc nnre nnge1 cuz nn0z adantl uzid peano2uz 3syl leexp2ad cz nnnn0 faclbnd wi nn0re peano2nn0 syl2an mpancom faccl nnred remulcl letr - syl3anc mp2and 0exp 0le1 syl6eqbr oveq2 0exp0e1 1le1 eqbrtri jaoi sylbi 1nn + syl3anc mp2and 0exp 0le1 eqbrtrdi oveq2 0exp0e1 1le1 eqbrtri jaoi sylbi 1nn nnmulcl sylancr nnge1d 0re 1re mp3an2 syl2anc wb oveq1 oveq12 anidms syl6eq mpan oveq1d breq12d mpbird jaoian sylanb ) ACDZAEDZAFGZHBCDZABIJZAAIJZBUAKZ TJZLMZAUBWPWRXCWQWPWRNZWSABOUCJZIJZLMZXFXBLMZXCXDABXEWPAPDZWRAUDQWPOALMWRAU @@ -146410,6 +146505,16 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( c0 chash cfv cc0 wceq eqid cvv wcel wb 0ex hasheq0 ax-mp mpbir ) ABCDEZAA EZAFAGHNOIJAGKLM $. + ${ + hashelne0d.1 $e |- ( ph -> B e. A ) $. + hashelne0d.2 $e |- ( ph -> A e. V ) $. + $( A set with an element has nonzero size. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + hashelne0d $p |- ( ph -> -. ( # ` A ) = 0 ) $= + ( chash cfv cc0 wceq c0 ne0d neneqd wcel wb hasheq0 syl mtbird ) ABGHIJZB + KJZABKABCELMABDNSTOFBDPQR $. + $} + $( The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.) $) hashsng $p |- ( A e. V -> ( # ` { A } ) = 1 ) $= @@ -146428,6 +146533,20 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ IAQCZVCQCVEVFTVAVEVJVAVDRCVEVJUEVDFRVHUFUGAVDBUHUIUJSUKAVCULUMUNUOAVCUPUQVF VGTVAVCKAIKVCUROUSPUT $. + ${ + $d ph x $. $d x A $. $d x B $. + hash1elsn.1 $e |- ( ph -> ( # ` A ) = 1 ) $. + hash1elsn.2 $e |- ( ph -> B e. A ) $. + hash1elsn.3 $e |- ( ph -> A e. V ) $. + $( A set of size 1 with a known element is the singleton of that element. + (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + hash1elsn $p |- ( ph -> A = { B } ) $= + ( vx cv csn wceq c1o cen wbr wex chash cfv c1 wcel syl wb mpbid en1 sylib + hashen1 wa simpr adantr eleqtrd elsni sneqd eqtr4d exlimddv ) ABHIZJZKZBC + JZKHABLMNZUPHOABPQRKZUREABDSUSURUAGBDUETUBHBUCUDAUPUFZBUOUQAUPUGZUTCUNUTC + UOSCUNKUTCBUOACBSUPFUHVAUICUNUJTUKULUM $. + $} + ${ $d x A $. $( The size of a restricted class abstraction restricted to a singleton is @@ -146973,7 +147092,7 @@ are used instead of sets because their representation is shorter (and more ( c0 wceq wcel c1 chash cfv clt wbr wa wne wrex wi cc0 cle wn wral biimpd hash0 fveq2 syl5eqr breq2 eqcoms 0le1 0re 1re lenlti pm2.21 sylbi syl6com cv ax-mp adantl com12 syl df-ne necom bitr3i weq ralnex nne equcom ralbii - bitri csn wb eqsn bicomd ralbidv hashsnle1 syl6eqbr a1i reximdva0 r19.36v + bitri csn wb eqsn bicomd ralbidv hashsnle1 eqbrtrdi a1i reximdva0 r19.36v sylbid cxr hashxrcl adantr xrlenlt sylancl sylibd syl5bi impancom pm2.61i 1xr con4d ) EAFZABGZHAIJZKLZMZCUNZDUNZNZDAOZCAOZPZWJQWLFZWTWJQEIJWLUBEAIU CUDWNXAWSWMXAWSPWKXAWMHQKLZWSWMXBPWLQWLQFWMXBWLQHKUEUAUFQHRLZXBWSPZUGXCXB @@ -147958,6 +148077,29 @@ are used instead of sets because their representation is shorter (and more YI $. $} + ${ + phphashd.1 $e |- ( ph -> A e. Fin ) $. + phphashd.2 $e |- ( ph -> B C_ A ) $. + phphashd.3 $e |- ( ph -> ( # ` A ) = ( # ` B ) ) $. + $( Corollary of the Pigeonhole Principle using equality. Equivalent of + ~ phpeqd expressed using the hash function. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + phphashd $p |- ( ph -> A = B ) $= + ( chash cfv wceq cen wbr cfn wcel wb ssfid hashen syl2anc mpbid phpeqd ) + ABCDEABGHCGHIZBCJKZFABLMCLMTUANDABCDEOBCPQRS $. + $} + + ${ + phphashrd.1 $e |- ( ph -> B e. Fin ) $. + phphashrd.2 $e |- ( ph -> A C_ B ) $. + phphashrd.3 $e |- ( ph -> ( # ` A ) = ( # ` B ) ) $. + $( Corollary of the Pigeonhole Principle using equality. Equivalent of + ~ phphashd with reversed arguments. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + phphashrd $p |- ( ph -> A = B ) $= + ( chash cfv eqcomd phphashd ) ACBACBDEABGHCGHFIJI $. + $} + $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- @@ -148081,7 +148223,7 @@ Proper unordered pairs and triples (sets of size 2 and 3) ( vc wcel c0 wa chash cfv c2 cle wbr cv cpr wceq wex wi syl6bi com12 syld wne cc0 c1 cxnn0 hashxnn0 xnn0le2is012 sylan ex hasheq0 eqneqall hash1snb w3o csn vex weq preq12 dfsn2 syl6eqr eqeq2d spc2ev exlimiv imp a1d expcom - hash2pr 3jaoi com23 fveq2 cfn hashprlei simpri syl6eqbr exlimivv impbid1 + hash2pr 3jaoi com23 fveq2 cfn hashprlei simpri eqbrtrdi exlimivv impbid1 ) ABFZAGUBZHAIJZKLMZACNZDNZOZPZDQCQZVPVQVSWDRVPVSVQWDVPVSVRUCPZVRUDPZVRKP ZUMZVQWDRZVPVSWHVPVRUEFVSWHABUFVRUGUHUIWHVPWIWEVPWIRWFWGVPWEWIVPWEAGPWIAB UJWDAGUKSTVPWFWIVPWFHWDVQVPWFWDVPWFAENZUNZPZEQWDABEULWLWDEWCWLCDWJWJEUOZW @@ -157883,7 +158025,7 @@ reflection about the origin (under the map ` x |-> -u x ` ). (Contributed ( ( 0 <_ ( Re ` A ) /\ ( _i x. A ) e/ RR+ ) <-> -. ( 0 <_ ( Re ` -u A ) /\ ( _i x. -u A ) e/ RR+ ) ) ) $= ( wcel cc0 wne wa cre cfv cle wbr ci cmul co crp cneg wn wb wceq ax-icn syl - cr wnel df-nel simpr 0le0 syl6eqbr biantrurd syl5bbr con1bid cim mulcl mpan + cr wnel df-nel simpr 0le0 eqbrtrdi biantrurd syl5bbr con1bid cim mulcl mpan cc reim0b imre cdiv c1 ine0 divrec2 mp3an23 irec oveq1i syl6eq eqtr3d eqtrd divcan3 fveq2d eqeq1d biimpar adantlr mulne0 mpanl12 adantr syl2anc con2bid bitrd rpneg syl6bbr breqtrrid necon3bbid rpre nsyl sylibr biantrud 0re recl @@ -168720,7 +168862,7 @@ attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof vm caddc cseq cli cdm wceq cuz csn cxp cmpt cv cn elnn0 simpr oveq1d 0exp wo sylan9eq oveq2d nncn mul01d simplll 0nn0 syl6eqel expcld mul02d jaodan eqtrd sylan2b mpteq2dva syl5eq fconstmpt nn0uz xpeq1i eqtr3i syl6eq cz 0z - seqeq3d serclim0 ax-mp syl6eqbr seqex c0ex breldm syl cdiv wral wrex 1red + seqeq3d serclim0 ax-mp eqbrtrdi seqex c0ex breldm syl cdiv wral wrex 1red wne abscl peano2re rehalfcld crp absrpcl adantlr rerpdivcld recnd mulid2d c2 wb 1re avglt1 sylancl mpbid eqbrtrd ltmuldivd expmulnbnd syl3anc nn0cn eluznn0 ad2antrr rpne0d expdivd breq12d nn0re reexpcl sylan rpgt0d expgt0 @@ -176543,7 +176685,7 @@ infinite descent (here implemented by strong induction). This is Carneiro, 2-Jul-2015.) $) dvds1 $p |- ( M e. NN0 -> ( M || 1 <-> M = 1 ) ) $= ( cn0 wcel c1 cdvds wbr wceq wa simpl 1nn0 a1i simpr cz 1dvds adantr dvdseq - nn0z syl syl22anc ex id 1z iddvds ax-mp syl6eqbr impbid1 ) ABCZADEFZADGZUGU + nn0z syl syl22anc ex id 1z iddvds ax-mp eqbrtrdi impbid1 ) ABCZADEFZADGZUGU HUIUGUHHZUGDBCZUHDAEFZUIUGUHIUKUJJKUGUHLUGULUHUGAMCULAQANROADPSTUIADDEUIUAD MCDDEFUBDUCUDUEUF $. @@ -180021,6 +180163,31 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact ( cz wcel c1 cgcd co wceq 1z gcdcom mpan gcd1 eqtrd ) ABCZDAEFZADEFZDDBCMNO GHDAIJAKL $. + ${ + gcdmultipled.1 $e |- ( ph -> M e. NN0 ) $. + gcdmultipled.2 $e |- ( ph -> N e. ZZ ) $. + $( The greatest common divisor of a nonnegative integer ` M ` and a + multiple of it is ` M ` itself. (Contributed by Rohan Ridenour, + 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 + CKLBKLFKLUFUIMEABDNZAOCBFPQABSLUFBMDBRTAUHUGBGAUGAUGACBEUJUAUBUCUDUE $. + $} + + ${ + dvdsgcdidd.1 $e |- ( ph -> M e. NN ) $. + dvdsgcdidd.2 $e |- ( ph -> N e. ZZ ) $. + dvdsgcdidd.3 $e |- ( ph -> M || N ) $. + $( The greatest common divisor of a positive integer and another integer it + divides is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + dvdsgcdidd $p |- ( ph -> ( M gcd N ) = M ) $= + ( cdiv co cmul cgcd zcnd nncnd nnne0d divcan1d oveq2d nnnn0d cdvds wbr cz + wcel cc0 wne wb nnzd dvdsval2 syl3anc mpbid gcdmultipled eqtr3d ) ABCBGHZ + BIHZJHBCJHBAUKCBJACBACEKABDLABDMZNOABUJABDPABCQRZUJSTZFABSTBUAUBCSTUMUNUC + ABDUDULEBCUEUFUGUHUI $. + $} + $( The greatest common divisor of six and four is two. To calculate this gcd, a simple form of Euclid's algorithm is used: ` ( 6 gcd 4 ) = ( ( 4 + 2 ) gcd 4 ) = ( 2 gcd 4 ) ` and @@ -190178,7 +190345,7 @@ with complex numbers (gaussian integers) instead, so that we only have ( C e. R /\ ( F ` C ) = 0 ) ) -> ( M Ramsey F ) = 0 ) $= ( vx va vi vb wcel cn0 cfv cc0 wceq wa cle wbr cv chash c0 vf vs vc cn wf w3a cram cvv cpw crab cmpo eqid simpl1 nnnn0d simpl2 simpl3 0nn0 a1i ccnv - csn cima wss wrex simplrl 0elpw simplrr 0le0 syl6eqbr simpll1 0hashbc syl + csn cima wss wrex simplrl 0elpw simplrr 0le0 eqbrtrdi simpll1 0hashbc syl 0ss syl6eqss fveq2 breq1d sneq imaeq2d sseq2d anbi12d hash0 syl6eq breq2d co oveq1 sseq1d rspc2ev syl112anc ramub ramubcl syl32anc nn0le0eq0 mpbid wb ) DUDJZBEJZBKCUEZUFZABJZACLZMNZOZOZDCUGWCZMPQZXCMNZXBFGHUHKIRSLHRNIGRU @@ -190197,7 +190364,7 @@ with complex numbers (gaussian integers) instead, so that we only have ( vc vy vx cn0 wcel cc0 cram co wceq wa cv a1i cr clt cle wral cz wne csn c0 cxp cn wo wi elnn0 wex n0 wf cfv simpll simplr fconst6 simpr fvconst2g 0nn0 sylancr ramz2 syl32anc ex exlimdv syl5bi expimpd crn csup wrex simpl - wbr 0z 0le0 syl6eqbr rgen rnxp adantl raleqdv mpbiri brralrspcev syl31anc + wbr 0z 0le0 eqbrtrdi rgen rnxp adantl raleqdv mpbiri brralrspcev syl31anc elsni 0ram supeq1d wor ltso supsn mp2an 3eqtrd oveq1 eqeq1d syl5ibr sylbi 0re jaoi 3impib ) BGHZACHZAUCUAZBAIUBZUDZJKZILZWPBUEHZBILZUFWQWRMZXBUGZBU HXCXFXDXCWQWRXBWRDNZAHZDUIXCWQMZXBDAUJXIXHXBDXIXHXBXIXHMZXCWQAGWTUKZXHXGW @@ -215791,6 +215958,18 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is PMN $. $} + ${ + hashfinmndnn.1 $e |- B = ( Base ` G ) $. + hashfinmndnn.2 $e |- ( ph -> G e. Mnd ) $. + hashfinmndnn.3 $e |- ( ph -> B e. Fin ) $. + $( A finite monoid has positive integer size. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + hashfinmndnn $p |- ( ph -> ( # ` B ) e. NN ) $= + ( chash cfv cn0 wcel cc0 wne cn cfn hashcl syl c0g cmnd eqid mndidcl + hashelne0d neqned elnnne0 sylanbrc ) ABGHZIJZUEKLUEMJABNJUFFBOPAUEKABCQHZ + NACRJUGBJEBCUGDUGSTPFUAUBUEUCUD $. + $} + ${ mndplusf.1 $e |- B = ( Base ` G ) $. mndplusf.2 $e |- .+^ = ( +f ` G ) $. @@ -218175,6 +218354,16 @@ with the same (relevant) properties is also a group. (Contributed by ( cgrp wcel cbs cfv c0 wne eqid grpbn0 wceq fveq2 base0 syl6eqr necon3i syl ) ABCADEZFGAFGPAPHIAFPFAFJPFDEFAFDKLMNO $. + ${ + hashfingrpnn.1 $e |- B = ( Base ` G ) $. + hashfingrpnn.2 $e |- ( ph -> G e. Grp ) $. + hashfingrpnn.3 $e |- ( ph -> B e. Fin ) $. + $( A finite group has positive integer size. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + hashfingrpnn $p |- ( ph -> ( # ` B ) e. NN ) $= + ( cgrp wcel cmnd grpmnd syl hashfinmndnn ) ABCDACGHCIHECJKFL $. + $} + ${ $d u v w y B $. $d u v w y G $. $d u v w y .+ $. $d u v w y X $. $d u v w y Y $. $d u v w y Z $. @@ -219881,6 +220070,18 @@ since the target of the homomorphism (operator ` O ` in our model) need UIEQUEABCEFGUCUAUBUD $. $} + ${ + mulgcld.1 $e |- B = ( Base ` G ) $. + mulgcld.2 $e |- .x. = ( .g ` G ) $. + mulgcld.3 $e |- ( ph -> G e. Grp ) $. + mulgcld.4 $e |- ( ph -> N e. ZZ ) $. + mulgcld.5 $e |- ( ph -> X e. B ) $. + $( Deduction associated with ~ mulgcl . (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + mulgcld $p |- ( ph -> ( N .x. X ) e. B ) $= + ( cgrp wcel cz co mulgcl syl3anc ) ADLMENMFBMEFCOBMIJKBCDEFGHPQ $. + $} + ${ mulgaddcom.b $e |- B = ( Base ` G ) $. mulgaddcom.t $e |- .x. = ( .g ` G ) $. @@ -220813,6 +221014,33 @@ by a normal subgroup (resp. two-sided ideal). (Contributed by Mario WBWJWKVCVF $. $} + ${ + trivsubgd.1 $e |- B = ( Base ` G ) $. + trivsubgd.2 $e |- .0. = ( 0g ` G ) $. + trivsubgd.3 $e |- ( ph -> G e. Grp ) $. + trivsubgd.4 $e |- ( ph -> B = { .0. } ) $. + trivsubgd.5 $e |- ( ph -> A e. ( SubGrp ` G ) ) $. + $( The only subgroup of a trivial group is itself. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + trivsubgd $p |- ( ph -> A = B ) $= + ( csn csubg cfv wcel wss subgss syl sseqtrd subg0cl snssd eqssd eqtr4d ) + ABEKZCABUCABCUCABDLMNZBCOJCBDFPQIRAEBAUDEBNJBDEGSQTUAIUB $. + $} + + ${ + $d x B $. $d x G $. $d x ph $. $d x .0. $. + trivsubgsnd.1 $e |- B = ( Base ` G ) $. + trivsubgsnd.2 $e |- .0. = ( 0g ` G ) $. + trivsubgsnd.3 $e |- ( ph -> G e. Grp ) $. + trivsubgsnd.4 $e |- ( ph -> B = { .0. } ) $. + $( The only subgroup of a trivial group is itself. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + trivsubgsnd $p |- ( ph -> ( SubGrp ` G ) = { B } ) $= + ( vx csubg cfv csn cv wcel wa wceq cgrp adantr simpr trivsubgd sylibr syl + velsn ex ssrdv subgid snssd eqssd ) ACJKZBLZAIUIUJAIMZUINZUKUJNZAULOZUKBP + UMUNUKBCDEFACQNZULGRABDLPULHRAULSTIBUCUAUDUEABUIAUOBUINGBCEUFUBUGUH $. + $} + ${ $d m n s x A $. $d m n s u v x G $. $d x S $. $d x .x. $. $d m n x X $. $d m n s u v F $. @@ -221140,6 +221368,56 @@ by a normal subgroup (resp. two-sided ideal). (Contributed by Mario G $. $} + ${ + 0idnsgd.1 $e |- B = ( Base ` G ) $. + 0idnsgd.2 $e |- .0. = ( 0g ` G ) $. + 0idnsgd.3 $e |- ( ph -> G e. Grp ) $. + $( The whole group and the zero subgroup are normal subgroups of a group. + (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + 0idnsgd $p |- ( ph -> { { .0. } , B } C_ ( NrmSGrp ` G ) ) $= + ( csn cnsg cfv cgrp wcel 0nsg syl nsgid prssd ) ADHZBCIJZACKLZQRLGCDFMNAS + BRLGBCEONP $. + $} + + ${ + $d x B $. $d x G $. $d x ph $. $d x .0. $. + trivnsgd.1 $e |- B = ( Base ` G ) $. + trivnsgd.2 $e |- .0. = ( 0g ` G ) $. + trivnsgd.3 $e |- ( ph -> G e. Grp ) $. + trivnsgd.4 $e |- ( ph -> B = { .0. } ) $. + $( The only normal subgroup of a trivial group is itself. (Contributed by + Rohan Ridenour, 3-Aug-2023.) $) + trivnsgd $p |- ( ph -> ( NrmSGrp ` G ) = { B } ) $= + ( vx cnsg cfv csn csubg cv wcel wi nsgsubg a1i ssrdv trivsubgsnd cgrp syl + sseqtrd nsgid snssd eqssd ) ACJKZBLZAUGCMKZUHAIUGUIINZUGOUJUIOPAUJCQRSABC + DEFGHTUCABUGACUAOBUGOGBCEUDUBUEUF $. + $} + + ${ + triv1nsgd.1 $e |- B = ( Base ` G ) $. + triv1nsgd.2 $e |- .0. = ( 0g ` G ) $. + triv1nsgd.3 $e |- ( ph -> G e. Grp ) $. + triv1nsgd.4 $e |- ( ph -> B = { .0. } ) $. + $( A trivial group has exactly one normal subgroup. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + triv1nsgd $p |- ( ph -> ( NrmSGrp ` G ) ~~ 1o ) $= + ( cnsg cfv csn c1o cen trivnsgd cvv wcel wbr snex syl6eqel ensn1g eqbrtrd + syl ) ACIJBKZLMABCDEFGHNABOPUCLMQABDKOHDRSBOTUBUA $. + $} + + ${ + 1nsgtrivd.1 $e |- B = ( Base ` G ) $. + 1nsgtrivd.2 $e |- .0. = ( 0g ` G ) $. + 1nsgtrivd.3 $e |- ( ph -> G e. Grp ) $. + 1nsgtrivd.4 $e |- ( ph -> ( NrmSGrp ` G ) ~~ 1o ) $. + $( A group with exactly one normal subgroup is trivial. (Contributed by + Rohan Ridenour, 3-Aug-2023.) $) + 1nsgtrivd $p |- ( ph -> B = { .0. } ) $= + ( csn wcel wceq cnsg cfv cgrp nsgid syl c1o cen wbr cvv 0nsg en1eqsn snex + syl2anc eleqtrd wb elsn2g mp1i mpbid ) ABDIZIZJZBUJKZABCLMZUKACNJZBUNJGBC + EOPAUJUNJZUNQRSUNUKKAUOUPGCDFUAPHUJUNUBUDUEUJTJULUMUFADUCBUJTUGUHUI $. + $} + ${ $d g s x y $. releqg.r $e |- R = ( G ~QG S ) $. @@ -226315,6 +226593,16 @@ operation is a permutation group (group consisting of permutations), see L $. $} + ${ + odcld.1 $e |- B = ( Base ` G ) $. + odcld.2 $e |- O = ( od ` G ) $. + odcld.3 $e |- ( ph -> A e. B ) $. + $( The order of a group element is always a nonnegative integer, deduction + form of ~ odcl . (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + odcld $p |- ( ph -> ( O ` A ) e. NN0 ) $= + ( wcel cfv cn0 odcl syl ) ABCIBEJKIHBDECFGLM $. + $} + ${ $d x y A $. $d x y G $. $d x y N $. $d x y O $. $d x y .x. $. $d x y X $. @@ -226909,7 +227197,7 @@ operation is a permutation group (group consisting of permutations), see gex1 $p |- ( G e. Mnd -> ( E = 1 <-> X ~~ 1o ) ) $= ( vx cmnd wcel c1 wceq c1o cen wbr wa c0g cfv csn co eqid adantl cv gexid cmg simplr oveq1d mulg1 3eqtr3rd velsn sylibr ex ssrdv adantr snssd eqssd - mndidcl fvex ensn1 syl6eqbr cfz cn wral simpl 1nn a1i sylan eleq2d biimpa + mndidcl fvex ensn1 eqbrtrdi cfz cn wral simpl 1nn a1i sylan eleq2d biimpa en1eqsn sylib eqtrd ralrimiva gexlem2 syl3anc elfz1eq syl impbida ) BGHZA IJZCKLMZVQVRNZCBOPZQZKLVTCWBVTFCWBVTFUAZCHZWCWBHZVTWDNZWCWAJZWEWFAWCBUCPZ RZIWCWHRZWAWCWFAIWCWHVQVRWDUDUEWDWIWAJVTWCWHABCWADEWHSZWASZUBTWDWJWCJZVTC @@ -227591,7 +227879,7 @@ P pGrp ( G |`s S ) ) $= c0 sylib cga wer gaorber qsss ssfid wa adantr sselda elpwid hashcl nn0cnd syl cn0 fsumsplit qshash c1 cmul cc wss inss1 ssfi sylancl fsumconst cuni ax-1cn csn c1o cen wbr elin cec wi eqid sseq1 velpw syl6bbr breq1 imbi12d - simpr erref vex elec sylibr syl5com sylow2alem1 ensn1 syl6eqbr ex ectocld + simpr erref vex elec sylibr syl5com sylow2alem1 ensn1 eqbrtrdi ex ectocld ssel syld impr sylan2b en1b fveq2d cvv vuniex ax-mp syl6eq sumeq2dv ssexg hashsng sylancr eqeltrd sseldi syl2anc wb mpbird oveq1d 3eqtr4rd wral cdm ssrab3 mulid1d rabexd inss2 crn cxp wrel cpr wrex relopabi relssdmrn erdm @@ -231056,16 +231344,16 @@ an extension of the previous (inserting an element and its inverse at $( -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Abelian groups -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Definition and basic properties -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) $c CMnd $. @@ -231098,6 +231386,14 @@ an extension of the previous (inserting an element and its inverse at ablgrp $p |- ( G e. Abel -> G e. Grp ) $= ( cabl wcel cgrp ccmn isabl simplbi ) ABCADCAECAFG $. + ${ + ablgrpd.1 $e |- ( ph -> G e. Abel ) $. + $( An Abelian group is a group, deduction form of ~ ablgrp . (Contributed + by Rohan Ridenour, 3-Aug-2023.) $) + ablgrpd $p |- ( ph -> G e. Grp ) $= + ( cabl wcel cgrp ablgrp syl ) ABDEBFECBGH $. + $} + $( An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.) $) ablcmn $p |- ( G e. Abel -> G e. CMnd ) $= @@ -231728,6 +232024,29 @@ an extension of the previous (inserting an element and its inverse at ) BDGZAHZCIJZBAKZLBTKZABCDEFMUAUCUDUBUAUCUDUCUDNATATBOPQRS $. $} + ${ + $d M x y $. $d Z x y $. + cntrcmnd.z $e |- Z = ( M |`s ( Cntr ` M ) ) $. + $( The center of a monoid is a commutative submonoid. (Contributed by + Thierry Arnoux, 21-Aug-2023.) $) + cntrcmnd $p |- ( M e. Mnd -> Z e. CMnd ) $= + ( vx vy cmnd wcel ccntr cfv cplusg cbs wss wceq eqid cntrss ressbas2 mp1i + cvv cv co fvex ressplusg csubmnd ccntz cntrval cntzsubm eqeltrrid submmnd + ssid mpan2 syl w3a simp2 simp3 sseldi cntri syl2anc iscmnd ) AFGZDEAHIZAJ + IZBUTAKIZLUTBKIMUSVBAVBNZOZUTVBBACVCPQUTRGVABJIMUSAHUAUTVAABRCVANZUBQUSUT + AUCIZGBFGUSUTVBAUDIZIZVFVBAVGVCVGNZUEUSVBVBLVHVFGVBUIVBVBAVGVCVIUFUJUGUTB + ACUHUKUSDSZUTGZESZUTGZULZVKVLVBGVJVLVATVLVJVATMUSVKVMUMVNUTVBVLVDUSVKVMUN + UOVBVAAVJVLUTVCVEUTNUPUQUR $. + + $( The center of a group is an abelian group. (Contributed by Thierry + Arnoux, 21-Aug-2023.) $) + cntrabl $p |- ( M e. Grp -> Z e. Abel ) $= + ( cgrp wcel ccmn cabl ccntr cfv csubg cbs ccntz eqid cntrval wss cntzsubg + ssid mpan2 eqeltrrid syl subggrp cmnd grpmnd cntrcmnd isabl sylanbrc ) AD + EZBDEZBFEZBGEUGAHIZAJIZEUHUGUJAKIZALIZIZUKULAUMULMZUMMZNUGULULOUNUKEULQUL + ULAUMUOUPPRSUJABCUATUGAUBEUIAUCABCUDTBUEUF $. + $} + ${ cntzspan.z $e |- Z = ( Cntz ` G ) $. cntzspan.k $e |- K = ( mrCls ` ( SubMnd ` G ) ) $. @@ -232395,9 +232714,9 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements $( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Cyclic groups -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) $c CycGrp $. $( Cyclic group $) @@ -232764,9 +233083,9 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements $( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Group sum operation -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) ${ @@ -234685,9 +235004,9 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements $( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Group sums over (ranges of) integers -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) ${ @@ -235002,9 +235321,9 @@ Group sums over (ranges of) integers $( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Internal direct products -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) $( Introduce new constant symbols. $) @@ -236659,9 +236978,9 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $. $( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- The Fundamental Theorem of Abelian Groups -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) ${ @@ -237803,6 +238122,460 @@ factorization into prime power factors (even if the exponents are ACDIUAJKLMNUUBUUC $. $} +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Simple groups +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + $c SimpGrp $. + + $( Extend class notation with the class of simple groups. $) + csimpg $a class SimpGrp $. + + $( Define class of all simple groups. A simple group is a group ( ~ df-grp ) + with exactly two normal subgroups. These are always the subgroup of all + elements and the subgroup containing only the identity ( ~ simpgnsgbid ). + (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + df-simpg $a |- SimpGrp = { g e. Grp | ( NrmSGrp ` g ) ~~ 2o } $. + + ${ + $d g G $. + $( The predicate "is a simple group". (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + issimpg $p |- ( G e. SimpGrp <-> ( G e. Grp /\ ( NrmSGrp ` G ) ~~ 2o ) ) $= + ( vg cnsg cfv c2o cen wbr cgrp csimpg wceq fveq2 breq1d df-simpg elrab2 + cv ) BOZCDZEFGACDZEFGBAHIPAJQREFPACKLBMN $. + $} + + ${ + issimpgd.1 $e |- ( ph -> G e. Grp ) $. + issimpgd.2 $e |- ( ph -> ( NrmSGrp ` G ) ~~ 2o ) $. + $( Deduce a simple group from its properties. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + issimpgd $p |- ( ph -> G e. SimpGrp ) $= + ( cgrp wcel cnsg cfv c2o cen wbr csimpg issimpg sylanbrc ) ABEFBGHIJKBLFC + DBMN $. + $} + + $( A simple group is a group. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + simpggrp $p |- ( G e. SimpGrp -> G e. Grp ) $= + ( csimpg wcel cgrp cnsg cfv c2o cen wbr issimpg simplbi ) ABCADCAEFGHIAJK + $. + + ${ + simpggrpd.1 $e |- ( ph -> G e. SimpGrp ) $. + $( A simple group is a group. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + simpggrpd $p |- ( ph -> G e. Grp ) $= + ( csimpg wcel cgrp simpggrp syl ) ABDEBFECBGH $. + $} + + $( A simple group has two normal subgroups. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + simpg2nsg $p |- ( G e. SimpGrp -> ( NrmSGrp ` G ) ~~ 2o ) $= + ( csimpg wcel cgrp cnsg cfv c2o cen wbr issimpg simprbi ) ABCADCAEFGHIAJK + $. + + ${ + trivnsimpgd.1 $e |- B = ( Base ` G ) $. + trivnsimpgd.2 $e |- .0. = ( 0g ` G ) $. + trivnsimpgd.3 $e |- ( ph -> G e. Grp ) $. + trivnsimpgd.4 $e |- ( ph -> B = { .0. } ) $. + $( Trivial groups are not simple. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + trivnsimpgd $p |- ( ph -> -. G e. SimpGrp ) $= + ( cnsg cfv c2o cen wbr csimpg wcel csn snnen2o trivnsgd breq1d mtbiri + simpg2nsg nsyl ) ACIJZKLMZCNOAUDBPZKLMBQAUCUEKLABCDEFGHRSTCUAUB $. + $} + + ${ + simpgntrivd.1 $e |- B = ( Base ` G ) $. + simpgntrivd.2 $e |- .0. = ( 0g ` G ) $. + simpgntrivd.3 $e |- ( ph -> G e. SimpGrp ) $. + $( Simple groups are nontrivial. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + simpgntrivd $p |- ( ph -> -. B = { .0. } ) $= + ( csn wceq csimpg wcel adantr cgrp simpggrpd simpr trivnsimpgd pm2.65da + wa ) ABDHIZCJKZATSGLASRBCDEFACMKSACGNLASOPQ $. + $} + + ${ + $d x .0. $. $d x B $. + simpgnideld.1 $e |- B = ( Base ` G ) $. + simpgnideld.2 $e |- .0. = ( 0g ` G ) $. + simpgnideld.3 $e |- ( ph -> G e. SimpGrp ) $. + $( A simple group contains a nonidentity element. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + simpgnideld $p |- ( ph -> E. x e. B -. x = .0. ) $= + ( cv wceq wral wn wrex csn simpgntrivd c0 wne wb cgrp wcel cmnd simpggrpd + grpmnd mndidcl 3syl ne0d eqsn syl mtbid rexnal sylibr ) ABIEJZBCKZLULLBCM + ACENJZUMACDEFGHOACPQUNUMRACEADSTDUATECTADHUBDUCCDEFGUDUEUFBCEUGUHUIULBCUJ + UK $. + $} + + ${ + simpgnsgd.1 $e |- B = ( Base ` G ) $. + simpgnsgd.2 $e |- .0. = ( 0g ` G ) $. + simpgnsgd.3 $e |- ( ph -> G e. SimpGrp ) $. + $( The only normal subgroups of a simple group are the group itself and the + trivial group. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + simpgnsgd $p |- ( ph -> ( NrmSGrp ` G ) = { { .0. } , B } ) $= + ( cnsg cfv csn c2o cfn wcel cen wbr a1i syl syl2anc cvv cbs cpr 2onn nnfi + com simpg2nsg enfii simpggrpd 0idnsgd snex wceq fvex syl6eqel simpgntrivd + csimpg neqcomd enpr2d ensymd entr phpeqd ) ACHIZDJZBUAZAKLMZUTKNOZUTLMAKU + DMZVCVEAUBPKUCQACUNMVDGCUEQZUTKUFRABCDEFACGUGUHAVDKVBNOUTVBNOVFAVBKAVABSS + VASMADUIPABCTIZSBVGUJAEPCTUKULABVAABCDEFGUMUOUPUQUTKVBURRUS $. + $} + + ${ + simpgnsgeqd.1 $e |- B = ( Base ` G ) $. + simpgnsgeqd.2 $e |- .0. = ( 0g ` G ) $. + simpgnsgeqd.3 $e |- ( ph -> G e. SimpGrp ) $. + simpgnsgeqd.4 $e |- ( ph -> A e. ( NrmSGrp ` G ) ) $. + $( A normal subgroup of a simple group is either the whole group or the + trivial subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + simpgnsgeqd $p |- ( ph -> ( A = { .0. } \/ A = B ) ) $= + ( csn cpr wcel wceq wo cnsg cfv simpgnsgd eleqtrd elpri syl ) ABEJZCKZLBU + AMBCMNABDOPUBIACDEFGHQRBUACST $. + $} + + ${ + $d ph x $. $d x .0. $. $d x B $. $d x G $. + 2nsgsimpgd.1 $e |- B = ( Base ` G ) $. + 2nsgsimpgd.2 $e |- .0. = ( 0g ` G ) $. + 2nsgsimpgd.3 $e |- ( ph -> G e. Grp ) $. + 2nsgsimpgd.4 $e |- ( ph -> -. { .0. } = B ) $. + 2nsgsimpgd.5 $e |- ( ( ph /\ x e. ( NrmSGrp ` G ) ) -> + ( x = { .0. } \/ x = B ) ) $. + $( If any normal subgroup of a nontrivial group is either the trivial + subgroup or the whole group, the group is simple. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + 2nsgsimpgd $p |- ( ph -> G e. SimpGrp ) $= + ( wcel wa wceq adantl simpr syl adantr eqeltrd adantlr cvv cfv csn cpr cv + cnsg c2o cen wo elprg mpbird cgrp 0nsg nsgid elpri mpjaodan impbida eqrdv + wb snex a1i cbs fvexi enpr2d eqbrtrd issimpgd ) ADHADUEUAZEUBZCUCZUFUGABV + FVHABUDZVFKZVIVHKZAVJLVKVIVGMZVICMZUHZJVJVKVNURAVIVGCVFUINUJAVKLVLVJVMAVL + VJVKAVLLVIVGVFAVLOAVGVFKZVLADUKKZVOHDEGULPQRSAVMVJVKAVMLVICVFAVMOACVFKZVM + AVPVQHCDFUMPQRSVKVNAVIVGCUNNUOUPUQAVGCTTVGTKAEUSUTCTKACDVAFVBUTIVCVDVE $. + $} + + ${ + $d ph x $. $d x .0. $. $d x B $. $d x G $. + simpgnsgbid.1 $e |- B = ( Base ` G ) $. + simpgnsgbid.2 $e |- .0. = ( 0g ` G ) $. + simpgnsgbid.3 $e |- ( ph -> G e. Grp ) $. + simpgnsgbid.4 $e |- ( ph -> -. { .0. } = B ) $. + $( A nontrivial group is simple if and only if its normal subgroups are + exactly the group itself and the trivial subgroup. (Contributed by + Rohan Ridenour, 4-Aug-2023.) $) + simpgnsgbid $p |- ( ph -> + ( G e. SimpGrp <-> ( NrmSGrp ` G ) = { { .0. } , B } ) ) $= + ( vx csimpg wcel cnsg cfv csn cpr wceq wa simpr simpgnsgd adantr wn cv wo + cgrp simplr eleqtrd elpri syl 2nsgsimpgd impbida ) ACJKZCLMZDNZBOZPZAUKQB + CDEFAUKRSAUOQZIBCDEFACUDKUOGTAUMBPUAUOHTUPIUBZULKZQZUQUNKUQUMPUQBPUCUSUQU + LUNUPURRAUOURUEUFUQUMBUGUHUIUJ $. + $} + +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Classification of abelian simple groups +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + ${ + $d ph n $. $d A n $. + cycsubggend.1 $e |- B = ( Base ` G ) $. + cycsubggend.2 $e |- .x. = ( .g ` G ) $. + cycsubggend.3 $e |- F = ( n e. ZZ |-> ( n .x. A ) ) $. + cycsubggend.4 $e |- ( ph -> A e. B ) $. + $( The cyclic subgroup generated by ` A ` includes its generator. Although + this theorem holds for any class ` G ` , the definition of ` F ` is only + meaningful if ` G ` is a group. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + cycsubggend $p |- ( ph -> A e. ran F ) $= + ( cz cv co c1 1zzd wceq wa simpr oveq1d adantr mulg1 syl eqtr2d elrnmptdv + wcel ) AELEMZBDNZOBFCJAPKAUGOQZRZUHOBDNZBUJUGOBDAUISTUJBCUFZUKBQAULUIKUAC + DGBHIUBUCUDUE $. + $} + + ${ + $d G n $. $d .x. n $. $d B n $. $d A n $. + cycsubgcld.1 $e |- B = ( Base ` G ) $. + cycsubgcld.2 $e |- .x. = ( .g ` G ) $. + cycsubgcld.3 $e |- F = ( n e. ZZ |-> ( n .x. A ) ) $. + cycsubgcld.4 $e |- ( ph -> G e. Grp ) $. + cycsubgcld.5 $e |- ( ph -> A e. B ) $. + $( The cyclic subgroup generated by ` A ` is a subgroup. Deduction related + to ~ cycsubgcl . (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + cycsubgcld $p |- ( ph -> ran F e. ( SubGrp ` G ) ) $= + ( crn csubg cfv wcel cgrp wa cycsubgcl syl2anc simpld ) AFMZGNOPZBUBPZAGQ + PBCPUCUDRKLEBDFGCHIJSTUA $. + $} + + ${ + ablsimpnosubgd.1 $e |- B = ( Base ` G ) $. + ablsimpnosubgd.2 $e |- .0. = ( 0g ` G ) $. + ablsimpnosubgd.3 $e |- ( ph -> G e. Abel ) $. + ablsimpnosubgd.4 $e |- ( ph -> G e. SimpGrp ) $. + ablsimpnosubgd.5 $e |- ( ph -> S e. ( SubGrp ` G ) ) $. + ablsimpnosubgd.6 $e |- ( ph -> A e. S ) $. + ablsimpnosubgd.7 $e |- ( ph -> -. A = .0. ) $. + $( A subgroup of an abelian simple group containing a nonidentity element + is the whole group. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + ablsimpnosubgd $p |- ( ph -> S = B ) $= + ( csn wceq wcel elsni nsyl eleq2 cfv syl5ibcom mtod pm2.21d idd cnsg cabl + csubg ablnsg eqcomd syl eleqtrd simpgnsgeqd mpjaod ) ADFNZOZDCOZUPAUOUPAU + OBUNPZABFOUQMBFQRABDPUOUQLDUNBSUAUBUCAUPUDADCEFGHJADEUGTZEUETZKAEUFPZURUS + OIUTUSUREUHUIUJUKULUM $. + $} + + ${ + $d ph n $. $d .x. n $. $d A n $. $d C n $. $d B n $. $d G n $. + ablsimpg1gend.1 $e |- B = ( Base ` G ) $. + ablsimpg1gend.2 $e |- .0. = ( 0g ` G ) $. + ablsimpg1gend.3 $e |- .x. = ( .g ` G ) $. + ablsimpg1gend.4 $e |- ( ph -> G e. Abel ) $. + ablsimpg1gend.5 $e |- ( ph -> G e. SimpGrp ) $. + ablsimpg1gend.6 $e |- ( ph -> A e. B ) $. + ablsimpg1gend.7 $e |- ( ph -> -. A = .0. ) $. + ablsimpg1gend.8 $e |- ( ph -> C e. B ) $. + $( An abelian simple group is generated by any non-identity element. + (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + ablsimpg1gend $p |- ( ph -> E. n e. ZZ C = ( n .x. A ) ) $= + ( cz cv co cmpt simpggrpd cycsubgcld cycsubggend ablsimpnosubgd elrnmpt2d + eqid crn eleqtrrd ) AFQFRBESZDFQUITZUJUFZADCUJUGZPABCULGHIJLMABCEFUJGIKUK + AGMUANUBABCEFUJGIKUKNUCOUDUHUE $. + $} + + ${ + $d ph x y z $. $d x y z G $. + ablsimpgcygd.1 $e |- ( ph -> G e. Abel ) $. + ablsimpgcygd.2 $e |- ( ph -> G e. SimpGrp ) $. + $( An abelian simple group is cyclic. (Contributed by Rohan Ridenour, + 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.) $) + ablsimpgcygd $p |- ( ph -> G e. CycGrp ) $= + ( vx vy vz cv c0g cfv wceq wn ccyg wcel cbs eqid simpgnideld cmg ad2antrr + wa cgrp simpggrpd adantr simprl cabl csimpg simplrl simplrr ablsimpg1gend + simpr iscygd rexlimddv ) AEHZBIJZKLZBMNEBOJZAEUPBUNUPPZUNPZDQAUMUPNZUOTZT + ZFUPBRJZGBUMUQVBPZABUANUTABDUBUCAUSUOUDVAFHZUPNZTUMUPVDVBGBUNUQURVCABUENU + TVECSABUFNUTVEDSAUSUOVEUGAUSUOVEUHVAVEUJUIUKUL $. + $} + + ${ + $d y G $. $d y .x. $. $d y O $. $d ph x y $. $d x y .0. $. $d x y B $. + ablsimpgfindlem1.1 $e |- B = ( Base ` G ) $. + ablsimpgfindlem1.2 $e |- .0. = ( 0g ` G ) $. + ablsimpgfindlem1.3 $e |- .x. = ( .g ` G ) $. + ablsimpgfindlem1.4 $e |- O = ( od ` G ) $. + ablsimpgfindlem1.5 $e |- ( ph -> G e. Abel ) $. + ablsimpgfindlem1.6 $e |- ( ph -> G e. SimpGrp ) $. + $( Lemma for ~ ablsimpgfind . An element of an abelian finite simple group + which doesn't square to the identity has finite order. (Contributed by + Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, + 31-Oct-2023.) $) + ablsimpgfindlem1 $p |- ( ( ( ph /\ x e. B ) /\ ( 2 .x. x ) =/= .0. ) + -> ( O ` x ) =/= 0 ) $= + ( vy wcel co wne wceq 3ad2ant1 wa cv c2 cfv cc0 w3a cabl csimpg simpggrpd + cz cgrp 2z a1i simp2 mulgcld simp3 neneqd ablsimpg1gend cmul c1 cdvds wbr + cmin wn simprr simpl2 mulg1 syl adantr mulgassr syl13anc 3eqtr4rd zmulcld + simprl wb 1zzd odcong syl112anc mpbird 0zd caddc 2t0e0 oveq1i 0p1e1 eqtri + zneo neeqtrd oveq1 syl6req adantl cc 2cnd mulcld 1cnd npcan syl2an eqtr2d + zcn ex necon3ad syl5 anabsi5 syl2anc zsubcld 0dvds mtbird rexlimddv 3expa + nbrne2 ) ABUAZCOZUBZXIDPZGQZXIFUCZUDZQZAXJXMUEZXINUAZXLDPZRZXPNUIZXQXLCXI + DNEGHIJAXJEUFOXMLSAXJEUGOXMMSXQCDEXKXIHJAXJEUJOZXMAEMUHSZXKYAOZXQUKZULAXJ + XMUMZUNXQXLGAXJXMUOUPYFUQXQXRYAOZXTTZTZXNXKXRURZPZUSZVBPZUTZVAZXOYMYNVAZV + CXPYIYOYKXIDPZYLXIDPZRZYIXIXSYRYQXQYGXTVDYIXJYRXIRAXJXMYHVEZCDEXIHJVFVGYI + YBYGYDXJYQXSRXQYBYHYCVHZXQYGXTVMZYDYIYEULZYTCDEXRXKXIHJVIVJVKYIYBXJYKYAOY + LYAOYOYSVNUUAYTYIXKXRUUCUUBVLZYIVOZXIDEYKYLFCGHKJIVPVQVRYIYPYMXORZYIYGXOY + AOZUUFVCZUUBYIVSYGUUGUUHYGUUGTZYKYLQYGUUHUUIYKXKXOYJPZYLVTZPZYLXRXOWEUULY + LRUUIUULXOYLUUKPZYLUUJXOYLUUKWAWBWCZWDULWFYGUUFYKYLYGUUFYKYLRYGUUFTYLYMYL + UUKPZYKUUFYLUUORYGUUFUUOUUMYLYMXOYLUUKWGUUNWHWIYGYKWJZOYLUUPOUUOYKRUUFYGX + KXRYGWKXRWQWLUUFWMYKYLWNWOWPWRWSWTXAXBYIYMYAOYPUUFVNYIYKYLUUDUUEXCYMXDVGX + EXNXOYMYNXHXBXFXG $. + + $( Lemma for ~ ablsimpgfind . An element of an abelian finite simple group + which squares to the identity has finite order. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + ablsimpgfindlem2 $p |- ( ( ( ph /\ x e. B ) /\ ( 2 .x. x ) = .0. ) -> + ( O ` x ) =/= 0 ) $= + ( wcel wa c2 wceq cdvds wbr cc0 cv co cfv wn wne simpr cgrp w3a simpggrpd + cz wb adantr 2z a1i 3jca oddvds syl mpbird 2ne0 neneqd 0dvds ax-mp nbrne2 + sylnibr syl2anc ) ABUAZCNZOZPVFDUBGQZOZVFFUCZPRSZTPRSZUDVKTUEVJVLVIVHVIUF + VJEUGNZVGPUJNZUHZVLVIUKVHVPVIVHVNVGVOAVNVGAEMUIULAVGUFVOVHUMUNUOULVFDEPFC + GHKJIUPUQURVJPTQZVMVJPTPTUEVJUSUNUTVOVMVQUKUMPVAVBVDVKTPRVCVE $. + $} + + ${ + $d .x. n $. $d A n $. $d B n $. $d n G $. $d n O $. + cycsubggenodd.1 $e |- B = ( Base ` G ) $. + cycsubggenodd.2 $e |- .x. = ( .g ` G ) $. + cycsubggenodd.3 $e |- O = ( od ` G ) $. + cycsubggenodd.4 $e |- ( ph -> G e. Grp ) $. + cycsubggenodd.5 $e |- ( ph -> A e. B ) $. + cycsubggenodd.6 $e |- ( ph -> C = ran ( n e. ZZ |-> ( n .x. A ) ) ) $. + $( Relationship between the order of a subgroup and the order of a + generator of the subgroup. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + cycsubggenodd $p |- ( ph -> + ( O ` A ) = if ( C e. Fin , ( # ` C ) , 0 ) ) $= + ( cfv cfn wcel chash cc0 cif cz cv cmpt crn cgrp wceq eqid syl2anc eqcomd + co dfod2 eleq1d fveq2d ifbieq1d eqtrd ) ABHOZFUAFUBBEUJUCZUDZPQZURROZSTZD + PQZDROZSTAGUEQBCQUPVAUFLMFBEUQGHCIKJUQUGUKUHAUSVBUTVCSAURDPADURNUIZULAURD + RVDUMUNUO $. + $} + + ${ + $d B n $. $d n G $. $d ph x n $. $d x y G $. $d ph x y $. $d x y B $. + $d y n $. + ablsimpgfind.1 $e |- B = ( Base ` G ) $. + ablsimpgfind.2 $e |- ( ph -> G e. Abel ) $. + ablsimpgfind.3 $e |- ( ph -> G e. SimpGrp ) $. + $( An abelian simple group is finite. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + ablsimpgfind $p |- ( ph -> B e. Fin ) $= + ( vx vn vy wcel wa cfv cc0 wne cv wrex eqid adantr cz ad2antrr wfal chash + cfn cif simpr iffalsed c0g wceq simpgnideld neqne reximi syl cod cmg cgrp + wn simpggrpd simprl cab cmpt crn cabl csimpg simplrr neneqd ablsimpg1gend + co simprr mulgcld eqeltrd rexlimdvaa impbid abbi2dv syl6eqr cycsubggenodd + ex rnmpt c2 ablsimpgfindlem2 ablsimpgfindlem1 pm2.61dane adantrr eqnetrrd + rexlimddv pm2.21ddne efald ) ABUCJZAWGUPZKZUAWGBUBLZMUDZMWIWGWJMAWHUEUFAW + KMNZWHAGOZCUGLZNZWLGBAWMWNUHUPZGBPWOGBPAGBCWNDWNQZFUIWPWOGBWMWNUJUKULAWMB + JZWOKZKZWMCUMLZLZWKMWTWMBBCUNLZHCXADXCQZXAQZACUOJZWSACFUQZRAWRWOURZWTBIOZ + HOZWMXCVGZUHZHSPZIUSHSXKUTZVAWTXMIBWTXIBJZXMWTXOXMWTXOKZWMBXIXCHCWNDWQXDA + CVBJWSXOETACVCJWSXOFTWTWRXOXHRXPWMWNAWRWOXOVDVEWTXOUEVFVPWTXLXOHSWTXJSJZX + LKZKZXIXKBWTXQXLVHXSBXCCXJWMDXDAXFWSXRXGTWTXQXLURWTWRXRXHRVIVJVKVLVMHISXK + XNXNQVQVNVOAWRXBMNZWOAWRKXTVRWMXCVGWNAGBXCCXAWNDWQXDXEEFVSAGBXCCXAWNDWQXD + XEEFVTWAWBWCWDRWEWF $. + $} + + ${ + $d .x. n $. $d A n $. $d C n $. $d B n $. $d G n $. + fincygsubgd.1 $e |- B = ( Base ` G ) $. + fincygsubgd.2 $e |- .x. = ( .g ` G ) $. + fincygsubgd.3 $e |- H = ( n e. ZZ |-> ( n .x. ( C .x. A ) ) ) $. + fincygsubgd.4 $e |- ( ph -> G e. Grp ) $. + fincygsubgd.5 $e |- ( ph -> A e. B ) $. + fincygsubgd.6 $e |- ( ph -> C e. NN ) $. + $( The subgroup referenced in ~ fincygsubgodd is a subgroup. (Contributed + by Rohan Ridenour, 3-Aug-2023.) $) + fincygsubgd $p |- ( ph -> ran H e. ( SubGrp ` G ) ) $= + ( co nnzd mulgcld cycsubgcld ) ADBEOCEFHGIJKLACEGDBIJLADNPMQR $. + $} + + ${ + $d .x. n $. $d A n $. $d B n $. $d C n $. $d n G $. + fincygsubgodd.1 $e |- B = ( Base ` G ) $. + fincygsubgodd.2 $e |- .x. = ( .g ` G ) $. + fincygsubgodd.3 $e |- D = ( ( # ` B ) / C ) $. + fincygsubgodd.4 $e |- F = ( n e. ZZ |-> ( n .x. A ) ) $. + fincygsubgodd.5 $e |- H = ( n e. ZZ |-> ( n .x. ( C .x. A ) ) ) $. + fincygsubgodd.6 $e |- ( ph -> G e. Grp ) $. + fincygsubgodd.7 $e |- ( ph -> A e. B ) $. + fincygsubgodd.8 $e |- ( ph -> ran F = B ) $. + fincygsubgodd.9 $e |- ( ph -> C || ( # ` B ) ) $. + fincygsubgodd.10 $e |- ( ph -> B e. Fin ) $. + fincygsubgodd.11 $e |- ( ph -> C e. NN ) $. + $( Calculate the order of a subgroup of a finite cyclic group. + (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + fincygsubgodd $p |- ( ph -> ( # ` ran H ) = D ) $= + ( co cod cfv crn cfn wcel chash cc0 cdiv eqid cz cmpt rneqi cycsubggenodd + cif syl5reqr iftrued eqtrd oveq1d wceq cmul cgcd cgrp nnzd odmulg syl3anc + cv cn0 odcl nn0z 3syl cdvds breqtrrd dvdsgcdidd odcld nn0cnd mulgcld zcnd + nnne0d divmul2d mpbird eqtr3d syl5eq a1i wn iffalse sylan9eq hashcl nn0cn + cc hashelne0d neqned divne0d eqnetrd neneqd adantr condan 3eqtrrd ) AEDBF + UBZIUCUDZUDZJUEZUFUGZXCUHUDZUIUPZXEAECUHUDZDUJUBZXBMABXAUDZDUJUBZXHXBAXIX + GDUJAXICUFUGZXGUIUPXGABCCFGIXAKLXAUKZPQAGULGVHZBFUBUMZUEHUECHXNNUNRUQUOAX + KXGUITURUSZUTAXJXBVAXIDXBVBUBZVAAXIDXIVCUBZXBVBUBZXPAIVDUGBCUGZDULUGXIXRV + APQADUAVEZBFIDXACKXLLVFVGAXQDXBVBADXIUAAXSXIVIUGXIULUGQBIXACKXLVJXIVKVLAD + XGXIVMSXOVNVOUTUSAXIXBDAXIABCIXAKXLQVPVQAXBAWTCIXAKXLACFIDBKLPXTQVRZVPVQA + DXTVSZADUAVTZWAWBWCWDZAWTCXCFGIXAKLXLPYAXCGULXMWTFUBUMZUEVAAJYEOUNWEUOZAX + DXEUIAXDEUIVAZAXDWFZEXFUIAEXBXFYDYFUSXDXEUIWGWHAYGWFYHAEUIAEXHUIEXHVAAMWE + AXGDAXKXGVIUGXGWKUGTCWIXGWJVLYBAXGUIACBUFQTWLWMYCWNWOWPWQWRURWS $. + $} + + ${ + $d ph x y $. $d x y B n $. $d x y C n $. $d x y n G $. + fincygsubgodexd.1 $e |- B = ( Base ` G ) $. + fincygsubgodexd.2 $e |- ( ph -> G e. CycGrp ) $. + fincygsubgodexd.3 $e |- ( ph -> C || ( # ` B ) ) $. + fincygsubgodexd.4 $e |- ( ph -> B e. Fin ) $. + fincygsubgodexd.5 $e |- ( ph -> C e. NN ) $. + $( A finite cyclic group has subgroups of every possible order. + (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + fincygsubgodexd $p |- ( ph -> E. x e. ( SubGrp ` G ) ( # ` x ) = C ) $= + ( vn vy cv cfv co wceq chash wcel eqid adantr cz cmg cmpt csubg wrex ccyg + crn cgrp iscyg simprbi syl wa cdiv cyggrp simprl cn cdvds wb hashfingrpnn + wbr nndivdvds syl2anc mpbid fincygsubgd fveq2d simprr cc0 wne divconjdvds + simpr nnne0d cfn fincygsubgodd nncnd ddcand 3eqtrd rspcedeq1vd rexlimddv + ad2antrr ) AKUAKMZLMZEUBNZOUCZUGCPZBMZQNZDPBEUDNZUELCAEUFRZWDLCUEZGWHEUHR + ZWILCWBKEFWBSZUIUJUKAWACRZWDULZULZBKUAVTCQNZDUMOZWAWBOWBOUCZUGZWGWFDWNWAC + WPWBKEWQFWKWQSZAWJWMAWHWJGEUNUKZTZAWLWDUOZAWPUPRZWMADWOUQUTZXCHAWOUPRDUPR + XDXCURACEFWTIUSZJWODVAVBVCTZVDWNWEWRPZULZWFWRQNZWOWPUMOZDXHWEWRQWNXGVJVEW + NXIXJPXGWNWACWPXJWBKWCEWQFWKXJSWCSWSXAXBAWLWDVFAWPWOUQUTZWMAXDDVGVHXKHADJ + VKZDWOVIVBTACVLRWMITXFVMTAXJDPWMXGAWODAWOXEVNADJVNAWOXEVKXLVOVSVPVQVR $. + $} + + ${ + $d ph x $. $d x B $. $d x G $. + prmgrpsimpgd.1 $e |- B = ( Base ` G ) $. + prmgrpsimpgd.2 $e |- ( ph -> G e. Grp ) $. + prmgrpsimpgd.3 $e |- ( ph -> ( # ` B ) e. Prime ) $. + $( A group of prime order is simple. (Contributed by Rohan Ridenour, + 3-Aug-2023.) $) + prmgrpsimpgd $p |- ( ph -> G e. SimpGrp ) $= + ( vx c0g cfv eqid wceq c1 cprime wcel wa chash adantl cvv adantr a1i mp1i + csn fveq2 fvexi hashsng eqtr3d eqeltrrd wn 1nprm pm2.65da cv cnsg nsgsubg + csubg wo cfn cn0 wi cbs cn prmnn nnnn0d hashvnfin syl2anc ad2antrr subgss + syl mpi wss ad2antlr simpr phphashrd olcd subg0cl vex hash1elsn cdvds wbr + orcd lagsubg sylan2 ancoms cc0 wne ssfid hashcl hashelne0d neqned elnnne0 + wb sylanbrc dvdsprime mpbid mpjaodan 2nsgsimpgd ) AGBCCHIZDWPJZEAWPUBZBKZ + LMNZAWSOZBPIZLMXAWRPIZXBLWSXCXBKAWRBPUCQWPRNXCLKXAWPCHWQUDWPRUEUAUFAXBMNZ + WSFSUGWTUHXAUITUJGUKZCULINAXECUNINZXEWRKZXEBKZUOZXECUMAXFOZXEPIZXBKZXIXKL + KZXJXLOZXHXGXNXEBABUPNZXFXLAXBXBKZXOXBJABRNZXBUQNXPXOURXQABCUSDUDTAXBAXDX + BUTNFXBVAVGVBBXBRVCVDVHZVEXFXEBVIZAXLBXECDVFZVJXJXLVKVLVMXJXMOZXGXHYAXEWP + RXJXMVKXFWPXENZAXMXECWPWQVNZVJXERNZYAGVOZTVPVSXJXKXBVQVRZXLXMUOZXFAYFAXFX + OYFXRCBXEDVTWAWBXJXDXKUTNZYFYGWJAXDXFFSXJXKUQNZXKWCWDYHXJXEUPNYIXJBXEAXOX + FXRSXFXSAXTQWEXEWFVGXJXKWCXJXEWPRXFYBAYCQYDXJYETWGWHXKWIWKXBXKWLVDWMWNWAW + O $. + $} + + ${ + $d x G $. $d ph x y $. $d x y B $. + ablsimpgprmd.1 $e |- B = ( Base ` G ) $. + ablsimpgprmd.2 $e |- ( ph -> G e. Abel ) $. + ablsimpgprmd.3 $e |- ( ph -> G e. SimpGrp ) $. + $( An abelian simple group has prime order. (Contributed by Rohan + Ridenour, 3-Aug-2023.) $) + ablsimpgprmd $p |- ( ph -> ( # ` B ) e. Prime ) $= + ( vy vx chash cfv wcel cv c1 wceq wo cn c0g wa simpr adantr cuz cdvds wbr + c2 wi wral cprime wn csn cfn cgrp simpggrpd eqid grpidcl syl ablsimpgfind + hash1elsn csimpg simpgntrivd pm2.65da hashfingrpnn elnn1uz2 sylib ord mpd + w3a csubg ablsimpgcygd 3ad2ant1 simp3 simp2 fincygsubgodexd simpl1 simprl + ccyg cnsg cabl ablnsg 3syl eleqtrrd simpgnsgeqd simplrr cvv fvexi hashsng + fveq2d mp1i 3eqtr3d ex eqtr3d orim12d rexlimddv ralrimiv isprm2 sylanbrc + 3exp ) ABIJZUDUAJKZGLZWQUBUCZWSMNZWSWQNZOZUEZGPUFWQUGKAWQMNZUHWRAXEBCQJZU + IZNAXERZBXFUJAXESAXFBKZXEACUKKXIACFULZBCXFDXFUMZUNUOTABUJKZXEABCDEFUPZTUQ + XHBCXFDXKACURKZXEFTUSUTAXEWRAWQPKXEWROABCDXJXMVAWQVBVCVDVEAXDGPAWSPKZWTXC + AXOWTVFZHLZIJZWSNZXCHCVGJZXPHBWSCDAXOCVOKWTACEFVHVIAXOWTVJAXOXLWTXMVIAXOW + TVKVLXPXQXTKZXSRZRZXQXGNZXQBNZOXCYCXQBCXFDXKYCAXNAXOWTYBVMZFUOYCXQXTCVPJZ + XPYAXSVNYCACVQKYGXTNYFECVRVSVTWAYCYDXAYEXBYCYDXAYCYDRZXRXGIJZWSMYHXQXGIYC + YDSWFXPYAXSYDWBXFWCKYIMNYHXFCQXKWDXFWCWEWGWHWIYCYEXBYCYERZXRWSWQXPYAXSYEW + BYJXQBIYCYESWFWJWIWKVEWLWPWMGWQWNWO $. + $} + + ${ + ablsimpgd.1 $e |- B = ( Base ` G ) $. + ablsimpgd.2 $e |- ( ph -> G e. Abel ) $. + $( An abelian group is simple if and only if its order is prime. + (Contributed by Rohan Ridenour, 3-Aug-2023.) $) + ablsimpgd $p |- ( ph -> ( G e. SimpGrp <-> ( # ` B ) e. Prime ) ) $= + ( csimpg wcel chash cprime wa cabl adantr simpr ablsimpgprmd cgrp ablgrpd + cfv prmgrpsimpgd impbida ) ACFGZBHQIGZATJBCDACKGTELATMNAUAJBCDACOGUAACEPL + AUAMRS $. + $} + $( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# @@ -281063,7 +281836,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on $( ` { (/) } ` is the only topology with one element. (Contributed by FL, 18-Aug-2008.) $) en1top $p |- ( J e. Top -> ( J ~~ 1o <-> J = { (/) } ) ) $= - ( ctop wcel c1o cen wbr c0 csn wceq wi en1eqsn ex syl id 0ex ensn1 syl6eqbr + ( ctop wcel c1o cen wbr c0 csn wceq wi en1eqsn ex syl id 0ex ensn1 eqbrtrdi 0opn impbid1 ) ABCZADEFZAGHZIZTGACZUAUCJARUDUAUCGAKLMUCAUBDEUCNGOPQS $. ${ @@ -281075,7 +281848,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on ( J ~~ 2o <-> ( J = { (/) , X } /\ X =/= (/) ) ) ) $= ( vx wcel c2o cen wbr c0 wceq wne wa c1o simprl sylibr syl 0ex mpd necomd adantr cvv ctopon cfv cpr simpr wn csdm wo csn cv toponss ad2ant2rl sseq0 - syl2anc velsn expr ssrdv ctop topontop 0opn ad2antrr snssd eqssd syl6eqbr + syl2anc velsn expr ssrdv ctop topontop 0opn ad2antrr snssd eqssd eqbrtrdi wss ensn1 olcd sdom2en01 sdomnen ex necon2ad toponmax en2eqpr syl3anc jca wi simprr pr2nelem mp3an2ani eqbrtrd impbida ) ABUAUBDZAEFGZAHBUCZIZBHJZK ZWAWBKZWDWEWGHBJZWDWGBHWGWBWEWAWBUDZWGWBBHWGBHIZWBUEZWGWJKZAEUFGZWKWLAHIZ @@ -287609,7 +288382,7 @@ require the space to be Hausdorff (which would make it the same as T_3), t1connperf $p |- ( ( J e. Fre /\ J e. Conn /\ -. X ~~ 1o ) -> J e. Perf ) $= ( vx ct1 wcel cconn c1o cen wbr wn cperf wa cv wral wrex simplr simprr c0 - csn wne vex snnz a1i ccld cfv t1sncld ad2ant2r ensn1 syl6eqbrr rexlimdvaa + csn wne vex snnz a1i ccld cfv t1sncld ad2ant2r ensn1 eqbrtrrdi rexlimdvaa connclo con3d ralnex syl6ibr ctop wb t1top adantr isperf3 baib syl 3impia sylibrd ) AEFZAGFZBHIJZKZALFZVEVFMZVHDNZTZAFZKDBOZVIVJVHVMDBPZKVNVJVOVGVJ VMVGDBVJVKBFZVMMZMZBVLHIVRVLABCVEVFVQQVJVPVMRVLSUAVRVKDUBZUCUDVEVPVLAUEUF @@ -294739,7 +295512,7 @@ Kolmogorov quotient is regular Hausdorff (T_3). (Contributed by Mario 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) $) hmph0 $p |- ( J ~= { (/) } <-> J = { (/) } ) $= ( c0 csn chmph wbr wceq c1o cen hmphen df1o2 breqtrrdi ctop hmphtop1 en1top - wcel wb syl mpbid id sn0top hmphref ax-mp syl6eqbr impbii ) ABCZDEZAUEFZUFA + wcel wb syl mpbid id sn0top hmphref ax-mp eqbrtrdi impbii ) ABCZDEZAUEFZUFA GHEZUGUFAUEGHAUEIJKUFALOUHUGPAUEMANQRUGAUEUEDUGSUELOUEUEDETUEUAUBUCUD $. ${ @@ -299713,7 +300486,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the fveq2 unieqd cbvixpv cufl elin2d adantr eqeltrrid ssrab2 eqnetrrid eqid eqtri resixpfo sylancr fonum syl2anc cdif crn wfn cvv wral difexg dmexg vex mp1i uniexg ralrimivw fnmpt dffn4 csn ssdif0 simpr syl6eleq simprbi - 3syl elixp r19.21bi snssd eqssd fvex ensn1 syl6eqbr ex con3d cif simplr + 3syl elixp r19.21bi snssd eqssd fvex ensn1 eqbrtrdi ex con3d cif simplr syl5bir iftrue syl5ibrcom ad2antrr iffalse wb cab ifex rgenw neq0 unisn syl6ib eldifi eleq1d ralrimiva ad3antrrr mptelixpg mpbird sylan2 eleq1w eleq12d pm2.61d syl6eleqr pm4.71rd equequ1 ifbieq2d fvmpt neeq1d adantl @@ -309045,7 +309818,7 @@ the half element (corresponding to half the distance) is also in this cxp wb caddc wral cif 0re 1re ifcli rgen2w fmpo mpbi equequ1 equequ2 0nn0 elexi ovmpo eqeq1d wn iffalse wne ax-1ne0 a1i eqnetrd neneqd con4i iftrue 1nn0 impbii syl6bb cn wo nn0addcli elnn0 breq1 leidi keephyp nnge1 nn0rei - 0le1 letri sylancr nn0ge0i add20i bibi12d chvarv syl2anb iftrued syl6eqbr + 0le1 letri sylancr nn0ge0i add20i bibi12d chvarv syl2anb iftrued eqbrtrdi mp2an eqtr2 sylbi breqtrrd jaoi mp1i adantl eqeq12 ovmpoa adantrr adantrl id oveq12d 3brtr4d expcom ralrimiv jca rgen2a pm3.2i ismet mpbiri ) EDJCE UAUBJEEUFUCCUDZGKZHKZCUEZLMZGHNZUGZXRIKZXPCUEZYBXQCUEZUHUEZOPZIEUIZQZHEUI @@ -309071,7 +309844,7 @@ the half element (corresponding to half the distance) is also in this ( vu vv vw wcel cfv cv wss wa wb c1 weq clt wbr cc0 cpw cbl crn wrex wral cmopn cxmet cmet dscmet metxmet syl eqid elmopn simpll adantll jca csn co ssel2 velsn eleq1a wi simpl a1i cif eqeq12 ifbid cr 0re 1re ovmpoa breq1d - ifcli elexi ltnri iffalse mtbiri con4i iftrue 0lt1 syl6eqbr impbii equcom + ifcli elexi ltnri iffalse mtbiri con4i iftrue 0lt1 eqbrtrdi impbii equcom wn bitri syl6rbb simpr biantrurd bitrd ex pm5.21ndd adantl cxr 1xr mp3an3 elbl sylan bitr4d syl5bb eqrdv eqeltrd snssi vsnid jctil wceq eleq2 sseq1 blelrn anbi12d rspcev syl2an sylancom ralrimiva pm4.71d velpw syl6bbr ) E @@ -313096,7 +313869,7 @@ Normed space homomorphisms (bounded linear operators) xrge0tsms2 $p |- ( ( A e. V /\ F : A --> ( 0 [,] +oo ) ) -> ( G tsums F ) ~~ 1o ) $= ( vx wcel cc0 cpnf cicc co wf wa ctsu cpw cfn cin cv cxr clt cres crn csn - cgsu cmpt csup c1o simpl simpr eqid xrge0tsms xrltso supex ensn1 syl6eqbr + cgsu cmpt csup c1o simpl simpr eqid xrge0tsms xrltso supex ensn1 eqbrtrdi cen ) ADGZAHIJKBLZMZCBNKFAOPQCBFRUAUDKUEUBZSTUFZUCUGUPUSAVABCDFEUQURUHUQU RUIVAUJUKVASUTTULUMUNUO $. $} @@ -315899,7 +316672,7 @@ topological space to the reals is bounded (above). (Boundedness below htpyi eqtr4d ralrimiva weq oveq1 eqeq12d rspccva sylan adantrl simprl 2cn oveq2d 2ne0 recidi syl6eq 1m1e0 3eqtr4d wss retopon 0re iccssre resttopon oveq1d mp2an cnmpt2nd cnmpt1st cmpt iihalf1cn oveq2 cnmpt21 htpycn sseldd - chtpy cnmpt22f iihalf2cn cnmpopc cnmptcom eqeltrid simpr 0elunit syl6eqbr + chtpy cnmpt22f iihalf2cn cnmpopc cnmptcom eqeltrid simpr 0elunit eqbrtrdi iftrued simpl 2t0e0 oveq12d eqtrd ovex ovmpoa sylancl 1elunit clt wn mpbi ltnlei breq1d mtbiri iffalsed 2t1e2 2m1e1 ishtpyd ) ADFKGHLUANOQAKBCLUCTU DUEZCUQZTURUFUEZUGUHZBUQZURUUEUIUEZIUEZUUHUUITUJUEZJUEZUKZULGUMUNUEHUOUEZ @@ -316514,7 +317287,7 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` ) oveq1d cnmpt2nd phtpycn sseldd cnmpt22f adantll phtpyhtpy htpyi syl2anc weq simpll wn ifeq12da simpr 0elunit breq1d oveq12d ifbieq12d ovex ifex simpl ovmpoa sylancl pcovalg 1elunit eqtrd sylancr oveq2 eqeltrid elii1 - iihalf1cn iihalf2cn cnmpopc iihalf1 sylbir chtpy elii2 iihalf2 syl6eqbr + iihalf1cn iihalf2cn cnmpopc iihalf1 sylbir chtpy elii2 iihalf2 eqbrtrdi syl iftrued 2t0e0 pco0 clt ltnlei mpbi mtbiri iffalsed 2t1e2 2m1e1 pco1 isphtpy2d ) AEFHUCUDZRZGIUVFRZDHUAAEFHAEUEHUFRZUNZGUVIUNZEGHUGUDZRZUHUI ZAEGHUJUDZUKUVJUVKUVNULMEGHUMUOZUPZAFUVIUNZIUVIUNZFIUVLRZUHUIZAFIUVOUKU @@ -316607,7 +317380,7 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` ) syl fvif syl6eq fmptco 3eqtr4d iitopon cnmptid cnmptc crn ctg crest dfii2 cioo 0re 1re halfre halfge0 halflt1 ltleii elicc01 mpbir3an simprl oveq2d 2cn recidi oveq1d 1m1e0 syl6req wss retopon iccssre mp2an resttopon oveq2 - 2ne0 weq breq1 fvmpt mp1i cnmpt2c cnmpt1st iihalf2cn ifbieq2d id syl6eqbr + 2ne0 weq breq1 fvmpt mp1i cnmpt2c cnmpt1st iihalf2cn ifbieq2d id eqbrtrdi cnmpt21 cnmpopc cnmpt12 c0ex 1elunit ltnlei mpbi mtbiri 2t1e2 2m1e1 eqtrd clt 1ex reparpht eqbrtrd ) BHCUAIZJZKBLZDMZUEZABCUBLIZBFKNUCIZFUFZNOUGIZP UHZKOUVIQIZNRIZUIZUJZUKZBCULLUVFFUVHUVKUVLALZUVMBLZUIZUJFUVHUVKUVDUVRUIZU @@ -316649,7 +317422,7 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` ) sylbir ex syl6eq fmptco 3eqtr4d iitopon cnmptid 0elunit cnmptc cioo crest crn ctg dfii2 0re 1re halfre halfge0 halflt1 ltleii elicc01 simprl oveq2d mpbir3an 2cn 2ne0 recidi wss retopon iccssre resttopon cnmpt1st iihalf1cn - mp2an oveq2 cnmpt21 breq1 fvmpt mp1i cnmpt2c cnmpopc ifbieq1d id syl6eqbr + mp2an oveq2 cnmpt21 breq1 fvmpt mp1i cnmpt2c cnmpopc ifbieq1d id eqbrtrdi cnmpt12 2t0e0 eqtrd c0ex clt ltnlei mpbi mtbiri 1ex reparpht eqbrtrd ) BH CUEIZJZKBLZDMZNZBACUBLIZBFOKUCIZFUDZKPUFIZQUGZPUVDRIZKUHZUIZUJZBCUKLUVAFU VCUVFUVGBLZUVGKULIZALZUHZUIFUVCUVFUVKUUSUHZUIUVBUVJUVAFUVCUVNUVOUVAUVDUVC @@ -316711,7 +317484,7 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` ) ovex eqcomi iihalf1cn cnmpt1res ccnfld ctopn unitssre cnfldtopon cnmpt12f sseldi ctx addcn cnmptre cnmpopc divccn ifbieq12d equcoms eqcomd eqeltrid weq cnmpt12 cnf fmpt sylibr cuni feqmptd fveq2 fmptcof pcoval 3eqtr4rd id - syl6eqbr iftrued 2t0e0 c0ex fvmpt 1ex reparpht eqbrtrd ) ADEGUUANZOZFVUQO + eqbrtrdi iftrued 2t0e0 c0ex fvmpt 1ex reparpht eqbrtrd ) ADEGUUANZOZFVUQO ZDEFVUQOZVUQOZCUUBZVVAGUUCNABPQUCOZBUDZQRUEOZUFUGZVVDQUHUEOZUFUGZRVVDUIOZ VVDVVGUJOZULZVVDRUEOZVVEUJOZULZVVANZUKBVVCVVFVVIVURNZVVIQUMOFNZULZUKVVBVU SABVVCVVOVVRAVVDVVCSZUNZVVFVVOVVRUOVVTVVFUNZVVKVVANZVVPVVOVVRVWAVVHVWBVVP @@ -316826,7 +317599,7 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` ) cc mpan mpbir3an simprl 2ne0 recidi iihalf1cn iimulcn iihalf2cn cnmpopc 0re cnmpt21f iftrue elii1 pcoval1 iihalf1 iffalse elii2 pcoval2 iihalf2 sylan2br nncan eqtr2d sylan2 pm2.61dan ifeq12d ifid csn fveq1i fvconst2 - mulcl cxp syl5eq 3eqtr4d syl6eqbr iftrued 2t0e0 mul01d pco0 syl6req clt + mulcl cxp syl5eq 3eqtr4d eqbrtrdi iftrued 2t0e0 mul01d pco0 syl6req clt ax-mp ltnlei mpbi mtbiri iffalsed 2t1e2 2m1e1 pco1 eqcomd isphtpy2d wne c0 ne0d isphtpc syl3anbrc jca ) DLGUBMZNZFEDGUCOMZCGUDOMZNUWRCGUEOUFZUW QUWRCFGUAUWQEDGUWQEAPQUGMZQAUHZRMZDOZUIUWPIUWQAUXCDLLGUXALUXAUJONUWQUKU @@ -329368,7 +330141,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by inss1 qssre sstri fss sylancl ffvelrnda shftmbl mblss csu caddc cmpt cseq eqid cli cxp nnuz syl6eq breqtrd cxr syl2an2r fmptd iunmbl ovolss syl2anc ralrimiva vitalilem4 cuz csn mpteq2dva fconstmpt xpeq1i eqtr3i seqeq3d cz - syl6eqel 1z serclim0 syl6eqbr seqex c0ex breldm ovoliun2 sumeq2dv eqimssi + syl6eqel 1z serclim0 eqbrtrdi seqex c0ex breldm ovoliun2 sumeq2dv eqimssi cfn wo orci sumz ovolge0 wb ovolcl 0xr xrletri3 mpbir2and mto ) ASUBUCUDZ UEUFZSUGUHZSUVRUIUHUVSUJSUBUVRUIUKUVRUBSULUDZUBSUMUNUBUMUNSUBUGUHUVRUVTUQ UOUPURSUBUSUTVAVBZVCSUVRUOUVRUBUMUWAUPVDVEVFAUVRUATUAVIZGUFZVGZUEUFZSUGAU @@ -338350,7 +339123,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by pncan3 fveq2d eqtr3d mpteq2dva cnex fvexd fconstmpt eqidd offval2 feqmptd eff 3eqtr4d oveq2d ccnfld ctopn efcl fconstg snssd fssd ssidd fmpttd 0cnd 1cnd cop c0ex snid opelxpi mpan2 dvconst eleqtrrd df-br sylibr ccom oveq1 - cofmpt eqid ovex fvmpt subid eqtrd dveflem syl6eqbr 1ex cr cpr cnelprrecn + cofmpt eqid ovex fvmpt subid eqtrd dveflem eqbrtrdi 1ex cr cpr cnelprrecn simpr dvmptid simpl id dvmptc dvmptsub 1m0e1 mpteq2i eqtr4i syl6eq dvcobr 1t1e1 breqtrdi breqdi dvmulbr mul02d fvex mulid2d oveq12d addid2d breqtrd ffvelrnd fvconst2 vex breldm ssriv eqssi feq2i mpbi wfun ffun ax-mp mpsyl @@ -339322,7 +340095,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by vy wf cncff syl ffnd cvv wfn fvex fnconstg mp1i fvconst2 adantl cxr rexrd cv w3a wb elicc2 syl2anc biimpa simp1d simp2d simp3d letrd lbicc2 syl3anc ffvelrnd ffvelrnda cmin subcld cmul simpr jca cdv cdm cioo dmeqd wne c0ex - snnz dmxp ax-mp syl6eq 0red fveq1d sylan9eq abs00bd syl6eqbr dvlip syldan + snnz dmxp ax-mp syl6eq 0red fveq1d sylan9eq abs00bd eqbrtrdi dvlip syldan c0 0le0 abscld mul02d breqtrd absge0d 0re letri3 sylancl mpbir2and abs00d recnd subeq0d eqtr2d eqfnfvd ) AUABCUBIZDXKBDJZUCUDZAXKKDADXKKUEILXKKDUGZ GXKKDUHUIZUJXLUKLXMXKULABDUMZXKXLUKUNUOAUAUTZXKLZMZXQXMJZXLXQDJZXRXTXLNAX @@ -339357,7 +340130,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cdm feqmptd offval2 oveq2d cr cnelprrecn fvexd dvfcn feq2d mpbii 3eqtr3rd cpr wf eqtr3d dvmptsub subidd mpteq2dva fconstmpt syl6eqr 3eqtrd dmeqd c0 wne snnzg dmxp mp2b syl6eq eqimss2 syl 0red fveq1d c0ex fvconst2 sylan9eq - abs00bd 0le0 syl6eqbr dvlipcn syldan fveq2 oveq12d eqid ffvelrnd subeq0bd + abs00bd 0le0 eqbrtrdi dvlipcn syldan fveq2 oveq12d eqid ffvelrnd subeq0bd ovex fvmpt adantr subid1d eqtrd fveq2d sselda sseldd subcld abscld mul02d recnd 3brtr3d absge0d wb 0re letri3 sylancl mpbir2and abs00d eqtr4d mpbid eqfnfvd ofsubeq0 ) AEFUBUCRZGSUDZUEZUFZEFUFZAQGUUIUUKAGGUBGEFUGUGAGTEKUHA @@ -342666,7 +343439,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by deg1lt0 $p |- ( ( R e. Ring /\ F e. B ) -> ( ( D ` F ) < 0 <-> F = .0. ) ) $= ( crg wcel wa cfv cc0 clt wbr wceq wne wn w3a cn0 nn0nlt0 3expia necon4ad - deg1nn0cl syl deg1z mnflt0 syl6eqbr adantr fveq2 breq1d syl5ibrcom impbid + deg1nn0cl syl deg1z mnflt0 eqbrtrdi adantr fveq2 breq1d syl5ibrcom impbid cmnf ) DKLZEALZMZEBNZOPQZEFRZUSVAEFUQUREFSZVATZUQURVCUAUTUBLVDABCDEFGHIJU FUTUCUGUDUEUSVAVBFBNZOPQZUQVFURUQVEUPOPBCDFGHIUHUIUJUKVBUTVEOPEFBULUMUNUO $. @@ -344011,7 +344784,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( co cc0 cco1 cfv cdg1 cle wbr wceq cn0 wcel csn c1 caddc clt crg cuc1p cmnf cnzr nzrring syl cmn1 ccnv c0g cima eqid ply1remlem simp1d syl2anc mon1puc1p r1pdeglt syl3anc simp2d breqtrd 1e0p1 breqtrdi 0nn0 nn0leltp1 - wb mpan2 syl5ibrcom wi elsni cxr 0xr mnfle ax-mp syl6eqbr a1i cun r1pcl + wb mpan2 syl5ibrcom wi elsni cxr 0xr mnfle ax-mp eqbrtrdi a1i cun r1pcl deg1cl elun sylib mpjaod deg1le0 mpbid cq1p cmulr cplusg cof cpws r1pid wo fveq2d cghm ccrg evl1rhm rhmghm ply1ring q1pcl mon1pcl ringcl ghmlin crh cbs cvv fvexi rhmf ffvelrnd pwsplusgval 3eqtrd fveq1d pwselbas ffnd @@ -344183,7 +344956,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by wi fveq2 imaeq1d fveq2d breq12d imbi12d cn0 wcel cidom wral crg wne cdomn ccrg isidom simplbi crngring 3syl deg1nn0cl syl3anc cc0 c1 caddc co eqeq2 imbi1d ralbidv imbi2d wa c0 wn simprr 0nn0 syl6eqel syl deg1nn0clb syl2an - wb simpl mpbird cco1 cascl simplrr 0le0 syl6eqbr ad2antrr simplrl deg1le0 + wb simpl mpbird cco1 cascl simplrr 0le0 eqbrtrdi ad2antrr simplrl deg1le0 syl2anc mpbid cbs cxp adantr wf coe1f ffvelrn sylancl evl1sca fveq1d cpws eqtrd wfn cvv crh evl1rhm rhmf simprl ffvelrnd pwselbas fniniseg simplbda fvexd ffn simprbda fvex fvconst2 3eqtr3rd ply1scl0 3eqtrd ex necon3ad mpd @@ -347135,7 +347908,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cply ccnv cima cxp cof wss ssid ax-1cn plyid mp2an plyconst mpan plysubcl sylancr eqeltrid cneg cv negcl addcom sylan negsub ancoms eqtrd mpteq2dva cnex negex simpr fconstmpt cid cres df-idp mptresid eqtri offval2 3eqtr4d - simpl syl6eqr fveq2d clt wbr 0dgr 0lt1 syl6eqbr eqid dgrid eqcomi dgradd2 + simpl syl6eqr fveq2d clt wbr 0dgr 0lt1 eqbrtrdi eqid dgrid eqcomi dgradd2 syl syl3anc eqtr3d syl5eq fveq1d adantr ovex fvmpt2 sylancl eqeq1d subeq0 wb bitrd pm5.32da wf wfn plyf fniniseg 4syl eleq1a pm4.71rd 3bitr4d velsn ffn syl6bbr eqrdv 3jca ) AEFZBEUAGZFZBHGZIJBUBKLUCZALZJXOBMEXTUDZNUEOZXPC @@ -348131,7 +348904,7 @@ of all kernels (preimages of ` { 0 } ` ) of all polynomials in mp2an sseldi cpnres sylancr cima df-ima wf zssre ax-resscn plyss plyreres frnd eqsstrid cdm iccssre syl2anc syl6ss plyf fdmd sseqtr4d c1lip3 wa w3a simp2 recnd adantr 3ad2ant1 abssubd eqbrtrd 1red elicc4abs syl3anc mpbird - simp3 wb subidd fveq2d abs0 0le1 eqbrtri syl6eqbr wceq fveq2 oveq2d oveq2 + simp3 wb subidd fveq2d abs0 0le1 eqbrtri eqbrtrdi wceq fveq2 oveq2d oveq2 breq12d fvoveq1d fvoveq1 rspc2v simp1l ffvelrnd eqtrd breq1d mpd cdiv wne adantl oveq1 ad2antrr abscld absge0d syl5ibrcom expimpd remulcld syl6eqel 0cn subid1d sylibd 3exp com34 com23 ralrimdv reximdva cif 1rp a1i wn recn @@ -352744,7 +353517,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is efcl sylan2 sylancl ctopon cnfldtopon toponmax cin df-ss sylib adantl eff dvef feqmptd oveq2d 3eqtr3a fveq2 dvmptco 1re oveq2 fveq2d fvmpt3i absefi mulcli c2 eqtr3i fveq2i 3brtr3i dvmptres3 tgioo2 cnt iccntr fveq1d oveq1d - dvmptres2 ovex sylan9eq ioossre sselda recnd absmul absi oveq12d syl6eqbr + dvmptres2 ovex sylan9eq ioossre sselda recnd absmul absi oveq12d eqbrtrdi 3eqtrd 1le1 dvlip it0e0 fvex oveq12i recni negicn caddc ccos cosval csqrt csin sincos3rdpi simpri addcli 2ne0 div11i mpbi subaddrii mulneg12 3eqtri ef0 2cn absnegi df-neg rprege0 absid mp2b oveq2i renegcli clt wb lemuldiv @@ -353641,7 +354414,7 @@ imaginary part lies in the interval (-pi, pi]. See Carneiro, 1-Apr-2015.) $) relogrn $p |- ( A e. RR -> A e. ran log ) $= ( cr wcel cpi cneg cim cfv clt wbr cle clog crn recn cc0 pipos pire lt0neg2 - cc wb ax-mp mpbi reim0 breqtrrid 0re ltleii syl6eqbr ellogrn syl3anbrc ) AB + cc wb ax-mp mpbi reim0 breqtrrid 0re ltleii eqbrtrdi ellogrn syl3anbrc ) AB CZARCDEZAFGZHIUKDJIAKLCAMUIUJNUKHNDHIZUJNHIZODBCULUMSPDQTUAAUBZUCUIUKNDJUNN DUDPOUEUFAUGUH $. @@ -353842,7 +354615,7 @@ imaginary part lies in the interval (-pi, pi]. See ( wcel clog cfv ci cpi cmul co ce cneg cc wceq sylancl clt wbr cle pipos cr cc0 pire crp caddc c1 relogcl recnd ax-icn picn mulcli efadd oveq2i reeflog efipi oveq1d syl5eq rpcn neg1cn mulcom mulm1d eqtrd 3eqtrd fveq2d crn addcl - cim wb lt0neg2 ax-mp mpbi renegcli 0re lttri mp2an breqtrrid leidi syl6eqbr + cim wb lt0neg2 ax-mp mpbi renegcli 0re lttri mp2an breqtrrid leidi eqbrtrdi crim ellogrn syl3anbrc logef syl eqtr3d ) AUABZACDZEFGHZUBHZIDZCDZAJZCDWEWB WFWHCWBWFWCIDZWDIDZGHZAUCJZGHZWHWBWCKBZWDKBZWFWKLWBWCAUDZUEZEFUFUGUHZWCWDUI MWBWKWIWLGHWMWJWLWIGULUJWBWIAWLGAUKUMUNWBWMWLAGHZWHWBAKBWLKBWMWSLAUOZUPAWLU @@ -356513,7 +357286,7 @@ the complex square root function (the branch cut is in the same place for ( cc0 wbr ccxp co cle wceq wa c1 cr wcel adantr ad2antrr clt cmin resubcl cmul 1re rpred sylancr recxpcld 1red w3a recxpcl jca syl3anc 0le1 a1i crp cxpge0 difrp sylancl biimpa cxple2d mpbid recnd 1cxpd breqtrd simpr 1m1e0 - wb oveq2d cc cxp0d eqtrd 1le1 syl6eqbr wo leloe mpjaodan lemul1a syl31anc + wb oveq2d cc cxp0d eqtrd 1le1 eqbrtrdi wo leloe mpjaodan lemul1a syl31anc syl6eq caddc ax-1cn npcan anim1i elrp sylibr rpne0d cxpaddd cxp1d 3eqtr3d cxpcld mulid2d 3brtr3d cxpge0d breq1 syl5ibcom imp 0re ) AIBUAJZBBCKLZMJZ IBNZAWSOZBPCUBLZKLZWTUDLZPWTUDLZBWTMXCXEQRZPQRZWTQRZIWTMJZOZXEPMJZXFXGMJA @@ -365838,7 +366611,7 @@ sum notation (which we used for its unordered summing capabilities) into nnne0i cc recn recni climconst2 ax-resscn sylancl basellem7 ffnd fnconstg c3 wbr wfn syl offn eqidd ofval climmul breqtrdi eqbrtrid 3cn mul01i ofc1 2cn eqbrtrrd cle npcand mpteq2dva w3a subdi syl3an caofdi oveq12i syl6eqr - zrei cuz eqimss2i ofnegsub mp3an2i neg1cn syl6eqbrr mulid1i basellem6 3ex + zrei cuz eqimss2i ofnegsub mp3an2i neg1cn eqbrtrrdi mulid1i basellem6 3ex 3re mulid2d mulneg1 negeqd mulcl negnegd eqtr2d oveq12d eqtrd df-3 ax-1cn negsubd addcomi eqtri oveq1i 1cnd adddird syl5eq pnpcand basellem8 simprd 3eqtr4rd lesub1dd 3brtr4d simpld subge0d mpbird breqtrrd climsqz2 climadd @@ -366457,7 +367230,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 Carneiro, 22-Sep-2014.) $) mule1 $p |- ( A e. NN -> ( abs ` ( mmu ` A ) ) <_ 1 ) $= ( vp wcel cexp co cdvds wbr cprime cfv cabs c1 cle wa cc0 sylan9eq fveq2d - syl6eqbr wceq syl eqtrd cn cv c2 wrex cmu cneg crab chash cif iftrue abs0 + eqbrtrdi wceq syl eqtrd cn cv c2 wrex cmu cneg crab chash cif iftrue abs0 muval 0le1 eqbrtri iffalse cn0 neg1cn cfn prmdvdsfi hashcl absexp sylancr wn cc ax-1cn absnegi abs1 eqtri oveq1i nn0zd 1exp syl5eq adantr pm2.61dan cz 1le1 ) AUACZBUBZUCDEAFGBHUDZAUEIZJIZKLGVQVSMZWANJIZKLWBVTNJVQVSVTVSNKU @@ -366981,7 +367754,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 wf1o ovex ralrimiva issqf mpbird pcelnn syl2anc simpr cvv biantrurd wo id adantrr bitrd con4i ssriv elpwi rabss2 dvdsssfz1 sstrd ssfi elrabd 1arith prmssnn f1ocnv f1of f1ocnvfv2 1arithlem1 fveq1d sylan9req oveq1 1le1 0le1 - ffvelrni keephyp syl6eqbrr iftrue sselda elrab simpll nnge1d pcge0 breq1d + ffvelrni keephyp eqbrtrrdi iftrue sselda elrab simpll nnge1d pcge0 breq1d eqbrtrd ex syl5ibrcom pm2.61d eqbrtrrd pc2dvds jca sylanbrc eqcom simplbi ad2antrl mptex fvmpt2 sylancl eqeq1d adantrl f1ocnvfvb csn 0cnd 1cnd 0ne1 cpr cc pw2f1olem ssrab2 sspwb simprr sseldi syl2anr elnn0 orcomd r19.21bi @@ -367626,7 +368399,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 3cn recnd mpanr12 syl2anc oveq1d sylancr mp3an23 3eqtr3d adantlr ad2antrr cn0 fzfi hashcl nn0rei ppifl flcl flge biimpa eluz2 syl3anbrc ppidif df-4 3nn ineq1i syl6eqr eqtr3d cdom dfin5 wi elfzle1 ppiublem2 ss2rabi eqsstri - expcom ssdomg mp2 inss1 hashdom syl6eqbr leadd1dd le2addd cun ovex rabbii + 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 syl6eleqr df-3 mvrraddi nnne0i redivcli divgt0ii 4z 2pos 2lt6 mulid1i breqtrri ltdivmul 1e0p1 subid1d 5pos 5lt6 negsubdi2i @@ -369440,7 +370213,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 $( A Dirichlet character takes values inside the unit circle. (Contributed by Mario Carneiro, 3-May-2016.) $) dchrabs2 $p |- ( ph -> ( abs ` ( X ` A ) ) <_ 1 ) $= - ( cfv c1 cle cc0 wa syl6eqbr cabs wbr wceq simpr abs00bd 0le1 wcel adantr + ( cfv c1 cle cc0 wa eqbrtrdi cabs wbr wceq simpr abs00bd 0le1 wcel adantr wne cui eqid dchrn0 biimpa dchrabs 1le1 pm2.61dane ) ABGOZUAOZPQUBUQRAUQR UCZSZURRPQUTUQAUSUDUEUFTAUQRUIZSZURPPQVBBDHUJOZEFGHIJAGDUGVAMUHKVCUKZAVAB VCUGABCDVCEFGHIKJLVDMNULUMUNUOTUP $. @@ -370171,7 +370944,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 lemul2a eqbrtrrid ltletrd rpdivcld 3bitr4rd notbid ltnled mulid1d biimprd elfz bitr4d nngt0d ltaddrp2d 2timesd breqtrrd wi lttr mpand sylibrd 2t0e0 oveq2d 0m0e0 0le0 expr sylbid iffalse mpbidi pm2.61d fsumle pcbcctr chash - syl6eqbr cfn bernneq3 reeflogd reexplog relogcld remulcld efle ledivmul2d + eqbrtrdi cfn bernneq3 reeflogd reexplog relogcld remulcld efle ledivmul2d wss ltled letrd eluz fzss2 sumhash rprege0d flge0nn0 hashfz1 eqtr2d simpr nnnn0 fzctr bccl2 pccld nn0zd syl211anc ) BCDZAUBDZEZAAFBGHZBUCHZUDHZUEHU XRIJZUXTUXRUFKZAUFKZUGHZUHKZIJZUXQLUXRUIHZUXRAUAUPZUEHZUGHZUHKZFBUYIUGHZU @@ -370858,7 +371631,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 most ` 1 ` . (Contributed by AV, 13-Jul-2021.) $) zabsle1 $p |- ( Z e. ZZ -> ( Z e. { -u 1 , 0 , 1 } <-> ( abs ` Z ) <_ 1 ) ) $= - ( cz wcel c1 cneg cc0 cabs cfv cle wbr wceq fveq2 eqbrtri syl6eqbr wa cr wi + ( cz wcel c1 cneg cc0 cabs cfv cle wbr wceq fveq2 eqbrtri eqbrtrdi wa cr wi w3o adantl com12 ctp eltpi ax-1cn absnegi abs1 eqtri 1le1 abs0 0le1 syl zre 3jaoi 1red absled cn elz 3mix2 a1d nnle1eq1 biimpac 3mix3d elnnz1 lenegcon2 ex wb mpancom neg1rr a1i id letri3d 3mix1 eqcoms syl6bir adantr sylbid impd @@ -373032,7 +373805,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and cn neg1ne0 wa nndivred adantr 2re elfznn remulcl sylancr remulcld fsumzcl adantl flcld reexpclzd 1re nnrpd cmgp czrh cmpt eqid lgseisenlem4 modadd1 czn syl221anc clt peano2re df-neg cabs cc neg1cn absexpz mp3an12i absnegi - ax-1cn abs1 eqtri oveq1i 1exp syl5eq eqtrd 1le1 syl6eqbr wb absle sylancl + ax-1cn abs1 eqtri oveq1i 1exp syl5eq eqtrd 1le1 eqbrtrdi wb absle sylancl mpbid simpld eqbrtrrid 0red lesubaddd peano2rem simprd df-2 prmuz2 eluzle cuz eldifsni leneltd ltaddsubd lelttrd mpbird modid syl22anc oveq1d recnd 3syl pncan ) ADCUBHZDCIUCHZJUEHZKHZIUFHCLHZIUCHZIUGZIYSUDHZDCUEHZJBUHZMHZ @@ -373224,7 +373997,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and cop cgcd prmrp coprmdvds nnz dvdsmul2 breq2 syl5ibrcom necon3bd mpbiran2d mpd ltlend nnre lemuldiv nnne0d div23d breq2d 3bitrd ltmul2 2cnd 2ne0 w3a divass div23 eqtr3d breqtrd lttr ltmul1 sylibrd divsubdird syl6eqr ax-1cn - halflt1 syl6eqbr eqbrtrrd 1red ltsubadd2d peano2re syld nnleltp1 pm4.71rd + halflt1 eqbrtrdi eqbrtrrd 1red ltsubadd2d peano2re syld nnleltp1 pm4.71rd recnd nncan syl2an 3bitr3d pm5.32da syl5bb elfz5 eqeltrd biantrurd bitr3d nnuz fznn mulcomd sylan9eqr anbi12d 3bitr4d vex op1std eqeq2d eqcom elrab syl6bb biancomi opelxp velsn anbi1i 3bitr4g eqrelrdv eqcomd hashfz1 wdisj @@ -376115,7 +376888,7 @@ to the second component (see, for example, ~ 2sqreunnltb and resubcl sylancr remulcld eqidd wceq 1cnd oveq1d mulid2d sumeq2dv eqtr3d eqtrd cdvds wbr cprime crab wss 1red cfn nnrpd relogcld sylan2 fsumrecl cuz simpr 0re syl6eqel wne ad3antrrr ad2antrr adantlr dchrn0 pm2.61dane - biimpa dchr1 0le1 syl6eqbr leidi wb mpbird vmage0 nnred nngt0d syl22anc + biimpa dchr1 0le1 eqbrtrdi leidi wb mpbird vmage0 nnred nngt0d syl22anc clt divge0 sseli cexp oveq2d ad2antrl simprr nncnd nnne0d ad2antll nnzd oveq12d anassrs syl2anc mpbid syl3anc nnrecred cof ssexi relogcl subcld reex cbs dchr1re wfo znzrhfo 3syl ffvelrn syl2an offval2 sub32d fsumsub @@ -383187,7 +383960,7 @@ a multiplicative function (but not completely multiplicative). 10-Sep-2014.) $) qabvle $p |- ( ( F e. A /\ N e. NN0 ) -> ( F ` N ) <_ N ) $= ( wcel cfv cle wbr wi cc0 c1 caddc wceq fveq2 id breq12d cq cr cn0 imbi2d - vk vn cv co qrng0 abv0 0le0 syl6eqbr wa cn nn0p1nn ad2antrl nnq syl abvcl + vk vn cv co qrng0 abv0 0le0 eqbrtrdi wa cn nn0p1nn ad2antrl nnq syl abvcl qrngbas syldan cz nn0z zq peano2re zred simpl 1z mp1i cvv cplusg cnfldadd qex ccnfld ressplusg ax-mp abvtri syl3anc wne ax-1ne0 qrng1 adantr oveq2d abv1z mpan2 breqtrd 1red simprr leadd1dd letrd expr expcom nn0ind impcom @@ -383698,7 +384471,7 @@ a multiplicative function (but not completely multiplicative). simplr eqeq1d simprrl simprrr qaddcl readdcld rpexpcl remulcl sylancr simpr rppwr cplusg cnfldadd ressplusg abvtri cnfldmul ressmulr abvmul ax-mp cmulr qabvexp reexpcld remulcld w3o elz simprbi abv0 syl5ibrcom - 0le1 syl6eqbr ralbidva rspccv cminusg abvneg qrngneg eqeltrd eqbrtrrd + 0le1 eqbrtrdi ralbidva rspccv cminusg abvneg qrngneg eqeltrd eqbrtrrd lenlt expr ralrimiva rsp sylc expgt0 lemul2 syl112anc mulid1d breqtrd 3jaod rpge0d max1 leexp1a syl32anc le2addd 2timesd anassrs rexlimdvva max2 sylbid rpregt0d ledivmul2 mp3an12i reexpcl pm2.21d breq2d notbid @@ -394519,7 +395292,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to ( vp cfv wcel w3a wbr cmin co cmul cc0 cle wral wceq wa wb cee cop w3o cv cbtwn c1 brbtwn2 3comr colinearalglem3 anbi2d bitrd colinearalglem2 3coml 3orbi123d wi cc fveecn subid oveq2d adantl subcl mul01d eqtrd syl2an 0le0 - anandirs syl6eqbr ralrimiva 3adant1 fveq1 oveq12d ralbidv syl5ibcom 3mix1 + anandirs eqbrtrdi ralrimiva 3adant1 fveq1 oveq12d ralbidv syl5ibcom 3mix1 cfz breq1d syl6 a1dd wne wrex simp3 simp1 eqeefv syl2anc necon3abid df-ne wn rexbii rexnal bitr2i syl6bb ralcom fveq2 oveq1d eqeq12d rspcv ad2antrl weq cr fveere 3ad2antl1 3ad2antl2 3ad2antl3 3jca anim1i anasss cdiv caddc @@ -397308,7 +398081,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to ( vx vy vi vz cfv cnx cop c1 co cv c2 cmpo cbtwn wbr c7 cdc 1nn0 decnncl c6 cn wcel ceeng cbs cee cds cfz cmin cexp csu cpr citv crab clng csn w3o cdif cun cstr eengv 1nn basendx 2nn0 1lt10 declti 2nn dsndx strle2 itvndx - 6nn 6nn0 7nn 6lt7 declt lngndx 2lt6 strleun syl6eqbr ) AUAUBAUCFGUDFZAUEF + 6nn 6nn0 7nn 6lt7 declt lngndx 2lt6 strleun eqbrtrdi ) AUAUBAUCFGUDFZAUEF ZHGUFFZBCVTVTIAUGJDKZBKZFWBCKZFUHJLUIJDUJMZHUKZGULFZBCVTVTEKZWCWDHNOZEVTU MMZHGUNFZBCVTVTWCUOUQWIWCWHWDHNOWDWCWHHNOUPEVTUMMZHUKZURIIPQZHUSBCEDAUTII LQZITQZWNWFWMVSWAIWOVTWEVAVBILIVAVCRVDVEILRVFSVGVHWGWKWPWNWJWLITRVJSVIITP @@ -415308,7 +416081,7 @@ prime number equals this length (in an undirected simple graph). over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.) $) clwwlknon1le1 $p |- ( # ` ( X ( ClWWalksNOn ` G ) 1 ) ) <_ 1 $= - ( cfv wcel c1 chash cle csn wa wceq eqid fveq2 syl6eq syl6eqbr syl c0 hash0 + ( cfv wcel c1 chash cle csn wa wceq eqid fveq2 syl6eq eqbrtrdi syl c0 hash0 cc0 0le1 wn cvtx cclwwlknon wbr cedg cs1 clwwlknon1loop cword s1cli hashsng co cvv ax-mp 1le1 clwwlknon1nloop adantl pm2.61danel cn intnanrd clwwlknon0 wnel id fveq2d pm2.61i ) BAUACZDZBEAUBCZUJZFCZEGUCZVEVIBHZAUDCZVEVJVKDIVGBU @@ -417788,7 +418561,7 @@ edge remains odd if it was odd before (regarding the subgraphs induced cv cmpt cres adantl eqcomi oveq2i oveq1 cfzo wcel cn0 clt w3a elfzo0 nncn cc 3ad2ant2 sylbi npcan1 sylan9eq oveq2d cdm cword ccrcts cwlks crctiswlk cn 3syl wlkf syl cshwn eqtrd syl5eqr csn cun crctcshlem1 fz0sn0fz1 eleq2d - eqid elun syl6bb elsni 0le0 syl6eqbr iftrued fveq2d ctrls crctprop eqcomd + eqid elun syl6bb elsni 0le0 eqbrtrdi iftrued fveq2d ctrls crctprop eqcomd wo simpr adantr addid2d sylan9eqr fveq2 ex wn sylbid syl5eq breq2d fveq2i 3eqtr4d oveq1d mpteq2dv 3brtr4d cz cima 3jca syl6eq reseq2d eucrct2eupth1 imp a1i sseqtrid resmptd wi 3ad2ant1 eqcom cr wb ad2antlr sylan2 iffalsed @@ -418126,45 +418899,33 @@ edge remains odd if it was odd before (regarding the subgraphs induced _friendship graph_ if every pair of its vertices has exactly one common neighbor. This condition is called the _friendship condition_ , see definition in [MertziosUnger] p. 152. (Contributed by Alexander van der - Vekens and Mario Carneiro, 2-Oct-2017.) (Revised by AV, - 29-Mar-2021.) $) - df-frgr $a |- FriendGraph = { g | ( g e. USGraph - /\ [. ( Vtx ` g ) / v ]. [. ( Edg ` g ) / e ]. - A. k e. v A. l e. ( v \ { k } ) - E! x e. v { { x , k } , { x , l } } C_ e ) } $. + Vekens and Mario Carneiro, 2-Oct-2017.) (Revised by AV, 29-Mar-2021.) + (Revised by AV, 3-Jan-2024.) $) + df-frgr $a |- FriendGraph = { g e. USGraph | + [. ( Vtx ` g ) / v ]. [. ( Edg ` g ) / e ]. + A. k e. v A. l e. ( v \ { k } ) + E! x e. v { { x , k } , { x , l } } C_ e } $. $} ${ - $d h e g k l x v $. $d E h $. $d G h k l x $. $d V h k l x $. + $d E e g k l v x $. $d G e g v $. $d V e g k l v x $. isfrgr.v $e |- V = ( Vtx ` G ) $. isfrgr.e $e |- E = ( Edg ` G ) $. $( The property of being a friendship graph. (Contributed by Alexander van - der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) $) - isfrgr $p |- ( G e. U -> ( G e. FriendGraph <-> ( G e. USGraph - /\ A. k e. V A. l e. ( V \ { k } ) - E! x e. V { { x , k } , { x , l } } C_ E ) ) ) $= - ( vg wcel cv cusgr wreu wral cedg cfv cvtx wa wceq ve vv vh cfrgr cpr wss - csn cdif wsbc df-frgr eleq2i eleq1 fveq2 syl6eqr difeq1d wb reueq1 sseq2d - cab syl reubidv bitrd raleqbidv anbi12d weq cvv fvexd adantr simpr difeq1 - ad2antlr sbcied2 cbvabv elab2g syl5bb ) EUDKEJLZMKZALZCLZUEVRGLUEUEZUALZU - FZAUBLZNZGWCVSUGZUHZOZCWCOZUAVPPQZUIZUBVPRQZUIZSZJUSZKEBKEMKZVTDUFZAFNZGF - WEUHZOZCFOZSZUDWNEAUBUAJCGUJUKUCLZMKZVTXBPQZUFZAXBRQZNZGXFWEUHZOZCXFOZSZX - AUCEWNBXBETZXCWOXJWTXBEMULXLXIWSCXFFXLXFERQFXBERUMHUNZXLXGWQGXHWRXLXFFWEX - MUOXLXGXEAFNZWQXLXFFTXGXNUPXMXEAXFFUQUTXLXEWPAFXLXDDVTXLXDEPQDXBEPUMIUNUR - VAVBVCVCVDWMXKJUCJUCVEZVQXCWLXJVPXBMULXOWJXJUBWKXFVFXOVPRVGVPXBRUMXOWCXFT - ZSZWHXJUAWIXDVFXQVPPVGXOWIXDTXPVPXBPUMVHXQWAXDTZSZWGXICWCXFXQXPXRXOXPVIVH - XSWDXGGWFXHXPWFXHTXOXRWCXFWEVJVKXSWDWBAXFNZXGXPWDXTUPXOXRWBAWCXFUQVKXSWBX - EAXFXSWAXDVTXQXRVIURVAVBVCVCVLVLVDVMVNVO $. - - $( A friendship graph is a simple graph which fulfils the friendship - condition. (Contributed by Alexander van der Vekens, 4-Oct-2017.) - (Revised by AV, 29-Mar-2021.) $) - frgrusgrfrcond $p |- ( G e. FriendGraph <-> ( G e. USGraph + der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) (Revised by AV, + 3-Jan-2024.) $) + isfrgr $p |- ( G e. FriendGraph <-> ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) ) $= - ( cfrgr wcel cusgr cv cpr wss wreu csn cdif wral wa isfrgr syl6bi pm2.43i - simpl pm5.21nii ) DIJZDKJZUFALZBLZMUGFLMMCNAEOFEUHPQRBERZSZUEUFUEUEUJUFAI - BCDEFGHTUFUIUCZUAUBUKAKBCDEFGHTUD $. + ( ve vv vg cv cpr wreu wral cedg cfv cvtx wceq wb wss csn cdif wsbc cusgr + cfrgr fvex fveq2 eqeq2d eqcomi eqeq2i syl6bb anbi12d difeq1 adantr reueq1 + simpl sseq2 adantl reubidv bitrd raleqbidv syl6bi sbc2iedv df-frgr elrab2 + wa ) ALZBLZMVHFLMMZILZUAZAJLZNZFVMVIUBZUCZOZBVMOZIKLZPQZUDJVSRQZUDVJCUAZA + ENZFEVOUCZOZBEOZKDUEUFVSDSZVRWFJIWAVTVSRUGVSPUGWGVMWASZVKVTSZVGVMESZVKCSZ + VGZVRWFTWGWHWJWIWKWGWHVMDRQZSWJWGWAWMVMVSDRUHUIWMEVMEWMGUJUKULWGWIVKDPQZS + WKWGVTWNVKVSDPUHUIWNCVKCWNHUJUKULUMWLVQWEBVMEWJWKUQWLVNWCFVPWDWJVPWDSWKVM + EVOUNUOWLVNVLAENZWCWJVNWOTWKVLAVMEUPUOWLVLWBAEWKVLWBTWJVKCVJURUSUTVAVBVBV + CVDAJIKBFVEVF $. $} ${ @@ -418173,23 +418934,20 @@ edge remains odd if it was odd before (regarding the subgraphs induced Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) $) frgrusgr $p |- ( G e. FriendGraph -> G e. USGraph ) $= ( vx vk vl cfrgr wcel cusgr cpr cedg cfv wss cvtx wreu csn cdif wral eqid - cv frgrusgrfrcond simplbi ) AEFAGFBRZCRZHUADRHHAIJZKBALJZMDUDUBNOPCUDPBCU - CAUDDUDQUCQST $. - $} + cv isfrgr simplbi ) AEFAGFBRZCRZHUADRHHAIJZKBALJZMDUDUBNOPCUDPBCUCAUDDUDQ + UCQST $. - ${ - $d G k l x $. $( Any null graph (set with no vertices) is a friendship graph iff its edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) $) frgr0v $p |- ( ( G e. W /\ ( Vtx ` G ) = (/) ) -> ( G e. FriendGraph <-> ( iEdg ` G ) = (/) ) ) $= ( vx vk vl cfrgr wcel cusgr cv cpr cedg cfv wss cvtx wral wa c0 wceq eqid - adantr wreu csn ciedg frgrusgrfrcond cuhgr usgruhgr uhgr0vb syl5ib simpll - cdif simpr usgr0e ral0 wb raleq adantl mpbiri jca ex impbid syl5bb ) AFGA - HGZCIZDIZJVCEIJJAKLZMCANLZUAEVFVDUBUJOZDVFOZPZABGZVFQRZPZAUCLQRZCDVEAVFEV - FSVESUDVLVIVMVIAUEGZVLVMVBVNVHAUFTABUGUHVLVMVIVLVMPZVBVHVOABVJVKVMUIVLVMU - KULVLVHVMVLVHVGDQOZVGDUMVKVHVPUNVJVGDVFQUOUPUQTURUSUTVA $. + adantr wreu cdif ciedg isfrgr cuhgr usgruhgr uhgr0vb syl5ib simpll usgr0e + csn simpr ral0 wb raleq adantl mpbiri jca ex impbid syl5bb ) AFGAHGZCIZDI + ZJVCEIJJAKLZMCANLZUAEVFVDUKUBOZDVFOZPZABGZVFQRZPZAUCLQRZCDVEAVFEVFSVESUDV + LVIVMVIAUEGZVLVMVBVNVHAUFTABUGUHVLVMVIVLVMPZVBVHVOABVJVKVMUIVLVMULUJVLVHV + MVLVHVGDQOZVGDUMVKVHVPUNVJVGDVFQUOUPUQTURUSUTVA $. $( Any null graph (without vertices and edges) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Revised by AV, @@ -418213,30 +418971,12 @@ edge remains odd if it was odd before (regarding the subgraphs induced (Contributed by AV, 29-Mar-2021.) $) frgr0 $p |- (/) e. FriendGraph $= ( vx vk vl c0 cfrgr wcel cusgr cpr cedg cfv wss wreu cdif wral usgr0 ral0 - cv csn cvtx vtxval0 eqcomi eqid frgrusgrfrcond mpbir2an ) DEFDGFAQZBQZHUE - CQHHDIJZKADLCDUFRMNZBDNOUHBPABUGDDCDSJDTUAUGUBUCUD $. - $} - - ${ - $d A x y $. $d B y $. $d C x $. $d D y $. $d E x $. $d ph x $. - $d ch x $. $d ps y $. - rspc2vd.a $e |- ( x = A -> ( th <-> ch ) ) $. - rspc2vd.b $e |- ( y = B -> ( ch <-> ps ) ) $. - rspc2vd.c $e |- ( ph -> A e. C ) $. - rspc2vd.d $e |- ( ( ph /\ x = A ) -> D = E ) $. - rspc2vd.e $e |- ( ph -> B e. E ) $. - $( Deduction version of 2-variable restricted specialization, using - implicit substitution. Notice that the class ` D ` for the second set - variable ` y ` may depend on the first set variable ` x ` . - (Contributed by AV, 29-Mar-2021.) $) - rspc2vd $p |- ( ph -> ( A. x e. C A. y e. D th -> ps ) ) $= - ( csb wcel wral csbied eleqtrrd wi nfcsb1v nfv nfral cv csbeq1a raleqbidv - wceq rspc syl rspcv sylsyld ) AHEGJQZRDFJSZEISZCFUNSZBAHKUNPAEGJKINOTUAAG - IRUPUQUBNUOUQEGICEFUNEGJUCCEUDUEEUFGUIDCFJUNEGJUGLUHUJUKCBFHUNMULUM $. + cv csn cvtx vtxval0 eqcomi eqid isfrgr mpbir2an ) DEFDGFAQZBQZHUECQHHDIJZ + KADLCDUFRMNZBDNOUHBPABUGDDCDSJDTUAUGUBUCUD $. $} ${ - $d A b k l $. $d C b k l $. $d E k l $. $d G b k l $. $d V b k l $. + $d A b k l $. $d C b k l $. $d E b k l $. $d G b k l $. $d V b k l $. frcond1.v $e |- V = ( Vtx ` G ) $. frcond1.e $e |- E = ( Edg ` G ) $. $( The friendship condition: any two (different) vertices in a friendship @@ -418244,15 +418984,15 @@ edge remains odd if it was odd before (regarding the subgraphs induced Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.) $) frcond1 $p |- ( G e. FriendGraph -> ( ( A e. V /\ C e. V /\ A =/= C ) -> E! b e. V { { A , b } , { b , C } } C_ E ) ) $= - ( vk vl wcel cv cpr wss wreu csn cdif wral wne wceq w3a wi frgrusgrfrcond - cfrgr cusgr preq2 preq1d sseq1d reubidv preq2d simp1 difeq2d adantl necom - sneq wa biimpi anim2i 3adant1 eldifsn sylibr rspc2vd preq1i sseq1i reubii - prcom syl6com simplbiim ) DUDKDUEKFLZILZMZVIJLZMZMZCNZFEOZJEVJPZQZRIERZAE - KZBEKZABSZUAZAVIMZVIBMZMZCNZFEOZUBFICDEJGHUCWCVSVIAMZWEMZCNZFEOZWHWCWLWIV - MMZCNZFEOVPIJABEVREAPZQZVJATZVOWNFEWQVNWMCWQVKWIVMVJAVIUFUGUHUIVLBTZWNWKF - EWRWMWJCWRVMWEWIVLBVIUFUJUHUIVTWAWBUKWQVRWPTWCWQVQWOEVJAUOULUMWCWABASZUPZ - BWPKWAWBWTVTWBWSWAWBWSABUNUQURUSBEAUTVAVBWLWHWKWGFEWJWFCWIWDWEVIAVFVCVDVE - UQVGVH $. + ( vk vl wcel cv cpr wss wreu csn cdif wral wne wceq cfrgr cusgr wi isfrgr + w3a preq2 preq1d sseq1d reubidv preq2d simp1 sneq difeq2d adantl wa necom + biimpi anim2i 3adant1 eldifsn sylibr rspc2vd preq1i sseq1i reubii syl6com + prcom simplbiim ) DUAKDUBKFLZILZMZVIJLZMZMZCNZFEOZJEVJPZQZRIERZAEKZBEKZAB + SZUEZAVIMZVIBMZMZCNZFEOZUCFICDEJGHUDWCVSVIAMZWEMZCNZFEOZWHWCWLWIVMMZCNZFE + OVPIJABEVREAPZQZVJATZVOWNFEWQVNWMCWQVKWIVMVJAVIUFUGUHUIVLBTZWNWKFEWRWMWJC + WRVMWEWIVLBVIUFUJUHUIVTWAWBUKWQVRWPTWCWQVQWOEVJAULUMUNWCWABASZUOZBWPKWAWB + WTVTWBWSWAWBWSABUPUQURUSBEAUTVAVBWLWHWKWGFEWJWFCWIWDWEVIAVGVCVDVEUQVFVH + $. $( The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der @@ -418325,12 +419065,11 @@ edge remains odd if it was odd before (regarding the subgraphs induced ( vx vk vl wcel cfv csn wceq cv cpr wss wreu cdif cvv c0 raleqbidv mpbiri wral eqid cusgr cvtx wa cedg cfrgr simpl ral0 difeq2d difid syl6eq preq1d sneq preq2 sseq1d reubidv ralsng wn snprc rzal sylbi pm2.61i wb id difeq1 - reueq1 adantl frgrusgrfrcond sylanbrc ) AUAFZAUBGZBHZIZUCZVICJZDJZKZVNEJK - ZKZAUDGZLZCVJMZEVJVOHZNZSZDVJSZAUEFVIVLUFVMWEVTCVKMZEVKWBNZSZDVKSZBOFZWIW - JWIVNBKZVQKZVSLZCVKMZEPSZWNEUGWHWODBOVOBIZWFWNEWGPWPWGVKVKNPWPWBVKVKVOBUL - UHVKUIUJWPVTWMCVKWPVRWLVSWPVPWKVQVOBVNUMUKUNUOQUPRWJUQVKPIWIBURWHDVKUSUTV - AVLWEWIVBVIVLWDWHDVJVKVLVCVLWAWFEWCWGVJVKWBVDVTCVJVKVEQQVFRCDVSAVJEVJTVST - VGVH $. + reueq1 adantl isfrgr sylanbrc ) AUAFZAUBGZBHZIZUCZVICJZDJZKZVNEJKZKZAUDGZ + LZCVJMZEVJVOHZNZSZDVJSZAUEFVIVLUFVMWEVTCVKMZEVKWBNZSZDVKSZBOFZWIWJWIVNBKZ + VQKZVSLZCVKMZEPSZWNEUGWHWODBOVOBIZWFWNEWGPWPWGVKVKNPWPWBVKVKVOBULUHVKUIUJ + WPVTWMCVKWPVRWLVSWPVPWKVQVOBVNUMUKUNUOQUPRWJUQVKPIWIBURWHDVKUSUTVAVLWEWIV + BVIVLWDWHDVJVKVLVCVLWAWFEWCWGVJVKWBVDVTCVJVKVEQQVFRCDVSAVJEVJTVSTVGVH $. $} ${ @@ -418347,29 +419086,28 @@ edge remains odd if it was odd before (regarding the subgraphs induced preq12d 3adant3 mtbird reurex rexnal bicomi a1i difprsn1 3ad2ant3 rexeqdv nsyl preq2 preq2d reubidv notbid rexsng 3ad2ant2 3bitrd difprsn2 3ad2ant1 orcd orbi12d mpbird sneq difeq2d preq1d raleqbidv sylib adantlr id difeq1 - reueq1 anbi2d df-nel frgrusgrfrcond xchbinx sylibr frgrusgr con3i pm2.61i - expcom a1d ) CUCIZADIZBEIZABUAZUDZCUEUFZABJZKZLZCUGUHZUBXSXKXTXSXKLZXKFUI - ZGUIZJZYBHUIZJZJZCUJUFZMZFXPNZHXPYCUKZULZOZGXPOZLZPZXTYAYPXKYIFXQNZHXQYKU - LZOZGXQOZLZPZXOXKUUBXRXOXKLZYTXKUUCYSPZGXQQZYTPUUCUUEYBAJZYFJZYHMZFXQNZHX - QAUKZULZOZPZYBBJZYFJZYHMZFXQNZHXQBUKZULZOZPZUMZUUCUVBUUFUUNJZYHMZFXQNZPZU - UNUUFJZYHMZFXQNZPZUMUUCUVFUVJUUCUVDFXQQZUVEUUCUVKAAJZXQJZYHMZBAJZBBJZJZYH - MZUMZUUCUVNPUVRPUVSPUUCUVLYHIZXQYHIZLUVNUUCUVTUWAXKUVTPXOXKUVTAAUAZAUNXKU - VTUWBYHCAAYHUOZUPUQURUSUTUVLXQYHAAVAABVAVBVCUUCUVOYHIZUVPYHIZLUVRUUCUWEUW - DXKUWEPXOXKUWEBBUAZBUNXKUWEUWFYHCBBUWCUPUQURUSVDUVOUVPYHBAVABBVAVBVCUVNUV - RVEVFXOUVKUVSRZXKXLXMUWGXNUVDUVNUVRFABDEYBAKZUVCUVMYHUWHUUFUVLUUNXQYBAAVG - YBABVGVISYBBKZUVCUVQYHUWIUUFUVOUUNUVPYBBAVGYBBBVGVISVHVJTVKUVDFXQVLVSWIUU - CUUMUVFUVAUVJUUCUUMUUIPZHUUKQZUWJHUURQZUVFUUMUWKRUUCUWKUUMUUIHUUKVMVNVOUU - CUWJHUUKUURXOUUKUURKZXKXNXLUWMXMABVPVQTVRXOUWLUVFRZXKXMXLUWNXNUWJUVFHBEYE - BKZUUIUVEUWOUUHUVDFXQUWOUUGUVCYHUWOYFUUNUUFYEBYBVTWASWBWCWDWETWFUUCUVAUUQ - PZHUUSQZUWPHUUJQZUVJUVAUWQRUUCUWQUVAUUQHUUSVMVNVOUUCUWPHUUSUUJXOUUSUUJKZX - KXNXLUWSXMABWGVQTVRXOUWRUVJRZXKXLXMUWTXNUWPUVJHADYEAKZUUQUVIUXAUUPUVHFXQU - XAUUOUVGYHUXAYFUUFUUNYEAYBVTWASWBWCWDWHTWFWJWKXOUUEUVBRZXKXLXMUXBXNUUDUUM - UVAGABDEYCAKZYSUULUXCYQUUIHYRUUKUXCYKUUJXQYCAWLWMUXCYIUUHFXQUXCYGUUGYHUXC - YDUUFYFYCAYBVTWNSWBWOWCYCBKZYSUUTUXDYQUUQHYRUUSUXDYKUURXQYCBWLWMUXDYIUUPF - XQUXDYGUUOYHUXDYDUUNYFYCBYBVTWNSWBWOWCVHVJTWKYSGXQVMWPVDWQXSYPUUBRZXKXRUX - EXOXRYOUUAXRYNYTXKXRYMYSGXPXQXRWRXRYJYQHYLYRXPXQYKWSYIFXPXQWTWOWOXAWCUSTW - KXTCUGIZYOCUGXBZFGYHCXPHXPUOUWCXCXDXEXIXKPZXTXSUXHUXFPXTUXFXKCXFXGUXGXEXJ - XH $. + reueq1 anbi2d df-nel isfrgr xchbinx sylibr expcom frgrusgr con3i pm2.61i + a1d ) CUCIZADIZBEIZABUAZUDZCUEUFZABJZKZLZCUGUHZUBXSXKXTXSXKLZXKFUIZGUIZJZ + YBHUIZJZJZCUJUFZMZFXPNZHXPYCUKZULZOZGXPOZLZPZXTYAYPXKYIFXQNZHXQYKULZOZGXQ + OZLZPZXOXKUUBXRXOXKLZYTXKUUCYSPZGXQQZYTPUUCUUEYBAJZYFJZYHMZFXQNZHXQAUKZUL + ZOZPZYBBJZYFJZYHMZFXQNZHXQBUKZULZOZPZUMZUUCUVBUUFUUNJZYHMZFXQNZPZUUNUUFJZ + YHMZFXQNZPZUMUUCUVFUVJUUCUVDFXQQZUVEUUCUVKAAJZXQJZYHMZBAJZBBJZJZYHMZUMZUU + CUVNPUVRPUVSPUUCUVLYHIZXQYHIZLUVNUUCUVTUWAXKUVTPXOXKUVTAAUAZAUNXKUVTUWBYH + CAAYHUOZUPUQURUSUTUVLXQYHAAVAABVAVBVCUUCUVOYHIZUVPYHIZLUVRUUCUWEUWDXKUWEP + XOXKUWEBBUAZBUNXKUWEUWFYHCBBUWCUPUQURUSVDUVOUVPYHBAVABBVAVBVCUVNUVRVEVFXO + UVKUVSRZXKXLXMUWGXNUVDUVNUVRFABDEYBAKZUVCUVMYHUWHUUFUVLUUNXQYBAAVGYBABVGV + ISYBBKZUVCUVQYHUWIUUFUVOUUNUVPYBBAVGYBBBVGVISVHVJTVKUVDFXQVLVSWIUUCUUMUVF + UVAUVJUUCUUMUUIPZHUUKQZUWJHUURQZUVFUUMUWKRUUCUWKUUMUUIHUUKVMVNVOUUCUWJHUU + KUURXOUUKUURKZXKXNXLUWMXMABVPVQTVRXOUWLUVFRZXKXMXLUWNXNUWJUVFHBEYEBKZUUIU + VEUWOUUHUVDFXQUWOUUGUVCYHUWOYFUUNUUFYEBYBVTWASWBWCWDWETWFUUCUVAUUQPZHUUSQ + ZUWPHUUJQZUVJUVAUWQRUUCUWQUVAUUQHUUSVMVNVOUUCUWPHUUSUUJXOUUSUUJKZXKXNXLUW + SXMABWGVQTVRXOUWRUVJRZXKXLXMUWTXNUWPUVJHADYEAKZUUQUVIUXAUUPUVHFXQUXAUUOUV + GYHUXAYFUUFUUNYEAYBVTWASWBWCWDWHTWFWJWKXOUUEUVBRZXKXLXMUXBXNUUDUUMUVAGABD + EYCAKZYSUULUXCYQUUIHYRUUKUXCYKUUJXQYCAWLWMUXCYIUUHFXQUXCYGUUGYHUXCYDUUFYF + YCAYBVTWNSWBWOWCYCBKZYSUUTUXDYQUUQHYRUUSUXDYKUURXQYCBWLWMUXDYIUUPFXQUXDYG + UUOYHUXDYDUUNYFYCBYBVTWNSWBWOWCVHVJTWKYSGXQVMWPVDWQXSYPUUBRZXKXRUXEXOXRYO + UUAXRYNYTXKXRYMYSGXPXQXRWRXRYJYQHYLYRXPXQYKWSYIFXPXQWTWOWOXAWCUSTWKXTCUGI + ZYOCUGXBZFGYHCXPHXPUOUWCXCXDXEXFXKPZXTXSUXHUXFPXTUXFXKCXGXHUXGXEXJXI $. $} ${ @@ -418456,64 +419194,64 @@ edge remains odd if it was odd before (regarding the subgraphs induced <-> ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. 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$d G a b c $. $d V a b c $. + $d E a b c $. $d G a b c $. $d V a b c $. 2pthfrgrrn.v $e |- V = ( Vtx ` G ) $. 2pthfrgrrn.e $e |- E = ( Edg ` G ) $. $( Between any two (different) vertices in a friendship graph is a 2-path @@ -418680,11 +419418,11 @@ edge remains odd if it was odd before (regarding the subgraphs induced 15-Nov-2017.) (Revised by AV, 1-Apr-2021.) $) 2pthfrgrrn $p |- ( G e. FriendGraph -> A. a e. V A. c e. ( V \ { a } ) E. b e. V ( { a , b } e. E /\ { b , c } e. E ) ) $= - ( cfrgr wcel cusgr cv cpr wss wreu csn wral wa wrex zfpair2 cdif wi prcom - frgrusgrfrcond reurex eleq1i anbi1i sylbbr reximi syl a1i ralimdvva sylbi - prss imp ) BIJBKJZELZDLZMZUQFLZMZMANZECOZFCURPUAZQDCQZRURUQMZAJZVAAJZRZEC - SZFVDQDCQZEDABCFGHUDUPVEVKUPVCVJDFCVDVCVJUBUPURCJUTVDJRRVCVBECSVJVBECUEVB - VIECVIUSAJZVHRVBVGVLVHVFUSAURUQUCUFUGUSVAAEDTEFTUNUHUIUJUKULUOUM $. + ( cfrgr wcel cusgr cv cpr wss wreu csn wral wa wrex zfpair2 isfrgr reurex + cdif prcom eleq1i anbi1i prss sylbbr reximi syl a1i ralimdvva imp sylbi + wi ) BIJBKJZELZDLZMZUQFLZMZMANZECOZFCURPUCZQDCQZRURUQMZAJZVAAJZRZECSZFVDQ + DCQZEDABCFGHUAUPVEVKUPVCVJDFCVDVCVJUOUPURCJUTVDJRRVCVBECSVJVBECUBVBVIECVI + USAJZVHRVBVGVLVHVFUSAURUQUDUEUFUSVAAEDTEFTUGUHUIUJUKULUMUN $. $( Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A @@ -418855,25 +419593,25 @@ edge remains odd if it was odd before (regarding the subgraphs induced -> ( # ` F ) =/= 4 ) $= ( va vb vc vd vx vk vl wcel cfv wa wne wi cv cpr wrex wss wreu ccycls wbr cfrgr chash c4 wceq cupgr cusgr frgrusgr usgrupgr syl cedg upgr4cycl4dv4e - w3a cvtx eqid csn cdif wral frgrusgrfrcond necom biimpi 3ad2ant2 ad2antrl - simplrl adantl eldifsn sneq difeq2d preq2 preq1d sseq1d reubidv raleqbidv - rspcv ad3antrrr preq2d sylsyld prcom preq1i sseq1i reubii simprll simprlr - sylanbrc simpllr simplrr simprr2 4cycl2vnunb syl113anc pm2.21d com12 syl6 - wn sylbi pm2.43b expdcom rexlimdvva rexlimivv com34 com23 mpcom imp neqne - 3exp pm2.61d1 ) CUCKZBACUALUBZMBUDLZUEUFZXIUENZXGXHXJXKOZCUGKZXGXHXLOXGCU - HKZXMCUICUJUKXMXHXGXLXMXHXJXGXKXMXHXJXGXKOZXMXHXJUNDPZEPZQCULLZKXQFPZQXRK - MZXSGPZQXRKYAXPQXRKMZMZXPXQNZXPXSNZXPYANZUNZXQXSNZXQYANZXSYANZUNZMZMZGCUO - LZRFYNRZEYNRDYNRXOAXRBCYNDEFGYNUPZXRUPZUMYOXODEYNYNXPYNKZXQYNKZMZYMXOFGYN - YNXGYTXSYNKZYAYNKZMZMZYMXKXGXNHPZIPZQZUUEJPZQZQZXRSZHYNTZJYNUUFUQZURZUSZI - YNUSZMUUDYMMZXKOZHIXRCYNJYPYQUTUUPUURXNUUPUUQXKUUQUUPUUEXPQZUUEXSQZQZXRSZ - HYNTZUURUUQXSYNXPUQZURZKZUUPUUSUUIQZXRSZHYNTZJUVEUSZUVCUUQUUAXSXPNZUVFYTU - UAUUBYMVEYMUVKUUDYGUVKYCYKYEYDUVKYFYEUVKXPXSVAVBVCVDVFXSYNXPVGWEYRUUPUVJO - YSUUCYMUUOUVJIXPYNUUFXPUFZUULUVIJUUNUVEUVLUUMUVDYNUUFXPVHVIUVLUUKUVHHYNUV - LUUJUVGXRUVLUUGUUSUUIUUFXPUUEVJVKVLVMVNVOVPUVIUVCJXSUVEUUHXSUFZUVHUVBHYNU - VMUVGUVAXRUVMUUIUUTUUSUUHXSUUEVJVQVLVMVOVRUVCXPUUEQZUUTQZXRSZHYNTZUURUVBU - VPHYNUVAUVOXRUUSUVNUUTUUEXPVSVTWAWBUUQUVQXKUUQUVQXKUUQXTYBYSUUBYIUVQWNUUD - XTYBYLWCUUDXTYBYLWDYRYSUUCYMWFYTUUAUUBYMWGYMYIUUDYHYIYJYGYCWHVFHXPXQXSYAX - RYNWIWJWKWLWOWMWPVFWOWQWRWSUKXEWTXAXBXCXIUEXDXF $. + w3a cvtx eqid csn cdif wral isfrgr simplrl necom biimpi 3ad2ant2 ad2antrl + adantl eldifsn sylanbrc sneq difeq2d preq2 preq1d reubidv raleqbidv rspcv + sseq1d preq2d sylsyld prcom preq1i sseq1i simprll simprlr simpllr simplrr + ad3antrrr reubii simprr2 4cycl2vnunb syl113anc pm2.21d com12 syl6 pm2.43b + wn sylbi expdcom rexlimdvva rexlimivv 3exp com34 com23 mpcom imp pm2.61d1 + neqne ) CUCKZBACUALUBZMBUDLZUEUFZXIUENZXGXHXJXKOZCUGKZXGXHXLOXGCUHKZXMCUI + CUJUKXMXHXGXLXMXHXJXGXKXMXHXJXGXKOZXMXHXJUNDPZEPZQCULLZKXQFPZQXRKMZXSGPZQ + XRKYAXPQXRKMZMZXPXQNZXPXSNZXPYANZUNZXQXSNZXQYANZXSYANZUNZMZMZGCUOLZRFYNRZ + EYNRDYNRXOAXRBCYNDEFGYNUPZXRUPZUMYOXODEYNYNXPYNKZXQYNKZMZYMXOFGYNYNXGYTXS + YNKZYAYNKZMZMZYMXKXGXNHPZIPZQZUUEJPZQZQZXRSZHYNTZJYNUUFUQZURZUSZIYNUSZMUU + DYMMZXKOZHIXRCYNJYPYQUTUUPUURXNUUPUUQXKUUQUUPUUEXPQZUUEXSQZQZXRSZHYNTZUUR + UUQXSYNXPUQZURZKZUUPUUSUUIQZXRSZHYNTZJUVEUSZUVCUUQUUAXSXPNZUVFYTUUAUUBYMV + AYMUVKUUDYGUVKYCYKYEYDUVKYFYEUVKXPXSVBVCVDVEVFXSYNXPVGVHYRUUPUVJOYSUUCYMU + UOUVJIXPYNUUFXPUFZUULUVIJUUNUVEUVLUUMUVDYNUUFXPVIVJUVLUUKUVHHYNUVLUUJUVGX + RUVLUUGUUSUUIUUFXPUUEVKVLVPVMVNVOWFUVIUVCJXSUVEUUHXSUFZUVHUVBHYNUVMUVGUVA + XRUVMUUIUUTUUSUUHXSUUEVKVQVPVMVOVRUVCXPUUEQZUUTQZXRSZHYNTZUURUVBUVPHYNUVA + UVOXRUUSUVNUUTUUEXPVSVTWAWGUUQUVQXKUUQUVQXKUUQXTYBYSUUBYIUVQWOUUDXTYBYLWB + UUDXTYBYLWCYRYSUUCYMWDYTUUAUUBYMWEYMYIUUDYHYIYJYGYCWHVFHXPXQXSYAXRYNWIWJW + KWLWPWMWNVFWPWQWRWSUKWTXAXBXCXDXIUEXFXE $. $} ${ @@ -419042,7 +419780,7 @@ follows that deg(v) >= 2 for every node v of a friendship graph". ESVEKEUDVDKEUGEVCKNUHUIUNGKJUJUKVBJUSUFZVAHUSULVBVFUOOHUSJUMUPJUSUGUQUR $. - $d G y $. $d V y $. $d Y y $. $d x y $. + $d E y $. $d G y $. $d V y $. $d Y y $. $d x y $. $( Lemma 2 for ~ frgrncvvdeq . In a friendship graph, for each neighbor of a vertex there is exactly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van @@ -419062,7 +419800,7 @@ follows that deg(v) >= 2 for every node v of a friendship graph". TWAWBXKXRXOXAXOWPGKULUPZUCXKXRHYAWPOUMFGKWPMWCWJWDWEWFWGWSCIWHXACHWHWKWIU KWL $. - $d D n $. $d E n y $. $d G n $. $d N n y $. $d V n $. $d Y n $. + $d D n $. $d E n $. $d G n $. $d N n y $. $d V n $. $d Y n $. $d ph n $. $d n x $. $( Lemma 3 for ~ frgrncvvdeq . The unique neighbor of a vertex (expressed by a restricted iota) is the intersection of the corresponding @@ -428987,7 +429725,7 @@ a normed complex vector space (normally a Hilbert space). @) (New usage is discouraged.) $) nmosetn0 $p |- ( U e. NrmCVec -> ( N ` ( T ` Z ) ) e. { x | E. y e. X ( ( M ` y ) <_ 1 /\ x = ( N ` ( T ` y ) ) ) } ) $= - ( wcel cv cfv c1 cle wbr wceq wa wrex cnv cab cc0 nvz0 0le1 syl6eqbr eqid + ( wcel cv cfv c1 cle wbr wceq wa wrex cnv cab cc0 nvz0 0le1 eqbrtrdi eqid nvzcl jctir fveq2 breq1d 2fveq3 eqeq2d anbi12d rspcev syl2anc fvex anbi2d eqeq1 rexbidv elab sylibr ) DUALZBMZENZOPQZHCNZFNZVDCNFNZRZSZBGTZVHVFAMZV IRZSZBGTZAUBLVCHGLHENZOPQZVHVHRZSZVLDGHIJUHVCVRVSVCVQUCOPDEHJKUDUEUFVHUGU @@ -429354,7 +430092,7 @@ a normed complex vector space (normally a Hilbert space). @) nmoo0 $p |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( N ` Z ) = 0 ) $= ( vz vx cnv wcel wa cfv cc0 cxr clt c1 cle wceq wrex eqid csn csup cv wbr - cnmcv cba cab wf 0oo nmooval mpd3an3 df-sn cn0v nvzcl nvz0 syl6eqbr fveq2 + cnmcv cba cab wf 0oo nmooval mpd3an3 df-sn cn0v nvzcl nvz0 eqbrtrdi fveq2 wb 0le1 breq1d rspcev syl2anc biantrurd adantr 0oval 3expa ad2antlr eqtrd fveq2d eqeq2d anbi2d rexbidva r19.41v syl6rbb bitrd abbidv syl5req xrltso supeq1d wor 0xr supsn mp2an syl6eq ) AIJZCIJZKZDBLZMUAZNOUBZMWGWHGUCZAUEL @@ -430730,7 +431468,7 @@ orthogonal vectors (i.e. whose inner product is 0) is the sum of the (New usage is discouraged.) $) siii $p |- ( abs ` ( A P B ) ) <_ ( ( N ` A ) x. ( N ` B ) ) $= ( co cfv cmul cle wbr wceq cc0 wcel cabs cn0v oveq2 cnv phnvi dip0r mp2an - eqid syl6eq abs00bd nvge0 nvcli mulge0i syl6eqbr c2 cexp cdiv csqrt recni + eqid syl6eq abs00bd nvge0 nvcli mulge0i eqbrtrdi c2 cexp cdiv csqrt recni wne ccj sqeq0i wb bitri necon3bii cc dipcl mp3an resqcli divcan1zi sylbir nvz dipcj syl6eqr oveq2d fveq2d absval ax-mp syl6reqr eqcomd cr wi ipipcj divclzi mulcomli oveq1i wa mp3an12 mpan syl5reqr abscli redivclzi eqeltrd @@ -435086,11 +435824,11 @@ orthogonal vectors (i.e. whose inner product is 0) is the sum of the htmldef "VtxDeg" as 'VtxDeg'; althtmldef "VtxDeg" as 'VtxDeg'; latexdef "VtxDeg" as "\mathrm{VtxDeg}"; -htmldef "RegGraph" as 'RegGraph'; - althtmldef "RegGraph" as 'RegGraph'; +htmldef "RegGraph" as ' RegGraph '; + althtmldef "RegGraph" as ' RegGraph '; latexdef "RegGraph" as "\mathrm{RegGraph}"; -htmldef "RegUSGraph" as 'RegUSGraph'; - althtmldef "RegUSGraph" as 'RegUSGraph'; +htmldef "RegUSGraph" as ' RegUSGraph '; + althtmldef "RegUSGraph" as ' RegUSGraph '; latexdef "RegUSGraph" as "\mathrm{RegUSGraph}"; htmldef "EdgWalks" as ' EdgWalks '; althtmldef "EdgWalks" as ' EdgWalks '; @@ -441311,7 +442049,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). bcsiALT $p |- ( abs ` ( A .ih B ) ) <_ ( ( normh ` A ) x. ( normh ` B ) ) $= ( csp co cc0 wceq cabs cfv cmul cle wbr wcel ax-mp c2 caddc cc c1 cr abs0 - cno fveq2 chba normge0 normcli mulge0i mp2an eqbrtri syl6eqbr csqrt df-ne + cno fveq2 chba normge0 normcli mulge0i mp2an eqbrtri eqbrtrdi csqrt df-ne wn wne cdiv ccj his1i oveq2i abslem2 mpan syl5req abs00i necon3bii abscli hicli wa divclzi divreczi fveq2d recclzi absmul sylancr rerecclzi clt 0re recni jctil absgt0i bitri recgt0i sylbi ltle absidd oveq2d recidzi 3eqtrd @@ -452056,7 +452794,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. csup nmfnval syl wss wral wi nmfnsetre ressxr syl6ss normcl rexrd jca vex eqeq1 anbi2d rexbidv elab id braval fveq2d sylan9eqr bcs2 ancom1s eqbrtrd csp 3expa exp41 imp4a rexlimdv imp sylan2b ralrimiva cdiv normne0 biimpar - recnd reccld simpl hvmulcl syl2anc 1le1 syl6eqbr ax-his3 syl3anc rereccld + 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 @@ -457867,6 +458605,25 @@ Class abstractions (a.k.a. class builders) -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) + $( Elementhood to a union with a singleton. (Contributed by Thierry Arnoux, + 14-Dec-2023.) $) + elunsn $p |- ( A e. V -> ( A e. ( B u. { C } ) <-> ( A e. B \/ A = C ) ) ) + $= + ( csn cun wcel wo wceq elun elsng orbi2d syl5bb ) ABCEZFGABGZANGZHADGZOACIZ + HABNJQPROACDKLM $. + + $( Negated membership for a union. (Contributed by Thierry Arnoux, + 13-Dec-2023.) $) + nelun $p |- ( A = ( B u. C ) -> ( -. X e. A <-> ( -. X e. B /\ -. X e. C ) + ) ) $= + ( cun wceq wcel wn wo wa eleq2 elun syl6bb notbid ioran ) ABCEZFZDAGZHDBGZD + CGZIZHSHTHJQRUAQRDPGUAAPDKDBCLMNSTOM $. + + $( A class and its relative complement are disjoint. (Contributed by Thierry + Arnoux, 29-Nov-2023.) $) + disjdifr $p |- ( ( B \ A ) i^i A ) = (/) $= + ( cdif cin c0 incom disjdif eqtr3i ) ABACZDIADEAIFABGH $. + ${ $d x A $. $d x B $. $d x ph $. rabss3d.1 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> x e. B ) $. @@ -461897,6 +462654,24 @@ its graph has a given second element (that is, function value). EIKUHUIUOWIWK $. $} + $( Elementhood in a finite set of sequential integers, except its lower + bound. (Contributed by Thierry Arnoux, 1-Jan-2024.) $) + fzne1 $p |- ( ( K e. ( M ... N ) /\ K =/= M ) -> K e. ( ( M + 1 ) ... N ) ) + $= + ( wne cfz co wcel wceq wn c1 caddc df-ne wo cuz cfv elfzuz2 elfzp12 syl ibi + wb orcanai sylan2b ) ABDABCEFGZABHZIABJKFCEFGZABLUCUDUEUCUDUEMZUCCBNOGUCUFT + ABCPABCQRSUAUB $. + + $( Elementhood of an integer and its predecessor in finite intervals of + integers. (Contributed by Thierry Arnoux, 1-Jan-2024.) $) + fzm1ne1 $p |- ( ( K e. ( M ... N ) /\ K =/= M ) + -> ( K - 1 ) e. ( M ... ( N - 1 ) ) ) $= + ( cfz co wcel wne wa c1 cmin caddc fzne1 cz elfzel1 elfzel2 elfzelz 1zzd id + fzsubel biimp3a syl221anc syl adantr zcnd 1cnd pncand oveq1d eleqtrd ) ABCD + EFZABGZHZAIJEZBIKEZIJEZCIJEZDEZBUODEUKAUMCDEFZULUPFZABCLUQUMMFZCMFZAMFZIMFZ + UQURAUMCNAUMCOAUMCPUQQUQRUSUTHVAVBHUQURAIUMCSTUAUBUKUNBUODUKBIUKBUIBMFUJABC + NUCUDUKUEUFUGUH $. + $( Split the last element of a finite set of sequential integers. (more generic than ~ fzsuc ) (Contributed by Thierry Arnoux, 7-Nov-2016.) $) fzspl $p |- ( N e. ( ZZ>= ` M ) -> @@ -462073,6 +462848,24 @@ its graph has a given second element (that is, function value). BUCHIUEEGKUKSTUJULEUIUKRBCDEFHIUDEGUIUKSTUFUG $. $} + $( Elementhood in a half-open interval, except its lower bound. (Contributed + by Thierry Arnoux, 1-Jan-2024.) $) + fzone1 $p |- ( ( K e. ( M ..^ N ) /\ K =/= M ) -> K e. ( ( M + 1 ) ..^ N ) + ) $= + ( cfzo co wcel wne wa c1 caddc wo w3o cfz elfzofz adantr elfzlmr syl df-3or + wceq neneqd sylib cz elfzelz clt wbr elfzolt2 ltned unitresl simpr unitresr + zred ) ABCDEFZABGZHZABSZABIJECDEFZUNUOUPKZACSZUNUOUPURLZUQURKUNABCMEFZUSULU + TUMABCNOZABCPQUOUPURRUAUNACUNACUNAUNUTAUBFVAABCUCQUKULACUDUEUMABCUFOUGTUHUN + ABULUMUITUJ $. + + $( Elementhood in a half-open interval, except the lower bound, shifted by + one. (Contributed by Thierry Arnoux, 1-Jan-2024.) $) + fzom1ne1 $p |- ( ( K e. ( M ..^ N ) /\ K =/= M ) + -> ( K - 1 ) e. ( M ..^ ( N - 1 ) ) ) $= + ( cfzo co wcel wne wa c1 cmin caddc fzone1 fzosubel sylancl elfzoel1 adantr + cz 1z zcnd 1cnd pncand oveq1d eleqtrd ) ABCDEFZABGZHZAIJEZBIKEZIJEZCIJEZDEZ + BUJDEUFAUHCDEFIQFUGUKFABCLRAUHCIMNUFUIBUJDUFBIUFBUDBQFUEABCOPSUFTUAUBUC $. + ${ $( TODO shorten theorems using ~ iundisjcnt or ~ iundisj2cnt with ~ f1ocnt $) @@ -462339,6 +463132,18 @@ its graph has a given second element (that is, function value). WJXAWRFOWPEWKWLWMWQFOWNWO $. $} + ${ + subne0nn.1 $e |- ( ph -> M e. CC ) $. + subne0nn.2 $e |- ( ph -> N e. CC ) $. + subne0nn.3 $e |- ( ph -> ( M - N ) e. NN0 ) $. + subne0nn.4 $e |- ( ph -> M =/= N ) $. + $( A nonnegative difference is positive if the two numbers are not equal. + (Contributed by Thierry Arnoux, 17-Dec-2023.) $) + subne0nn $p |- ( ph -> ( M - N ) e. NN ) $= + ( cmin co cn0 wcel cc0 wne cn subne0d elnnne0 sylanbrc ) ABCHIZJKRLMRNKFA + BCDEGORPQ $. + $} + ${ ltesubnnd.1 $e |- ( ph -> M e. ZZ ) $. ltesubnnd.2 $e |- ( ph -> N e. NN ) $. @@ -463519,6 +464324,47 @@ its graph has a given second element (that is, function value). 2re eqtrd ) ACDZBCDZEZABFZGHIZJUQKZLZALUPUQCMDZUQNOZURUTPABCQVARUQSKZTUAVBR RVCTRULUBABUCUDCUQUEUFCUQUGUHUPUSAUNUSAPUOABCUIUJUKUM $. + $( The range of a prefix of a word is a subset of the range of that word. + Stronger version of ~ pfxrn . (Contributed by Thierry Arnoux, + 12-Dec-2023.) $) + pfxrn2 $p |- ( ( W e. Word S /\ L e. ( 0 ... ( # ` W ) ) ) + -> ran ( W prefix L ) C_ ran W ) $= + ( cword wcel cc0 chash cfv cfz co wa cpfx crn cfzo pfxres rneqd resss rnssi + cres syl6eqss ) CADEBFCGHIJEKZCBLJZMCFBNJZSZMCMUAUBUDACBOPUDCCUCQRT $. + + $( Express the range of a prefix of a word. Stronger version of ~ pfxrn2 . + (Contributed by Thierry Arnoux, 13-Dec-2023.) $) + pfxrn3 $p |- ( ( W e. Word S /\ L e. ( 0 ... ( # ` W ) ) ) + -> ran ( W prefix L ) = ( W " ( 0 ..^ L ) ) ) $= + ( cword wcel cc0 chash cfv cfz co wa cpfx crn cfzo cres pfxres rneqd df-ima + cima syl6eqr ) CADEBFCGHIJEKZCBLJZMCFBNJZOZMCUCSUAUBUDACBPQCUCRT $. + + ${ + pfxf1.1 $e |- ( ph -> W e. Word S ) $. + pfxf1.2 $e |- ( ph -> W : dom W -1-1-> S ) $. + pfxf1.3 $e |- ( ph -> L e. ( 0 ... ( # ` W ) ) ) $. + $( Condition for a prefix to be injective. (Contributed by Thierry Arnoux, + 13-Dec-2023.) $) + pfxf1 $p |- ( ph -> ( W prefix L ) : dom ( W prefix L ) -1-1-> S ) $= + ( cpfx co cdm wf1 cc0 cfzo wss wf cfv wcel wceq syl syl2anc chash cfz cuz + cres elfzuz3 fzoss2 3syl cword wrddm sseqtr4d wrdf fssresd f1resf1 pfxres + syl3anc wfn pfxfn fndmd eqidd f1eq123d mpbird ) ADCHIZJZBVBKLCMIZBDVDUDZK + ZADJZBDKVDVGNVDBVEOVFFAVDLDUAPZMIZVGACLVHUBIQZVHCUCPQVDVINGCLVHUECLVHUFUG + ZADBUHQZVGVIREBDUISUJAVIBVDDAVLVIBDOEBDUKSVKULVGBVDBDUMUOAVCVDBBVBVEAVLVJ + VBVEREGBDCUNTAVDVBAVLVJVBVDUPEGDCBUQTURABUSUTVA $. + $} + + ${ + s1f1.1 $e |- ( ph -> I e. D ) $. + $( Conditions for a length 1 string to be a one-to-one function. + (Contributed by Thierry Arnoux, 11-Dec-2023.) $) + s1f1 $p |- ( ph -> <" I "> : dom <" I "> -1-1-> D ) $= + ( cs1 cdm wf1 cc0 csn cop wss wf1o cn0 wcel 0nn0 a1i f1osng syl2anc wceq + syl f1of1 snssd f1ss s1val s1dm eqidd f1eq123d mpbird ) ACEZFZBUIGHIZBHCJ + IZGZAUKCIZULGZUNBKUMAUKUNULLZUOAHMNZCBNZUPUQAOPDHCMBQRUKUNULUATACBDUBUKUN + BULUCRAUJUKBBUIULAURUIULSDCBUDTUJUKSACUEPABUFUGUH $. + $} + ${ s2rn.i $e |- ( ph -> I e. D ) $. s2rn.j $e |- ( ph -> J e. D ) $. @@ -463611,6 +464457,59 @@ its graph has a given second element (that is, function value). JZUJELGMZUJKNABCOPABCSKQRHTZEUAEKUBUKULUMUCUDUEKUNEUFKGUJEUGUHUI $. $} + ${ + $d A i j $. $d B i j $. $d i j ph $. + ccatf1.s $e |- ( ph -> S e. V ) $. + ccatf1.a $e |- ( ph -> A e. Word S ) $. + ccatf1.b $e |- ( ph -> B e. Word S ) $. + ccatf1.1 $e |- ( ph -> A : dom A -1-1-> S ) $. + ccatf1.2 $e |- ( ph -> B : dom B -1-1-> S ) $. + ccatf1.3 $e |- ( ph -> ( ran A i^i ran B ) = (/) ) $. + $( Conditions for a concatenation to be injective. (Contributed by Thierry + Arnoux, 11-Dec-2023.) $) + ccatf1 $p |- ( ph -> ( A ++ B ) : dom ( A ++ B ) -1-1-> S ) $= + ( co cfv wceq wcel syl2anc syl wa ad5antr syl3anc vi vj cconcat cdm wf cv + wi wral wf1 cc0 chash cfzo cword ccatcl wrdf ffdmd simpllr ccatval1 simpr + simplr 3eqtr3d wrddm f1eq2 biimpa dff13 simprbi ad3antrrr r19.21bi mpd c0 + crn wfun f1fun eleqtrrd fvelrn cmin caddc ccatlen oveq2d eleqtrd ccatval2 + cin cz cn0 lencl nn0zd fzosubel3 eqeltrd elind wn noel pm2.21dd wo eleq2d + a1i adantr fzospliti ad5ant13 mpjaodan 3eqtr3rd fzossz sseldi zcnd nn0cnd + cc f1veqaeq anassrs imp syl1111anc subcan2d ad2antrr ex anasss ralrimivva + sylanbrc ) ABCUCLZUDZDXPUEUAUFZXPMZUBUFZXPMZNZXRXTNZUGZUBXQUHUAXQUHXQDXPU + IAUJXPUKMZULLZDXPAXPDUMZOZYFDXPUEABYGOZCYGOZYHGHDBCUNPZDXPUOQUPAYDUAUBXQX + QAXRXQOZXTXQOZYDAYLRZYMRZYBYCYOYBRZXRUJBUKMZULLZOZYCXRYQYEULLZOZYPYSRZXTY + ROZYCXTYTOZUUBUUCRZXRBMZXTBMZNZYCUUEXSYAUUFUUGYOYBYSUUCUQUUEYIYJYSXSUUFNZ + AYIYLYMYBYSUUCGSZAYJYLYMYBYSUUCHSZYPYSUUCUTDBCXRURZTUUEYIYJUUCYAUUGNZUUJU + UKUUBUUCUSDBCXTURZTVAUUBUUHYCUGZUBYRYPUUOUBYRUHZUAYRAUUPUAYRUHZYLYMYBAYRD + BUIZUUQABUDZYRNZUUSDBUIZUURAYIUUTGDBVBZQZIUUTUVAUURUUSYRDBVCVDPUURYRDBUEU + UQUAUBYRDBVEVFQVGVHVHVIUUBUUDRZUUFVJOZYCUVDUUFBVKZCVKZWBZVJUVDUVFUVGUUFUV + DBVLZXRUUSOUUFUVFOAUVIYLYMYBYSUUDAUVAUVIIUUSDBVMQZSUVDXRYRUUSYPYSUUDUTZUV + DYIUUTAYIYLYMYBYSUUDGSZUVBQVNXRBVOPUVDUUFXTYQVPLZCMZUVGUVDXSYAUUFUVNYOYBY + SUUDUQUVDYIYJYSUUIUVLAYJYLYMYBYSUUDHSZUVKUULTUVDYIYJXTYQYQCUKMZVQLZULLZOZ + YAUVNNZUVLUVOUVDXTYTUVRUUBUUDUSAYTUVRNZYLYMYBYSUUDAYEUVQYQULAYIYJYEUVQNGH + DBCVRPVSZSVTZDBCXTWAZTVAUVDCVLZUVMCUDZOZUVNUVGOAUWEYLYMYBYSUUDAUWFDCUIZUW + EJUWFDCVMQZSUVDUVMUJUVPULLZUWFUVDUVSUVPWCOZUVMUWJOZUWCAUWKYLYMYBYSUUDAUVP + AYJUVPWDOHDCWEQWFZSXTYQUVPWGZPUVDYJUWFUWJNZUVODCVBZQVNUVMCVOPWHWIAUVHVJNZ + YLYMYBYSUUDKSVTUVEWJUVDUUFWKWOWLAYMUUCUUDWMZYLYBYSAYMRXTYFOZYQWCOZUWRAYMU + WSAXQYFXTAYHXQYFNYKDXPVBQZWNVDAUWTYMAYQAYIYQWDOGDBWEQZWFZWPXTUJYEYQWQPZWR + WSYPUUARZUUCYCUUDUXEUUCRZUUGVJOZYCUXFUUGUVHVJUXFUVFUVGUUGUXFUVIXTUUSOUUGU + VFOAUVIYLYMYBUUAUUCUVJSUXFXTYRUUSUXEUUCUSZAUUTYLYMYBUUAUUCUVCSVNXTBVOPUXF + UUGXRYQVPLZCMZUVGUXFXSYAUXJUUGYOYBUUAUUCUQUXFYIYJXRUVROZXSUXJNZAYIYLYMYBU + UAUUCGSZAYJYLYMYBUUAUUCHSZUXFXRYTUVRYPUUAUUCUTAUWAYLYMYBUUAUUCUWBSVTZDBCX + RWAZTUXFYIYJUUCUUMUXMUXNUXHUUNTWTUXFUWEUXIUWFOZUXJUVGOAUWEYLYMYBUUAUUCUWI + SUXFUXIUWJUWFUXFUXKUWKUXIUWJOZUXOAUWKYLYMYBUUAUUCUWMSXRYQUVPWGZPAUWOYLYMY + BUUAUUCAYJUWOHUWPQZSVNUXICVOPWHWIAUWQYLYMYBUUAUUCKSVTUXGWJUXFUUGWKWOWLUXE + UUDRZXRXTYQUYAXRUYAYTWCXRYQYEXAZYPUUAUUDUTZXBXCUYAXTUYAYTWCXTUYBUXEUUDUSZ + XBXCAYQXEOYLYMYBUUAUUDAYQUXBXDSUYAUWHUXQUWGUXJUVNNZUXIUVMNZAUWHYLYMYBUUAU + UDJSUYAUXIUWJUWFUYAUXKUWKUXRUYAXRYTUVRUYCAUWAYLYMYBUUAUUDUWBSZVTZAUWKYLYM + YBUUAUUDUWMSZUXSPAUWOYLYMYBUUAUUDUXTSZVNUYAUVMUWJUWFUYAUVSUWKUWLUYAXTYTUV + RUYDUYGVTZUYIUWNPUYJVNUYAXSYAUXJUVNYOYBUUAUUDUQUYAYIYJUXKUXLAYIYLYMYBUUAU + UDGSZAYJYLYMYBUUAUUDHSZUYHUXPTUYAYIYJUVSUVTUYLUYMUYKUWDTVAUWHUXQRUWGRUYEU + YFUWHUXQUWGUYEUYFUGUWFDUXIUVMCXFXGXHXIXJAYMUWRYLYBUUAUXDWRWSYNYSUUAWMZYMY + BYNXRYFOZUWTUYNAYLUYOAXQYFXRUXAWNVDAUWTYLUXCWPXRUJYEYQWQPXKWSXLXMXNUAUBXQ + DXPVEXO $. + $} + ${ pfxlsw2ccat.n $e |- N = ( # ` W ) $. $( Reconstruct a word from its prefix and its last two symbols. @@ -463733,6 +464632,129 @@ its graph has a given second element (that is, function value). KYTUWIXRUWKUWLUWRUWSULUEZUWTUXCUAULUWMZIHVSVXFUWTVXGUAUWSUWNYBUWOUWP $. $} + ${ + $d M x y $. $d N x y $. $d V x y $. $d W x y $. + $( The range of a subword is a subset of the range of that word. Stronger + version of ~ swrdrn . (Contributed by Thierry Arnoux, 12-Dec-2023.) $) + swrdrn2 $p |- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) + ) ) -> ran ( W substr <. M , N >. ) C_ ran W ) $= + ( vx wcel cc0 cfz co cfv crn cfzo wss cuz adantr syl fzssz sseldi sseldd + cz cword chash w3a csubstr cmin cv caddc cmpt swrdval2 rneqd wral wa wfun + cop cdm eqidd simpl1 wrdfd elfzuz3 3ad2ant3 fzoss2 elfzuz 3ad2ant2 fzoss1 + ffund simpr simpl3 simpl2 fzoaddel2 syl3anc wceq 3ad2ant1 eleqtrrd fvelrn + wrddm syl2anc ralrimiva eqid rnmptss eqsstrd ) DCUAFZAGBHIZFZBGDUBJZHIZFZ + UCZDABUNUDIZKEGBAUEILIZEUFZAUGIZDJZUHZKZDKZWGWHWMECDABUIUJWGWLWOFZEWIUKWN + WOMWGWPEWIWGWJWIFZULZDUMWKDUOZFWPWRGWDLIZCDWRCWDDWRWDUPWAWCWFWQUQURVEWRWK + WTWSWRGBLIZWTWKWRWDBNJFZXAWTMWGXBWQWFWAXBWCBGWDUSUTOBGWDVAPWRABLIZXAWKWRA + GNJFZXCXAMWGXDWQWCWAXDWFAGBVBVCOAGBVDPWRWQBTFATFWKXCFWGWQVFWRWETBGWDQWAWC + WFWQVGRWRWBTAGBQWAWCWFWQVHRWJBAVIVJSSWGWSWTVKZWQWAWCXEWFCDVOVLOVMWKDVNVPV + QEWIWLWOWMWMVRVSPVT $. + $} + + ${ + $d M i j y $. $d N i j y $. $d V i j y $. $d W i j y $. + $( Express the range of a subword. Stronger version of ~ swrdrn2 . + (Contributed by Thierry Arnoux, 13-Dec-2023.) $) + swrdrn3 $p |- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) + ) ) -> ran ( W substr <. M , N >. ) = ( W " ( M ..^ N ) ) ) $= + ( vy vj vi wcel cc0 co cfv cfzo cv wceq caddc wa cz simpr sseldi zcnd cfz + cword chash w3a csubstr cima wrex cmin cmpt fzssz simpl3 simpl2 fzoaddel2 + cop syl3anc pncan3d oveq2d zsubcld fzosubel3 syl2anc oveq1d eqeq2d fzossz + crn eleqtrrd npcand eqcomd rspcedvd eqcom fveq2d syl5bbr rexxfrd elrnmpti + eqid fvex syl6bbr wf wrdf 3ad2ant1 cuz wss elfzuz 3ad2ant2 fzoss1 elfzuz3 + ffnd 3ad2ant3 fzoss2 sstrd fvelimabd swrdval2 rneqd eleq2d 3bitr4rd eqrdv + syl ) DCUBHZAIBUAJZHZBIDUCKZUAJZHZUDZEDABUNUEJZVDZDABLJZUFZXCFMZDKZEMZNZF + XFUGZXJGIBAUHJZLJZGMZAOJZDKZUIZVDZHZXJXGHXJXEHXCXLXJXQNZGXNUGXTXCXKYAFGXP + XFXNXCXOXNHZPZYBBQHAQHXPXFHXCYBRYCXAQBIWTUJZWQWSXBYBUKSYCWRQAIBUJZWQWSXBY + BULSXOBAUMUOXCXHXFHZPZXHXPNZXHXHAUHJZAOJZNGYIXNYGXHAAXMOJZLJZHXMQHYIXNHYG + XHXFYLXCYFRZYGYKBALYGABYGAYGWRQAYEWQWSXBYFULSZTZYGBYGXAQBYDWQWSXBYFUKSZTU + PUQVEYGBAYPYNURXHAXMUSUTYGXOYINZPZXPYJXHYRXOYIAOYGYQRVAVBYGYJXHYGXHAYGXHY + GXFQXHABVCYMSTYOVFVGVHXKXJXINXCYHPZYAXJXIVIYSXIXQXJYSXHXPDXCYHRVJVBVKVLGX + NXQXJXRXRVNXPDVOVMVPXCFIWTLJZXFXJDXCYTCDWQWSYTCDVQXBCDVRVSWFXCXFIBLJZYTXC + AIVTKHZXFUUAWAWSWQUUBXBAIBWBWCAIBWDWPXCWTBVTKHZUUAYTWAXBWQUUCWSBIWTWEWGBI + WTWHWPWIWJXCXEXSXJXCXDXRGCDABWKWLWMWNWO $. + $} + + ${ + $d D x $. $d M i j $. $d M x $. $d N i j $. $d N x $. $d W i j $. + $d W x $. $d i j ph $. + swrdf1.w $e |- ( ph -> W e. Word D ) $. + swrdf1.m $e |- ( ph -> M e. ( 0 ... N ) ) $. + swrdf1.n $e |- ( ph -> N e. ( 0 ... ( # ` W ) ) ) $. + swrdf1.1 $e |- ( ph -> W : dom W -1-1-> D ) $. + $( Condition for a subword to be injective. (Contributed by Thierry + Arnoux, 12-Dec-2023.) $) + swrdf1 $p |- ( ph + -> ( W substr <. M , N >. ) : dom ( W substr <. M , N >. ) -1-1-> D ) $= + ( vi vj co cfv wceq cc0 cfzo wcel wa cz ad3antrrr cop csubstr cdm wf wral + cv wi wf1 cmin cword cfz chash swrdf syl3anc ffdmd fzossz simpllr eleqtrd + fdmd sseldi zcnd simplr fzssz caddc wss elfzuz fzoss1 3syl elfzuz3 fzoss2 + cuz sstrd elfzelz fzoaddel2 sseldd wrddm eleqtrrd swrdfv syl31anc 3eqtr3d + simpr f1veqaeq anassrs syl1111anc addcan2ad ex anasss ralrimivva sylanbrc + syl imp dff13 ) AECDUAUBLZUCZBWMUDJUFZWMMZKUFZWMMZNZWOWQNZUGZKWNUEJWNUEWN + BWMUHAODCUILZPLZBWMAEBUJQZCODUKLZQZDOEULMZUKLQZXCBWMUDFGHCDBEUMUNZUOAXAJK + WNWNAWOWNQZWQWNQZXAAXJRZXKRZWSWTXMWSRZWOWQCXNWOXNXCSWOOXBUPZXNWOWNXCAXJXK + WSUQAWNXCNXJXKWSAXCBWMXIUSTZURZUTVAXNWQXNXCSWQXOXNWQWNXCXLXKWSVBXPURZUTVA + XNCACSQZXJXKWSAXESCODVCGUTTZVAXNEUCZBEUHZWOCVDLZYAQZWQCVDLZYAQZYCEMZYEEMZ + NZYCYENZAYBXJXKWSITXNYCOXGPLZYAXNCDPLZYKYCAYLYKVEXJXKWSAYLODPLZYKAXFCOVKM + QYLYMVEGCODVFCODVGVHAXHXGDVKMQYMYKVEHDOXGVIDOXGVJVHVLTZXNWOXCQZDSQZXSYCYL + QXQAYPXJXKWSAXHYPHDOXGVMWJTZXTWODCVNUNVOAYAYKNZXJXKWSAXDYRFBEVPWJTZVQXNYE + YKYAXNYLYKYEYNXNWQXCQZYPXSYEYLQXRYQXTWQDCVNUNVOYSVQXNWPWRYGYHXMWSWAXNXDXF + XHYOWPYGNAXDXJXKWSFTZAXFXJXKWSGTZAXHXJXKWSHTZXQBECDWOVRVSXNXDXFXHYTWRYHNU + UAUUBUUCXRBECDWQVRVSVTYBYDRYFRYIYJYBYDYFYIYJUGYABYCYEEWBWCWKWDWEWFWGWHJKW + NBWMWLWI $. + + $d M x y $. $d N x y $. $d O x y $. $d P x y $. $d W x y $. + $d ph x y $. + swrdrndisj.1 $e |- ( ph -> O e. ( N ... P ) ) $. + swrdrndisj.2 $e |- ( ph -> P e. ( N ... ( # ` W ) ) ) $. + $( Condition for the range of two subwords of an injective word to be + disjoint. (Contributed by Thierry Arnoux, 13-Dec-2023.) $) + swrdrndisj $p |- ( ph -> ( ran ( W substr <. M , N >. ) + i^i ran ( W substr <. O , P >. ) ) = (/) ) $= + ( co c0 wcel cc0 cfz wss 3syl cop csubstr crn cin cfzo cima cword swrdrn3 + chash cfv wceq syl3anc cuz elfzuz fzss1 sseldd ineq12d cdm wf1 ccnv df-f1 + wfun simprbi imain fzoss1 elfzuz3 fzoss2 sstrd sslin syl fzodisj syl6sseq + wf ss0 imaeq2d ima0 syl6eq 3eqtr2d ) AGDEUAUBNUCZGFCUAUBNUCZUDGDEUENZUFZG + FCUENZUFZUDZGWAWCUDZUFZOAVSWBVTWDAGBUGPZDQERNPEQGUIUJZRNZPZVSWBUKHIJDEBGU + HULAWHFQCRNZPCWJPVTWDUKHAECRNZWLFAWKEQUMUJPZWMWLSJEQWIUNZEQCUOTLUPAEWIRNZ + WJCAWKWNWPWJSJWOEQWIUOTMUPFCBGUHULUQAGURZBGUSZGUTVBZWGWEUKKWRWQBGVMWSWQBG + VAVCWAWCGVDTAWGGOUFOAWFOGAWFOSWFOUKAWFWAEWIUENZUDZOAWCWTSWFXASAWCECUENZWT + AFWMPFEUMUJPWCXBSLFECUNFECVETACWPPWICUMUJPXBWTSMCEWIVFCEWIVGTVHWCWTWAVIVJ + DEWIVKVLWFVNVJVOGVPVQVR $. + $} + + +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Splicing words (substring replacement) +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + ${ + splfv3.s $e |- ( ph -> S e. Word A ) $. + splfv3.f $e |- ( ph -> F e. ( 0 ... T ) ) $. + splfv3.t $e |- ( ph -> T e. ( 0 ... ( # ` S ) ) ) $. + splfv3.r $e |- ( ph -> R e. Word A ) $. + splfv3.x $e |- ( ph -> X e. ( 0 ..^ ( ( # ` S ) - T ) ) ) $. + splfv3.k $e |- ( ph -> K = ( F + ( # ` R ) ) ) $. + $( Symbols to the right of a splice are unaffected. (Contributed by + Thierry Arnoux, 14-Dec-2023.) $) + splfv3 $p |- ( ph -> ( ( S splice <. F , T , R >. ) ` ( X + K ) ) + = ( S ` ( X + T ) ) ) $= + ( caddc co cfv wcel cc0 wceq cotp csplice cconcat chash cop csubstr cword + cpfx cfz splval syl13anc cuz wss elfzuz3 fzss2 3syl sseldd pfxlen syl2anc + oveq1d pfxcl syl ccatlen 3eqtr4rd oveq2d fveq12d cfzo ccatcl swrdcl lencl + cn0 nn0fz0 sylib swrdlen syl3anc eleqtrrd ccatval3 swrdfv syl31anc 3eqtrd + cmin ) AHGOPZDFECUAUBPZQHDFUHPZCUCPZUDQZOPZWEDEDUDQZUEUFPZUCPZQZHWIQZHEOP + DQZAWBWGWCWJADBUGZRZFSEUIPZRESWHUIPZRZCWNRZWCWJTIJKLCDEFWNWPWQWNUJUKAGWFH + OAWDUDQZCUDQZOPZFXAOPWFGAWTFXAOAWOFWQRWTFTIAWPWQFAWRWHEULQRWPWQUMKESWHUNE + SWHUOUPJUQBDFURUSUTAWDWNRZWSWFXBTAWOXCIBDFVAVBZLBWDCVCUSNVDVEVFAWEWNRZWIW + NRZHSWIUDQZVGPZRWKWLTAXCWSXEXDLBWDCVHUSAWOXFIBDEWHVIVBAHSWHEWAPZVGPZXHMAX + GXISVGAWOWRWHWQRZXGXITIKAWOXKIWOWHVKRXKBDVJWHVLVMVBZBDEWHVNVOVEVPBWEWIHVQ + VOAWOWRXKHXJRWLWMTIKXLMBDEWHHVRVSVT $. + $} + $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Cyclic shift of words @@ -464888,30 +465910,6 @@ real number multiplication operation (this has to be defined in the main ULUQAUOBULCEUTGUAAUOBULCEUTGUBUEUFUGUHUI $. $} - - ${ - $d M x y $. $d Z x y $. - cntrcmnd.z $e |- Z = ( M |`s ( Cntr ` M ) ) $. - $( The center of a monoid is a commutative submonoid. (Contributed by - Thierry Arnoux, 21-Aug-2023.) $) - cntrcmnd $p |- ( M e. Mnd -> Z e. CMnd ) $= - ( vx vy cmnd wcel ccntr cfv cplusg cbs wss wceq eqid cntrss ressbas2 mp1i - cvv cv co fvex ressplusg csubmnd ccntz cntrval cntzsubm eqeltrrid submmnd - ssid mpan2 syl w3a simp2 simp3 sseldi cntri syl2anc iscmnd ) AFGZDEAHIZAJ - IZBUTAKIZLUTBKIMUSVBAVBNZOZUTVBBACVCPQUTRGVABJIMUSAHUAUTVAABRCVANZUBQUSUT - AUCIZGBFGUSUTVBAUDIZIZVFVBAVGVCVGNZUEUSVBVBLVHVFGVBUIVBVBAVGVCVIUFUJUGUTB - ACUHUKUSDSZUTGZESZUTGZULZVKVLVBGVJVLVATVLVJVATMUSVKVMUMVNUTVBVLVDUSVKVMUN - UOVBVAAVJVLUTVCVEUTNUPUQUR $. - - $( The center of a group is an abelian group. (Contributed by Thierry - Arnoux, 21-Aug-2023.) $) - cntrabl $p |- ( M e. Grp -> Z e. Abel ) $= - ( cgrp wcel ccmn cabl ccntr cfv csubg cbs ccntz eqid cntrval wss cntzsubg - ssid mpan2 eqeltrrid syl subggrp cmnd grpmnd cntrcmnd isabl sylanbrc ) AD - EZBDEZBFEZBGEUGAHIZAJIZEUHUGUJAKIZALIZIZUKULAUMULMZUMMZNUGULULOUNUKEULQUL - ULAUMUOUPPRSUJABCUATUGAUBEUIAUCABCUDTBUEUF $. - $} - ${ cntrcrng.z $e |- Z = ( R |`s ( Cntr ` ( mulGrp ` R ) ) ) $. $( The center of a ring is a commutative ring. (Contributed by Thierry @@ -464928,7 +465926,7 @@ real number multiplication operation (this has to be defined in the main $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- - Ordered monoids and groups + Totally ordered monoids and groups -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) @@ -466284,6 +467282,469 @@ C_ dom ( ( ( T ` { I , J } ) o. F ) \ _I ) ) $= XALJBUUBGKUUFWKVCWOWLWMXAUUIUUJWPWTKXQXECUUGDUUGWCWNWQWRWS $. $} + ${ + $d D i $. $d D w $. $d I i $. $d J i $. $d M i $. $d U i $. $d W i $. + $d W w $. $d i ph $. + cycpmco2.c $e |- M = ( toCyc ` D ) $. + cycpmco2.s $e |- S = ( SymGrp ` D ) $. + cycpmco2.d $e |- ( ph -> D e. V ) $. + cycpmco2.w $e |- ( ph -> W e. dom M ) $. + cycpmco2.i $e |- ( ph -> I e. ( D \ ran W ) ) $. + cycpmco2.j $e |- ( ph -> J e. ran W ) $. + cycpmco2.e $e |- E = ( ( `' W ` J ) + 1 ) $. + cycpmco2.1 $e |- U = ( W splice <. E , E , <" I "> >. ) $. + $( The word U used in ~ cycpmco2 is injective, so it can represent a cycle + and form a cyclic permutation ` ( M `` U ) ` . (Contributed by Thierry + Arnoux, 4-Jan-2024.) $) + cycpmco2f1 $p |- ( ph -> U : dom U -1-1-> D ) $= + ( wcel c0 vw cdm wf1 cpfx co cs1 cconcat chash cfv cop csubstr cword crab + cv ssrab2 cbs eqid tocycf syl fdmd eleqtrd sseldi pfxcl crn eldifad s1cld + wf ccatcl syl2anc swrdcl wa wceq id dmeq eqidd f1eq123d elrab simprd ccnv + sylib caddc cc0 cfzo wf1o f1cnv f1of 3syl ffvelrnd wrddm fzofzp1 eqeltrid + c1 cfz pfxf1 s1f1 cin csn s1rn ineq2d wss pfxrn2 ssrind wn eldifbd disjsn + sylibr sseqtrd ss0 eqtrd ccatf1 cn0 lencl nn0fz0 biimpi swrdf1 cun ccatrn + ineq1d indir syl6eq pfxval rneqd 0elfz elfzuz3 eluzfz1 eluzfz2 swrdrndisj + fz0ssnn0 cuz incom swrdrn2 syl3anc syl5eq uneq12d a1i 3eqtrd cotp csplice + unidm cvv ovexd splval syl13anc dmeqd mpbird ) ADUBZBDUCJEUDUEZFUFZUGUEZJ + EJUHUIZUJUKUEZUGUEZUBZBUULUCAUUIUUKBIMAUUGBULZSZUUHUUNSZUUIUUNSAJUUNSZUUO + AUAUNZUBZBUURUCZUAUUNUMZUUNJUUTUAUUNUOAJHUBZUVANAUVACUPUIZHABISUVAUVCHVGM + UAUVCHBCIKLUVCUQURUSUTVAZVBZBJEVCUSZAFBAFBJVDZOVEZVFZBUUGUUHVHVIAUUQUUKUU + NSUVEBJEUUJVJUSAUUGUUHBIMUVFUVIABEJUVEAUUQJUBZBJUCZAJUVASUUQUVKVKUVDUUTUV + KUAJUUNUURJVLZUUSUVJBBUURJUVLVMUURJVNUVLBVOVPVQVTVRZAEGJVSZUIZWLWAUEZWBUU + JWMUEZQAUVOWBUUJWCUEZSUVPUVQSAUVOUVJUVRAUVGUVJGUVNAUVKUVGUVJUVNWDUVGUVJUV + NVGUVMUVJBJWEUVGUVJUVNWFWGPWHAUUQUVJUVRVLUVEBJWIUSVAWBUUJUVOWJUSWKZWNABFU + VHWOAUUGVDZUUHVDZWPUVTFWQZWPZTAUWAUWBUVTAFBSUWAUWBVLUVHFBWRUSZWSAUWCTWTUW + CTVLAUWCUVGUWBWPZTAUVTUVGUWBAUUQEUVQSZUVTUVGWTUVEUVSBEJXAVIXBAFUVGSXCUWET + VLAFBUVGOXDUVGFXEXFZXGUWCXHUSXIXJABEUUJJUVEUVSAUUQUUJXKSZUUJUVQSZUVEBJXLU + WHUWIUUJXMXNWGZUVMXOAUUIVDZUUKVDZWPZUVTUWLWPZUWAUWLWPZXPZTTXPZTAUWMUVTUWA + XPZUWLWPUWPAUWKUWRUWLAUUOUUPUWKUWRVLUVFUVIBUUGUUHXQVIXRUVTUWAUWLXSXTAUWNT + UWOTAUWNJWBEUJUKUEZVDZUWLWPTAUVTUWTUWLAUUGUWSAUUQEXKSZUUGUWSVLUVEAUVQXKEU + UJYHUVSVBZJEUUNYAVIYBXRABUUJWBEEJUVEAUXAWBWBEWMUESUXBEYCUSUVSUVMAUWFUUJEY + IUISZEEUUJWMUEZSUVSEWBUUJYDZEUUJYEWGAUWFUXCUUJUXDSUVSUXEEUUJYFWGYGXIAUWOU + WLUWAWPZTUWAUWLYJAUXFUWLUWBWPZTAUWAUWBUWLUWDWSAUXGTWTUXGTVLAUXGUWETAUWLUV + GUWBAUUQUWFUWIUWLUVGWTUVEUVSUWJEUUJBJYKYLXBUWGXGUXGXHUSXIYMYNUWQTVLATYSYO + YPXJAUUFUUMBBDUULADJEEUUHYQYRUEZUULRAJUVBSEYTSZUXIUUPUXHUULVLNAEUVPYTQAUV + OWLWAUUAWKZUXJUVIUUHJEEUVBYTYTUUNUUBUUCYMZADUULUXKUUDABVOVPUUE $. + + $( The orbit of the composition of a cyclic permutation and a well-chosen + transposition is one element more than the orbit of the original + permutation. (Contributed by Thierry Arnoux, 4-Jan-2024.) $) + cycpmco2rn $p |- ( ph -> ran U = ( ran W u. { I } ) ) $= + ( wcel wceq vw cc0 cfzo co cima csn cun cfv crn un23 cpfx cs1 cconcat cop + chash csubstr cotp csplice cdm cword ccnv c1 caddc ovexd eqeltrid eldifad + cvv s1cld splval syl13anc syl5eq rneqd cv wf1 crab ssrab2 cbs eqid tocycf + wf syl fdmd eleqtrd sseldi pfxcl ccatcl syl2anc swrdcl ccatrn cfz wf1o wa + id dmeq eqidd f1eq123d elrab sylib f1cnv f1of 3syl ffvelrnd wrddm fzofzp1 + simprd pfxrn3 s1rn uneq12d eqtrd cn0 nn0fz0 biimpi swrdrn3 syl3anc 3eqtrd + lencl fzosplit imaeq2d wfn wrdf ffnd fnima wss cuz elfzuz3 fz0ssnn0 nn0uz + fzoss2 syl6eleq fzoss1 unima 3eqtr3d uneq1d 3eqtr4a ) AJUBEUCUDZUEZFUFZUG + ZJEJUOUHZUCUDZUEZUGZYPUUAUGZYQUGDUIZJUIZYQUGYPYQUUAUJAUUDJEUKUDZFULZUMUDZ + JEYSUNUPUDZUMUDZUIZUUHUIZUUIUIZUGZUUBADUUJADJEEUUGUQURUDZUUJRAJHUSZSEVGSZ + UUQUUGBUTZSZUUOUUJTNAEGJVAZUHZVBVCUDZVGQAUVAVBVCVDVEZUVCAFBAFBUUEOVFZVHZU + UGJEEUUPVGVGUURVIVJVKVLAUUHUURSZUUIUURSZUUKUUNTAUUFUURSZUUSUVFAJUURSZUVHA + UAVMZUSZBUVJVNZUAUURVOZUURJUVLUAUURVPAJUUPUVMNAUVMCVQUHZHABISUVMUVNHVTMUA + UVNHBCIKLUVNVRVSWAWBWCZWDZBJEWEWAZUVEBUUFUUGWFWGAUVIUVGUVPBJEYSWHWABUUHUU + IWIWGAUULYRUUMUUAAUULUUFUIZUUGUIZUGZYRAUVHUUSUULUVTTUVQUVEBUUFUUGWIWGAUVR + YPUVSYQAUVIEUBYSWJUDZSZUVRYPTUVPAEUVBUWAQAUVAUBYSUCUDZSUVBUWASAUVAJUSZUWC + AUUEUWDGUUTAUWDBJVNZUUEUWDUUTWKUUEUWDUUTVTAUVIUWEAJUVMSUVIUWEWLUVOUVLUWEU + AJUURUVJJTZUVKUWDBBUVJJUWFWMUVJJWNUWFBWOWPWQWRXEUWDBJWSUUEUWDUUTWTXAPXBAU + VIUWDUWCTUVPBJXCWAWCUBYSUVAXDWAVEZBEJXFWGAFBSUVSYQTUVDFBXGWAXHXIAUVIUWBYS + UWASZUUMUUATUVPUWGAUVIYSXJSZUWHUVPBJXPUWIUWHYSXKXLXAEYSBJXMXNXHXOAUUEUUCY + QAJUWCUEZJYOYTUGZUEZUUEUUCAUWCUWKJAUWBUWCUWKTUWGUBYSEXQWAXRAJUWCXSZUWJUUE + TAUWCBJAUVIUWCBJVTUVPBJXTWAYAZUWCJYBWAAUWMYOUWCYCZYTUWCYCZUWLUUCTUWNAUWBY + SEYDUHSUWOUWGEUBYSYEEUBYSYHXAAEUBYDUHZSUWPAEXJUWQAUWAXJEYSYFUWGWDYGYIEUBY + SYJWAUWCYOYTJYKXNYLYMYN $. + + $( Lemma for ~ cycpmco2 . (Contributed by Thierry Arnoux, 4-Jan-2024.) $) + cycpmco2lem1 $p |- ( ph -> ( ( M ` W ) ` ( ( M ` <" I J "> ) ` I ) ) = ( ( + M ` W ) ` J ) ) $= + ( vw wcel cs2 cfv crn eldifad cdm wf1 wf cword cv crab ssrab2 eqid tocycf + cbs fdmd eleqtrd sseldi wceq id dmeq eqidd f1eq123d elrab3 biimpa syl2anc + syl f1f frnd sseldd wn wne eldifbd nelne2 necomd cyc2fv1 fveq2d ) AFFGUAH + UBUBGJHUBAHBCFGIKMAFBJUCZOUDAVQBGAJUEZBJAVRBJUFZVRBJUGAJBUHZTZJSUIZUEZBWB + UFZSVTUJZTZVSAWEVTJWDSVTUKAJHUEWENAWECUNUBZHABITWEWGHUGMSWGHBCIKLWGULUMVF + UOUPZUQWHWAWFVSWDVSSJVTWBJURZWCVRBBWBJWIUSWBJUTWIBVAVBVCVDVEVRBJVGVFVHPVI + AGFAGVQTFVQTVJGFVKPAFBVQOVLGFVQVMVEVNLVOVP $. + + $( Lemma for ~ cycpmco2 . (Contributed by Thierry Arnoux, 4-Jan-2024.) $) + cycpmco2lem2 $p |- ( ph -> ( U ` E ) = I ) $= + ( co wcel vw cfv cpfx chash cmin cs1 cc0 cconcat cop csubstr cotp csplice + cdm cvv cword wceq ccnv c1 caddc ovexd eqeltrid crn s1cld splval syl13anc + eldifad syl5eq fveq1d cfzo cv wf1 crab ssrab2 cbs wf eqid tocycf syl fdmd + eleqtrd sseldi pfxcl ccatcl syl2anc swrdcl cn0 cfz fz0ssnn0 wf1o wa eqidd + id dmeq f1eq123d elrab sylib simprd f1cnv ffvelrnd wrddm fzofzp1 fzonn0p1 + f1of 3syl ccatws1len pfxlen oveq1d eqtrd oveq2d eleqtrrd ccatval1 syl3anc + cz nn0zd elfzomin s1len oveq12d ccatval2 3eqtrd nn0cnd subidd fveq2d cdif + a1i s1fv ) AEDUBZEJEUCSZUDUBZUESZFUFZUBZUGYJUBZFAYFEYGYJUHSZJEJUDUBZUIUJS + ZUHSZUBZEYMUBZYKAEDYPADJEEYJUKULSZYPRAJHUMZTEUNTZUUAYJBUOZTZYSYPUPNAEGJUQ + ZUBZURUSSZUNQAUUEURUSUTVAZUUGAFBAFBJVBZOVFVCZYJJEEYTUNUNUUBVDVEVGVHAYMUUB + TZYOUUBTZEUGYMUDUBZVISZTYQYRUPAYGUUBTZUUCUUJAJUUBTZUUNAUAVJZUMZBUUPVKZUAU + UBVLZUUBJUURUAUUBVMAJYTUUSNAUUSCVNUBZHABITUUSUUTHVOMUAUUTHBCIKLUUTVPVQVRV + SVTZWAZBJEWBVRZUUIBYGYJWCWDAUUOUUKUVBBJEYNWEVRAEUGEURUSSZVISZUUMAEWFTEUVE + TAUGYNWGSZWFEYNWHAEUUFUVFQAUUEUGYNVISZTUUFUVFTAUUEJUMZUVGAUUHUVHGUUDAUVHB + JVKZUUHUVHUUDWIUUHUVHUUDVOAUUOUVIAJUUSTUUOUVIWJUVAUURUVIUAJUUBUUPJUPZUUQU + VHBBUUPJUVJWLUUPJWMUVJBWKWNWOWPWQUVHBJWRUUHUVHUUDXCXDPWSAUUOUVHUVGUPUVBBJ + WTVRVTUGYNUUEXAVRVAZWAZEXBVRAUULUVDUGVIAUULYHURUSSZUVDAUUNUULUVMUPUVCBYGF + XEVRAYHEURUSAUUOEUVFTYHEUPUVBUVKBJEXFWDZXGXHXIXJBYMYOEXKXLAUUNUUCEYHYHYJU + DUBZUSSZVISZTYRYKUPUVCUUIAEEUVDVISZUVQAEXMTEUVRTAEUVLXNEXOVRAYHEUVPUVDVIU + VNAYHEUVOURUSUVNUVOURUPAFXPYDXQXQXJBYGYJEXRXLXSAYIUGYJAYIEEUESUGAYHEEUEUV + NXIAEAEUVLXTYAXHYBAFBUUHYCZTYLFUPOFUVSYEVRXS $. + + $( Lemma for ~ cycpmco2 . (Contributed by Thierry Arnoux, 4-Jan-2024.) $) + cycpmco2lem3 $p |- ( ph -> ( ( # ` U ) - 1 ) = ( # ` W ) ) $= + ( co wcel vw chash cfv c1 cc0 cfz cz cword cn0 cv cdm wf1 crab ssrab2 cbs + eqid tocycf syl fdmd eleqtrd sseldi lencl nn0fz0 biimpi 3syl elfzelz zcnd + 1cnd caddc cmin cpfx cs1 cconcat cop csubstr cotp csplice wceq ccnv ovexd + cvv eqeltrid crn eldifad s1cld splval syl13anc syl5eq fveq2d pfxcl ccatcl + wf syl2anc swrdcl ccatlen ccatws1len cfzo wf1o wa id eqidd f1eq123d elrab + dmeq sylib simprd f1cnv f1of ffvelrnd wrddm fzofzp1 pfxlen oveq1d swrdlen + eqtrd syl3anc oveq12d 3eqtrd fz0ssnn0 peano2zd addsubassd addassd 3eqtr2d + nn0zd nn0cnd addcld pncan2d addcomd mvrraddd ) ADUBUCZJUBUCZUDAYKAYKUEYKU + FSZTZYKUGTAJBUHZTZYKUITZYMAUAUJZUKZBYQULZUAYNUMZYNJYSUAYNUNAJHUKZYTNAYTCU + OUCZHABITYTUUBHWLMUAUUBHBCIKLUUBUPUQURUSUTZVAZBJVBYPYMYKVCVDVEZYKUEYKVFUR + VGZAVHZAYJEUDYKVISZVISZEVJSZUUHYKUDVISAYJEUDVISZYKEVJSZVISZUUKYKVISZEVJSU + UJAYJJEVKSZFVLZVMSZJEYKVNVOSZVMSZUBUCZUUQUBUCZUURUBUCZVISZUUMADUUSUBADJEE + UUPVPVQSZUUSRAJUUATEWATZUVEUUPYNTZUVDUUSVRNAEGJVSZUCZUDVISZWAQAUVHUDVIVTW + BZUVJAFBAFBJWCZOWDWEZUUPJEEUUAWAWAYNWFWGWHWIAUUQYNTZUURYNTZUUTUVCVRAUUOYN + TZUVFUVMAYOUVOUUDBJEWJURZUVLBUUOUUPWKWMAYOUVNUUDBJEYKWNURBUUQUURWOWMAUVAU + UKUVBUULVIAUVAUUOUBUCZUDVISZUUKAUVOUVAUVRVRUVPBUUOFWPURAUVQEUDVIAYOEYLTZU + VQEVRUUDAEUVIYLQAUVHUEYKWQSZTUVIYLTAUVHJUKZUVTAUVKUWAGUVGAUWABJULZUVKUWAU + VGWRUVKUWAUVGWLAYOUWBAJYTTYOUWBWSUUCYSUWBUAJYNYQJVRZYRUWABBYQJUWCWTYQJXDU + WCBXAXBXCXEXFUWABJXGUVKUWAUVGXHVEPXIAYOUWAUVTVRUUDBJXJURUTUEYKUVHXKURWBZB + JEXLWMXMXOAYOUVSYMUVBUULVRUUDUWDUUEBJEYKXNXPXQXRAUUKYKEAUUKAEAEAYLUIEYKXS + UWDVAZYDXTVGUUFAEUWEYEZYAAUUNUUIEVJAEUDYKUWFUUGUUFYBXMYCAEUUHUWFAUDYKUUGU + UFYFYGAUDYKUUGUUFYHXRYI $. + + $( Lemma for ~ cycpmco2 . (Contributed by Thierry Arnoux, 4-Jan-2024.) $) + cycpmco2lem4 $p |- ( ph -> ( ( M ` W ) ` ( ( M ` <" I J "> ) ` I ) ) = ( ( + M ` U ) ` I ) ) $= + ( cc0 co vw cs2 cfv cycpmco2lem1 chash cfzo wcel wa c1 caddc adantr cword + wceq cs1 cotp csplice cv cdm wf1 crab ssrab2 cbs eqid tocycf fdmd eleqtrd + syl sseldi crn eldifad s1cld splcl syl2anc eqeltrid cycpmco2f1 cmin simpr + cycpmco2lem3 oveq2d eleqtrrd cycpmfv1 cycpmco2lem2 fveq2d cop csubstr cfz + cn0 lencl nn0fz0 biimpi 3syl swrdfv0 syl3anc cpfx cconcat cvv ccnv splval + wf ovexd syl13anc syl5eq fveq1d pfxcl ccatcl swrdcl cz fzoaddel elfzolt2b + 1zzd ccatws1len wf1o id eqidd f1eq123d elrab3 biimpa f1cnv ffvelrnd wrddm + dmeq f1of fzofzp1 oveq1d eqtrd oveq12d 3eqtrd nn0cnd 3eqtr2d 3eqtr3rd clt + zcnd wbr nn0p1gt0 cycpmfv2 cn nn0p1nn lbfzo0 sylibr ccatval1 pfxlen nn0zd + ccatlen swrdlen fz0ssnn0 peano2zd elfzelz addsubassd 1cnd addassd pncan2d + addcld addcomd ccatval2 subidd a1i pncand eqtr2d wne breqtrrdi fzo1fzo0n0 + elfzonn0 gt0ne0d sylanbrc elfzo1elm1fzo0 eqeltrd f1f1orn 3eqtr2rd 3eqtr4d + f1ocnvfv2 breqtrd eqtr3d breqtrrd eqtr4d fzne1 addid2d pfxfv0 wo mpjaodan + elfzr ) AFFGUBHUCUCJHUCZUCGUWAUCZFDHUCZUCZABCDEFGHIJKLMNOPQRUDAESJUEUCZUF + TZUGZUWBUWDUMEUWEUMZAUWGUHZEDUCZUWCUCZEUIUJTZDUCZUWDUWBUWIHBEIDKABIUGZUWG + MUKZADBULZUGZUWGADJEEFUNZUOUPTZUWPRAJUWPUGZUWRUWPUGZUWSUWPUGAUAUQZURZBUXB + USZUAUWPUTZUWPJUXDUAUWPVAAJHURZUXENAUXECVBUCZHAUWNUXEUXGHWSMUAUXGHBCIKLUX + GVCVDVGVEVFZVHZAFBAFBJVIZOVJVKZBUWRJEEVLVMVNZUKADURBDUSZUWGABCDEFGHIJKLMN + OPQRVOZUKUWIEUWFSDUEUCZUIVPTZUFTZAUWGVQZAUXQUWFUMUWGAUXPUWESUFABCDEFGHIJK + LMNOPQRVRZVSUKVTWAAUWKUWDUMZUWGAUWJFUWCABCDEFGHIJKLMNOPQRWBWCZUKUWISJEUWE + WDWETZUCZEJUCZUWMUWBUWIUWTUWGUWESUWEWFTZUGZUYCUYDUMAUWTUWGUXIUKZUXRUWIUWT + UWEWGUGZUYFUYGBJWHZUYHUYFUWEWIWJZWKBJEUWEWLWMUWIUWMUWLJEWNTZUWRWOTZUYBWOT + ZUCZUWLUYLUEUCZVPTZUYBUCZUYCAUWMUYNUMUWGAUWLDUYMADUWSUYMRAJUXFUGEWPUGZUYR + UXAUWSUYMUMNAEGJWQZUCZUIUJTZWPQAUYTUIUJWTVNZVUBUXKUWRJEEUXFWPWPUWPWRXAXBZ + XCUKUWIUYLUWPUGZUYBUWPUGZUWLUYOUYOUYBUEUCZUJTZUFTZUGUYNUYQUMAVUDUWGAUYKUW + PUGZUXAVUDAUWTVUIUXIBJEXDVGZUXKBUYKUWRXEVMZUKAVUEUWGAUWTVUEUXIBJEUWEXFVGZ + UKUWIUWLUWLUWEUIUJTZUFTZVUHUWIUWLSUIUJTZVUMUFTUGZUWLVUNUGUWIUWGUIXGUGVUPU + XRUWIXJESUWEUIXHVMUWLVUOVUMXIVGUWIUYOUWLVUGVUMUFAUYOUWLUMUWGAUYOUYKUEUCZU + IUJTZUWLAVUIUYOVURUMVUJBUYKFXKVGAVUQEUIUJAUWTEUYEUGZVUQEUMUXIAEVUAUYEQAUY + TUWFUGZVUAUYEUGAUYTJURZUWFAUXJVVAGUYSAVVABJUSZUXJVVAUYSXLUXJVVAUYSWSAUWTJ + UXEUGZVVBUXIUXHUWTVVCVVBUXDVVBUAJUWPUXBJUMZUXCVVABBUXBJVVDXMUXBJYAVVDBXNX + OXPXQVMZVVABJXRUXJVVAUYSYBWKPXSAUWTVVAUWFUMUXIBJXTVGVFZSUWEUYTYCVGVNZBJEU + UAVMZYDYEZUKAVUGVUMUMUWGAUXOUYMUEUCZVUMVUGADUYMUEVUCWCZAUXOEUIUWEUJTZUJTZ + EVPTZVVLVUMAUXOUWLUWEEVPTZUJTZUWLUWEUJTZEVPTVVNAUXOVVJVUGVVPVVKAVUDVUEVVJ + VUGUMVUKVULBUYLUYBUUCVMZAUYOUWLVUFVVOUJVVIAUWTVUSUYFVUFVVOUMUXIVVGAUWTUYH + UYFUXIUYIUYJWKZBJEUWEUUDWMYFYGAUWLUWEEAUWLAEAEAUYEWGEUWEUUEVVGVHZUUBUUFYL + ZAUWEAUYFUWEXGUGVVSUWESUWEUUGVGYLZAEVVTYHZUUHAVVQVVMEVPAEUIUWEVWCAUUIZVWB + UUJYDYIAEVVLVWCAUIUWEVWDVWBUULUUKAUIUWEVWDVWBUUMYGZVVRYJUKYFVTBUYLUYBUWLU + UNWMAUYQUYCUMUWGAUYPSUYBAUYPUWLUWLVPTSAUYOUWLUWLVPVVIVSAUWLVWAUUOYEWCUKYG + UWIUYDVUAJUCZUYTJUCZUWAUCZUWBAUYDVWFUMUWGAEVUAJEVUAUMAQUUPZWCUKUWIHBUYTIJ + KUWOUYGAVVBUWGVVEUKUWIUYTEUIVPTZSUWEUIVPTZUFTZAUYTVWJUMZUWGAVWJVUAUIVPTUY + TAEVUAUIVPVWIYDAUYTUIAUYTAVUTUYTWGUGZVVFUYTUWEUVBZVGZYHVWDUUQUURZUKUWIEUI + UWEUFTUGZVWJVWLUGUWIUWGESUUSZVWRUXRAVWSUWGAEASVUAEYKAVUTVWNSVUAYKYMVVFVWO + UYTYNWKQUUTZUVCZUKEUWEUVAUVDEUWEUVEVGUVFWAAVWHUWBUMZUWGAVWGGUWAAVVAUXJJXL + ZGUXJUGVWGGUMAVVBVXCVVEVVABJUVGVGPVVAUXJGJUVJVMWCZUKUVHUVIYJAUWHUHZUWBSJU + CZUWKUWDVXEVWHUWBVXFAVXBUWHVXDUKVXEHBUYTIJKAUWNUWHMUKZAUWTUWHUXIUKAVVBUWH + VVEUKVXESEUWEYKASEYKYMUWHVWTUKAUWHVQZUVKVXEUYTVWJVWKAVWMUWHVWQUKVXEEUWEUI + VPVXHYDYEYOUVLVXEUWKSDUCZSUYMUCZVXFVXEHBEIDKVXGAUWQUWHUXLUKAUXMUWHUXNUKAS + UXOYKYMUWHASVUMUXOYKAUWTUYHSVUMYKYMUXIUYIUWEYNWKVWEUVMUKVXEEUWEUXPVXHAUXP + UWEUMUWHUXSUKUVNYOAVXIVXJUMUWHASDUYMVUCXCUKAVXJVXFUMUWHAVXJSUYLUCZSUYKUCZ + VXFAVUDVUESSUYOUFTZUGVXJVXKUMVUKVULASSUWLUFTZVXMAUWLYPUGZSVXNUGAEWGUGVXOV + VTEYQVGUWLYRYSAUYOUWLSUFVVIVSVTBUYLUYBSYTWMAVUIUXASSVUQUFTZUGVXKVXLUMVUJU + XKASSEUFTZVXPAEYPUGSVXQUGAEVUAYPQAVWNVUAYPUGVWPUYTYQVGVNEYRYSAVUQESUFVVHV + SVTBUYKUWRSYTWMAUWTEUIUWEWFTZUGVXLVXFUMUXIAEVUOUWEWFTZVXRAVUSVWSEVXSUGVVG + VXAESUWEUVOVMAVUOUIUWEWFAUIVWDUVPYDVFEBJUVQVMYGUKYGAUXTUWHUYAUKYIAVUSUWGU + WHUVRVVGESUWEUVTVGUVSYE $. + + ${ + cycpmco2lem.1 $e |- ( ph -> K e. ran W ) $. + ${ + cycpmco2lem5.1 $e |- ( ph -> ( `' U ` K ) = ( ( # ` U ) - 1 ) ) $. + $( Lemma for ~ cycpmco2 . (Contributed by Thierry Arnoux, + 4-Jan-2024.) $) + cycpmco2lem5 $p |- ( ph -> ( ( M ` U ) ` K ) = ( ( M ` W ) ` K ) ) $= + ( vw chash cfv c1 cmin co wceq crn wcel adantr caddc cpfx cs1 cconcat + wa cop csubstr cotp csplice cdm cvv cword ccnv ovexd eqeltrid eldifad + s1cld splval syl13anc syl5eq fveq2d cv wf1 crab ssrab2 wf eqid tocycf + cbs syl eleqtrd sseldi pfxcl ccatcl syl2anc swrdcl ccatlen ccatws1len + fdmd cc0 cfz cfzo wf1o id dmeq eqidd f1eq123d elrab sylib simprd f1of + f1cnv 3syl ffvelrnd wrddm pfxlen oveq1d eqtrd cn0 lencl nn0fz0 biimpi + fzofzp1 swrdlen syl3anc oveq12d 3eqtrd fz0ssnn0 addcomd simpr 3eqtr3d + zcnd 1cnd wne clt wbr nn0p1gt0 breqtrrd cycpmfv2 a1i oveq2d npcand cc + cn nn0p1nn lbfzo0 sylibr eleqtrrd ccatval1 peano2zd cz elfzelz nn0cnd + nn0zd addsubassd addassd 3eqtr2d addcld pncan2d pncand cycpmco2f1 csn + cun ssun1 cycpmco2rn sseqtrrid sselda f1ocnvfv2 syl2an2r mpdan eqtr3d + f1f1orn cycpmco2lem2 wn eldifbd eqneltrd pm2.21dd splcl c0 frnd ssexd + f1f ne0d hashgt0 hashf1rn cycpmco2lem3 subcld nppcan3d eqtr4d fveq12d + dmexd sub32d fznn0sub subne0nn fzo0end eqeltrd eqcomd splfv3 elfzonn0 + s1len fveq1d breqtrrdi gt0ne0d addid2d pfxfv0 3eqtr4rd pm2.61dane + fzne1 ) ADUCUDZUEUFUGZDUDZDIUDZUDZUXBKIUDZUDZHUXCUDHUXEUDAUXDUXFUHZKU + CUDZEAUXHEUHZUPZHKUIZUJZUXGAUXLUXITUKUXJHFUXKUXJUXBEDUDZHFUXJUXAEDUXJ + UXAEUEULUGZUEUFUGZEUXJUWTUXNUEUFUXJUWTUXHUEULUGZUXNAUWTUXPUHUXIAUWTEU + EUXHULUGZULUGZEUFUGZUXQUXPAUWTUXNUXHEUFUGZULUGZUXNUXHULUGZEUFUGUXSAUW + TKEUMUGZFUNZUOUGZKEUXHUQURUGZUOUGZUCUDZUYEUCUDZUYFUCUDZULUGZUYAADUYGU + CADKEEUYDUSUTUGZUYGSAKIVAZUJEVBUJZUYNUYDBVCZUJZUYLUYGUHOAEGKVDZUDZUEU + LUGZVBRAUYRUEULVEVFZUYTAFBAFBUXKPVGVHZUYDKEEUYMVBVBUYOVIVJVKZVLAUYEUY + OUJZUYFUYOUJZUYHUYKUHAUYCUYOUJZUYPVUCAKUYOUJZVUEAUBVMZVAZBVUGVNZUBUYO + VOZUYOKVUIUBUYOVPAKUYMVUJOAVUJCVTUDZIABJUJVUJVUKIVQNUBVUKIBCJLMVUKVRV + SWAWJWBZWCZBKEWDWAZVUABUYCUYDWEWFZAVUFVUDVUMBKEUXHWGWAZBUYEUYFWHWFAUY + IUXNUYJUXTULAUYIUYCUCUDZUEULUGZUXNAVUEUYIVURUHVUNBUYCFWIWAAVUQEUEULAV + UFEWKUXHWLUGZUJZVUQEUHVUMAEUYSVUSRAUYRWKUXHWMUGZUJZUYSVUSUJAUYRKVAZVV + AAUXKVVCGUYQAVVCBKVNZUXKVVCUYQWNUXKVVCUYQVQAVUFVVDAKVUJUJVUFVVDUPVULV + UIVVDUBKUYOVUGKUHZVUHVVCBBVUGKVVEWOVUGKWPVVEBWQWRWSWTXAZVVCBKXCUXKVVC + UYQXBXDQXEAVUFVVCVVAUHVUMBKXFWAWBZWKUXHUYRXNWAVFZBKEXGWFZXHXIZAVUFVUT + UXHVUSUJZUYJUXTUHVUMVVHAVUFUXHXJUJZVVKVUMBKXKZVVLVVKUXHXLXMXDZBKEUXHX + OXPXQXRAUXNUXHEAUXNAEAEAVUSXJEUXHXSVVHWCZUUEUUAYCAUXHAVVKUXHUUBUJVVNU + XHWKUXHUUCWAYCZAEVVOUUDZUUFAUYBUXREUFAEUEUXHVVQAYDZVVPUUGXHUUHAEUXQVV + QAUEUXHVVRVVPUUIUUJAUEUXHVVRVVPXTXRZUKUXJUXHEUEULAUXIYAXHXIXHAUXOEUHU + XIAEUEVVQVVRUUKUKXIVLAUXBHUHUXIAHDVDUDZDUDZUXBHAVVTUXADUAVLAUXLVWAHUH + ZTADVAZDUIZDWNZUXLHVWDUJVWBAVWCBDVNVWEABCDEFGIJKLMNOPQRSUULZVWCBDUVCW + AAUXKVWDHAUXKFUUMZUUNUXKVWDUXKVWGUUOABCDEFGIJKLMNOPQRSUUPUUQUURVWCVWD + HDUUSUUTUVAUVBZUKAUXMFUHUXIABCDEFGIJKLMNOPQRSUVDUKYBAFUXKUJUVEUXIAFBU + XKPUVFUKUVGUVHAUXHEYEZUPZUXDWKDUDZUXFAUXDVWKUHVWIAIBUXAJDLNADUYLUYOSA + VUFUYPUYLUYOUJVUMVUABUYDKEEUVIWFVFVWFAWKUXPUWTYFAVUFVVLWKUXPYFYGVUMVV + MUXHYHXDVVSYIAUXAWQYJUKVWJUXHUEUFUGZKUDZUXEUDZWKKUDZUXFVWKAVWNVWOUHVW + IAIBVWLJKLNVUMVVFAWKUXKUCUDZUXHYFAUXKVBUJUXKUVJYEWKVWPYFYGAUXKBJNAVVC + BKAVVDVVCBKVQVVFVVCBKUVMWAUVKUVLAUXKGQUVNUXKVBUVOWFAVVCVBUJVVDUXHVWPU + HAKUYMOUWBVVFVVCBKVBUVPWFYIAVWLWQYJUKVWJUXBVWMUXEVWJUXBVWLEUFUGZUXNUL + UGZUYLUDVWQEULUGZKUDZVWMVWJUXAVWRDUYLDUYLUHVWJSYKAUXAVWRUHVWIAUXAUXHV + WRABCDEFGIJKLMNOPQRSUVQAVWRVWQUEEULUGZULUGVWLUEULUGUXHAUXNVXAVWQULAEU + EVVQVVRXTYLAVWLEUEAUXHUEVVPVVRUVRZVVQVVRUVSAUXHUEVVPVVRYMXRUVTUKUWAVW + JBUYDKEEUXNVWQAVUFVWIVUMUKAEWKEWLUGUJZVWIAEXJUJZVXCVVOEXLWTUKAVUTVWIV + VHUKZAUYPVWIVUAUKVWJVWQUXTUEUFUGZWKUXTWMUGZVWJUXHUEEAUXHYNUJVWIVVPUKZ + VWJYDAEYNUJVWIVVQUKZUWCVWJUXTYOUJVXFVXGUJVWJUXHEVXHVXIVWJVUTUXTXJUJVX + EEWKUXHUWDWAAVWIYAUWEUXTUWFWAUWGVWJUEUYDUCUDZEULAUEVXJUHVWIAVXJUEVXJU + EUHAFUWKYKUWHUKYLUWIAVWTVWMUHVWIAVWSVWLKAVWLEVXBVVQYMVLUKXRVLAVWKVWOU + HVWIAVWKWKUYGUDZVWOAWKDUYGVUBUWLAVXKWKUYEUDZWKUYCUDZVWOAVUCVUDWKWKUYI + WMUGZUJVXKVXLUHVUOVUPAWKWKUXNWMUGZVXNAUXNYOUJZWKVXOUJAVXDVXPVVOEYPWAU + XNYQYRAUYIUXNWKWMVVJYLYSBUYEUYFWKYTXPAVUEUYPWKWKVUQWMUGZUJVXLVXMUHVUN + VUAAWKWKEWMUGZVXQAEYOUJWKVXRUJAEUYSYORAUYRXJUJZUYSYOUJAVVBVXSVVGUYRUX + HUWJZWAUYRYPWAVFEYQYRAVUQEWKWMVVIYLYSBUYCUYDWKYTXPAVUFEUEUXHWLUGZUJVX + MVWOUHVUMAEWKUEULUGZUXHWLUGZVYAAVUTEWKYEEVYCUJVVHAEAWKUYSEYFAVVBVXSWK + UYSYFYGVVGVXTUYRYHXDRUWMUWNEWKUXHUWSWFAVYBUEUXHWLAUEVVRUWOXHWBEBKUWPW + FXRXIUKUWQXIUWRAUXBHUXCVWHVLAUXBHUXEVWHVLYB $. + $} + + cycpmco2lem6.2 $e |- ( ph -> K =/= I ) $. + cycpmco2lem6.1 $e |- ( ph -> ( `' U ` K ) e. ( E ..^ ( ( # ` U ) - 1 ) ) + ) $. + $( Lemma for ~ cycpmco2 . (Contributed by Thierry Arnoux, + 4-Jan-2024.) $) + cycpmco2lem6 $p |- ( ph -> ( ( M ` U ) ` K ) = ( ( M ` W ) ` K ) ) $= + ( vw cfv ccnv cmin co c1 caddc cs1 cotp csplice cword wcel cdm wf1 crab + cv ssrab2 cbs wf eqid tocycf syl eleqtrd sseldi crn eldifad s1cld splcl + fdmd syl2anc eqeltrid cycpmco2f1 cfzo cc0 cuz wss cn0 cfz fz0ssnn0 wf1o + chash wa wceq id dmeq eqidd f1eq123d elrab sylib simprd f1cnv f1of 3syl + ffvelrnd wrddm fzofzp1 nn0uz syl6eleq fzoss1 cycpmfv1 f1f1orn csn ssun1 + sseldd cun cycpmco2rn sseqtrrid sselda f1ocnvfv2 syl2an2r fveq2d a1i cz + mpdan zcnd fveq12d 3eqtr3d npcand nn0fz0 cconcat oveq1d 3eqtrd peano2zd + cvv addcld addcomd oveq2d eqtr4d splfv3 wne cle cn zred fzoss2 eleqtrrd + wbr eqeltrd zsubcld fzossz nn0cnd 1cnd eqcomd lencl biimpi elfzelz cpfx + nppcan3d cop csubstr ovexd splval syl13anc syl5eq ccatcl swrdcl ccatlen + pfxcl ccatws1len eqtrd swrdlen syl3anc oveq12d nn0zd addsubassd addassd + pfxlen 3eqtr2d pncan2d mvrraddd fzosubel subidd s1len f1ocnvdm elfzonn0 + eqsstrd nn0p1nn nnred 1red elfzle2 leadd1dd eluz2 fzonn0p1 cycpmco2lem2 + syl3anbrc 3netr4d necon3bid biimp3a syl121anc fzom1ne1 subsub4d pncan3d + f1fveq eqtr2d 1zzd ltm1d breqtrd ltled eluz1 biimpar syl12anc fzosubel3 + clt cr subcld eqtr3d 3eqtr4rd ) AHDIUDZUDZHDUEUDZEUFUGZUHEUIUGZUIUGZKEE + FUJZUKULUGZUDZHKIUDZUDZAUXKDUDZUXIUDUXKUHUIUGZDUDUXJUXQAIBUXKJDLNADUXPB + UMZSAKUYBUNZUXOUYBUNZUXPUYBUNAUCURZUOZBUYEUPZUCUYBUQZUYBKUYGUCUYBUSAKIU + OZUYHOAUYHCUTUDZIABJUNUYHUYJIVANUCUYJIBCJLMUYJVBVCVDVKVEZVFZAFBAFBKVGZP + VHVIZBUXOKEEVJVLVMZABCDEFGIJKLMNOPQRSVNZAEDWCUDZUHUFUGZVOUGZVPUYRVOUGZU + XKAEVPVQUDZUNZUYSUYTVRAEVSVUAAVPKWCUDZVTUGZVSEVUCWAAEGKUEZUDZUHUIUGZVUD + RAVUFVPVUCVOUGZUNZVUGVUDUNAVUFKUOZVUHAUYMVUJGVUEAVUJBKUPZUYMVUJVUEWBUYM + VUJVUEVAAUYCVUKAKUYHUNUYCVUKWDUYKUYGVUKUCKUYBUYEKWEZUYFVUJBBUYEKVULWFUY + EKWGVULBWHWIWJWKWLZVUJBKWMUYMVUJVUEWNWOQWPAUYCVUJVUHWEUYLBKWQVDVEZVPVUC + VUFWRVDVMZVFZWSWTZEVPUYRXAVDUBXFXBAUXTHUXIAHUYMUNZUXTHWEZTADUOZDVGZDWBZ + VURHVVAUNZVUSAVUTBDUPZVVBUYPVUTBDXCVDZAUYMVVAHAUYMFXDZXGUYMVVAUYMVVFXEA + BCDEFGIJKLMNOPQRSXHXIXJZVUTVVAHDXKXLXPZXMAUYAUXNDUXPDUXPWEASXNZAUXNUYAA + UXKEUHAUXKAUYSXOUXKEUYRUUAUBVFXQZAEVUPUUBZAUUCZUUIUUDXRXSAUXLEUIUGZKUDU + XKKUDZUXQUXSAVVMUXKKAUXKEVVJVVKXTXMABUXOKEEUXMUXLUYLAEVSUNZEVPEVTUGUNVU + PEYAWKZVUOUYNAUXLEEUFUGZVUCEUFUGZVOUGZVPVVRVOUGZAUXKEVUCVOUGZUNEXOUNUXL + VVSUNAUXKUYSVWAUBAUYRVUCEVOAUYQVUCUHAVUCAVUCVUDUNZVUCXOUNZAUYCVUCVSUNZV + WBUYLBKUUEVWDVWBVUCYAUUFWOZVUCVPVUCUUGVDZXQZVVLAUYQEUHVUCUIUGZUIUGZEUFU + GZVWHVUCUHUIUGZAUYQEUHUIUGZVVRUIUGZVWLVUCUIUGZEUFUGVWJAUYQKEUUHUGZUXOYB + UGZKEVUCUUJUUKUGZYBUGZWCUDZVWPWCUDZVWQWCUDZUIUGZVWMADVWRWCADUXPVWRSAKUY + IUNEYFUNZVXCUYDUXPVWRWEOAEVUGYFRAVUFUHUIUULVMZVXDUYNUXOKEEUYIYFYFUYBUUM + UUNUUOXMAVWPUYBUNZVWQUYBUNZVWSVXBWEAVWOUYBUNZUYDVXEAUYCVXGUYLBKEUUSZVDU + YNBVWOUXOUUPVLAUYCVXFUYLBKEVUCUUQVDBVWPVWQUURVLAVWTVWLVXAVVRUIAVWTVWOWC + UDZUHUIUGZVWLAUYCVXGVWTVXJWEUYLVXHBVWOFUUTWOAVXIEUHUIAUYCEVUDUNZVXIEWEU + YLVUOBKEUVHVLYCUVAAUYCVXKVWBVXAVVRWEUYLVUOVWEBKEVUCUVBUVCUVDYDAVWLVUCEA + VWLAEAEVUPUVEZYEZXQVWGVVKUVFAVWNVWIEUFAEUHVUCVVKVVLVWGUVGYCUVIAEVWHVVKA + UHVUCVVLVWGYGUVJAUHVUCVVLVWGYHYDZUVKZYIVEVXLUXKEVUCEUVLVLAVVQVPVVRVOAEV + VKUVMYCVEAUXMVWLEUXOWCUDZUIUGAUHEVVLVVKYHAVXPUHEUIVXPUHWEAFUVNXNYIYJZYK + AUXKUHUFUGZKUDZUXRUDVXRUHUIUGZKUDUXSVVNAIBVXRJKLNUYLVUMAEUYRUHUFUGZVOUG + ZVPVUCUHUFUGZVOUGZVXRAVYBEVYCVOUGZVYDAVYAVYCEVOAUYRVUCUHUFVXOYCYIAVUBVY + EVYDVRVUQEVPVYCXAVDUVQAUXKUYSUNUXKEYLZVXRVYBUNUBAVVDUXKVUTUNZEVUTUNZUXT + EDUDZYLZVYFUYPAVURVYGTAVVBVURVVCVYGVVEVVGVUTVVAHDUVOXLXPAEVPUYQVOUGZVUT + AEVPVWKVOUGZVYKAVPVWLVOUGZVYLEAVWKVWLVQUDUNZVYMVYLVRAVWLXOUNVWKXOUNVWLV + WKYMYRVYNVXMAVUCVWFYEAEVUCUHAEAEVUGYNRAVUIVUFVSUNVUGYNUNVUNVUFVUCUVPVUF + UVRWOVMUVSAVUCVWFYOZAUVTAVXKEVUCYMYRVUOEVPVUCUWAVDUWBVWLVWKUWCUWFVWLVPV + WKYPVDAVVOEVYMUNVUPEUWDVDXFAUYQVWKVPVOVXNYIYQADUYBUNVUTVYKWEUYOBDWQVDYQ + AHFUXTVYIUAVVHABCDEFGIJKLMNOPQRSUWEUWGVVDVYGVYHWDZVYJVYFVVDVYPWDUXTVYIU + XKEVUTBUXKEDUWNUWHUWIUWJUXKEUYRUWKVLZXFXBAVXSHUXRAUXTVXSHAUXTVXREUFUGZU + XMUIUGZUXPUDVYREUIUGZKUDVXSAUXKVYSDUXPVVIAVYSUXKUXMUFUGZUXMUIUGUXKAVYRW + UAUXMUIAUXKUHEVVJVVLVVKUWLYCAUXKUXMVVJAUHEVVLVVKYGXTUWOXRABUXOKEEUXMVYR + UYLVVPVUOUYNAVXREEVVRUIUGZVOUGZUNVVRXOUNVYRVVTUNAVYBWUCVXRAWUBVYAVQUDZU + NVYBWUCVRAWUBVUCWUDAEVUCVVKVWGUWMAVYAXOUNZVWCVYAVUCYMYRZVUCWUDUNZAUYRUH + AUYRVUCXOVXOVWFYSAUWPYTZVWFAVYAVUCAVYAWUHYOVYOAVYAUYRVUCUXDAUYRAUYRVUCU + XEVXOVYOYSUWQVXOUWRUWSWUEWUGVWCWUFWDVYAVUCUWTUXAUXBYSVYAEWUBYPVDVYQXFAV + UCEVWFVXLYTVXREVVRUXCVLVXQYKAVYTVXRKAVXREAUXKUHVVJVVLUXFVVKXTXMYDVVHUXG + XMAVXTUXKKAUXKUHVVJVVLXTXMXSUXHYJ $. + $} + + ${ + cycpmco2lem7.1 $e |- ( ph -> K e. ran W ) $. + cycpmco2lem7.2 $e |- ( ph -> K =/= J ) $. + cycpmco2lem7.3 $e |- ( ph -> ( `' U ` K ) e. ( 0 ..^ E ) ) $. + $( Lemma for ~ cycpmco2 . (Contributed by Thierry Arnoux, + 4-Jan-2024.) $) + cycpmco2lem7 $p |- ( ph -> ( ( M ` U ) ` K ) = ( ( M ` W ) ` K ) ) $= + ( vw cfv ccnv c1 caddc co cs1 cotp csplice cword wcel cv cdm wf1 ssrab2 + crab cbs wf eqid tocycf syl fdmd eleqtrd sseldi crn eldifad s1cld splcl + syl2anc eqeltrid cycpmco2f1 cc0 cfzo chash cmin cfz cuz wf1o wa wceq id + dmeq eqidd f1eq123d elrab sylib simprd f1cnv f1of 3syl ffvelrnd fzofzp1 + wss elfzuz3 fzoss2 cycpmco2lem3 oveq2d sseqtr4d sseldd cycpmfv1 f1f1orn + wrddm ssun1 cycpmco2rn sseqtrrid sselda f1ocnvfv2 syl2an2r fveq2d mpdan + csn cun f1ocnvfv1 fveq1i cn0 nn0fz0 splfv1 syl5eq eqtr3d oveq1d 3eqtr3d + fz0ssnn0 a1i fveq1d cz nn0zd simpr elfzonn0 nn0cnd 1cnd adantr 3eqtr2rd + cn cvv eqeltrrd cle wbr zsubcld pncand eqtr2d wne pm2.21ddne wo nn0p1nn + 0p1e1 fveq2i nnuz eqtr4i syl6eleqr fzosplitsnm1 wb fvex elunsn mpjaodan + ax-mp elfzom1elp1fzo eqtrd 1zzd lencl biimpi elfzelz nnred zred elfzle2 + 0zd 1red lesub1dd eluz biimpar syl21anc eqtr4d ) AHDIUDZUDZHKUEZUDZUFUG + UHZDUDZHKIUDZUDZAHDUEZUDZDUDZUVNUDZUWCUFUGUHZDUDUVOUVSAIBUWCJDLNADKEEFU + IZUJUKUHZBULZSAKUWIUMZUWGUWIUMUWHUWIUMAUCUNZUOZBUWKUPZUCUWIURZUWIKUWMUC + UWIUQAKIUOUWNOAUWNCUSUDZIABJUMUWNUWOIUTNUCUWOIBCJLMUWOVAVBVCVDVEZVFZAFB + AFBKVGZPVHVIZBUWGKEEVJVKVLABCDEFGIJKLMNOPQRSVMZAVNEVOUHZVNDVPUDUFVQUHZV + OUHZUWCAUXAVNKVPUDZVOUHZUXCAEVNUXDVRUHZUMZUXDEVSUDUMUXAUXEWOAEGUVPUDZUF + UGUHZUXFRAUXHUXEUMZUXIUXFUMAUXHKUOZUXEAUWRUXKGUVPAUXKBKUPZUWRUXKUVPVTUW + RUXKUVPUTAUWJUXLAKUWNUMUWJUXLWAUWPUWMUXLUCKUWIUWKKWBZUWLUXKBBUWKKUXMWCU + WKKWDUXMBWEWFWGWHWIZUXKBKWJUWRUXKUVPWKWLQWMAUWJUXKUXEWBUWQBKXDVCZVEZVNU + XDUXHWNVCVLZEVNUXDWPEVNUXDWQWLZAUXBUXDVNVOABCDEFGIJKLMNOPQRSWRWSWTUBXAX + BAHUWRUMZUWEUVOWBTAUXSWAUWDHUVNADUOZDVGZDVTZUXSHUYAUMUWDHWBZAUXTBDUPUYB + UWTUXTBDXCVCAUWRUYAHAUWRFXMZXNUWRUYAUWRUYDXEABCDEFGIJKLMNOPQRSXFXGXHUXT + UYAHDXIXJZXKXLAUWFUVRDAUWCUVQUFUGAUWCKUDZUVPUDZUWCUVQAUXKUWRKVTZUWCUXKU + MUYGUWCWBAUXLUYHUXNUXKBKXCVCZAUXAUXKUWCAUXAUXEUXKUXRUXOWTUBXAUXKUWRUWCK + XOVKAUYFHUVPAUWDUYFHAUWDUWCUWHUDUYFUWCDUWHSXPABUWGKEEUWCUWQAEXQUMEVNEVR + UHUMAUXFXQEUXDYDUXQVFZEXRWHZUXQUWSUBXSXTAUXSUYCTUYEXLYAXKYAZYBZXKYCAUVS + UVRKUDZUVQKUDZUVTUDUWAAUVSUVRUWHUDUYNAUVRDUWHDUWHWBASYEYFABUWGKEEUVRUWQ + UYKUXQUWSAUWFUVRUXAUYMAEYGUMUWCVNEUFVQUHZVOUHZUMZUWFUXAUMAEUYJYHZAUYRUY + RUWCUYPWBZAUYRYIAUYTWAZUYRHGVUAUYOUXHKUDZHGVUAUVQUXHKVUAUXHUYPUWCUVQAUX + HUYPWBUYTAUYPUXIUFVQUHUXHAEUXIUFVQEUXIWBARYEYBAUXHUFAUXHAUXJUXHXQUMZUXP + UXHUXDYJVCZYKAYLUUAUUBYMAUYTYIAUWCUVQWBUYTUYLYMYNXKAUYOHWBZUYTAUYHUXSVU + EUYITUXKUWRHKXIVKZYMAVUBGWBZUYTAUYHGUWRUMVUGUYIQUXKUWRGKXIVKYMYCAHGUUCU + YTUAYMUUDAUWCUYQUYPXMXNZUMZUYRUYTUUEZAUWCUXAVUHUBAVNYGUMEVNUFUGUHZVSUDZ + UMUXAVUHWBAUVGAEYOVULAEUXIYORAVUCUXIYOUMVUDUXHUUFVCVLZVULUFVSUDYOVUKUFV + SUUGUUHUUIUUJUUKVNEUULVKVEUWCYPUMVUIVUJUUMHUWBUUNUWCUYQUYPYPUUOUUQWHUUP + ZUWCEUURVKYQXSUUSAIBUVQJKLNUWQUXNAUWCUVQVNUXDUFVQUHZVOUHZUYLAUYQVUPUWCA + VUOUYPVSUDUMZUYQVUPWOAUYPYGUMZVUOYGUMZUYPVUOYRYSZVUQAEUFUYSAUUTZYTAUXDU + FAUXDUXFUMZUXDYGUMAUWJUXDXQUMZVVBUWQBKUVAVVCVVBUXDXRUVBWLUXDVNUXDUVCVCZ + VVAYTAEUXDUFAEVUMUVDAUXDVVDUVEAUVHAUXGEUXDYRYSUXQEVNUXDUVFVCUVIVURVUSWA + VUQVUTUYPVUOUVJUVKUVLUYPVNVUOWQVCVUNXAYQXBAUYOHUVTVUFXKYNUVM $. + $} + + $( The composition of a cyclic permutation and a transposition of one + element in the cycle and one outside the cycle results in a cyclic + permutation with one more element in its orbit. (Contributed by Thierry + Arnoux, 2-Jan-2024.) $) + cycpmco2 $p |- ( ph -> ( ( M ` W ) o. ( M ` <" I J "> ) ) = ( M ` U ) ) $= + ( cfv wcel vi vw cs2 ccom wfn crn wss cbs wf cv cdm wf1 cword crab tocycf + eqid fdmd eleqtrd ffvelrnd symgbasf ffnd eldifad ssrab2 sseldi wceq eqidd + syl id dmeq f1eq123d elrab3 biimpa syl2anc f1f frnd sseldd wn wne eldifbd + nelne2 necomd cycpm2cl fnco syl3anc cs1 cotp csplice s1cld splcl eqeltrid + co cycpmco2f1 cycpmcl wa fvco3 sylan cdif ccnv fveq2d c1 caddc chash cfzo + cc0 cmin wf1o 3syl wrddm cfz cz zcnd cconcat oveq1d eqtrd 3eqtrd peano2zd + cn0 cvv 3eqtr2d oveq2d eleqtrrd eqtr4d cycpmfv3 3eqtr4d ccatval1 ad2antrr + f1f1orn simpr sselda adantr nelprd eleq2d mpbird ad3antrrr simpllr simplr + wo cun pm2.61dane adantlr cyc2fv2 cycpmco2lem2 f1cnv lencl nn0fz0 elfzelz + f1of biimpi 1cnd cpfx cop csubstr ovexd splval syl5eq pfxcl ccatcl swrdcl + syl13anc ccatlen ccatws1len fzofzp1 pfxlen swrdlen oveq12d fz0ssnn0 nn0zd + nn0cnd addsubassd addassd addcld pncan2d addcomd mvrraddd cycpmfv1 fveq1d + a1i fzossfzop1 elfzonn0 fzonn0p1 syl6eleqr pfxfv f1ocnvfv2 s2cld s2f1 cpr + oveq2i notbid cycpmco2lem7 cycpmco2lem6 cycpmco2lem5 w3o ssun1 cycpmco2rn + wb csn sseqtrrid f1ocnvdm eqeltrd fzoval elfzr fzospliti ex orim1d df-3or + s2rn mpd sylibr mpjao3dan cycpmco2lem4 nelsn adantl biimpar syl12anc elun + nelun undif sylib syl5rbbr mpjaodan eqfnfvd ) AUABJHSZFGUCZHSZUDZDHSZAUYB + BUEUYDBUEUYDUFBUGUYEBUEABBUYBAUYBCUHSZTBBUYBUIAUBUJZUKZBUYHULZUBBUMZUNZUY + GJHABITZUYLUYGHUIMUBUYGHBCIKLUYGUPZUOVGZAJHUKZUYLNAUYLUYGHUYOUQURZUSBUYGU + YBCLUYNUTVGVAABBUYDAUYDUYGTBBUYDUIZAHBCFGIKMAFBJUFZOVBZAUYSBGAJUKZBJAVUAB + JULZVUABJUIAJUYKTZJUYLTZVUBAUYLUYKJUYJUBUYKVCUYQVDZUYQVUCVUDVUBUYJVUBUBJU + YKUYHJVEZUYIVUABBUYHJVUFVHUYHJVIVUFBVFVJVKVLVMZVUABJVNVGVOZPVPZAGFAGUYSTZ + FUYSTVQZGFVRPAFBUYSOVSZGFUYSVTVMWAZLWBBUYGUYDCLUYNUTVGZVAABBUYDVUNVOBBUYB + UYDWCWDABBUYFAUYFUYGTBBUYFUIAHBCIDKMADJEEFWEZWFWGWKZUYKRAVUCVUOUYKTZVUPUY + KTVUEAFBUYTWHZBVUOJEEWIVMWJZABCDEFGHIJKLMNOPQRWLZLWMBUYGUYFCLUYNUTVGVAAUA + UJZBTZWNZVVAUYESZVVAUYDSZUYBSZVVAUYFSZAUYRVVBVVDVVFVEVUNBBVVAUYBUYDWOWPVV + CVVAUYSTZVVFVVGVEZVVABUYSWQZTZAVVHVVIVVBAVVHWNZVVIVVAGVVLVVAGVEZWNZGUYDSZ + UYBSZGUYFSZVVFVVGAVVPVVQVEVVHVVMAVVPFUYBSZGJWRZSZDSZUYFSZVVQAVVOFUYBAHBCF + GIKMUYTVUIVUMLUUAWSAEDSZFVWBVVRABCDEFGHIJKLMNOPQRUUBAVWBVVTWTXAWKZDSVWCAH + BVVTIDKMVUSVUTAVVTXDJXBSZXCWKZXDDXBSZWTXEWKZXCWKZAVVTVUAVWFAUYSVUAGVVSAVU + BUYSVUAVVSXFUYSVUAVVSUIVUGVUABJUUCUYSVUAVVSUUGXGPUSAVUCVUAVWFVEVUEBJXHVGU + RZAVWHVWEXDXCAVWGVWEWTAVWEAVWEXDVWEXIWKZTZVWEXJTAVUCVWEXQTZVWLVUEBJUUDVWM + VWLVWEUUEUUHXGZVWEXDVWEUUFVGZXKZAUUIZAVWGEWTVWEXAWKZXAWKZEXEWKZVWRVWEWTXA + WKZAVWGEWTXAWKZVWEEXEWKZXAWKZVXBVWEXAWKZEXEWKVWTAVWGJEUUJWKZVUOXLWKZJEVWE + UUKUULWKZXLWKZXBSZVXGXBSZVXHXBSZXAWKZVXDADVXIXBADVUPVXIRAJUYPTZEXRTZVXOVU + QVUPVXIVENAEVWDXRQAVVTWTXAUUMWJZVXPVURVUOJEEUYPXRXRUYKUUNUUSUUOZWSAVXGUYK + TZVXHUYKTZVXJVXMVEAVXFUYKTZVUQVXRAVUCVXTVUEBJEUUPVGZVURBVXFVUOUUQVMZAVUCV + XSVUEBJEVWEUURVGZBVXGVXHUUTVMAVXKVXBVXLVXCXAAVXKVXFXBSZWTXAWKZVXBAVXTVXKV + YEVEVYABVXFFUVAVGAVYDEWTXAAVUCEVWKTZVYDEVEVUEAEVWDVWKQAVVTVWFTZVWDVWKTVWJ + XDVWEVVTUVBVGWJZBJEUVCVMZXMXNZAVUCVYFVWLVXLVXCVEVUEVYHVWNBJEVWEUVDWDUVEXO + AVXBVWEEAVXBAEAEAVWKXQEVWEUVFVYHVDZUVGZXPXKVWPAEVYKUVHZUVIAVXEVWSEXEAEWTV + WEVYMVWQVWPUVJXMXSAEVWRVYMAWTVWEVWQVWPUVKUVLAWTVWEVWQVWPUVMXOZUVNXTYAUVOA + EVWDDEVWDVEAQUVQWSYBAHBIJFKMVUEVUGUYTVULYCYDAVWAGUYFAVWAVVTVXFSZVVTJSZGAV + WAVVTVXISZVVTVXGSZVYOAVVTDVXIVXQUVPAVXRVXSVVTXDVXKXCWKZTVYQVYRVEVYBVYCAVV + TXDVXBXCWKZVYSAXDEXCWKZVYTVVTAEXQTWUAVYTUGVYKEUVRVGAVVTXDVWDXCWKZWUAAVVTX + QTZVVTWUBTAVYGWUCVWJVVTVWEUVSVGVVTUVTVGEVWDXDXCQUWGUWAZVPAVXKVXBXDXCVYJXT + YABVXGVXHVVTYEWDAVXTVUQVVTXDVYDXCWKZTVYRVYOVEVYAVURAVVTWUAWUEWUDAVYDEXDXC + VYIXTYABVXFVUOVVTYEWDXOAVUCVYFVVTWUATVYOVYPVEVUEVYHWUDVVTEBJUWBWDAVUAUYSJ + XFZVUJVYPGVEAVUBWUFVUGVUABJYGVGPVUAUYSGJUWCVMXOWSXSYFVVNVVEVVOUYBVVNVVAGU + YDVVLVVMYHZWSWSVVNVVAGUYFWUGWSYDVVLVVAGVRZWNZVVFVVAUYBSZVVGWUIVVEVVAUYBWU + IHBIUYCVVAKAUYMVVHWUHMYFAUYCUYKTZVVHWUHAFGBUYTVUIUWDZYFAUYCUKBUYCULZVVHWU + HABFGUYTVUIVUMUWEZYFVVLVVBWUHAUYSBVVAVUHYIYJWUIVVAUYCUFZTZVQZVVAFGUWFZTZV + QZWUIVVAFGVVLVVAFVRZWUHVVLVVHVUKWVAAVVHYHAVUKVVHVULYJVVAFUYSVTVMYJZVVLWUH + YHYKAWUQWUTUWOZVVHWUHAWUPWUSAWUOWURVVAABFGUYTVUIUXFYLUWHZYFYMYCWSWUIVVADW + RSZWUATZVVGWUJVEWVEEVWHXCWKTZWVEVWHVEZWUIWVFWNBCDEFGVVAHIJKLAUYMVVHWUHWVF + MYNAVXNVVHWUHWVFNYNAFVVJTZVVHWUHWVFOYNAVUJVVHWUHWVFPYNQRAVVHWUHWVFYOVVLWU + HWVFYPWUIWVFYHUWIWUIWVGWNBCDEFGVVAHIJKLAUYMVVHWUHWVGMYNAVXNVVHWUHWVGNYNAW + VIVVHWUHWVGOYNAVUJVVHWUHWVGPYNQRAVVHWUHWVGYOWUIWVAWVGWVBYJWUIWVGYHUWJWUIW + VHWNBCDEFGVVAHIJKLAUYMVVHWUHWVHMYNAVXNVVHWUHWVHNYNAWVIVVHWUHWVHOYNAVUJVVH + WUHWVHPYNQRAVVHWUHWVHYOWUIWVHYHUWKVVLWVFWVGWVHUWLZWUHVVLWVFWVGYQZWVHYQZWV + JVVLWVEVWITZWVHYQZWVLVVLWVEXDVWHXIWKZTWVNVVLWVEXDVWGXCWKZWVOVVLWVEDUKZWVP + VVLWVQDUFZDXFZVVAWVRTZWVEWVQTAWVSVVHAWVQBDULZWVSVUTWVQBDYGVGYJAUYSWVRVVAA + UYSFUWPZYRZUYSWVRUYSWWBUWMABCDEFGHIJKLMNOPQRUWNZUWQYIWVQWVRVVADUWRVMAWVQW + VPVEZVVHADUYKTZWWEVUSBDXHVGYJURAWVPWVOVEZVVHAVWGXJTWWGAVWGVXAXJVYNAVWEVWO + XPUWSXDVWGUWTVGYJURWVEXDVWHUXAVGVVLWVMWVKWVHVVLWVMWVKVVLWVMWNWVMEXJTZWVKV + VLWVMYHAWWHVVHWVMVYLYFWVEXDVWHEUXBVMUXCUXDUXGWVFWVGWVHUXEUXHYJUXIYBYSYTAV + VKVVIVVBAVVKWNZVVIVVAFWWIVVAFVEZWNZFUYDSZUYBSZFUYFSZVVFVVGAWWMWWNVEVVKWWJ + ABCDEFGHIJKLMNOPQRUXJYFWWKVVEWWLUYBWWKVVAFUYDWWIWWJYHZWSWSWWKVVAFUYFWWOWS + YDWWIWVAWNZWUJVVAVVFVVGWWPHBIJVVAKAUYMVVKWVAMYFZAVUCVVKWVAVUEYFAVUBVVKWVA + VUGYFWWPVVABUYSAVVKWVAYPZVBZWWPVVABUYSWWRVSZYCWWPVVEVVAUYBWWPHBIUYCVVAKWW + QAWUKVVKWVAWULYFAWUMVVKWVAWUNYFWWSWWPWUQWUTWWPVVAFGWWIWVAYHWWPGVVAWWPVUJV + VHVQZGVVAVRAVUJVVKWVAPYFWWTGVVAUYSVTVMWAYKAWVCVVKWVAWVDYFYMYCWSWWPHBIDVVA + KWWQAWWFVVKWVAVUSYFAWWAVVKWVAVUTYFWWSWWPWVRWWCVEZWXAVVAWWBTVQZWVTVQZAWXBV + VKWVAWWDYFWWTWVAWXCWWIVVAFUXKUXLWXBWXDWXAWXCWNWVRUYSWWBVVAUXPUXMUXNYCYDYS + YTAVVBVVHVVKYQZWXEVVAUYSVVJYRZTAVVBVVAUYSVVJUXOAWXFBVVAAUYSBUGWXFBVEVUHUY + SBUXQUXRYLUXSVLUXTXNUYA $. + $} + ${ cycpm3.c $e |- C = ( toCyc ` D ) $. cycpm3.s $e |- S = ( SymGrp ` D ) $. @@ -468317,6 +469778,24 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a $} $} +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Simple groups +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + ${ + prmsimpcyc.1 $e |- B = ( Base ` G ) $. + $( A group of prime order is cyclic if and only if it is simple. This is + the first family of finite simple groups. (Contributed by Thierry + Arnoux, 21-Sep-2023.) $) + prmsimpcyc $p |- ( ( # ` B ) e. Prime -> ( G e. SimpGrp <-> G e. CycGrp ) ) + $= + ( chash cprime wcel csimpg ccyg cgrp simpggrp id prmcyg syl2anr wa cyggrp + cfv adantl simpl prmgrpsimpgd impbida ) ADPEFZBGFZBHFZUBBIFZUAUCUABJUAKAB + CLMUAUCNABCUCUDUABOQUAUCRST $. + $} + $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- @@ -468609,7 +470088,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- - Ordered rings and fields + Totally ordered rings and fields -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) @@ -477252,7 +478731,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be snex biimpi nfre1 nfan simpllr simpr eqtr4d simp-4l simplr syl21anc eqtrd nfv r19.29af2 0e0iccpnf syl6eqel nfmpt1 nfrn nfel ad2antlr sylibr elrnmpt vex eqid ax-mp r19.29af ex ssrdv adantr esummono 0ex esumsn breqtrd rabn0 - nfn wne necon1bi mpteq12df mpt0 syl6eq rn0 esumeq1d esumnul 0le0 syl6eqbr + nfn wne necon1bi mpteq12df mpt0 syl6eq rn0 esumeq1d esumnul 0le0 eqbrtrdi rneqd wral mptexgf rnexg simplll adantlr syl2anc eqeltrd ralrimiva esumcl 3syl cxr sseldi oveq1d xaddid2 3eqtr4d cin esumsplit eqtri elxrge0 sselii pm2.61dan jca iccssxr xrletri3 mpbird simpl esumeq2d ssrind neqned necomd @@ -486930,7 +488409,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cfz cuz wne caddc ballotlemiex simpld elfznn adantr ballotlemi1 eluz2b3 syl sylanbrc uz2m1nn cle wbr elnnuz biimpi eluzfz1 3syl wi ballotlemfp1 1nn a1i imp 1m1e0 fveq2i oveq1i ballotlemfval0 oveq1d 3eqtrrd cr wb 0re - 0le1 1re suble0 mp2an mpbir syl6eqbrr fveq2 breq1d rspcev syl2anc 1p0e1 + 0le1 1re suble0 mp2an mpbir eqbrtrrdi fveq2 breq1d rspcev syl2anc 1p0e1 0lt1 simprd eqtr3d cz nnzd 1zzd zsubcld ballotlemfelz zcnd 1cnd subaddd 0cnd mpbid syl5eqr breqtrid adantlr ballotlemfc0 ballotlemimin condan clt ) BKFUAUBZUCBUBZUDZUEZBHUFZBUBZEUGBGUFZUFUHUIEUCYEUCUJUKZUQUKZULZYD @@ -486959,7 +488438,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry wn wi 1nn ballotlemfp1 simprd 1m1e0 fveq2i oveq1i ballotlemfval0 oveq1d a1i imp 3eqtrrd breqtrid fveq2 breq2d rspcev syl2anc cneg df-neg eqtr3d clt 0cnd 1cnd cz nnzd 1zzd zsubcld ballotlemfelz subadd2d mpbird syl5eq - zcnd neg1lt0 syl6eqbrr adantlr ballotlemfcc ballotlemimin pm2.65da ) BK + zcnd neg1lt0 eqbrtrrdi adantlr ballotlemfcc ballotlemimin pm2.65da ) BK FUAUBZUCBUBZUDZBHUEZBUBZEUFBGUEZUEUGUHEUCXTUCUIUJZUKUJZUQZXSYAUDZABCDEG YCIJKLMNOPQXQBKUBZXRYABKFULZUMXSYCUNUBZYAXSXTUOUPUEUBZYIXSXTUNUBZXTUCUR YJXQYKXRXQXTUCIJVFUJZUKUJUBZYKXQYMXTYBUEZUGUHZABCDEFGHIJKLMNOPQRSTUSZUT @@ -487415,7 +488894,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry $( Range of ` R ` . (Contributed by Thierry Arnoux, 19-Apr-2017.) $) ballotlemrc $p |- ( C e. ( O \ E ) -> ( R ` C ) e. ( O \ E ) ) $= ( vv vu cdif wcel cfv cv cc0 cle wbr c1 caddc cfz wrex ballotlemro wceq - ballotlemiex simpld cfn cin chash cmin cmpo eqid ballotlemfrci syl6eqbr + ballotlemiex simpld cfn cin chash cmin cmpo eqid ballotlemfrci eqbrtrdi co 0le0 fveq2 breq1d rspcev syl2anc ballotlemodife sylanbrc ) BMHUGZUHZ BDUIZMUHFUJZVTIUIZUIZUKULUMZFUNKLUOVJUPVJZUQZVTVRUHABCDEFGHIJKLMNOPQRST UAUBUCUDURVSBJUIZWEUHZWGWBUIZUKULUMZWFVSWHWGBIUIUIUKUSABCFGHIJKLMNOPQRS @@ -500537,7 +502016,7 @@ have become an indirect lemma of the theorem in question (i.e. a lemma wf ispth cz nn0z fzoval reseq2d resabs1 syl6eqr cnveqd funeqd mpbid df-f1 sylanbrc csn snsspr1 imass2 wb 0elfz snssd resima2 sseq1 imaeq2d syl6reqr mpbiri ineq2d simp3bi eqtrd ssdisj f1resfz0f1d cvv cvtx fvexi hashf1dmcdm - syl2anr mp3an2 syl2an2r eqbrtrrd wn 0nn0m1nnn0 biimpri syl6eqbr pm2.61dan + syl2anr mp3an2 syl2an2r eqbrtrrd wn 0nn0m1nnn0 biimpri eqbrtrdi pm2.61dan sylan hashge0 ) BACUCFGZBUDFZHUAIZUBJZYDDUDFZKGYCYFUEZLYEMIZUDFZYDYGKYFYC YJYEHUFIZYDYEUGYCYDUBJZYDUHJYKYDNYCBACUIFGZYLABCUJZABCUKOZYDULYDUOUMZUNYC AYIPZUPJZYFYIDYQUQZYJYGKGZYCALYDMIZURYIUPJYRYCUUADAYCYMUUADAVSYNABCDEUSOZ @@ -502195,7 +503674,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by nnnn0 subfacf ffvelrni syl nn0zd crp faccl nnred rerpdivcl sylancl halfre zred epr readdcl cmin cabs wa cuz wo elnn1uz2 fveq2 syl6eq oveq1d subfac1 fac1 oveq12d rpreccl ax-mp rpre recni subid1i fveq2d rpge0 absid mp2an c3 - egt2lt3 simpli 2re 2pos epos ltrecii mpbi syl6eqbr eluz2nn resubcld recnd + egt2lt3 simpli 2re 2pos epos ltrecii mpbi eqbrtrdi eluz2nn resubcld recnd ere cc abscld nnrecred a1i subfaclim eluzle wb nnre nngt0 mpanl12 syl2anc lerec mpbid ltletrd jaoi sylbi absdifltd simpld ltsubaddd simprd ltadd1dd ltled addassd ax-1cn 2halves oveq2i breqtrd cz flbi mpbir2and eqcomd ) GU @@ -503392,7 +504871,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by ccn ctopon toptopon sylib simpr elpi1 cicc cxp wer phtpcer simpllr simplr simprl simprr eqtr4d sconnpht syl3anc sneqd xpeq2d breqtrd erthi ad2antrr a1i pi1id eqtrd velsn eqeq1 syl5bb syl5ibrcom rexlimdva sylbid ssrdv cgrp - expimpd pi1grp grpidcl snssd eqssd fvex ensn1 syl6eqbr adantll wi simplll + expimpd pi1grp grpidcl snssd eqssd fvex ensn1 eqbrtrdi adantll wi simplll wral simpll pconntop wf iiuni 0elunit ffvelrn sylancl eqidd eqcomd elpi1i cnf pcoptcl simp1d simp2d simp3d cgic pconnpi1 gicen entr en1eqsn eleqtrd w3a elsni erth mpbird expr ralrimiva issconn sylanbrc impbida ) AUCGZCBGZ @@ -512130,7 +513609,7 @@ Real and complex numbers (cont.) fz0n $p |- ( N e. NN0 -> ( ( 0 ... ( N - 1 ) ) = (/) <-> N = 0 ) ) $= ( cn0 wcel c1 cmin co cc0 clt wbr cfz c0 wceq cz wb nn0z sylancr cle bitr3d 0z cr peano2zm syl fzn cn wo elnn0 wn nnge1 1re wa subge0 0re resubcl lenlt - nnre sylancl mpbid nnne0 neneqd 2falsed cneg oveq1 syl6eqr neg1lt0 syl6eqbr + nnre sylancl mpbid nnne0 neneqd 2falsed cneg oveq1 syl6eqr neg1lt0 eqbrtrdi df-neg id 2thd jaoi sylbi ) ABCZADEFZGHIZGVLJFKLZAGLZVKGMCVLMCZVMVNNSVKAMCV PAOAUAUBGVLUCPVKAUDCZVOUEVMVONZAUFVQVRVOVQVMVOVQDAQIZVMUGZAUHVQATCZDTCZVSVT NAUOUIWAWBUJZGVLQIZVSVTADUKWCGTCVLTCWDVTNULADUMGVLUNPRUPUQVQAGAURUSUTVOVMVO @@ -537933,7 +539412,7 @@ Complex numbers (supplements) ad2ant2lr simprbi cbvrexv breq1 anbi12d pweq ineq1d rexeqdv rspccv mp2and imbi12d elinel1 ssdif difun2 difss2d sseq2 uniexg eqeltrid difexg disjdif ineq2d syl6eq inunissunidif sylan9bbr biimpd impancom anim2d difss ssdomg - anim12d fvineqsneq mp2 syl6eqbrr endomtr syl2anc syl6 expdimp jcad syl5bi + anim12d fvineqsneq mp2 eqbrtrrdi endomtr syl2anc syl6 expdimp jcad syl5bi elinel2 anass1rs 3adant3 syl3anc domsdomtr exlimiv sdomnen pm2.65da imnan anasss neq0 sylib ancrd syl6ibr lpss3 3expb reximdv an42s ralrimiva fveq2 imp lpss eleq12d cbvrexdva cbvralv pibp21 sylanbrc ) EDMZEUEMZCUJZBUJZEUF @@ -544729,7 +546208,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= cz cc cn0 uzid peano2uz 3syl eqeltrrd syld imp vtocl bitri a1i notbid ovex ianor jaod cmap ctop cvv adantr adantl oveq2d eleq1d cima fconst wf eqeq1d cpr elpri simprr fz1ssfz0 sseli anim2i crab csup chvarv 1re - lenlt 0lt1 syl6eqbr nsyl wor w3a ltso snfi fzfi rabfi unfi snid elun1 + lenlt 0lt1 eqbrtrdi nsyl wor w3a ltso snfi fzfi rabfi unfi snid elun1 ne0i mp2b snssi ssrab2 ssriv zssre unssi fisupcl elfzuz eluzfz2 simpl ralimi rspcva sylbi orim2i orel1 syl2im reximdv sylan2i mpcom elfzm1b nnzd syl2anr ex pm2.43d nncnd nnm1nn0 nn0zd fzss2 sseld anim2d syldan @@ -546500,7 +547979,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= syl5eq eqeltrrd recnd ifbieq1d eqtr4d iffalse adantlrl halfcld oveq2d ex cc cicc cofr cab itg2val supeq1i wor xrltso simprr eqeltrd rexlimiva citg2 abssi supxrcl mp1i eqeq1d ifbieq2d eqeq2d cbvrexv 0le0 rpge0 i1ff - syl6eqbr rpre addge01d readdcld ifcl iccssxr fss syl3anc mpand ralimdva + eqbrtrdi rpre addge01d readdcld ifcl iccssxr fss syl3anc mpand ralimdva xrletr 3imtr4d rexlimdva anim1d reximdva syl5bi ss2abdv simp3r 3ad2ant2 sseld w3a rexlimdv3a abssdv xrsupss supub syld supxrlub simprrr simplll mpan wex csn cxp c0 2rp ne0ii ffvelrn elxrge0 sylib ralrimiva ralrimivw @@ -548857,7 +550336,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= cpw xrlelttr sylanl1 mpd wlogle sylanb ralrimiva ralnex nne ralbii bitr3i rspceaimv ffnd fnfvelrn eqeq1 rspcva elun2 it0e0 adantlll an32s 00id 0cnd oveq2d subid1d ad3antrrr fvif ifeq2 eqtri mpteq2dva com23 imp32 anasss c1 - abs0 1rp ne0ii adantlrr rpre rehalfcld leidd breq1 ifboth ssneld syl6eqbr + abs0 1rp ne0ii adantlrr rpre rehalfcld leidd breq1 ifboth ssneld eqbrtrdi ifbothda xrletrd simplr rphalflt a1d r19.2z sylan2b rexlimdvva ralrimivva lttrd pm2.61dan ssid elcncf2 sylancl mpbir2and ) AHDEUJUKZULUUAUKSZWUBULH UMZUAUNZUBUNZUOUKZUTUPZUCUNZUQURZWUEHUPZWUFHUPZUOUKZUTUPZUDUNZUQURZUSZUAW @@ -650390,13 +651869,6 @@ base set if and only if the neighborhoods (convergents) of every point $. $} - ${ - neqcomd.1 $e |- ( ph -> -. A = B ) $. - $( Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - neqcomd $p |- ( ph -> -. B = A ) $= - ( wceq eqcom sylnib ) ABCECBEDBCFG $. - $} - ${ $d ph x $. $d x A $. rr-spce.1 $e |- ( ( ph /\ x = A ) -> ps ) $. @@ -650407,16 +651879,6 @@ base set if and only if the neighborhoods (convergents) of every point TADEGMCDNOASBCASBFPQR $. $} - ${ - $d x ph $. $d x B $. $d x A $. - rr-rspce.1 $e |- ( ( ph /\ x = A ) -> ps ) $. - rr-rspce.2 $e |- ( ph -> A e. B ) $. - $( Prove a restricted existential. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - rr-rspce $p |- ( ph -> E. x e. B ps ) $= - ( cv wceq wa simpl 2thd id rspcedvd ) ABACDEGACHDIZJBAFAOKLAMN $. - $} - ${ $d A x $. $d A y $. $d ps y $. $d th x $. $d ph y $. $d ch y $. rexlimdvaacbv.1 $e |- ( x = y -> ( ps <-> th ) ) $. @@ -650442,29 +651904,6 @@ base set if and only if the neighborhoods (convergents) of every point ( wrex rexlimdvaacbv mpd ) ADEGKBHADBCEFGJILM $. $} - ${ - $d D x $. $d x A $. $d x C $. $d x ph $. - rr-elrnmptd.1 $e |- F = ( x e. A |-> B ) $. - rr-elrnmptd.2 $e |- ( ph -> C e. A ) $. - rr-elrnmptd.3 $e |- ( ph -> D e. V ) $. - rr-elrnmptd.4 $e |- ( ( ph /\ x = C ) -> D = B ) $. - $( Elementhood in an image set. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - rr-elrnmptd $p |- ( ph -> D e. ran F ) $= - ( crn wcel wceq wrex rr-rspce wb elrnmpt syl mpbird ) AFGMNZFDOZBCPZAUCBE - CLJQAFHNUBUDRKBCDFGHISTUA $. - $} - - ${ - $d x C $. - rr-elrnmpt2d.1 $e |- F = ( x e. A |-> B ) $. - rr-elrnmpt2d.2 $e |- ( ph -> C e. ran F ) $. - $( Elementhood in an image set. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - rr-elrnmpt2d $p |- ( ph -> E. x e. A C = B ) $= - ( crn wcel wceq wrex elrnmpt ibi syl ) AEFIZJZEDKBCLZHQRBCDEFPGMNO $. - $} - ${ $d D x $. $d x A $. $d x C $. $d x ph $. rr-elrnmpt3d.1 $e |- F = ( x e. A |-> B ) $. @@ -650474,21 +651913,7 @@ base set if and only if the neighborhoods (convergents) of every point $( Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.) $) rr-elrnmpt3d $p |- ( ph -> D e. ran F ) $= - ( cv wceq wa eqcomd rr-elrnmptd ) ABCDEFGHIJKABMENODFLPQ $. - $} - - ${ - enpr2d.1 $e |- ( ph -> A e. C ) $. - enpr2d.2 $e |- ( ph -> B e. D ) $. - enpr2d.3 $e |- ( ph -> -. A = B ) $. - $( A pair with distinct elements is equinumerous to ordinal two. - (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - enpr2d $p |- ( ph -> { A , B } ~~ 2o ) $= - ( c1o cen csn cun wbr cin c0 wceq wcel syl con0 1on cpr csuc ensn1g en2sn - c2o sylancl wne neqned disjsn2 wn onirri a1i disjsn sylibr syl22anc df-pr - unen df-suc 3brtr4g df-2o breqtrrdi ) ABCUAZIUBZUEJABKZCKZLZIIKZLZVBVCJAV - DIJMZVEVGJMZVDVENOPZIVGNOPZVFVHJMABDQVIFBDUCRACEQISQVJGTCIESUDUFABCUGVKAB - CHUHBCUIRAIIQUJZVLVMAITUKULIIUMUNVDIVEVGUQUOBCUPIURUSUTVA $. + ( cv wceq wa eqcomd elrnmptdv ) ABCDEFGHIJKABMENODFLPQ $. $} ${ @@ -650503,88 +651928,6 @@ base set if and only if the neighborhoods (convergents) of every point IAUKUGEOUJBCAUGPQCBRSBCUAUBUCUD $. $} - ${ - rr-php2d.1 $e |- ( ph -> A e. Fin ) $. - rr-php2d.2 $e |- ( ph -> B C_ A ) $. - rr-php2d.3 $e |- ( ph -> A ~~ B ) $. - $( Strengthening of ~ php expressed without negation. (Contributed by - Rohan Ridenour, 3-Aug-2023.) $) - rr-php2d $p |- ( ph -> A = B ) $= - ( cen wbr wceq wn wa csdm cfn wcel wpss wss adantr simpr neqcomd dfpss2 - sylanbrc php3 syl2an2r sdomnen ensym nsyl syl ex mt4d ) ABCGHZBCIZFAUKJZU - JJZAULKZCBLHZUMABMNULCBOZUODUNCBPZCBIJUPAUQULEQUNBCAULRSCBTUABCUBUCUOCBGH - UJCBUDBCUEUFUGUHUI $. - $} - - ${ - phphashd.1 $e |- ( ph -> A e. Fin ) $. - phphashd.2 $e |- ( ph -> B C_ A ) $. - phphashd.3 $e |- ( ph -> ( # ` A ) = ( # ` B ) ) $. - $( Equivalent of ~ rr-php2d expressed using the hash function. - (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - phphashd $p |- ( ph -> A = B ) $= - ( chash cfv wceq cen wbr cfn wcel wb ssfid hashen syl2anc mpbid rr-php2d - ) ABCDEABGHCGHIZBCJKZFABLMCLMTUANDABCDEOBCPQRS $. - $} - - ${ - phphash2d.1 $e |- ( ph -> B e. Fin ) $. - phphash2d.2 $e |- ( ph -> A C_ B ) $. - phphash2d.3 $e |- ( ph -> ( # ` A ) = ( # ` B ) ) $. - $( Equivalent of ~ phphashd with reversed arguments. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - phphash2d $p |- ( ph -> A = B ) $= - ( chash cfv eqcomd phphashd ) ACBACBDEABGHCGHFIJI $. - $} - - ${ - hashelne0d.1 $e |- ( ph -> B e. A ) $. - hashelne0d.2 $e |- ( ph -> A e. V ) $. - $( A set with an element has nonzero size. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - hashelne0d $p |- ( ph -> -. ( # ` A ) = 0 ) $= - ( chash cfv cc0 wceq c0 ne0d neneqd wcel wb hasheq0 syl mtbird ) ABGHIJZB - KJZABKABCELMABDNSTOFBDPQR $. - $} - - ${ - $d ph x $. $d x A $. $d x B $. - hash1elsn.1 $e |- ( ph -> ( # ` A ) = 1 ) $. - hash1elsn.2 $e |- ( ph -> B e. A ) $. - hash1elsn.3 $e |- ( ph -> A e. V ) $. - $( A set of size 1 with a known element is the singleton of that element. - (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - hash1elsn $p |- ( ph -> A = { B } ) $= - ( vx cv csn wceq c1o cen wbr wex chash cfv c1 wcel syl wb mpbid en1 sylib - hashen1 wa simpr adantr eleqtrd elsni sneqd eqtr4d exlimddv ) ABHIZJZKZBC - JZKHABLMNZUPHOABPQRKZUREABDSUSURUAGBDUETUBHBUCUDAUPUFZBUOUQAUPUGZUTCUNUTC - UOSCUNKUTCBUOACBSUPFUHVAUICUNUJTUKULUM $. - $} - - ${ - gcdmuld.1 $e |- ( ph -> M e. NN0 ) $. - gcdmuld.2 $e |- ( ph -> N e. ZZ ) $. - $( The greatest common divisor of a nonnegative integer and a multiple of - it is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - gcdmuld $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 - CKLBKLFKLUFUIMEABDNZAOCBFPQABSLUFBMDBRTAUHUGBGAUGAUGACBEUJUAUBUCUDUE $. - $} - - ${ - gcddvdsd.1 $e |- ( ph -> M e. NN ) $. - gcddvdsd.2 $e |- ( ph -> N e. ZZ ) $. - gdcdvdsd.3 $e |- ( ph -> M || N ) $. - $( The greatest common divisor of a positive integer and another integer it - divides is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - gcddvdsd $p |- ( ph -> ( M gcd N ) = M ) $= - ( cdiv co cmul cgcd zcnd nncnd nnne0d divcan1d oveq2d nnnn0d cdvds wbr cz - wcel cc0 wne wb nnzd dvdsval2 syl3anc mpbid gcdmuld eqtr3d ) ABCBGHZBIHZJ - HBCJHBAUKCBJACBACEKABDLABDMZNOABUJABDPABCQRZUJSTZFABSTBUAUBCSTUMUNUCABDUD - ULEBCUEUFUGUHUI $. - $} - ${ suceqd.1 $e |- ( ph -> A = B ) $. $( Deduction associated with ~ suceq . (Contributed by Rohan Ridenour, @@ -651274,12 +652617,12 @@ collection and union ( ~ mnuop3d ), from which closure under pairing $( Lemma for ~ mnuprd . (Contributed by Rohan Ridenour, 11-Aug-2023.) $) mnuprdlem3 $p |- ( ph -> A. i e. { (/) , { (/) } } E. v e. F i e. v ) $= ( va cv wcel wrex c0 csn cpr wa wceq prid1 a1i simplr elpri simpr 3eltr4d - wo 0ex prex eleqtrri rr-rspce prid2 jaodan sylan2 elequ2 cbvrexv sylib ex - p0ex ralrimi ) AEJZBJKZBFLZEMMNZOZHAURVBKZUTAVCPURIJZKZIFLZUTVCAURMQZURVA - QZUDVFURMVAUAAVGVFVHAVGPZVEIMCNZOZFVIVDVKQZPZMVKURVDMVKKVMMVJUERSAVGVLTVI - VLUBUCVKFKVIVKVKVADNZOZOZFVKVOMVJUFRGUGSUHAVHPZVEIVOFVQVDVOQZPZVAVOURVDVA - VOKVSVAVNUPRSAVHVRTVQVRUBUCVOFKVQVOVPFVKVOVAVNUFUIGUGSUHUJUKVEUSIBFIBEULU - MUNUOUQ $. + wo prex eleqtrri rspcime p0ex prid2 jaodan sylan2 elequ2 cbvrexv sylib ex + 0ex ralrimi ) AEJZBJKZBFLZEMMNZOZHAURVBKZUTAVCPURIJZKZIFLZUTVCAURMQZURVAQ + ZUDVFURMVAUAAVGVFVHAVGPZVEIMCNZOZFVIVDVKQZPZMVKURVDMVKKVMMVJUPRSAVGVLTVIV + LUBUCVKFKVIVKVKVADNZOZOZFVKVOMVJUERGUFSUGAVHPZVEIVOFVQVDVOQZPZVAVOURVDVAV + OKVSVAVNUHRSAVHVRTVQVRUBUCVOFKVQVOVPFVKVOVAVNUEUIGUFSUGUJUKVEUSIBFIBEULUM + UNUOUQ $. $} ${ @@ -651408,20 +652751,20 @@ collection and union ( ~ mnuop3d ), from which closure under pairing mnurndlem1 $p |- ( ph -> ran F C_ w ) $= ( cv wcel wral wss wa cpr wceq cvv wfn cfv crn ffnd cuni cmpt wrex wi vex prid1 simpr eleqtrrid eqid id prex a1i preq1d preq12d adantl rr-elrnmpt3d - fveq2 rr-rspce rgen ralim mpisyl rgenw eleq2 unieq anbi12d rexrnmpt ax-mp - wb sseq1d wn simplrl prid2 elnotel elnelneq2d elpri orcomd fveq2d simplrr - ord sylc unipr sseq1i unss bicomi simprbi sylbi fvex prss simplbi eqeltrd - cun 3syl ex rexlimiva com12 ralimia syl fnfvrnss syl2anc ) AHEUAGMZHUBZBM - ZNZGEOZHUCXFPAEFHJUDAXDDMZNZXIUEZXFPZQZDIEIMZXNHUBZERZRZUFZUCZUGZGEOZXHAX - DCMZNZCXSUGZXTUHGEOYDGEOYALYDGEXDENZYCCXDXEERZRZXSYEYBYGSZQXDYGYBXDYFGUIU - JYEYHUKULYEIEXQXDYGXRTXRUMZYEUNYGTNYEXDYFUOUPXNXDSZXQYGSYEYJXNXDXPYFYJUNY - JXOXEEXNXDHVAUQURUSUTVBVCYDXTGEVDVEXTXGGEXTYEXGXTXDXQNZXQUEZXFPZQZIEUGZYE - XGUHZXQTNZIEOXTYOVLYQIEXNXPUOVFXMYNIDEXQXRTYIXIXQSZXJYKXLYMXIXQXDVGYRXKYL - XFXIXQVHVMVIVJVKYNYPIEXNENZYNQZYEXGYTYEQZXEXOXFUUAXDXNHUUAYKXDXPSZVNXDXNS - ZYSYKYMYEVOUUAXDXPEYTYEUKXPENVNZUUAEXPNUUDXOEKVPEXPVQVKUPVRYKUUBUUCYKUUCU - UBXDXNXPVSVTWCWDWAUUAYMXPXFPZXOXFNZYSYKYMYEWBYMXNXPWOZXFPZUUEYLUUGXFXNXPI - UIXOEUOWEWFUUHXNXFPZUUEUUIUUEQUUHXNXPXFWGWHWIWJUUEUUFEXFNZUUFUUJQUUEXOEXF - XNHWKKWLWHWMWPWNWQWRWJWSWTXAGEXFHXBXC $. + fveq2 rspcime rgen ralim mpisyl rgenw eleq2 unieq sseq1d anbi12d rexrnmpt + wb ax-mp wn simplrl prid2 elnotel elnelneq2d elpri orcomd ord sylc fveq2d + simplrr cun unipr sseq1i unss bicomi simprbi sylbi fvex prss simplbi 3syl + eqeltrd ex rexlimiva com12 ralimia syl fnfvrnss syl2anc ) AHEUAGMZHUBZBMZ + NZGEOZHUCXFPAEFHJUDAXDDMZNZXIUEZXFPZQZDIEIMZXNHUBZERZRZUFZUCZUGZGEOZXHAXD + CMZNZCXSUGZXTUHGEOYDGEOYALYDGEXDENZYCCXDXEERZRZXSYEYBYGSZQXDYGYBXDYFGUIUJ + YEYHUKULYEIEXQXDYGXRTXRUMZYEUNYGTNYEXDYFUOUPXNXDSZXQYGSYEYJXNXDXPYFYJUNYJ + XOXEEXNXDHVAUQURUSUTVBVCYDXTGEVDVEXTXGGEXTYEXGXTXDXQNZXQUEZXFPZQZIEUGZYEX + GUHZXQTNZIEOXTYOVLYQIEXNXPUOVFXMYNIDEXQXRTYIXIXQSZXJYKXLYMXIXQXDVGYRXKYLX + FXIXQVHVIVJVKVMYNYPIEXNENZYNQZYEXGYTYEQZXEXOXFUUAXDXNHUUAYKXDXPSZVNXDXNSZ + YSYKYMYEVOUUAXDXPEYTYEUKXPENVNZUUAEXPNUUDXOEKVPEXPVQVMUPVRYKUUBUUCYKUUCUU + BXDXNXPVSVTWAWBWCUUAYMXPXFPZXOXFNZYSYKYMYEWDYMXNXPWEZXFPZUUEYLUUGXFXNXPIU + IXOEUOWFWGUUHXNXFPZUUEUUIUUEQUUHXNXPXFWHWIWJWKUUEUUFEXFNZUUFUUJQUUEXOEXFX + NHWLKWMWIWNWOWPWQWRWKWSWTXAGEXFHXBXC $. $} ${ @@ -651504,31 +652847,31 @@ collection and union ( ~ mnuop3d ), from which closure under pairing syl3an1 alrimiv pwss ccoll cun wceq w3a ssun1 simp3 sseqtrrid simp1l3 wex sylibr wbr simp1r simpr unieqd simpl eqtr4d adantll simpll3 simprd simpld weq eqeltrd eleqtrrd 3jca simpl2 rr-spce simp1l1 syl simp2 gruuni syl2anc - rr-rspce simpl1 gruel syl3anc 3ad2ant1 sylan rexbidva mpbird rexex adantr - wb cpcoll2d cxp copab cin inss2 eqsstri grucollcld syl2an2r mpbid rexcom4 - a1i rexlimiva exlimiv sylbi elssuni syl6ss adantl sseqtr4d anim2d reximdv - ssun2 ex sylc rexlimdv3a ralrimiva jca 3expa grupw gruun ismnu ) ALMUIZBU - JZUKZLULZYTUFUJZULZFUJZUGUJZUIZUGDUMZUNZUGLUOUUDCUJZUIZUUIUPZUUBULZUNZCDU - JZUOZUQZFYSURZUNZUFLUOZDUSZUNZBLURZAUVABLAYSLUIZUNZUUAUUTUVDUHUJZYSULZUVE - LUIZUQZUHUSUUAUVDUVHUHAUVCUVFUVGALUTUIZUVCUVFUVGUBYSUVELVAVCVBVDUHYSLVEVO - UVDUUSDUVDUURUFYTYSKVFZUPZVGZLAUVCUUBUVLVHZUURAUVCUVMVIZUUCUUQUVNUVLYTUUB - YTUVKVJAUVCUVMVKVLUVNUUPFYSUVNFBUMZUNZUUHUUOUGLUVPUUELUIZUUHVIZUVMUUJUUKU - VJUIZUNZCUUNUOZUUOAUVCUVMUVOUVQUUHVMUVRGUJZUPZEUJZVHZUWBUUNUIZUUDUWBUIZVI - ZGVNZEUVJUOZUWAUVRUUDUWDKVPZEUVJUOUWJUVRFEYSKUVNUVOUVQUUHVQUVRUWKELUOZUWK - EVNUVRUWLUWIELUOUVRUWIEUUEUPZLUVRUWDUWMVHZUNZUWHGUUELUWOGUGWFZUNZUWEUWFUW - GUWNUWPUWEUVRUWNUWPUNZUWCUWMUWDUWRUWBUUEUWNUWPVRVSUWNUWPVTWAWBUWQUWBUUEUU - NUWOUWPVRZUWQUUFUUGUVPUVQUUHUWNUWPWCZWDWGUWQUUDUUEUWBUWQUUFUUGUWTWEUWSWHW - IUVPUVQUUHUWNWJWKUVRUVIUVQUWMLUIUVRAUVIAUVCUVMUVOUVQUUHWLZUBWMZUVPUVQUUHW - NUUELWOWPWQUVRUWKUWIELUVRUUDLUIZUWDLUIZUWKUWIXGZUVPUVQUXCUUHUVPUVIUVCUVOU - XCUVPAUVIAUVCUVMUVOWRUBWMAUVCUVMUVOWJZUVNUVOVRYSUUDLWSWTXAZUDXBXCXDUWKELX - EWMXHUVRUWKUWIEUVJUVRUXCUWDUVJUIZUXDUXEUXGUVRUXHUNUVIUVJLUIZUXHUXDUVRUVIU - XHUXBXFUVRAUXHUVCUXIUXAUVRUVCUXHUVPUVQUVCUUHUXFXAXFUVDYSKLAUVIUVCUBXFZKLL - XIZULUVDKSUJZUPRUJVHUXLUUNUIQUJUXLUIVISVNQRXJZUXKXKUXKUCUXMUXKXLXMXRAUVCV - RXNZXOUVRUXHVRUVJUWDLWSWTUDXOXCXPUWJUWHEUVJUOZGVNUWAUWHEGUVJXQUXOUWAGUWHU - WAEUVJUEXSXTYAWMUVMUVTUUMCUUNUVMUVSUULUUJUVMUVSUULUVMUVSUNUUKUVLUUBUVSUUK - UVLULUVMUVSUUKUVKUVLUUKUVJYBUVKYTYHYCYDUVMUVSVTYEYIYFYGYJYKYLYMYNUVDUVIYT - LUIZUVKLUIZUVLLUIUXJAUVIUVCUXPUBYSLYOXBAUVIUVCUXIUXQUBUXNUVJLWOXOYTUVKLYP - WTWQVDYMYLAUVIYRUVBXGUBBUFUGCLDFHIJMUTNOPTUAYQWMXD $. + rspcime simpl1 gruel 3ad2ant1 sylan rexbidva mpbird rexex cpcoll2d adantr + syl3anc cxp copab cin inss2 eqsstri a1i grucollcld syl2an2r mpbid rexcom4 + wb rexlimiva exlimiv sylbi elssuni ssun2 syl6ss adantl sseqtr4d ex anim2d + reximdv sylc rexlimdv3a ralrimiva jca 3expa grupw gruun ismnu ) ALMUIZBUJ + ZUKZLULZYTUFUJZULZFUJZUGUJZUIZUGDUMZUNZUGLUOUUDCUJZUIZUUIUPZUUBULZUNZCDUJ + ZUOZUQZFYSURZUNZUFLUOZDUSZUNZBLURZAUVABLAYSLUIZUNZUUAUUTUVDUHUJZYSULZUVEL + UIZUQZUHUSUUAUVDUVHUHAUVCUVFUVGALUTUIZUVCUVFUVGUBYSUVELVAVCVBVDUHYSLVEVOU + VDUUSDUVDUURUFYTYSKVFZUPZVGZLAUVCUUBUVLVHZUURAUVCUVMVIZUUCUUQUVNUVLYTUUBY + TUVKVJAUVCUVMVKVLUVNUUPFYSUVNFBUMZUNZUUHUUOUGLUVPUUELUIZUUHVIZUVMUUJUUKUV + JUIZUNZCUUNUOZUUOAUVCUVMUVOUVQUUHVMUVRGUJZUPZEUJZVHZUWBUUNUIZUUDUWBUIZVIZ + GVNZEUVJUOZUWAUVRUUDUWDKVPZEUVJUOUWJUVRFEYSKUVNUVOUVQUUHVQUVRUWKELUOZUWKE + VNUVRUWLUWIELUOUVRUWIEUUEUPZLUVRUWDUWMVHZUNZUWHGUUELUWOGUGWFZUNZUWEUWFUWG + UWNUWPUWEUVRUWNUWPUNZUWCUWMUWDUWRUWBUUEUWNUWPVRVSUWNUWPVTWAWBUWQUWBUUEUUN + UWOUWPVRZUWQUUFUUGUVPUVQUUHUWNUWPWCZWDWGUWQUUDUUEUWBUWQUUFUUGUWTWEUWSWHWI + UVPUVQUUHUWNWJWKUVRUVIUVQUWMLUIUVRAUVIAUVCUVMUVOUVQUUHWLZUBWMZUVPUVQUUHWN + UUELWOWPWQUVRUWKUWIELUVRUUDLUIZUWDLUIZUWKUWIXRZUVPUVQUXCUUHUVPUVIUVCUVOUX + CUVPAUVIAUVCUVMUVOWRUBWMAUVCUVMUVOWJZUVNUVOVRYSUUDLWSXGWTZUDXAXBXCUWKELXD + WMXEUVRUWKUWIEUVJUVRUXCUWDUVJUIZUXDUXEUXGUVRUXHUNUVIUVJLUIZUXHUXDUVRUVIUX + HUXBXFUVRAUXHUVCUXIUXAUVRUVCUXHUVPUVQUVCUUHUXFWTXFUVDYSKLAUVIUVCUBXFZKLLX + HZULUVDKSUJZUPRUJVHUXLUUNUIQUJUXLUIVISVNQRXIZUXKXJUXKUCUXMUXKXKXLXMAUVCVR + XNZXOUVRUXHVRUVJUWDLWSXGUDXOXBXPUWJUWHEUVJUOZGVNUWAUWHEGUVJXQUXOUWAGUWHUW + AEUVJUEXSXTYAWMUVMUVTUUMCUUNUVMUVSUULUUJUVMUVSUULUVMUVSUNUUKUVLUUBUVSUUKU + VLULUVMUVSUUKUVKUVLUUKUVJYBUVKYTYCYDYEUVMUVSVTYFYGYHYIYJYKYLYMYNUVDUVIYTL + UIZUVKLUIZUVLLUIUXJAUVIUVCUXPUBYSLYOXAAUVIUVCUXIUXQUBUXNUVJLWOXOYTUVKLYPX + GWQVDYMYLAUVIYRUVBXRUBBUFUGCLDFHIJMUTNOPTUAYQWMXC $. $} ${ @@ -651544,14 +652887,14 @@ collection and union ( ~ mnuop3d ), from which closure under pairing ( vu vj vd vc vb cv wcel wa vz vf vh cuni wceq w3a wex copab cxp cin eqid vi wbr brxp brin rbaib sylbir vex weq unieqd simplr eqeq12d elequ1 adantl wb simpr eleq12 adantlr 3anbi123d cbvexdva braba simplr3 eleqtrrd simplr1 - syl6bb ccoll eqtrd simpll eqeltrd jca simpr2 rr-rspce grumnudlem ) AUAMUB - UCULNBCDORZUDZPRZUEZWDUBRZSZQRZWDSZUFZOUGZQPUHZEEUIZUJZEFGHIQPOJKLWPUKULR - ZESUCRZESTZWQWRWPUMZWQWRWNUMZNRZUDZWRUEZXBWHSZWQXBSZUFZNUGZWSWQWRWOUMZWTX - AVEWQWREEUNWTXAXIWQWRWNWOUOUPUQWMXHQPWQWRWNULURUCURWJWQUEZWFWRUEZTZWLXGON - XLONUSZTZWGXDWIXEWKXFXNWEXCWFWRXNWDXBXLXMVFUTXJXKXMVAVBXMWIXEVEXLONUBVCVD - XJXMWKXFVEXKWJWQWDXBVGVHVIVJWNUKVKVOWRUARWPVPZSZXGTZWQMRZSZXRUDZXOSZTMXBW - HXQMNUSZTZXSYAYCWQXBXRXDXEXFXPYBVLXQYBVFZVMYCXTWRXOYCXTXCWRYCXRXBYDUTXDXE - XFXPYBVNVQXPXGYBVRVSVTXPXDXEXFWAWBWC $. + syl6bb ccoll eqtrd simpll eqeltrd jca simpr2 rspcime grumnudlem ) AUAMUBU + CULNBCDORZUDZPRZUEZWDUBRZSZQRZWDSZUFZOUGZQPUHZEEUIZUJZEFGHIQPOJKLWPUKULRZ + ESUCRZESTZWQWRWPUMZWQWRWNUMZNRZUDZWRUEZXBWHSZWQXBSZUFZNUGZWSWQWRWOUMZWTXA + VEWQWREEUNWTXAXIWQWRWNWOUOUPUQWMXHQPWQWRWNULURUCURWJWQUEZWFWRUEZTZWLXGONX + LONUSZTZWGXDWIXEWKXFXNWEXCWFWRXNWDXBXLXMVFUTXJXKXMVAVBXMWIXEVEXLONUBVCVDX + JXMWKXFVEXKWJWQWDXBVGVHVIVJWNUKVKVOWRUARWPVPZSZXGTZWQMRZSZXRUDZXOSZTMXBWH + XQMNUSZTZXSYAYCWQXBXRXDXEXFXPYBVLXQYBVFZVMYCXTWRXOYCXTXCWRYCXRXBYDUTXDXEX + FXPYBVNVQXPXGYBVRVSVTXPXDXEXFWAWBWC $. $} ${ @@ -651696,11 +653039,11 @@ collection and union ( ~ mnuop3d ), from which closure under pairing inaex $p |- ( A e. On -> E. x e. Inacc A e. x ) $= ( con0 wcel cv cina csuc cdif cint wceq wa wss cwina inawina winaon ssriv syl onmindif c0 cvv mpan adantr simpr eleqtrrd difss sstri inaprc neli wi - wne ssdif0 sucexg ssexg expcom syl5bir mtoi neqned onint sylancr rr-rspce - eldifad ) BCDZBAEZDAFBGZHZIZFVBVCVFJZKBVFVCVBBVFDZVGFCLVBVHAFCVCFDVCMDVCC - DVCNVCOQPZFBRUAUBVBVGUCUDVBVFFVDVBVECLVESUJVFVEDVEFCFVDUEVIUFVBVESVBVESJZ - FTDZFTUGUHVJFVDLZVBVKFVDUKVBVDTDZVLVKUIBCULVLVMVKFVDTUMUNQUOUPUQVEURUSVAU - T $. + wne ssdif0 sucexg ssexg expcom syl5bir mtoi onint sylancr eldifad rspcime + neqned ) BCDZBAEZDAFBGZHZIZFVBVCVFJZKBVFVCVBBVFDZVGFCLVBVHAFCVCFDVCMDVCCD + VCNVCOQPZFBRUAUBVBVGUCUDVBVFFVDVBVECLVESUJVFVEDVEFCFVDUEVIUFVBVESVBVESJZF + TDZFTUGUHVJFVDLZVBVKFVDUKVBVDTDZVLVKUIBCULVLVMVKFVDTUMUNQUOUPVAVEUQURUSUT + $. $} ${ @@ -651710,9 +653053,9 @@ collection and union ( ~ mnuop3d ), from which closure under pairing gruex $p |- E. y e. Univ x e. y $= ( vz crnk cfv wcel cina wrex cgru con0 rankon inaex ax-mp cr1 wceq simplr cv wa wb syl cwina inawina winaon ad2antrr vex rankr1a mpbird simpr simpl - eleqtrrd inagrud rr-rspce rexlimiva ) AQZDEZCQZFZCGHZUNBQZFZBIHZUOJFURUNK - CUOLMUQVACGUPGFZUQRZUTBUPNEZIVCUSVDOZRZUNVDUSVFUNVDFZUQVBUQVEPVFUPJFZVGUQ - SVBVHUQVEVBUPUAFVHUPUBUPUCTUDUNUPAUEUFTUGVCVEUHUJVCUPVBUQUIUKULUMM $. + eleqtrrd inagrud rspcime rexlimiva ) AQZDEZCQZFZCGHZUNBQZFZBIHZUOJFURUNKC + UOLMUQVACGUPGFZUQRZUTBUPNEZIVCUSVDOZRZUNVDUSVFUNVDFZUQVBUQVEPVFUPJFZVGUQS + VBVHUQVEVBUPUAFVHUPUBUPUCTUDUNUPAUEUFTUGVCVEUHUJVCUPVBUQUIUKULUMM $. $} ${ @@ -651749,606 +653092,6 @@ collection and union ( ~ mnuop3d ), from which closure under pairing URAABTUAABCDEFGHIJKUBUCUD $. $} - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Groups -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - ${ - 0idnsgd.1 $e |- B = ( Base ` G ) $. - 0idnsgd.2 $e |- .0. = ( 0g ` G ) $. - 0idnsgd.3 $e |- ( ph -> G e. Grp ) $. - $( The whole group and the zero subgroup are normal subgroups of a group. - (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - 0idnsgd $p |- ( ph -> { { .0. } , B } C_ ( NrmSGrp ` G ) ) $= - ( csn cnsg cfv cgrp wcel 0nsg syl nsgid prssd ) ADHZBCIJZACKLZQRLGCDFMNAS - BRLGBCEONP $. - $} - - ${ - trivsubgd.1 $e |- B = ( Base ` G ) $. - trivsubgd.2 $e |- .0. = ( 0g ` G ) $. - trivsubgd.3 $e |- ( ph -> G e. Grp ) $. - trivsubgd.4 $e |- ( ph -> B = { .0. } ) $. - trivsubgd.5 $e |- ( ph -> A e. ( SubGrp ` G ) ) $. - $( The only subgroup of a trivial group is itself. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - trivsubgd $p |- ( ph -> A = B ) $= - ( csn csubg cfv wcel wss subgss syl sseqtrd subg0cl snssd eqssd eqtr4d ) - ABEKZCABUCABCUCABDLMNZBCOJCBDFPQIRAEBAUDEBNJBDEGSQTUAIUB $. - $} - - ${ - $d x B $. $d x G $. $d x ph $. $d x .0. $. - trivsubgd2.1 $e |- B = ( Base ` G ) $. - trivsubgd2.2 $e |- .0. = ( 0g ` G ) $. - trivsubgd2.3 $e |- ( ph -> G e. Grp ) $. - trivsubgd2.4 $e |- ( ph -> B = { .0. } ) $. - $( The only subgroup of a trivial group is itself. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - trivsubgd2 $p |- ( ph -> ( SubGrp ` G ) = { B } ) $= - ( vx csubg cfv csn cv wcel wa wceq cgrp adantr simpr trivsubgd sylibr syl - velsn ex ssrdv subgid snssd eqssd ) ACJKZBLZAIUIUJAIMZUINZUKUJNZAULOZUKBP - UMUNUKBCDEFACQNZULGRABDLPULHRAULSTIBUCUAUDUEABUIAUOBUINGBCEUFUBUGUH $. - $} - - ${ - $d x B $. $d x G $. $d x ph $. $d x .0. $. - trivnsgd.1 $e |- B = ( Base ` G ) $. - trivnsgd.2 $e |- .0. = ( 0g ` G ) $. - trivnsgd.3 $e |- ( ph -> G e. Grp ) $. - trivnsgd.4 $e |- ( ph -> B = { .0. } ) $. - $( The only normal subgroup of a trivial group is itself. (Contributed by - Rohan Ridenour, 3-Aug-2023.) $) - trivnsgd $p |- ( ph -> ( NrmSGrp ` G ) = { B } ) $= - ( vx cnsg cfv csn csubg cv wcel wi nsgsubg a1i ssrdv trivsubgd2 nsgid syl - sseqtrd cgrp snssd eqssd ) ACJKZBLZAUGCMKZUHAIUGUIINZUGOUJUIOPAUJCQRSABCD - EFGHTUCABUGACUDOBUGOGBCEUAUBUEUF $. - $} - - ${ - triv1nsgd.1 $e |- B = ( Base ` G ) $. - triv1nsgd.2 $e |- .0. = ( 0g ` G ) $. - triv1nsgd.3 $e |- ( ph -> G e. Grp ) $. - triv1nsgd.4 $e |- ( ph -> B = { .0. } ) $. - $( A trivial group has exactly one normal subgroup. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - triv1nsgd $p |- ( ph -> ( NrmSGrp ` G ) ~~ 1o ) $= - ( cnsg cfv csn c1o cen trivnsgd cvv wcel wbr snex syl6eqel ensn1g eqbrtrd - syl ) ACIJBKZLMABCDEFGHNABOPUCLMQABDKOHDRSBOTUBUA $. - $} - - ${ - 1nsgtrivd.1 $e |- B = ( Base ` G ) $. - 1nsgtrivd.2 $e |- .0. = ( 0g ` G ) $. - 1nsgtrivd.3 $e |- ( ph -> G e. Grp ) $. - 1nsgtrivd.4 $e |- ( ph -> ( NrmSGrp ` G ) ~~ 1o ) $. - $( A group with exactly one normal subgroup is trivial. (Contributed by - Rohan Ridenour, 3-Aug-2023.) $) - 1nsgtrivd $p |- ( ph -> B = { .0. } ) $= - ( csn wcel wceq cnsg cfv cgrp nsgid syl c1o cen wbr cvv 0nsg en1eqsn snex - syl2anc eleqtrd wb elsn2g mp1i mpbid ) ABDIZIZJZBUJKZABCLMZUKACNJZBUNJGBC - EOPAUJUNJZUNQRSUNUKKAUOUPGCDFUAPHUJUNUBUDUEUJTJULUMUFADUCBUJTUGUHUI $. - $} - - ${ - $d .x. n $. $d A n $. $d B n $. $d n G $. $d n O $. - cycsubggenodd.1 $e |- B = ( Base ` G ) $. - cycsubggenodd.2 $e |- .x. = ( .g ` G ) $. - cycsubggenodd.3 $e |- O = ( od ` G ) $. - cycsubggenodd.4 $e |- ( ph -> G e. Grp ) $. - cycsubggenodd.5 $e |- ( ph -> A e. B ) $. - cycsubggenodd.6 $e |- ( ph -> C = ran ( n e. ZZ |-> ( n .x. A ) ) ) $. - $( Relationship between the order of a subgroup and the order of a - generator of the subgroup. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - cycsubggenodd $p |- ( ph -> - ( O ` A ) = if ( C e. Fin , ( # ` C ) , 0 ) ) $= - ( cfv cfn wcel chash cc0 cif cz cv cmpt crn cgrp wceq eqid syl2anc eqcomd - co dfod2 eleq1d fveq2d ifbieq1d eqtrd ) ABHOZFUAFUBBEUJUCZUDZPQZURROZSTZD - PQZDROZSTAGUEQBCQUPVAUFLMFBEUQGHCIKJUQUGUKUHAUSVBUTVCSAURDPADURNUIZULAURD - RVDUMUNUO $. - $} - - ${ - mulgcld.1 $e |- B = ( Base ` G ) $. - mulgcld.2 $e |- .x. = ( .g ` G ) $. - mulgcld.3 $e |- ( ph -> G e. Grp ) $. - mulgcld.4 $e |- ( ph -> N e. ZZ ) $. - mulgcld.5 $e |- ( ph -> X e. B ) $. - $( Deduction associated with ~ mulgcl . (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - mulgcld $p |- ( ph -> ( N .x. X ) e. B ) $= - ( cgrp wcel cz co mulgcl syl3anc ) ADLMENMFBMEFCOBMIJKBCDEFGHPQ $. - $} - - ${ - odcld.1 $e |- B = ( Base ` G ) $. - odcld.2 $e |- O = ( od ` G ) $. - odcld.3 $e |- ( ph -> A e. B ) $. - $( Deduction associated with ~ odcl . (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - odcld $p |- ( ph -> ( O ` A ) e. NN0 ) $= - ( wcel cfv cn0 odcl syl ) ABCIBEJKIHBDECFGLM $. - $} - - ${ - ablgrpd.1 $e |- ( ph -> G e. Abel ) $. - $( Deduction associated with ~ ablgrp . (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - ablgrpd $p |- ( ph -> G e. Grp ) $= - ( cabl wcel cgrp ablgrp syl ) ABDEBFECBGH $. - $} - - ${ - hashfinmndnn.1 $e |- B = ( Base ` G ) $. - hashfinmndnn.2 $e |- ( ph -> G e. Mnd ) $. - hashfinmndnn.3 $e |- ( ph -> B e. Fin ) $. - $( A finite monoid has positive integer size. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - hashfinmndnn $p |- ( ph -> ( # ` B ) e. NN ) $= - ( chash cfv cn0 wcel cc0 wne cn cfn hashcl syl c0g cmnd eqid mndidcl - hashelne0d neqned elnnne0 sylanbrc ) ABGHZIJZUEKLUEMJABNJUFFBOPAUEKABCQHZ - NACRJUGBJEBCUGDUGSTPFUAUBUEUCUD $. - $} - - ${ - hashfingrpnn.1 $e |- B = ( Base ` G ) $. - hashfingrpnn.2 $e |- ( ph -> G e. Grp ) $. - hashfingrpnn.3 $e |- ( ph -> B e. Fin ) $. - $( A finite group has positive integer size. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - hashfingrpnn $p |- ( ph -> ( # ` B ) e. NN ) $= - ( cgrp wcel cmnd grpmnd syl hashfinmndnn ) ABCDACGHCIHECJKFL $. - $} - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Simple groups -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - $c SimpGrp $. - - $( Extend class notation with the class of simple groups. $) - csimpg $a class SimpGrp $. - - $( Define class of all simple groups. A simple group is a group ( ~ df-grp ) - with exactly two normal subgroups. These are always the subgroup of all - elements and the subgroup containing only the identity ( ~ simpgnsgbid ). - (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - df-simpg $a |- SimpGrp = { g e. Grp | ( NrmSGrp ` g ) ~~ 2o } $. - - ${ - $d g G $. - $( The predicate "is a simple group". (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - issimpg $p |- ( G e. SimpGrp <-> ( G e. Grp /\ ( NrmSGrp ` G ) ~~ 2o ) ) $= - ( vg cnsg cfv c2o cen wbr cgrp csimpg wceq fveq2 breq1d df-simpg elrab2 - cv ) BOZCDZEFGACDZEFGBAHIPAJQREFPACKLBMN $. - $} - - ${ - issimpgd.1 $e |- ( ph -> G e. Grp ) $. - issimpgd.2 $e |- ( ph -> ( NrmSGrp ` G ) ~~ 2o ) $. - $( Deduce a simple group from its properties. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - issimpgd $p |- ( ph -> G e. SimpGrp ) $= - ( cgrp wcel cnsg cfv c2o cen wbr csimpg issimpg sylanbrc ) ABEFBGHIJKBLFC - DBMN $. - $} - - $( A simple group is a group. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - simpggrp $p |- ( G e. SimpGrp -> G e. Grp ) $= - ( csimpg wcel cgrp cnsg cfv c2o cen wbr issimpg simplbi ) ABCADCAEFGHIAJK - $. - - ${ - simpggrpd.1 $e |- ( ph -> G e. SimpGrp ) $. - $( A simple group is a group. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - simpggrpd $p |- ( ph -> G e. Grp ) $= - ( csimpg wcel cgrp simpggrp syl ) ABDEBFECBGH $. - $} - - $( A simple group has two normal subgroups. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - simpg2nsg $p |- ( G e. SimpGrp -> ( NrmSGrp ` G ) ~~ 2o ) $= - ( csimpg wcel cgrp cnsg cfv c2o cen wbr issimpg simprbi ) ABCADCAEFGHIAJK - $. - - ${ - trivnsimpgd.1 $e |- B = ( Base ` G ) $. - trivnsimpgd.2 $e |- .0. = ( 0g ` G ) $. - trivnsimpgd.3 $e |- ( ph -> G e. Grp ) $. - trivnsimpgd.4 $e |- ( ph -> B = { .0. } ) $. - $( Trivial groups are not simple. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - trivnsimpgd $p |- ( ph -> -. G e. SimpGrp ) $= - ( cnsg cfv c2o cen wbr csimpg wcel csn snnen2o trivnsgd breq1d mtbiri - simpg2nsg nsyl ) ACIJZKLMZCNOAUDBPZKLMBQAUCUEKLABCDEFGHRSTCUAUB $. - $} - - ${ - simpgntrivd.1 $e |- B = ( Base ` G ) $. - simpgntrivd.2 $e |- .0. = ( 0g ` G ) $. - simpgntrivd.3 $e |- ( ph -> G e. SimpGrp ) $. - $( Simple groups are nontrivial. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - simpgntrivd $p |- ( ph -> -. B = { .0. } ) $= - ( csn wceq csimpg wcel adantr cgrp simpggrpd simpr trivnsimpgd pm2.65da - wa ) ABDHIZCJKZATSGLASRBCDEFACMKSACGNLASOPQ $. - $} - - ${ - $d x .0. $. $d x B $. - simpgnideld.1 $e |- B = ( Base ` G ) $. - simpgnideld.2 $e |- .0. = ( 0g ` G ) $. - simpgnideld.3 $e |- ( ph -> G e. SimpGrp ) $. - $( A simple group contains a nonidentity element. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - simpgnideld $p |- ( ph -> E. x e. B -. x = .0. ) $= - ( cv wceq wral wn wrex csn simpgntrivd c0 wne wb cgrp wcel cmnd simpggrpd - grpmnd mndidcl 3syl ne0d eqsn syl mtbid rexnal sylibr ) ABIEJZBCKZLULLBCM - ACENJZUMACDEFGHOACPQUNUMRACEADSTDUATECTADHUBDUCCDEFGUDUEUFBCEUGUHUIULBCUJ - UK $. - $} - - ${ - simpgnsgd.1 $e |- B = ( Base ` G ) $. - simpgnsgd.2 $e |- .0. = ( 0g ` G ) $. - simpgnsgd.3 $e |- ( ph -> G e. SimpGrp ) $. - $( The only normal subgroups of a simple group are the group itself and the - trivial group. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - simpgnsgd $p |- ( ph -> ( NrmSGrp ` G ) = { { .0. } , B } ) $= - ( cnsg cfv csn c2o cfn wcel cen wbr a1i syl syl2anc cvv cbs cpr 2onn nnfi - com simpg2nsg enfii simpggrpd 0idnsgd snex wceq fvex syl6eqel simpgntrivd - csimpg neqcomd enpr2d ensymd entr rr-php2d ) ACHIZDJZBUAZAKLMZUTKNOZUTLMA - KUDMZVCVEAUBPKUCQACUNMVDGCUEQZUTKUFRABCDEFACGUGUHAVDKVBNOUTVBNOVFAVBKAVAB - SSVASMADUIPABCTIZSBVGUJAEPCTUKULABVAABCDEFGUMUOUPUQUTKVBURRUS $. - $} - - ${ - simpgnsgeqd.1 $e |- B = ( Base ` G ) $. - simpgnsgeqd.2 $e |- .0. = ( 0g ` G ) $. - simpgnsgeqd.3 $e |- ( ph -> G e. SimpGrp ) $. - simpgnsgeqd.4 $e |- ( ph -> A e. ( NrmSGrp ` G ) ) $. - $( A normal subgroup of a simple group is either the whole group or the - trivial subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - simpgnsgeqd $p |- ( ph -> ( A = { .0. } \/ A = B ) ) $= - ( csn cpr wcel wceq wo cnsg cfv simpgnsgd eleqtrd elpri syl ) ABEJZCKZLBU - AMBCMNABDOPUBIACDEFGHQRBUACST $. - $} - - ${ - $d ph x $. $d x .0. $. $d x B $. $d x G $. - 2nsgsimpgd.1 $e |- B = ( Base ` G ) $. - 2nsgsimpgd.2 $e |- .0. = ( 0g ` G ) $. - 2nsgsimpgd.3 $e |- ( ph -> G e. Grp ) $. - 2nsgsimpgd.4 $e |- ( ph -> -. { .0. } = B ) $. - 2nsgsimpgd.5 $e |- ( ( ph /\ x e. ( NrmSGrp ` G ) ) -> - ( x = { .0. } \/ x = B ) ) $. - $( If any normal subgroup of a nontrivial group is either the trivial - subgroup or the whole group, the group is simple. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - 2nsgsimpgd $p |- ( ph -> G e. SimpGrp ) $= - ( wcel wa wceq adantl simpr syl adantr eqeltrd adantlr cvv cfv csn cpr cv - cnsg c2o cen wo elprg mpbird cgrp 0nsg nsgid elpri mpjaodan impbida eqrdv - wb snex a1i cbs fvexi enpr2d eqbrtrd issimpgd ) ADHADUEUAZEUBZCUCZUFUGABV - FVHABUDZVFKZVIVHKZAVJLVKVIVGMZVICMZUHZJVJVKVNURAVIVGCVFUINUJAVKLVLVJVMAVL - VJVKAVLLVIVGVFAVLOAVGVFKZVLADUKKZVOHDEGULPQRSAVMVJVKAVMLVICVFAVMOACVFKZVM - AVPVQHCDFUMPQRSVKVNAVIVGCUNNUOUPUQAVGCTTVGTKAEUSUTCTKACDVAFVBUTIVCVDVE $. - $} - - ${ - $d ph x $. $d x .0. $. $d x B $. $d x G $. - simpgnsgbid.1 $e |- B = ( Base ` G ) $. - simpgnsgbid.2 $e |- .0. = ( 0g ` G ) $. - simpgnsgbid.3 $e |- ( ph -> G e. Grp ) $. - simpgnsgbid.4 $e |- ( ph -> -. { .0. } = B ) $. - $( A nontrivial group is simple if and only if its normal subgroups are - exactly the group itself and the trivial subgroup. (Contributed by - Rohan Ridenour, 4-Aug-2023.) $) - simpgnsgbid $p |- ( ph -> - ( G e. SimpGrp <-> ( NrmSGrp ` G ) = { { .0. } , B } ) ) $= - ( vx csimpg wcel cnsg cfv csn cpr wceq wa simpr simpgnsgd adantr wn cv wo - cgrp simplr eleqtrd elpri syl 2nsgsimpgd impbida ) ACJKZCLMZDNZBOZPZAUKQB - CDEFAUKRSAUOQZIBCDEFACUDKUOGTAUMBPUAUOHTUPIUBZULKZQZUQUNKUQUMPUQBPUCUSUQU - LUNUPURRAUOURUEUFUQUMBUGUHUIUJ $. - $} - -$( --.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- - Classification of abelian simple groups --.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- -$) - - ${ - $d ph n $. $d A n $. - cycsubggend.1 $e |- B = ( Base ` G ) $. - cycsubggend.2 $e |- .x. = ( .g ` G ) $. - cycsubggend.3 $e |- F = ( n e. ZZ |-> ( n .x. A ) ) $. - cycsubggend.4 $e |- ( ph -> A e. B ) $. - $( The cyclic subgroup generated by ` A ` includes its generator. - (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - cycsubggend $p |- ( ph -> A e. ran F ) $= - ( cz cv co c1 1zzd wceq wa simpr oveq1d wcel adantr mulg1 syl rr-elrnmptd - eqtr2d ) AELEMZBDNZOBFCJAPKAUGOQZRZUHOBDNZBUJUGOBDAUISTUJBCUAZUKBQAULUIKU - BCDGBHIUCUDUFUE $. - $} - - ${ - $d G n $. $d .x. n $. $d B n $. $d A n $. - cycsubgcld.1 $e |- B = ( Base ` G ) $. - cycsubgcld.2 $e |- .x. = ( .g ` G ) $. - cycsubgcld.3 $e |- F = ( n e. ZZ |-> ( n .x. A ) ) $. - cycsubgcld.4 $e |- ( ph -> G e. Grp ) $. - cycsubgcld.5 $e |- ( ph -> A e. B ) $. - $( The cyclic subgroup generated by ` A ` is a subgroup. Deduction related - to ~ cycsubgcl . (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - cycsubgcld $p |- ( ph -> ran F e. ( SubGrp ` G ) ) $= - ( crn csubg cfv wcel cgrp wa cycsubgcl syl2anc simpld ) AFMZGNOPZBUBPZAGQ - PBCPUCUDRKLEBDFGCHIJSTUA $. - $} - - ${ - ablsimpnosubgd.1 $e |- B = ( Base ` G ) $. - ablsimpnosubgd.2 $e |- .0. = ( 0g ` G ) $. - ablsimpnosubgd.3 $e |- ( ph -> G e. Abel ) $. - ablsimpnosubgd.4 $e |- ( ph -> G e. SimpGrp ) $. - ablsimpnosubgd.5 $e |- ( ph -> S e. ( SubGrp ` G ) ) $. - ablsimpnosubgd.6 $e |- ( ph -> A e. S ) $. - ablsimpnosubgd.7 $e |- ( ph -> -. A = .0. ) $. - $( A subgroup of an abelian simple group containing a nonidentity element - is the whole group. (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - ablsimpnosubgd $p |- ( ph -> S = B ) $= - ( csn wceq wcel elsni nsyl eleq2 cfv syl5ibcom mtod pm2.21d idd cnsg cabl - csubg ablnsg eqcomd syl eleqtrd simpgnsgeqd mpjaod ) ADFNZOZDCOZUPAUOUPAU - OBUNPZABFOUQMBFQRABDPUOUQLDUNBSUAUBUCAUPUDADCEFGHJADEUGTZEUETZKAEUFPZURUS - OIUTUSUREUHUIUJUKULUM $. - $} - - ${ - $d ph n $. $d .x. n $. $d A n $. $d C n $. $d B n $. $d G n $. - ablsimpg1gend.1 $e |- B = ( Base ` G ) $. - ablsimpg1gend.2 $e |- .0. = ( 0g ` G ) $. - ablsimpg1gend.3 $e |- .x. = ( .g ` G ) $. - ablsimpg1gend.4 $e |- ( ph -> G e. Abel ) $. - ablsimpg1gend.5 $e |- ( ph -> G e. SimpGrp ) $. - ablsimpg1gend.6 $e |- ( ph -> A e. B ) $. - ablsimpg1gend.7 $e |- ( ph -> -. A = .0. ) $. - ablsimpg1gend.8 $e |- ( ph -> C e. B ) $. - $( An abelian simple group is generated by any non-identity element. - (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - ablsimpg1gend $p |- ( ph -> E. n e. ZZ C = ( n .x. A ) ) $= - ( cz cv co cmpt eqid crn simpggrpd cycsubgcld ablsimpnosubgd rr-elrnmpt2d - cycsubggend eleqtrrd ) AFQFRBESZDFQUITZUJUAZADCUJUBZPABCULGHIJLMABCEFUJGI - KUKAGMUCNUDABCEFUJGIKUKNUGOUEUHUF $. - $} - - ${ - $d ph x y z $. $d x y z G $. - ablsimpgcygd.1 $e |- ( ph -> G e. Abel ) $. - ablsimpgcygd.2 $e |- ( ph -> G e. SimpGrp ) $. - $( An abelian simple group is cyclic. (Contributed by Rohan Ridenour, - 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.) $) - ablsimpgcygd $p |- ( ph -> G e. CycGrp ) $= - ( vx vy vz cv c0g cfv wceq wn ccyg wcel cbs eqid simpgnideld cmg ad2antrr - wa cgrp simpggrpd adantr simprl cabl csimpg simplrl simplrr ablsimpg1gend - simpr iscygd rexlimddv ) AEHZBIJZKLZBMNEBOJZAEUPBUNUPPZUNPZDQAUMUPNZUOTZT - ZFUPBRJZGBUMUQVBPZABUANUTABDUBUCAUSUOUDVAFHZUPNZTUMUPVDVBGBUNUQURVCABUENU - TVECSABUFNUTVEDSAUSUOVEUGAUSUOVEUHVAVEUJUIUKUL $. - $} - - ${ - $d y G $. $d y .x. $. $d y O $. $d ph x y $. $d x y .0. $. $d x y B $. - ablsimpgfindlem1.1 $e |- B = ( Base ` G ) $. - ablsimpgfindlem1.2 $e |- .0. = ( 0g ` G ) $. - ablsimpgfindlem1.3 $e |- .x. = ( .g ` G ) $. - ablsimpgfindlem1.4 $e |- O = ( od ` G ) $. - ablsimpgfindlem1.5 $e |- ( ph -> G e. Abel ) $. - ablsimpgfindlem1.6 $e |- ( ph -> G e. SimpGrp ) $. - $( Lemma for ~ ablsimpgfind . An element of an abelian finite simple group - which doesn't square to the identity has finite order. (Contributed by - Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, - 31-Oct-2023.) $) - ablsimpgfindlem1 $p |- ( ( ( ph /\ x e. B ) /\ ( 2 .x. x ) =/= .0. ) - -> ( O ` x ) =/= 0 ) $= - ( vy wcel co wne wceq 3ad2ant1 wa cv c2 cfv cc0 w3a cabl csimpg simpggrpd - cz cgrp 2z a1i simp2 mulgcld simp3 neneqd ablsimpg1gend cmul c1 cdvds wbr - cmin wn simprr simpl2 mulg1 syl adantr mulgassr syl13anc 3eqtr4rd zmulcld - simprl wb 1zzd odcong syl112anc mpbird 0zd caddc 2t0e0 oveq1i 0p1e1 eqtri - zneo neeqtrd oveq1 syl6req adantl cc 2cnd mulcld 1cnd npcan syl2an eqtr2d - zcn ex necon3ad syl5 anabsi5 syl2anc zsubcld 0dvds mtbird rexlimddv 3expa - nbrne2 ) ABUAZCOZUBZXIDPZGQZXIFUCZUDZQZAXJXMUEZXINUAZXLDPZRZXPNUIZXQXLCXI - DNEGHIJAXJEUFOXMLSAXJEUGOXMMSXQCDEXKXIHJAXJEUJOZXMAEMUHSZXKYAOZXQUKZULAXJ - XMUMZUNXQXLGAXJXMUOUPYFUQXQXRYAOZXTTZTZXNXKXRURZPZUSZVBPZUTZVAZXOYMYNVAZV - CXPYIYOYKXIDPZYLXIDPZRZYIXIXSYRYQXQYGXTVDYIXJYRXIRAXJXMYHVEZCDEXIHJVFVGYI - YBYGYDXJYQXSRXQYBYHYCVHZXQYGXTVMZYDYIYEULZYTCDEXRXKXIHJVIVJVKYIYBXJYKYAOY - LYAOYOYSVNUUAYTYIXKXRUUCUUBVLZYIVOZXIDEYKYLFCGHKJIVPVQVRYIYPYMXORZYIYGXOY - AOZUUFVCZUUBYIVSYGUUGUUHYGUUGTZYKYLQYGUUHUUIYKXKXOYJPZYLVTZPZYLXRXOWEUULY - LRUUIUULXOYLUUKPZYLUUJXOYLUUKWAWBWCZWDULWFYGUUFYKYLYGUUFYKYLRYGUUFTYLYMYL - UUKPZYKUUFYLUUORYGUUFUUOUUMYLYMXOYLUUKWGUUNWHWIYGYKWJZOYLUUPOUUOYKRUUFYGX - KXRYGWKXRWQWLUUFWMYKYLWNWOWPWRWSWTXAXBYIYMYAOYPUUFVNYIYKYLUUDUUEXCYMXDVGX - EXNXOYMYNXHXBXFXG $. - - $( Lemma for ~ ablsimpgfind . An element of an abelian finite simple group - which squares to the identity has finite order. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - ablsimpgfindlem2 $p |- ( ( ( ph /\ x e. B ) /\ ( 2 .x. x ) = .0. ) -> - ( O ` x ) =/= 0 ) $= - ( wcel wa c2 wceq cdvds wbr cc0 cv co cfv wn wne simpr cgrp w3a simpggrpd - cz wb adantr 2z a1i 3jca oddvds syl mpbird 2ne0 neneqd 0dvds ax-mp nbrne2 - sylnibr syl2anc ) ABUAZCNZOZPVFDUBGQZOZVFFUCZPRSZTPRSZUDVKTUEVJVLVIVHVIUF - VJEUGNZVGPUJNZUHZVLVIUKVHVPVIVHVNVGVOAVNVGAEMUIULAVGUFVOVHUMUNUOULVFDEPFC - GHKJIUPUQURVJPTQZVMVJPTPTUEVJUSUNUTVOVMVQUKUMPVAVBVDVKTPRVCVE $. - $} - - ${ - $d B n $. $d n G $. $d ph x n $. $d x y G $. $d ph x y $. $d x y B $. - $d y n $. - ablsimpgfind.1 $e |- B = ( Base ` G ) $. - ablsimpgfind.2 $e |- ( ph -> G e. Abel ) $. - ablsimpgfind.3 $e |- ( ph -> G e. SimpGrp ) $. - $( An abelian simple group is finite. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - ablsimpgfind $p |- ( ph -> B e. Fin ) $= - ( vx vn vy wcel wa cfv cc0 wne cv wrex eqid adantr cz ad2antrr wfal chash - cfn cif simpr iffalsed c0g wceq simpgnideld neqne reximi syl cod cmg cgrp - wn simpggrpd simprl cab cmpt crn cabl csimpg simplrr neneqd ablsimpg1gend - co simprr mulgcld eqeltrd rexlimdvaa impbid abbi2dv syl6eqr cycsubggenodd - ex rnmpt c2 ablsimpgfindlem2 ablsimpgfindlem1 pm2.61dane adantrr eqnetrrd - rexlimddv pm2.21ddne efald ) ABUCJZAWGUPZKZUAWGBUBLZMUDZMWIWGWJMAWHUEUFAW - KMNZWHAGOZCUGLZNZWLGBAWMWNUHUPZGBPWOGBPAGBCWNDWNQZFUIWPWOGBWMWNUJUKULAWMB - JZWOKZKZWMCUMLZLZWKMWTWMBBCUNLZHCXADXCQZXAQZACUOJZWSACFUQZRAWRWOURZWTBIOZ - HOZWMXCVGZUHZHSPZIUSHSXKUTZVAWTXMIBWTXIBJZXMWTXOXMWTXOKZWMBXIXCHCWNDWQXDA - CVBJWSXOETACVCJWSXOFTWTWRXOXHRXPWMWNAWRWOXOVDVEWTXOUEVFVPWTXLXOHSWTXJSJZX - LKZKZXIXKBWTXQXLVHXSBXCCXJWMDXDAXFWSXRXGTWTXQXLURWTWRXRXHRVIVJVKVLVMHISXK - XNXNQVQVNVOAWRXBMNZWOAWRKXTVRWMXCVGWNAGBXCCXAWNDWQXDXEEFVSAGBXCCXAWNDWQXD - XEEFVTWAWBWCWDRWEWF $. - $} - - ${ - $d .x. n $. $d A n $. $d C n $. $d B n $. $d G n $. - fincygsubgd.1 $e |- B = ( Base ` G ) $. - fincygsubgd.2 $e |- .x. = ( .g ` G ) $. - fincygsubgd.3 $e |- H = ( n e. ZZ |-> ( n .x. ( C .x. A ) ) ) $. - fincygsubgd.4 $e |- ( ph -> G e. Grp ) $. - fincygsubgd.5 $e |- ( ph -> A e. B ) $. - fincygsubgd.6 $e |- ( ph -> C e. NN ) $. - $( The subgroup referenced in ~ fincygsubgodd is a subgroup. (Contributed - by Rohan Ridenour, 3-Aug-2023.) $) - fincygsubgd $p |- ( ph -> ran H e. ( SubGrp ` G ) ) $= - ( co nnzd mulgcld cycsubgcld ) ADBEOCEFHGIJKLACEGDBIJLADNPMQR $. - $} - - ${ - $d .x. n $. $d A n $. $d B n $. $d C n $. $d n G $. - fincygsubgodd.1 $e |- B = ( Base ` G ) $. - fincygsubgodd.2 $e |- .x. = ( .g ` G ) $. - fincygsubgodd.3 $e |- D = ( ( # ` B ) / C ) $. - fincygsubgodd.4 $e |- F = ( n e. ZZ |-> ( n .x. A ) ) $. - fincygsubgodd.5 $e |- H = ( n e. ZZ |-> ( n .x. ( C .x. A ) ) ) $. - fincygsubgodd.6 $e |- ( ph -> G e. Grp ) $. - fincygsubgodd.7 $e |- ( ph -> A e. B ) $. - fincygsubgodd.8 $e |- ( ph -> ran F = B ) $. - fincygsubgodd.9 $e |- ( ph -> C || ( # ` B ) ) $. - fincygsubgodd.10 $e |- ( ph -> B e. Fin ) $. - fincygsubgodd.11 $e |- ( ph -> C e. NN ) $. - $( Calculate the order of a subgroup of a finite cyclic group. - (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - fincygsubgodd $p |- ( ph -> ( # ` ran H ) = D ) $= - ( co cod cfv crn cfn wcel chash cc0 cdiv eqid cz cmpt rneqi cycsubggenodd - cif syl5reqr iftrued eqtrd oveq1d wceq cmul cgcd cgrp nnzd odmulg syl3anc - cv cn0 odcl nn0z 3syl cdvds breqtrrd gcddvdsd odcld nn0cnd mulgcld nnne0d - zcnd divmul2d mpbird eqtr3d syl5eq a1i iffalse sylan9eq hashcl hashelne0d - wn cc nn0cn neqned divne0d eqnetrd neneqd adantr condan 3eqtrrd ) AEDBFUB - ZIUCUDZUDZJUEZUFUGZXCUHUDZUIUPZXEAECUHUDZDUJUBZXBMABXAUDZDUJUBZXHXBAXIXGD - UJAXICUFUGZXGUIUPXGABCCFGIXAKLXAUKZPQAGULGVHZBFUBUMZUEHUECHXNNUNRUQUOAXKX - GUITURUSZUTAXJXBVAXIDXBVBUBZVAAXIDXIVCUBZXBVBUBZXPAIVDUGBCUGZDULUGXIXRVAP - QADUAVEZBFIDXACKXLLVFVGAXQDXBVBADXIUAAXSXIVIUGXIULUGQBIXACKXLVJXIVKVLADXG - XIVMSXOVNVOUTUSAXIXBDAXIABCIXAKXLQVPVQAXBAWTCIXAKXLACFIDBKLPXTQVRZVPVQADX - TVTZADUAVSZWAWBWCWDZAWTCXCFGIXAKLXLPYAXCGULXMWTFUBUMZUEVAAJYEOUNWEUOZAXDX - EUIAXDEUIVAZAXDWJZEXFUIAEXBXFYDYFUSXDXEUIWFWGAYGWJYHAEUIAEXHUIEXHVAAMWEAX - GDAXKXGVIUGXGWKUGTCWHXGWLVLYBAXGUIACBUFQTWIWMYCWNWOWPWQWRURWS $. - $} - - ${ - $d ph x y $. $d x y B n $. $d x y C n $. $d x y n G $. - fincygsubgodexd.1 $e |- B = ( Base ` G ) $. - fincygsubgodexd.2 $e |- ( ph -> G e. CycGrp ) $. - fincygsubgodexd.3 $e |- ( ph -> C || ( # ` B ) ) $. - fincygsubgodexd.4 $e |- ( ph -> B e. Fin ) $. - fincygsubgodexd.5 $e |- ( ph -> C e. NN ) $. - $( A finite cyclic group has subgroups of every possible order. - (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - fincygsubgodexd $p |- ( ph -> E. x e. ( SubGrp ` G ) ( # ` x ) = C ) $= - ( vn vy cv cfv co wceq chash wcel eqid adantr cz cmg cmpt csubg wrex ccyg - crn cgrp iscyg simprbi syl wa cdiv cyggrp simprl cn cdvds wb hashfingrpnn - wbr nndivdvds syl2anc mpbid fincygsubgd fveq2d simprr cc0 wne divconjdvds - simpr nnne0d cfn fincygsubgodd nncnd ddcand 3eqtrd rspcedeq1vd rexlimddv - ad2antrr ) AKUAKMZLMZEUBNZOUCZUGCPZBMZQNZDPBEUDNZUELCAEUFRZWDLCUEZGWHEUHR - ZWILCWBKEFWBSZUIUJUKAWACRZWDULZULZBKUAVTCQNZDUMOZWAWBOWBOUCZUGZWGWFDWNWAC - WPWBKEWQFWKWQSZAWJWMAWHWJGEUNUKZTZAWLWDUOZAWPUPRZWMADWOUQUTZXCHAWOUPRDUPR - XDXCURACEFWTIUSZJWODVAVBVCTZVDWNWEWRPZULZWFWRQNZWOWPUMOZDXHWEWRQWNXGVJVEW - NXIXJPXGWNWACWPXJWBKWCEWQFWKXJSWCSWSXAXBAWLWDVFAWPWOUQUTZWMAXDDVGVHXKHADJ - VKZDWOVIVBTACVLRWMITXFVMTAXJDPWMXGAWODAWOXEVNADJVNAWOXEVKXLVOVSVPVQVR $. - $} - - ${ - $d ph x $. $d x B $. $d x G $. - prmgrpsimpgd.1 $e |- B = ( Base ` G ) $. - prmgrpsimpgd.2 $e |- ( ph -> G e. Grp ) $. - prmgrpsimpgd.3 $e |- ( ph -> ( # ` B ) e. Prime ) $. - $( A group of prime order is simple. (Contributed by Rohan Ridenour, - 3-Aug-2023.) $) - prmgrpsimpgd $p |- ( ph -> G e. SimpGrp ) $= - ( vx c0g cfv eqid wceq c1 cprime wcel wa chash adantl cvv adantr a1i mp1i - csn fveq2 fvexi hashsng eqtr3d eqeltrrd wn 1nprm pm2.65da cv cnsg nsgsubg - csubg wo cfn cn0 wi cbs cn prmnn nnnn0d hashvnfin syl2anc ad2antrr subgss - syl mpi wss ad2antlr simpr phphash2d olcd subg0cl vex hash1elsn cdvds wbr - orcd lagsubg sylan2 ancoms cc0 wne ssfid hashcl hashelne0d neqned elnnne0 - wb sylanbrc dvdsprime mpbid mpjaodan 2nsgsimpgd ) AGBCCHIZDWPJZEAWPUBZBKZ - LMNZAWSOZBPIZLMXAWRPIZXBLWSXCXBKAWRBPUCQWPRNXCLKXAWPCHWQUDWPRUEUAUFAXBMNZ - WSFSUGWTUHXAUITUJGUKZCULINAXECUNINZXEWRKZXEBKZUOZXECUMAXFOZXEPIZXBKZXIXKL - KZXJXLOZXHXGXNXEBABUPNZXFXLAXBXBKZXOXBJABRNZXBUQNXPXOURXQABCUSDUDTAXBAXDX - BUTNFXBVAVGVBBXBRVCVDVHZVEXFXEBVIZAXLBXECDVFZVJXJXLVKVLVMXJXMOZXGXHYAXEWP - RXJXMVKXFWPXENZAXMXECWPWQVNZVJXERNZYAGVOZTVPVSXJXKXBVQVRZXLXMUOZXFAYFAXFX - OYFXRCBXEDVTWAWBXJXDXKUTNZYFYGWJAXDXFFSXJXKUQNZXKWCWDYHXJXEUPNYIXJBXEAXOX - FXRSXFXSAXTQWEXEWFVGXJXKWCXJXEWPRXFYBAYCQYDXJYETWGWHXKWIWKXBXKWLVDWMWNWAW - O $. - $} - - ${ - $d x G $. $d ph x y $. $d x y B $. - ablsimpgprmd.1 $e |- B = ( Base ` G ) $. - ablsimpgprmd.2 $e |- ( ph -> G e. Abel ) $. - ablsimpgprmd.3 $e |- ( ph -> G e. SimpGrp ) $. - $( An abelian simple group has prime order. (Contributed by Rohan - Ridenour, 3-Aug-2023.) $) - ablsimpgprmd $p |- ( ph -> ( # ` B ) e. Prime ) $= - ( vy vx chash cfv wcel cv c1 wceq wo cn c0g wa simpr adantr cuz cdvds wbr - c2 wi wral cprime wn csn cfn cgrp simpggrpd eqid grpidcl syl ablsimpgfind - hash1elsn csimpg simpgntrivd pm2.65da hashfingrpnn elnn1uz2 sylib ord mpd - w3a csubg ablsimpgcygd 3ad2ant1 simp3 simp2 fincygsubgodexd simpl1 simprl - ccyg cnsg cabl ablnsg 3syl eleqtrrd simpgnsgeqd simplrr cvv fvexi hashsng - fveq2d mp1i 3eqtr3d ex eqtr3d orim12d rexlimddv ralrimiv isprm2 sylanbrc - 3exp ) ABIJZUDUAJKZGLZWQUBUCZWSMNZWSWQNZOZUEZGPUFWQUGKAWQMNZUHWRAXEBCQJZU - IZNAXERZBXFUJAXESAXFBKZXEACUKKXIACFULZBCXFDXFUMZUNUOTABUJKZXEABCDEFUPZTUQ - XHBCXFDXKACURKZXEFTUSUTAXEWRAWQPKXEWROABCDXJXMVAWQVBVCVDVEAXDGPAWSPKZWTXC - AXOWTVFZHLZIJZWSNZXCHCVGJZXPHBWSCDAXOCVOKWTACEFVHVIAXOWTVJAXOXLWTXMVIAXOW - TVKVLXPXQXTKZXSRZRZXQXGNZXQBNZOXCYCXQBCXFDXKYCAXNAXOWTYBVMZFUOYCXQXTCVPJZ - XPYAXSVNYCACVQKYGXTNYFECVRVSVTWAYCYDXAYEXBYCYDXAYCYDRZXRXGIJZWSMYHXQXGIYC - YDSWFXPYAXSYDWBXFWCKYIMNYHXFCQXKWDXFWCWEWGWHWIYCYEXBYCYERZXRWSWQXPYAXSYEW - BYJXQBIYCYESWFWJWIWKVEWLWPWMGWQWNWO $. - $} - - ${ - ablsimpgd.1 $e |- B = ( Base ` G ) $. - ablsimpgd.2 $e |- ( ph -> G e. Abel ) $. - $( An abelian group is simple if and only if its order is prime. - (Contributed by Rohan Ridenour, 3-Aug-2023.) $) - ablsimpgd $p |- ( ph -> ( G e. SimpGrp <-> ( # ` B ) e. Prime ) ) $= - ( csimpg wcel chash cprime wa cabl adantr simpr ablsimpgprmd cgrp ablgrpd - cfv prmgrpsimpgd impbida ) ACFGZBHQIGZATJBCDACKGTELATMNAUAJBCDACOGUAACEPL - AUAMRS $. - $} - - ${ - prmsimpcyc.1 $e |- B = ( Base ` G ) $. - $( A group of prime order is cyclic if and only if it is simple. This is - the first family of finite simple groups. (Contributed by Thierry - Arnoux, 21-Sep-2023.) $) - prmsimpcyc $p |- ( ( # ` B ) e. Prime -> ( G e. SimpGrp <-> G e. CycGrp ) ) - $= - ( chash cprime wcel csimpg ccyg cgrp simpggrp id prmcyg syl2anr wa cyggrp - cfv adantl simpl prmgrpsimpgd impbida ) ADPEFZBGFZBHFZUBBIFZUAUCUABJUAKAB - CLMUAUCNABCUCUDUABOQUAUCRST $. - $} - $( (End of Rohan Ridenour's mathbox.) $) @@ -652623,7 +653366,7 @@ elements and the subgroup containing only the identity ( ~ simpgnsgbid ). cz crp fvres eqeltrrd absge0d breqtrd climge0 ne0gt0d elrpd ltmuldivd csn 1red cin wo cun elun inundif eleq2i bitr3i elin simprbi elsni abs0 syl6eq mul02d sylan9eqr 0lt1 radcnv0 eleq1 syl5ibrcom 2thd ax-resscn ssdif ax-mp - syl6eqbr sseli nn0uz oveq2 mulcld expne0d mulne0d eqnetrd fvoveq1 cbvmptv + eqbrtrdi sseli nn0uz oveq2 mulcld expne0d mulne0d eqnetrd fvoveq1 cbvmptv eldifsni syl5eqr 1nn0 nn0addcld divmuldivd nn0cnd pncan2d expsubd 3eqtr3d eqidd 3eqtr2d absmuld eqtr3d mulcomd climmulc2 eqbrtrrd cvgdvgrat seqeq3d exp1d eqcomd eleq1d elrab3 bitr4d jaodan bitr3d notbid bitrd con2d leabsd @@ -666173,19 +666916,6 @@ not even needed (it can be any class). (Contributed by Glauco =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - ${ - $d A y $. $d B x y $. $d C x y $. $d F x y $. - $( Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) - unima $p |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( F " ( B u. C ) ) = ( ( F - " B ) u. ( F " C ) ) ) $= - ( vy vx wfn wss w3a cun cima cv wcel wo cfv wceq wrex simp1 wb fvelimab - simpl simpr unssd 3adant1 fvelimabd syl6bb 3adant3 3adant2 orbi12d bitr4d - wa rexun elun syl6bbr eqrdv ) DAGZBAHZCAHZIZEDBCJZKZDBKZDCKZJZUSELZVAMZVE - VBMZVEVCMZNZVEVDMUSVFFLDOVEPZFBQZVJFCQZNZVIUSVFVJFUTQVMUSFAUTVEDUPUQURRUQ - URUTAHUPUQURUKBCAUQURUAUQURUBUCUDUEVJFBCULUFUSVGVKVHVLUPUQVGVKSURFABVEDTU - GUPURVHVLSUQFACVEDTUHUIUJVEVBVCUMUNUO $. - $} - ${ feq1dd.eq $e |- ( ph -> F = G ) $. feq1dd.f $e |- ( ph -> F : A --> B ) $. @@ -669353,7 +670083,7 @@ not even needed (it can be any class). (Contributed by Glauco infxr $p |- ( ph -> inf ( A , RR* , < ) = B ) $= ( cxr wcel clt wbr wi wa cpnf cmnf c1 cv wn wral wrex cr r19.21bi adantlr cinf wceq simplll simpllr simplr wne mnfxr a1i ad2antrr cle syl xrlelttrd - mnfle simpr xrgtned xrnmnfpnf w3a simpl id 1re mnflt syl6eqbr adantl 1red + mnfle simpr xrgtned xrnmnfpnf w3a simpl id 1re mnflt eqbrtrdi adantl 1red ax-mp breq2 rexbidv imbi12d rspcva syl2anc sylc nfv sselda ad4ant13 pnfxr 1xr syl6eqel ltpnf eqcomd breqtrd xrlttrd ex reximdai adantr mpd 3adantl3 nfan caddc co 3ad2antl1 necon3bi 3adant1 xrltned xrred readdcld jca ltp1d @@ -685781,7 +686511,7 @@ distinct definitions for the same symbol (limit of a sequence). ( x ` t ) <_ 1 ) /\ A. t e. D ( x ` t ) < E /\ A. t e. B ( 1 - E ) < ( x ` t ) ) ) $= ( c1 wcel cc0 cv cfv cle wbr wa wral clt cmin co wrex cmpt cr stoweidlem4 - w3a mpan2 eqeltrid 0le1 wceq simpr fvmpt2 sylancl breqtrrid 1le1 syl6eqbr + w3a mpan2 eqeltrid 0le1 wceq simpr fvmpt2 sylancl breqtrrid 1le1 eqbrtrdi 1re jca ex ralrimi c0 nfcv nfeq rzalf syl 1red ltsubrpd adantr ccld cldss sselda breqtrrd nfmpt1 nfcxfr fveq1 breq2d breq1d ralbid 3anbi123d rspcev wss anbi12d syl13anc ) AIDUAUBCUCZIUDZUEUFZWOTUEUFZUGZCGUHZWOHUIUFZCFUHZT @@ -688082,7 +688812,7 @@ used to represent the final q_n in the paper (the one with n large $= ( cdif c0 wceq cc0 cv cfv cle wbr wral clt w3a wrex cmpt wcel stoweidlem4 c1 wa cr 0re mpan2 adantr nfcv nfdif nfeq nfan 0le0 cc 0cn eqid breqtrrid - fvmpt2 adantl 0le1 syl6eqbr jca ralrimi cuni elunii syl6eleqr eqidd fvmpt + fvmpt2 adantl 0le1 eqbrtrdi jca ralrimi cuni elunii syl6eleqr eqidd fvmpt c0ex 3syl rzalf nfmpt1 fveq1 breq2d breq1d ralbid eqeq1d 3anbi123d rspcev ex anbi12d syl13anc wn nfn ccmp wss caddc 3adant1r cmul adantlr wne neqne co stoweidlem53 pm2.61dan ) AHIUOZUPUQZURDUSZSUSZUTZVAVBZYGVJVAVBZVKZDHVC @@ -695097,7 +695827,7 @@ approximated is nonnegative (this assumption is removed in a later 1div1e1 fveq1i syl5req 3eltr3g oveq1d feq1i iccssre syl6ss ltleii elicc2i lhop cun ssequn2 rerest fveq2i fveq12i resttopon topontopi pm3.2i 3pm3.2i snss ovex isopn3i 3eqtrri limcres iftrue 1cnd iffalse adantr neqne syl2an - restopnb divcld pm2.61dan eldifn sylnib 3eltr4d breqtrrid syl6eqbr eliccd + restopnb divcld pm2.61dan eldifn sylnib 3eltr4d breqtrrid eqbrtrdi eliccd velsn cnfldtop reex restabs cnplimc mpbir2and simpl notbii biimpri eleq2d ssdifssd divrecd eqeltrid cncffvrn mpbir divcncf restid cncfcn cnfldtopon unicntop cncnp vtoclri 3eltr4g ssdif sscon unssi elun1 elun2 eqssi cldopn @@ -702873,7 +703603,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u ex 2re clt sylibr orcd sylbi oveq1d 3eqtrd eqeltrid syl mp3an12 cle ax-mp 0re pm2.61dan cxr eliocd iftrued necon2bi iffalsed syl5eq 1pneg1e0 syl6eq 0xr rexri 2cn iftrue syl5req ax-1cn mulcomi oveq1i oveq2i leneltd 2div2e1 - eqtri mulcli eqtr2i syl6eqbr mp3an 3eqtr2i nsyl ltnled negeqi 2pos elrpii + eqtri mulcli eqtr2i eqbrtrdi mp3an 3eqtr2i nsyl ltnled negeqi 2pos elrpii mulgt0ii odd2np1 biimpa 1cnd adddird eqcomi oveq2d mulid2i 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 @@ -722015,6 +722745,29 @@ that every function in the sequence can have a different (partial) JKLMNOPQRUNUIUO $. $} +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Simple groups +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + ${ + simpcntrab.a $e |- B = ( Base ` G ) $. + simpcntrab.b $e |- .0. = ( 0g ` G ) $. + simpcntrab.c $e |- Z = ( Cntr ` G ) $. + simpcntrab.d $e |- ( ph -> G e. SimpGrp ) $. + $( The center of a simple group is trivial or the group is abelian. + (Contributed by SS, 3-Jan-2024.) $) + simpcntrab $p |- ( ph -> ( Z = { .0. } \/ G e. Abel ) ) $= + ( csn wceq wa wo cabl wcel cgrp cfv syl co adantr simpggrpd cntrnsg ancli + cnsg simpgnsgeqd andi biimpi simpr orim1i 3syl ccntr cress oveq2 syl5reqr + oveq2i adantl ressid eqtr3d eqid cntrabl eqeltrrd orim2i ) AEDJKZAEBKZLZM + ZVCCNZOZMAAVCVDMZLZAVCLZVEMZVFAVIAEBCDFGIACPZOZECUDQOACIUAZCEHUBRUEUCVJVL + AVCVDUFUGVKVCVEAVCUHUIUJVEVHVCVECCUKQZULZSZCVGVECBVQSZVRCVDVSVRKAVDVRCEVQ + SVSEVPCVQHUOEBCVQUMUNUPAVSCKZVDAVNVTVOBCVMFUQRTURAVRVGOZVDAVNWAVOCVRVRUSU + TRTVAVBR $. + $} + $( (End of Saveliy Skresanov's mathbox.) $) @@ -741634,7 +742387,7 @@ Symmetric groups (extension) than or equal to 2. (Contributed by AV, 16-Mar-2019.) $) exple2lt6 $p |- ( ( N e. NN0 /\ N <_ 2 ) -> ( N ^ N ) < 6 ) $= ( cn0 wcel c2 cle wbr wa cc0 wceq c1 w3o cexp co c6 id oveq12d 1lt6 eqbrtri - clt syl6eqbr nn0le2is012 0exp0e1 cc ax-1cn exp1 ax-mp c4 sq2 4lt6 3jaoi syl + clt eqbrtrdi nn0le2is012 0exp0e1 cc ax-1cn exp1 ax-mp c4 sq2 4lt6 3jaoi syl ) ABCADEFGAHIZAJIZADIZKAALMZNSFZAUAULUPUMUNULUOHHLMZNSULAHAHLULOZURPUQJNSUB QRTUMUOJJLMZNSUMAJAJLUMOZUTPUSJNSJUCCUSJIUDJUEUFQRTUNUODDLMZNSUNADADLUNOZVB PVAUGNSUHUIRTUJUK $.