From 1ee0e4d93c4dc497dcf316cd2c900e23aaa6e568 Mon Sep 17 00:00:00 2001 From: Wolf Lammen <30736367+wlammen@users.noreply.github.com> Date: Sat, 1 Mar 2025 22:31:09 +0100 Subject: [PATCH] abbi* -> eqab* (#4682) * abbi2dv -> eqabdv * abbi1dv -> eqabcdv * abbi2i -> eqabi --- changes-set.txt | 3 + nf.mm | 2688 +++++++++++++++++++++++------------------------ set.mm | 1040 +++++++++--------- 3 files changed, 1866 insertions(+), 1865 deletions(-) diff --git a/changes-set.txt b/changes-set.txt index af71db5a3..225c681c3 100644 --- a/changes-set.txt +++ b/changes-set.txt @@ -85,6 +85,9 @@ make a github issue.) DONE: Date Old New Notes +28-Feb-25 abbi1dv eqabcdv +28-Feb-25 abbi2dv eqabdv +28-Feb-25 abbi2i eqabi 27-Feb-25 elnelall pm2.24nel 25-Feb-25 eqimssd [same] Moved from SN's mathbox to main set.mm 25-Feb-25 eqimss2d eqimsscd Moved from SN's mathbox to main set.mm diff --git a/nf.mm b/nf.mm index dbbe7a9bf..f41a628bb 100644 --- a/nf.mm +++ b/nf.mm @@ -21421,7 +21421,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim abbiri.1 $e |- ( x e. A <-> ph ) $. $( Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.) $) - abbi2i $p |- A = { x | ph } $= + eqabi $p |- A = { x | ph } $= ( cab wceq cv wcel wb eqabb mpgbir ) CABEFBGCHAIBABCJDK $. $} @@ -21457,20 +21457,20 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim ${ $d x A $. $d ph x $. - abbirdv.1 $e |- ( ph -> ( x e. A <-> ps ) ) $. + eqabdv.1 $e |- ( ph -> ( x e. A <-> ps ) ) $. $( Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) $) - abbi2dv $p |- ( ph -> A = { x | ps } ) $= + eqabdv $p |- ( ph -> A = { x | ps } ) $= ( cv wcel wb wal cab wceq alrimiv eqabb sylibr ) ACFDGBHZCIDBCJKAOCELBCDM N $. $} ${ $d x A $. $d ph x $. - abbildv.1 $e |- ( ph -> ( ps <-> x e. A ) ) $. + eqabcdv.1 $e |- ( ph -> ( ps <-> x e. A ) ) $. $( Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) $) - abbi1dv $p |- ( ph -> { x | ps } = A ) $= + eqabcdv $p |- ( ph -> { x | ps } = A ) $= ( cv wcel wb wal cab wceq alrimiv eqabcb sylibr ) ABCFDGHZCIBCJDKAOCELBCD MN $. $} @@ -21480,7 +21480,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim $( A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.) $) abid2 $p |- { x | x e. A } = A $= - ( cv wcel cab biid abbi2i eqcomi ) BACBDZAEIABIFGH $. + ( cv wcel cab biid eqabi eqcomi ) BACBDZAEIABIFGH $. $} ${ @@ -21532,7 +21532,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim proper substitution of ` y ` for ` x ` into a class variable. (Contributed by NM, 14-Sep-2003.) $) sbab $p |- ( x = y -> A = { z | [ y / x ] z e. A } ) $= - ( cv wceq wcel wsb sbequ12 abbi2dv ) AEBEFCEDGZABHCDKABIJ $. + ( cv wceq wcel wsb sbequ12 eqabdv ) AEBEFCEDGZABHCDKABIJ $. $} @@ -26287,7 +26287,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.) $) abidnf $p |- ( F/_ x A -> { z | A. x z e. A } = A ) $= - ( wnfc cv wcel wal sp nfcr nfrd impbid2 abbi1dv ) ACDZBECFZAGZBCMONNAHMNA + ( wnfc cv wcel wal sp nfcr nfrd impbid2 eqabcdv ) ACDZBECFZAGZBCMONNAHMNA ABCIJKL $. $} @@ -27905,7 +27905,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $( Substitution doesn't affect a constant ` B ` (in which ` x ` is not free). (Contributed by Mario Carneiro, 14-Oct-2016.) $) csbtt $p |- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B ) $= - ( vy wcel wnfc wa csb cv wsbc cab df-csb wnf wb nfcr sbctt sylan2 abbi1dv + ( vy wcel wnfc wa csb cv wsbc cab df-csb wnf wb nfcr sbctt sylan2 eqabcdv syl5eq ) BDFZACGZHZABCIEJCFZABKZELCAEBCMUCUEECUBUAUDANUEUDOAECPUDABDQRST $. $} @@ -28050,7 +28050,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $( The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.) $) csbvarg $p |- ( A e. V -> [_ A / x ]_ x = A ) $= - ( vz vy wcel cvv cv csb wceq elex wsbc cab df-csb sbcel2gv abbi1dv syl5eq + ( vz vy wcel cvv cv csb wceq elex wsbc cab df-csb sbcel2gv eqabcdv syl5eq vex ax-mp csbeq2i csbco 3eqtr3i syl ) BCFBGFZABAHZIZBJBCKUDUFDHZEHZFEBLZD MZBEBAUHUEIZIEBUHIUFUJEBUKUHUHGFZUKUHJERULUKUGUEFAUHLZDMUHADUHUENULUMDUHA UGUHGOPQSTAEBUEUAEDBUHNUBUDUIDBEUGBGOPQUC $. @@ -28263,7 +28263,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use This version of ~ sbcie avoids a disjointness condition on ` x , A ` by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.) $) csbie2g $p |- ( A e. V -> [_ A / x ]_ B = D ) $= - ( vz wcel csb cv wsbc cab df-csb wceq eleq2d sbcie2g abbi1dv syl5eq ) CGK + ( vz wcel csb cv wsbc cab df-csb wceq eleq2d sbcie2g eqabcdv syl5eq ) CGK ZACDLJMZDKZACNZJOFAJCDPUBUEJFUDUCEKUCFKABCGAMBMZQDEUCHRUFCQEFUCIRSTUA $. $} @@ -28432,13 +28432,13 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use cbvralcsf $p |- ( A. x e. A ph <-> A. y e. B ps ) $= ( vz vv cv wcel wi wal wsbc nfcri wral csb nfv nfcsb1v nfsbc1v id csbeq1a nfim wceq eleq12d sbceq1a imbi12d cbval nfcsb nfsbc csbeq1 cab df-csb wsb - nfcv eleq2d sbsbc bitr3i abbi2i eqtr4i syl6eq dfsbcq syl6bb bitri 3bitr4i - weq sbie df-ral ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACWBSZQ - ZMRWAVPWFCMVPMUCWDWECCMWCCWBEUDTACWBUEUHVNWBUIZVOWDAWEWGVNWBEWCWGUFCWBEUG - UJACWBUKULUMWFVTMDWDWEDDMWCDCWBEDWBUTZGUNTADCWBWHIUOUHVTMUCWBVRUIZWDVSWEB - WIWBVRWCFWIUFWIWCCVREUBZFCWBVREUPWJNOZEPZCVRSZNUQFCNVREURWMNFWKFPZWLCDUSW - MWLWNCDCNFHTCDVKEFWKKVAVLWLCDVBVCVDVEVFUJWIWEACVRSZBACWBVRVGWOACDUSBACDVB - ABCDJLVLVCVHULUMVIACEVMBDFVMVJ $. + nfcv weq eleq2d sbie sbsbc bitr3i eqabi eqtr4i syl6eq dfsbcq syl6bb bitri + df-ral 3bitr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACWBSZQZ + MRWAVPWFCMVPMUCWDWECCMWCCWBEUDTACWBUEUHVNWBUIZVOWDAWEWGVNWBEWCWGUFCWBEUGU + JACWBUKULUMWFVTMDWDWEDDMWCDCWBEDWBUTZGUNTADCWBWHIUOUHVTMUCWBVRUIZWDVSWEBW + IWBVRWCFWIUFWIWCCVREUBZFCWBVREUPWJNOZEPZCVRSZNUQFCNVREURWMNFWKFPZWLCDUSWM + WLWNCDCNFHTCDVAEFWKKVBVCWLCDVDVEVFVGVHUJWIWEACVRSZBACWBVRVIWOACDUSBACDVDA + BCDJLVCVEVJULUMVKACEVLBDFVLVM $. $( A more general version of ~ cbvrexf that has no distinct variable restrictions. Changes bound variables using implicit substitution. @@ -28454,25 +28454,25 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use cbvreucsf $p |- ( E! x e. A ph <-> E! y e. B ps ) $= ( vz vv cv wcel wa weu wsb nfcri wreu csb nfcsb1v nfs1v nfan wceq csbeq1a nfv eleq12d sbequ12 anbi12d cbveu nfcv nfcsb nfsb csbeq1 wsbc sbsbc abbii - cab eleq2d bicomi abbi2i df-csb 3eqtr4ri syl6eq sbequ syl6bb bitri df-reu - id sbie 3bitr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACMSZQZ - MRWAVPWFCMVPMUHWDWECCMWCCWBEUCTACMUDUEVNWBUFZVOWDAWEWGVNWBEWCWGVKCWBEUGUI - ACMUJUKULWFVTMDWDWEDDMWCDCWBEDWBUMGUNTACMDIUOUEVTMUHWBVRUFZWDVSWEBWHWBVRW - CFWHVKWHWCCVREUBZFCWBVREUPNOZEPZCDSZNUTWKCVRUQZNUTFWIWLWMNWKCDURUSWLNFWLW - JFPZWKWNCDCNFHTVNVRUFEFWJKVAVLVBVCCNVREVDVEVFUIWHWEACDSBAMDCVGABCDJLVLVHU - KULVIACEVJBDFVJVM $. + id cab eleq2d sbie bicomi eqabi df-csb 3eqtr4ri syl6eq sbequ syl6bb bitri + df-reu 3bitr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACMSZQZM + RWAVPWFCMVPMUHWDWECCMWCCWBEUCTACMUDUEVNWBUFZVOWDAWEWGVNWBEWCWGUTCWBEUGUIA + CMUJUKULWFVTMDWDWEDDMWCDCWBEDWBUMGUNTACMDIUOUEVTMUHWBVRUFZWDVSWEBWHWBVRWC + FWHUTWHWCCVREUBZFCWBVREUPNOZEPZCDSZNVAWKCVRUQZNVAFWIWLWMNWKCDURUSWLNFWLWJ + FPZWKWNCDCNFHTVNVRUFEFWJKVBVCVDVECNVREVFVGVHUIWHWEACDSBAMDCVIABCDJLVCVJUK + ULVKACEVLBDFVLVM $. $( A more general version of ~ cbvrab with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.) $) cbvrabcsf $p |- { x e. A | ph } = { y e. B | ps } $= ( vz vv cv wcel wa cab wsb nfcri crab csb nfcsb1v nfs1v nfan wceq csbeq1a nfv id eleq12d sbequ12 anbi12d cbvab nfcv nfcsb nfsb csbeq1 df-csb eleq2d - wsbc weq sbie sbsbc bitr3i abbi2i eqtr4i syl6eq sbequ syl6bb eqtri df-rab - 3eqtr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVPMOZCWAEUBZPZACMSZQZMRVTVOWE - CMVOMUHWCWDCCMWBCWAEUCTACMUDUEVMWAUFZVNWCAWDWFVMWAEWBWFUICWAEUGUJACMUKULU - MWEVSMDWCWDDDMWBDCWAEDWAUNGUOTACMDIUPUEVSMUHWAVQUFZWCVRWDBWGWAVQWBFWGUIWG - WBCVQEUBZFCWAVQEUQWHNOZEPZCVQUTZNRFCNVQEURWKNFWIFPZWJCDSWKWJWLCDCNFHTCDVA - EFWIKUSVBWJCDVCVDVEVFVGUJWGWDACDSBAMDCVHABCDJLVBVIULUMVJACEVKBDFVKVL $. + wsbc weq sbie sbsbc bitr3i eqabi eqtr4i syl6eq sbequ syl6bb eqtri 3eqtr4i + df-rab ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVPMOZCWAEUBZPZACMSZQZMRVTVOWEC + MVOMUHWCWDCCMWBCWAEUCTACMUDUEVMWAUFZVNWCAWDWFVMWAEWBWFUICWAEUGUJACMUKULUM + WEVSMDWCWDDDMWBDCWAEDWAUNGUOTACMDIUPUEVSMUHWAVQUFZWCVRWDBWGWAVQWBFWGUIWGW + BCVQEUBZFCWAVQEUQWHNOZEPZCVQUTZNRFCNVQEURWKNFWIFPZWJCDSWKWJWLCDCNFHTCDVAE + FWIKUSVBWJCDVCVDVEVFVGUJWGWDACDSBAMDCVHABCDJLVBVIULUMVJACEVLBDFVLVK $. $} ${ @@ -28609,8 +28609,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $( Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.) $) dfdif2 $p |- ( A \ B ) = { x e. A | -. x e. B } $= - ( cdif cv wcel wn wa cab crab eldif abbi2i df-rab eqtr4i ) BCDZAEZBFPCFGZ - HZAIQABJRAOPBCKLQABMN $. + ( cdif cv wcel wn wa cab crab eldif eqabi df-rab eqtr4i ) BCDZAEZBFPCFGZH + ZAIQABJRAOPBCKLQABMN $. $} $( Membership in symmetric difference. (Contributed by SF, 10-Jan-2015.) $) @@ -30380,8 +30380,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $( Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.) $) rabbi2dva $p |- ( ph -> ( A i^i B ) = { x e. A | ps } ) $= - ( cin cv wcel crab wa cab elin abbi2i df-rab eqtr4i rabbidva syl5eq ) ADE - GZCHZEIZCDJZBCDJSTDIUAKZCLUBUCCSTDEMNUACDOPAUABCDFQR $. + ( cin cv wcel crab wa cab elin eqabi df-rab eqtr4i rabbidva syl5eq ) ADEG + ZCHZEIZCDJZBCDJSTDIUAKZCLUBUCCSTDEMNUACDOPAUABCDFQR $. $} ${ @@ -30784,9 +30784,9 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $) symdif2 $p |- ( ( A \ B ) u. ( B \ A ) ) = { x | -. ( x e. A <-> x e. B ) } $= - ( cv wcel wb wn cdif cun wo wa eldif orbi12i elun xor 3bitr4i abbi2i ) AD - ZBEZRCEZFGZABCHZCBHZIZRUBEZRUCEZJSTGKZTSGKZJRUDEUAUEUGUFUHRBCLRCBLMRUBUCN - STOPQ $. + ( cv wcel wb wn cdif cun wo wa eldif orbi12i elun xor 3bitr4i eqabi ) ADZ + BEZRCEZFGZABCHZCBHZIZRUBEZRUCEZJSTGKZTSGKZJRUDEUAUEUGUFUHRBCLRCBLMRUBUCNS + TOPQ $. $} ${ @@ -31039,8 +31039,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use p. 20. (Contributed by NM, 26-Dec-1996.) $) dfnul2 $p |- (/) = { x | -. x = x } $= ( cv wceq wn c0 wcel cvv cdif wa df-nul eleq2i eldif eqid pm3.24 2th 3bitri - con2bii abbi2i ) ABZSCZDZAESEFSGGHZFSGFZUCDIZUAEUBSJKSGGLTUDTUDDSMUCNOQPR - $. + con2bii eqabi ) ABZSCZDZAESEFSGGHZFSGFZUCDIZUAEUBSJKSGGLTUDTUDDSMUCNOQPR $. $( Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) $) @@ -31148,7 +31147,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use either the universal class or the empty set. (Contributed by Mario Carneiro, 29-Aug-2013.) $) abvor0 $p |- ( { x | ph } = _V \/ { x | ph } = (/) ) $= - ( cab cvv wceq c0 wn cv wcel vex a1i 2thd abbi1dv con3i noel 2falsed orri + ( cab cvv wceq c0 wn cv wcel vex a1i 2thd eqabcdv con3i noel 2falsed orri id syl ) ABCZDEZTFEZUAGAGZUBAUAAABDAABHZDIZARUEABJKLMNUCABFUCAUDFIZUCRUFG UCUDOKPMSQ $. $} @@ -32005,14 +32004,14 @@ fewer connectives (but probably less intuitive to understand). (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $) iftrue $p |- ( ph -> if ( ph , A , B ) = A ) $= - ( vx cv wcel wi wa cab cif dedlem0a abbi2dv dfif2 syl6reqr ) ABDEZCFZAGOB - FZAHGZDIABCJARDBAQPKLADBCMN $. + ( vx cv wcel wi wa cab cif dedlem0a eqabdv dfif2 syl6reqr ) ABDEZCFZAGOBF + ZAHGZDIABCJARDBAQPKLADBCMN $. $( Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) $) iffalse $p |- ( -. ph -> if ( ph , A , B ) = B ) $= - ( vx wn cv wcel wa wo cab cif dedlemb abbi2dv df-if syl6reqr ) AEZCDFZBGZ - AHQCGZPHIZDJABCKPTDCARSLMADBCNO $. + ( vx wn cv wcel wa wo cab cif dedlemb eqabdv df-if syl6reqr ) AEZCDFZBGZA + HQCGZPHIZDJABCKPTDCARSLMADBCNO $. $} $( When values are unequal, but an "if" condition checks if they are equal, @@ -32765,7 +32764,7 @@ fewer connectives (but probably less intuitive to understand). $( Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) $) dfpr2 $p |- { A , B } = { x | ( x = A \/ x = B ) } $= - ( cpr csn cun cv wceq cab df-pr wcel elun elsn orbi12i bitri abbi2i eqtri + ( cpr csn cun cv wceq cab df-pr wcel elun elsn orbi12i bitri eqabi eqtri wo ) BCDBEZCEZFZAGZBHZUBCHZRZAIBCJUEAUAUBUAKUBSKZUBTKZRUEUBSTLUFUCUGUDABM ACMNOPQ $. $} @@ -32944,7 +32943,7 @@ fewer connectives (but probably less intuitive to understand). Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.) $) dftp2 $p |- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) } $= - ( cv wceq w3o ctp vex eltp abbi2i ) AEZBFLCFLDFGABCDHLBCDAIJK $. + ( cv wceq w3o ctp vex eltp eqabi ) AEZBFLCFLDFGABCDHLBCDAIJK $. $} ${ @@ -35153,8 +35152,8 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and $( An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.) $) 0iin $p |- |^|_ x e. (/) A = _V $= - ( vy c0 ciin cv wcel wral cab cvv df-iin vex ral0 2th abbi2i eqtr4i ) ADB - ECFZBGZADHZCIJACDBKSCJQJGSCLRAMNOP $. + ( vy c0 ciin cv wcel wral cab cvv df-iin vex ral0 2th eqabi eqtr4i ) ADBE + CFZBGZADHZCIJACDBKSCJQJGSCLRAMNOP $. $( Indexed intersection with a universal index class. When ` A ` doesn't depend on ` x ` , this evaluates to ` A ` by ~ 19.3 and ~ abid2 . When @@ -35325,7 +35324,7 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) $) iinxsng $p |- ( A e. V -> |^|_ x e. { A } B = C ) $= - ( vy wcel csn ciin cv wral cab df-iin wceq eleq2d ralsng abbi1dv syl5eq ) + ( vy wcel csn ciin cv wral cab df-iin wceq eleq2d ralsng eqabcdv syl5eq ) BEHZABIZCJGKZCHZAUALZGMDAGUACNTUDGDUCUBDHABEAKBOCDUBFPQRS $. $} @@ -38650,23 +38649,23 @@ Definite description binder (inverted iota) 3bitr4i opkeq1 eleq1d ceqsexv eldif ndisjrelk necon2bbii otkelins3k incom elcompl eqeq1i wel wb wal dfcleq elpw141c 19.41v excom otkelins2k elssetk elsymdif wo opksnelsik bitri orbi12i bibi12i notbii exnal con2bii anbi12i - elun bitr2i rexbii abbi2i eqtr4i ) ABUACHZDHZUBZUCIZEHZWTXAUDZIZJZDBKZCAK - ZEUELUFZLUGZUBUHMMZUIZUNZUFZXKUGZXJUGZLUJZUJZUFZUDZUKZXLMMZUIZULZBMMZUIZA - UIZECDABUMXIEYHXDYHNWTXDOZYGNZCAKXICYGAXDEUOZUPYJXHCAYJFHZYIOZYENZFYFKZYL - XAPZPZIZYNJZFTZDBKZXHFYEYFYIWTXDUQZUPYLYFNZYNJZFTYSDBKZFTYOUUAUUDUUEFUUDY - RDBKZYNJUUEUUCUUFYNDYLBURUSYRYNDBUTVAVBYNFYFVCYSDFBVDVEYTXGDBYTYQYIOZYENZ - UUGXONZUUGYDNZQZJXGYNUUHFYQYPRZYRYMUUGYEYLYQYIVFVGVHUUGXOYDVIUUIXCUUKXFXA - WTOZXNNZXAWTUBZUCIZUUIXCUUNUUMXMNZQUUPUUMXMXAWTUQVNUUQUUOUCXAWTDUOZCUOZVJ - VKVAXAWTXDXNUURUUSYKVLXBUUOUCWTXAVMVOVEXFGEVPZGHZXENZVQZGVRZUUKGXDXEVSUUJ - UVDUUJYLUVAPZPZPZPZPZIZYLUUGOZYBNZJZFTZGTZUVCQZGTUVDQUUJUVLFYCKYLYCNZUVLJ - ZFTZUVOFYBYCUUGYQYIUQUPUVLFYCVCUVSUVMGTZFTUVOUVRUVTFUVRUVJGTZUVLJUVTUVQUW - AUVLGYLVTUSUVJUVLGWAVAVBUVMGFWBVASUVNUVPGUVNUVIUUGOZYBNZUWBXPNZUWBYANZVQZ - QUVPUVLUWCFUVIUVHRUVJUVKUWBYBYLUVIUUGVFVGVHUWBXPYAWEUWFUVCUWDUUTUWEUVBUWD - UVGYIOZXKNUVEXDOLNUUTUVGYQYIXKUVFRZUULUUBWCUVEWTXDLUVARZUUSYKWCUVAXDGUOZY - KWDSUWBXQNZUWBXTNZWFGCVPZGDVPZWFUWEUVBUWKUWMUWLUWNUWKUWGXJNUVEWTOLNUWMUVG - YQYIXJUWHUULUUBWCUVEWTXDLUWIUUSYKVLUVAWTUWJUUSWDSUWLUVGYQOXSNUVFYPOXRNZUW - NUVGYQYIXSUWHUULUUBVLUVFYPXRUVERXARWGUWOUVEXAOLNUWNUVEXALUWIUURWGUVAXAUWJ - UURWDWHSWIUWBXQXTWOUVAWTXAWOVEWJWKSVBUVCGWLSWMWPWNSWQSWQWHWRWS $. + elun bitr2i rexbii eqabi eqtr4i ) ABUACHZDHZUBZUCIZEHZWTXAUDZIZJZDBKZCAKZ + EUELUFZLUGZUBUHMMZUIZUNZUFZXKUGZXJUGZLUJZUJZUFZUDZUKZXLMMZUIZULZBMMZUIZAU + IZECDABUMXIEYHXDYHNWTXDOZYGNZCAKXICYGAXDEUOZUPYJXHCAYJFHZYIOZYENZFYFKZYLX + APZPZIZYNJZFTZDBKZXHFYEYFYIWTXDUQZUPYLYFNZYNJZFTYSDBKZFTYOUUAUUDUUEFUUDYR + DBKZYNJUUEUUCUUFYNDYLBURUSYRYNDBUTVAVBYNFYFVCYSDFBVDVEYTXGDBYTYQYIOZYENZU + UGXONZUUGYDNZQZJXGYNUUHFYQYPRZYRYMUUGYEYLYQYIVFVGVHUUGXOYDVIUUIXCUUKXFXAW + TOZXNNZXAWTUBZUCIZUUIXCUUNUUMXMNZQUUPUUMXMXAWTUQVNUUQUUOUCXAWTDUOZCUOZVJV + KVAXAWTXDXNUURUUSYKVLXBUUOUCWTXAVMVOVEXFGEVPZGHZXENZVQZGVRZUUKGXDXEVSUUJU + VDUUJYLUVAPZPZPZPZPZIZYLUUGOZYBNZJZFTZGTZUVCQZGTUVDQUUJUVLFYCKYLYCNZUVLJZ + FTZUVOFYBYCUUGYQYIUQUPUVLFYCVCUVSUVMGTZFTUVOUVRUVTFUVRUVJGTZUVLJUVTUVQUWA + UVLGYLVTUSUVJUVLGWAVAVBUVMGFWBVASUVNUVPGUVNUVIUUGOZYBNZUWBXPNZUWBYANZVQZQ + UVPUVLUWCFUVIUVHRUVJUVKUWBYBYLUVIUUGVFVGVHUWBXPYAWEUWFUVCUWDUUTUWEUVBUWDU + VGYIOZXKNUVEXDOLNUUTUVGYQYIXKUVFRZUULUUBWCUVEWTXDLUVARZUUSYKWCUVAXDGUOZYK + WDSUWBXQNZUWBXTNZWFGCVPZGDVPZWFUWEUVBUWKUWMUWLUWNUWKUWGXJNUVEWTOLNUWMUVGY + QYIXJUWHUULUUBWCUVEWTXDLUWIUUSYKVLUVAWTUWJUUSWDSUWLUVGYQOXSNUVFYPOXRNZUWN + UVGYQYIXSUWHUULUUBVLUVFYPXRUVERXARWGUWOUVEXAOLNUWNUVEXALUWIUURWGUVAXAUWJU + URWDWHSWIUWBXQXTWOUVAWTXAWOVEWJWKSVBUVCGWLSWMWPWNSWQSWQWHWRWS $. $} $( The expression at the heart of ~ dfaddc2 is a set. (Contributed by SF, @@ -38780,22 +38779,21 @@ Definite description binder (inverted iota) eleq1d ceqsexv elssetk opkeq2 opksnelsik syl6bb opkelimagekg mp2an eqeq2i anbi12d dfaddc2 bitr4i anbi12i opkelcok el1c anbi1i 19.41v cxpk sikss1c1c wb sseli opkelxpk simprbi syl pm4.71ri anass excom 3bitr4i df-clel notbii - elimak df-rex rexnal bitr2i con1bii abbi2i inteqi eqtr4i ) UAUBBFZGZCFZHU - CZXDGZCXDUDZIZBUEZUFUBAFZGZAUEZJJJUGZJUHZUIHKKZLUMUGXPUHXOUHJUJUJUGUNUKXQ - KKLULXQLZUOZUJZUPZULZHLZULZUFCBUQYDXKXJBYDXDYDGXDXNGZXDYCGZMZIXJXDXNYCURY - EXEYGXIXMXEAXDBNZXLXDUBUSUTXIYFYFXHMZCXDVAZXIMDFZXFVBZOZYKXDPZYBGZIZDQZCQ - ZCBVCZYIIZCQYFYJYQYTCYQYLXDPZYBGZYTYOUUBDYLXFVDZYMYNUUAYBYKYLXDVEVFVGUUBU - UAJGZUUAYAGZMZIYTUUAJYAURUUDYSUUFYIXFXDCNZYHVHUUEXHYKEFZVBZOZYLYKPZXTGZYN - JGZIZIZDQZEQZUUHXGOZEBVCZIZEQUUEXHUUPUUTEUUPXFUUHPXSGZUUIXDPZJGZIZUUTUUNU - VDDUUIUUHVDUUJUULUVAUUMUVCUUJUULYLUUIPZXTGUVAUUJUUKUVEXTYKUUIYLVIVFXFUUHX - SUUGENZVJVKUUJYNUVBJYKUUIXDVEVFVOVGUVAUURUVCUUSUVAUUHXRXFLZOZUURXFSGUUHSG - UVAUVHWEUUGUVFXFUUHXRSSVLVMXGUVGUUHXFHVPVNVQUUHXDUVFYHVHVRRTUUEUUNDQZUUQD - YLXDJXTUUCYHVSYKHGZUUNIZDQUUOEQZDQUVIUUQUVKUVLDUVKUUJEQZUUNIUVLUVJUVMUUNE - YKVTWAUUJUUNEWBVQTUUNUVKDUUNUVJUULIZUUMIUVKUULUVNUUMUULUVJUULUUKHHWCZGZUV - JXTUVOUUKXSWDWFUVPYLHGUVJYLYKHHUUCDNWGWHWIWJWAUVJUULUUMWKRTUUOEDWLWMREXGX - DWNWMWOVRRRTYFYODHVAZYRDYBHXDYHWPUVJYOIZDQYPCQZDQUVQYRUVRUVSDUVRYMCQZYOIU - VSUVJUVTYOCYKVTWAYMYOCWBVQTYODHWQYPCDWLWMRYICXDWQWMXHCXDWRWSWTVRRXAXBXC - $. + elimak df-rex rexnal bitr2i con1bii eqabi inteqi eqtr4i ) UAUBBFZGZCFZHUC + ZXDGZCXDUDZIZBUEZUFUBAFZGZAUEZJJJUGZJUHZUIHKKZLUMUGXPUHXOUHJUJUJUGUNUKXQK + KLULXQLZUOZUJZUPZULZHLZULZUFCBUQYDXKXJBYDXDYDGXDXNGZXDYCGZMZIXJXDXNYCURYE + XEYGXIXMXEAXDBNZXLXDUBUSUTXIYFYFXHMZCXDVAZXIMDFZXFVBZOZYKXDPZYBGZIZDQZCQZ + CBVCZYIIZCQYFYJYQYTCYQYLXDPZYBGZYTYOUUBDYLXFVDZYMYNUUAYBYKYLXDVEVFVGUUBUU + AJGZUUAYAGZMZIYTUUAJYAURUUDYSUUFYIXFXDCNZYHVHUUEXHYKEFZVBZOZYLYKPZXTGZYNJ + GZIZIZDQZEQZUUHXGOZEBVCZIZEQUUEXHUUPUUTEUUPXFUUHPXSGZUUIXDPZJGZIZUUTUUNUV + DDUUIUUHVDUUJUULUVAUUMUVCUUJUULYLUUIPZXTGUVAUUJUUKUVEXTYKUUIYLVIVFXFUUHXS + UUGENZVJVKUUJYNUVBJYKUUIXDVEVFVOVGUVAUURUVCUUSUVAUUHXRXFLZOZUURXFSGUUHSGU + VAUVHWEUUGUVFXFUUHXRSSVLVMXGUVGUUHXFHVPVNVQUUHXDUVFYHVHVRRTUUEUUNDQZUUQDY + LXDJXTUUCYHVSYKHGZUUNIZDQUUOEQZDQUVIUUQUVKUVLDUVKUUJEQZUUNIUVLUVJUVMUUNEY + KVTWAUUJUUNEWBVQTUUNUVKDUUNUVJUULIZUUMIUVKUULUVNUUMUULUVJUULUUKHHWCZGZUVJ + XTUVOUUKXSWDWFUVPYLHGUVJYLYKHHUUCDNWGWHWIWJWAUVJUULUUMWKRTUUOEDWLWMREXGXD + WNWMWOVRRRTYFYODHVAZYRDYBHXDYHWPUVJYOIZDQYPCQZDQUVQYRUVRUVSDUVRYMCQZYOIUV + SUVJUVTYOCYKVTWAYMYOCWBVQTYODHWQYPCDWLWMRYICXDWQWMXHCXDWRWSWTVRRXAXBXC $. $} $( The class of all finite cardinals is a set. (Contributed by SF, @@ -38997,18 +38995,18 @@ Definite description binder (inverted iota) ( vn vm cnnc wcel c0c wceq cv c1c wrex cssetk cins3k cpw1 cimak cvv eqeq1 wo rexbidv orbi12d cplc csn cins2k cin ccompl csik cun csymdif cdif df-sn cimagek cab vex elimak wb opkelimagekg mp2an dfaddc2 eqeq2i bitr4i rexbii - copk bitri abbi2i uneq12i unab eqtri snex addcexlem pw1ex imakex imagekex - 1cex nncex unex eqeltrri eqid orci addceq1 eqeq2d rspcev mpan2 olcd finds - weq a1d peano1 eleq1 mpbiri peano2 syl5ibrcom rexlimiv jaoi impbii ) BEFZ - BGHZBAIZJUAZHZAEKZRZCIZGHZXBWRHZAEKZRZGGHZGWRHZAEKZRDIZGHZXJWRHZAEKZRZXJJ - UAZGHZXOWRHZAEKZRZXACDBGUBZLMZLUCZUDJNZNZOUEMYBUCYAUCLUFUFMUGUHYDNNOUIZYD - OZUKZEOZUGZXFCULZPYIXCCULZXECULZUGYJXTYKYHYLCGUJXECYHXBYHFWQXBVBYGFZAEKXE - AYGEXBCUMZUNYMXDAEYMXBYFWQOZHZXDWQPFXBPFYMYPUOAUMYNWQXBYFPPUPUQWRYOXBWQJU - RUSUTVAVCVDVEXCXECVFVGXTYHGVHYGEYFYEYDVIYCJVMVJVJVKVLVNVKVOVPXCXCXGXEXIXB - GGQXCXDXHAEXBGWRQSTCDWEZXCXKXEXMXBXJGQYQXDXLAEXBXJWRQSTXBXOHZXCXPXEXRXBXO - GQYRXDXQAEXBXOWRQSTXBBHZXCWPXEWTXBBGQYSXDWSAEXBBWRQSTXGXIGVQVRXJEFZXSXNYT - XRXPYTXOXOHZXRXOVQXQUUAAXJEADWEWRXOXOWQXJJVSVTWAWBWCWFWDWPWOWTWPWOGEFWGBG - EWHWIWSWOAEWQEFWOWSWREFWQWJBWREWHWKWLWMWN $. + copk bitri eqabi uneq12i unab eqtri addcexlem pw1ex imakex imagekex nncex + snex 1cex unex eqeltrri weq eqid orci addceq1 eqeq2d mpan2 olcd a1d finds + rspcev peano1 eleq1 mpbiri peano2 syl5ibrcom rexlimiv jaoi impbii ) BEFZB + GHZBAIZJUAZHZAEKZRZCIZGHZXBWRHZAEKZRZGGHZGWRHZAEKZRDIZGHZXJWRHZAEKZRZXJJU + AZGHZXOWRHZAEKZRZXACDBGUBZLMZLUCZUDJNZNZOUEMYBUCYAUCLUFUFMUGUHYDNNOUIZYDO + ZUKZEOZUGZXFCULZPYIXCCULZXECULZUGYJXTYKYHYLCGUJXECYHXBYHFWQXBVBYGFZAEKXEA + YGEXBCUMZUNYMXDAEYMXBYFWQOZHZXDWQPFXBPFYMYPUOAUMYNWQXBYFPPUPUQWRYOXBWQJUR + USUTVAVCVDVEXCXECVFVGXTYHGVMYGEYFYEYDVHYCJVNVIVIVJVKVLVJVOVPXCXCXGXEXIXBG + GQXCXDXHAEXBGWRQSTCDVQZXCXKXEXMXBXJGQYQXDXLAEXBXJWRQSTXBXOHZXCXPXEXRXBXOG + QYRXDXQAEXBXOWRQSTXBBHZXCWPXEWTXBBGQYSXDWSAEXBBWRQSTXGXIGVRVSXJEFZXSXNYTX + RXPYTXOXOHZXRXOVRXQUUAAXJEADVQWRXOXOWQXJJVTWAWFWBWCWDWEWPWOWTWPWOGEFWGBGE + WHWIWSWOAEWQEFWOWSWREFWQWJBWREWHWKWLWMWN $. $} ${ @@ -39103,19 +39101,19 @@ Definite description binder (inverted iota) ( va vb vc vx cnnc wcel cplc cv wceq eleq1d imbi2d c1c cab cpw1 cimak cvv wi addceq2 addceq1 c0c wn cssetk cins3k cins2k cin ccompl csymdif cimagek csik cun cdif ccnvk wo unab copk wrex wb vex opkelimagekg mp2an opkelcnvk - addccom dfaddc2 eqeq2i 3bitr4i rexbii elimak risset abbi2i uneq2i 3eqtr4i - eqtri imor abbii abexv addcexlem pw1ex imakex imagekex nncex eqeltrri weq - cnvkex unex addcid1 id syl5eqel addcass peano2 syl5eqelr imim2i a1i finds - com12 vtoclga imp ) AGHBGHZABIZGHZWSCJZBIZGHZSWSXASCAGXBAKZXDXAWSXEXCWTGX - BABUALMWSXBGHZXDXFXBDJZIZGHZSZXFXBUBIZGHZSXFXBEJZIZGHZSZXFXBXMNIZIZGHZSZX - FXDSDEBXFUCZDOZUDUEZUDUFZUGNPPZQUHUEYDUFYCUFUDUKUKUEULUIYEPPQUMZXBPZPZQZU - JZUNZGQZULZXJDOZRYBXIDOZULYAXIUOZDOYMYNYAXIDUPYLYOYBXIDYLFJZXGUQYKHZFGURY - QXHKZFGURXGYLHXIYRYSFGXGYQUQYJHZYQYIXGQZKZYRYSXGRHYQRHYTUUBUSDUTZFUTZXGYQ - YIRRVAVBYQXGYJUUDUUCVCXHUUAYQXHXGXBIUUAXBXGVDXGXBVEVNVFVGVHFYKGXGUUCVIFXH - GVJVGVKVLXJYPDXFXIVOVPVMYBYLYADVQYKGYJYIYFYHVRYGXBCUTVSVSVTWAWEWBVTWFWCXG - UBKZXIXLXFUUEXHXKGXGUBXBTLMDEWDZXIXOXFUUFXHXNGXGXMXBTLMXGXQKZXIXSXFUUGXHX - RGXGXQXBTLMXGBKZXIXDXFUUHXHXCGXGBXBTLMXFXKXBGXBWGXFWHWIXPXTSXMGHXOXSXFXOX - RXNNIGXBXMNWJXNWKWLWMWNWOWPWQWR $. + addccom dfaddc2 eqtri eqeq2i rexbii elimak risset eqabi uneq2i imor abbii + 3bitr4i 3eqtr4i abexv addcexlem pw1ex imakex imagekex nncex unex eqeltrri + cnvkex weq addcid1 syl5eqel addcass peano2 syl5eqelr imim2i finds vtoclga + id a1i com12 imp ) AGHBGHZABIZGHZWSCJZBIZGHZSWSXASCAGXBAKZXDXAWSXEXCWTGXB + ABUALMWSXBGHZXDXFXBDJZIZGHZSZXFXBUBIZGHZSXFXBEJZIZGHZSZXFXBXMNIZIZGHZSZXF + XDSDEBXFUCZDOZUDUEZUDUFZUGNPPZQUHUEYDUFYCUFUDUKUKUEULUIYEPPQUMZXBPZPZQZUJ + ZUNZGQZULZXJDOZRYBXIDOZULYAXIUOZDOYMYNYAXIDUPYLYOYBXIDYLFJZXGUQYKHZFGURYQ + XHKZFGURXGYLHXIYRYSFGXGYQUQYJHZYQYIXGQZKZYRYSXGRHYQRHYTUUBUSDUTZFUTZXGYQY + IRRVAVBYQXGYJUUDUUCVCXHUUAYQXHXGXBIUUAXBXGVDXGXBVEVFVGVOVHFYKGXGUUCVIFXHG + VJVOVKVLXJYPDXFXIVMVNVPYBYLYADVQYKGYJYIYFYHVRYGXBCUTVSVSVTWAWEWBVTWCWDXGU + BKZXIXLXFUUEXHXKGXGUBXBTLMDEWFZXIXOXFUUFXHXNGXGXMXBTLMXGXQKZXIXSXFUUGXHXR + GXGXQXBTLMXGBKZXIXDXFUUHXHXCGXGBXBTLMXFXKXBGXBWGXFWOWHXPXTSXMGHXOXSXFXOXR + XNNIGXBXMNWIXNWJWKWLWPWMWQWNWR $. $} ${ @@ -39158,52 +39156,52 @@ Definite description binder (inverted iota) wi elin wb elsymdif otkelins3k opksnelsik elssetk otkelins2k wo weq sneqb opkelidkg mp2an elsnc 3bitr4i orbi12i bibi12i notbii elcompl dfcleq ancom elun opkelcnvk opkelimagekg dfaddc2 eqeq2i bicomi anbi12i opkelcok addcex - 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df-ne exbii elimak addcex eqvinc abbi2i addcexlem imakex imagekex complex - cnvkex snex inex vvex eqeltrri abexv unex eqeltri vtocl2g ) BEZUAEZFZGUCZ - XIXGUBEZFZHZIZCAHZUDZBJZKLXGCFZGUCZXRXLHZIZXOUDZBJZKLXSXRXGAFZHZIZXOUDZBJ - ZKLUAUBCAUEUEXHCHZXQYCKYIXPYBBYIXNYAXOYIXJXSXMXTYIXIXRGXHCXGUFZUGYIXIXRXL - YJUHUMUIUJUKXKAHZYCYHKYKYBYGBYKYAYFXOYKXTYEXSYKXLYDXRXKAXGUFULUNUIUJUKXQX - NMZBJZXOBJZUOZKXQYLXOUPZBJYOXPYPBXNXOUQURYLXOBUSUTYMYNVAVBZVAVCZVDVENNZOV - FVBYRVCYQVCVAVGVGVBUOVHYSNNOVIZXHNZNZOZVJZVKZGVLZOZVFZUUDYTXKNZNZOZVJZVDZ - VKZKOZVDZVFZYMKYLBUUQXGUUQLXGUUPLZMYLXGUUPBPZVMUURXNUURXGUUHLZXGUUOLZIXNX - GUUHUUOVQUUTXJUVAXMXGUUGLZMXIGHZMUUTXJUVBUVCGXGQUUELXGGQUUDLZUVBUVCGXGUUD - VNUUSVOUUEGXGVNUUSVPGXIHGUUCXGOZHUVCUVDXIUVEGXGXHVRZVSXIGVTXGGUUCUUSVNWAR - RWBXGUUGUUSVMXIGWHRDEZXGQUUNLZDKWCZUVGXIHZUVGXLHZIZDWDZUVAXMUVIUVHDWDUVMU - VHDWEUVHUVLDUVHXGUVGQZUUMLZUVLUVGXGUUMDPZUUSVOUVOUVNUUDLZUVNUULLZIUVLUVNU - UDUULVQUVQUVJUVRUVKUVQUVGUVEHUVJXGUVGUUCUUSUVPWAXIUVEUVGUVFVSWFUVRUVGUUKX - GOZHUVKXGUVGUUKUUSUVPWAXLUVSUVGXGXKVRVSWFWGSSWISDUUNKXGUUSWJDXIXLXGXHUUSU - 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FUUGUUEYLUUFGYMYNYOCDUUHGYPYQWP $. $} ${ @@ -42206,8 +42204,8 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \ wsfin eqeltrd ncfinex sylibr simprl sfintfin ad2antll spfinsfincl syl2anc wss ex vex weq anbi1i 3imtr4g alrimiv ralrimiva c0 csn ccnvk cnnc csymdif csik cpw cpw1 cdif cin cidk ccompl cuni1 wrex copk snex elimak eqtfinrelk - cun opkelcnvk bitri rexbii eluni1 risset 3bitr4i abbi2i tfinrelkex cnvkex - spfinex imakex uni1ex eqeltrri spfininduct mp3an1 sselda wb elabg adantl + cun opkelcnvk bitri rexbii eluni1 risset 3bitr4i eqabi tfinrelkex spfinex + cnvkex imakex uni1ex eqeltrri spfininduct mp3an1 sselda wb elabg adantl mpbid ) EUAFZAGFZHABIZJZGFZBUBZFZAJZGFZXEGXJAXEEUCZXJFZCIZXJFZDIZXPUIZHZX RXJFZUDZDUEZCGUFZGXJURZXEXNJZGFZXOXEYFKUCGUGUHUJXIYGBXNEUKXGXNLXHYFGXGXNM NOULXEYCCGXEXPGFHZYBDYHXPJZGFZXSHZXRJZGFZXTYAYHYKYMYHYKHYJYLYIUIZYMYHYJXS @@ -42216,7 +42214,7 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \ ZVMZVMZTVNVKSYRVOUUATSVOUUBTVORVPSVJYTTRVNYTTSVJUUATVQYQEQVNWDZVHZGTZVRZX JEXIBUUFXGVGZUUEFZXPXHLZCGVSZXGUUFFXIUUHXPUUGVTUUDFZCGVSUUJCUUDGUUGXGWAZW BUUKUUICGUUKUUGXPVTUUCFUUIXPUUGUUCYOUULWEXGXPBUTZYOWCWFWGWFXGUUEUUMWHCXHG - WIWJWKUUEUUDGUUCWLWMWNWOWPWQCDXJEWRWSUQWTXFXKXMXAXEXIXMBAGXGALXHXLGXGAMNX + WIWJWKUUEUUDGUUCWLWNWMWOWPWQCDXJEWRWSUQWTXFXKXMXAXEXIXMBAGXGALXHXLGXGAMNX BXCXD $. $} @@ -42246,20 +42244,20 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \ vt cxpk cnnc cpw1 cimak wa cfin cv ctfin cab cun vfinspeqtncv ncfineq syl cplc c0 vfinncvntsp disjsn sylibr ccnvk csik cpw csymdif cdif cidk ccompl wn copk wex elimak df-rex elpw1 anbi1i r19.41v bitr4i exbii rexcom4 bitri - vex snex opkeq1 eleq1d ceqsexv eqtfinrelk rexbii tfinrelkex spfinex pw1ex - abbi2i imakex eqeltrri ncfindi mp3an2 mpanl2 mpdan ncfinprop mpan2 simpld - tfincl simprd wss vfinspnn syl6ss tfinnn syl3anc nnceleq syl22anc ncfinex - difss ncfinsn addceq12d 3eqtrd ) CUADZEFZAUBZBUBZUCGZBEHZAUDZCFZIZUEZFZXM - FZXOFZUIZXHUCZJUIXGEXPGXHXQGBAUFEXPUGUHXGXMXOKUJGZXQXTGZXGXNXMDVAYBBAUKXM - XNULUMXGXMCDZYBYCUJIZIZYEPLMZLUNNQCPLUOMZJUPCPLNYHUQJRZRZRZSURUONYGKYJSNK - YKSKMUSNUQYISMURYISNUQYJSUTYFCPURUEZERZSZXMCXLAYNXIYNDZOUBZXJIZGZYPXIVBZY - LDZTZOVCZBEHZXLYOYTOYMHZUUCOYLYMXIAVMZVDUUDYPYMDZYTTZOVCZUUCYTOYMVEUUHUUA - BEHZOVCUUCUUGUUIOUUGYRBEHZYTTUUIUUFUUJYTBYPEVFVGYRYTBEVHVIVJUUABOEVKVIVLV - LUUBXKBEUUBYQXIVBZYLDZXKYTUULOYQXJVNYRYSUUKYLYPYQXIVOVPVQXJXIBVMUUEVRVLVS - VLWCYLYMVTEWAWBWDWEZXGYDTXOCDYBYCXNVNXMXOCCWFWGWHWIXGXRYAXSJXGXRQDZYAQDZX - MXRDZXMYADZXRYAGXGUUNUUPXGYDUUNUUPTUUMXMCWJWKZWLXGXHQDZUUOXGUUSEXHDZXGECD - UUSUUTTWAECWJWKZWLZXHWMUHXGUUNUUPUURWNXGUUSEQWOUUTUUQUVBXGEQYEURQWPQYEXCW - QXGUUSUUTUVAWNBEXHAWRWSXMXRYAWTXAXGXNCDXSJGCXBXNCXDWKXEXF $. + vex snex opkeq1 eleq1d ceqsexv eqtfinrelk rexbii eqabi tfinrelkex spfinex + pw1ex imakex eqeltrri ncfindi mp3an2 mpanl2 mpdan ncfinprop simpld tfincl + mpan2 simprd vfinspnn difss syl6ss tfinnn syl3anc nnceleq ncfinex ncfinsn + wss syl22anc addceq12d 3eqtrd ) CUADZEFZAUBZBUBZUCGZBEHZAUDZCFZIZUEZFZXMF + ZXOFZUIZXHUCZJUIXGEXPGXHXQGBAUFEXPUGUHXGXMXOKUJGZXQXTGZXGXNXMDVAYBBAUKXMX + NULUMXGXMCDZYBYCUJIZIZYEPLMZLUNNQCPLUOMZJUPCPLNYHUQJRZRZRZSURUONYGKYJSNKY + KSKMUSNUQYISMURYISNUQYJSUTYFCPURUEZERZSZXMCXLAYNXIYNDZOUBZXJIZGZYPXIVBZYL + DZTZOVCZBEHZXLYOYTOYMHZUUCOYLYMXIAVMZVDUUDYPYMDZYTTZOVCZUUCYTOYMVEUUHUUAB + EHZOVCUUCUUGUUIOUUGYRBEHZYTTUUIUUFUUJYTBYPEVFVGYRYTBEVHVIVJUUABOEVKVIVLVL + UUBXKBEUUBYQXIVBZYLDZXKYTUULOYQXJVNYRYSUUKYLYPYQXIVOVPVQXJXIBVMUUEVRVLVSV + LVTYLYMWAEWBWCWDWEZXGYDTXOCDYBYCXNVNXMXOCCWFWGWHWIXGXRYAXSJXGXRQDZYAQDZXM + XRDZXMYADZXRYAGXGUUNUUPXGYDUUNUUPTUUMXMCWJWMZWKXGXHQDZUUOXGUUSEXHDZXGECDU + USUUTTWBECWJWMZWKZXHWLUHXGUUNUUPUURWNXGUUSEQXCUUTUUQUVBXGEQYEURQWOQYEWPWQ + XGUUSUUTUVAWNBEXHAWRWSXMXRYAWTXDXGXNCDXSJGCXAXNCXBWMXEXF $. $} $( The universe is infinite. Theorem X.1.63 of [Rosser] p. 536. @@ -42484,9 +42482,9 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \ = { x | E. y e. B x = ( Phi y u. { 0c } ) } $= ( cv csn cun wrex cssetk cins2k cins3k cin cpw1 cimak ccompl csik csymdif cimagek cnnc cvv cxpk cphi c0c wceq c1c cdif cidk ccnvk ccomk wcel elimak - copk vex dfop2lem1 rexbii bitri abbi2i ) ADZBDZUAUBEZFUCZBCGZAHIZHJZVBKUD - LLZMNJVBIVCIHOOJFPVDLLMUEVDMQRSTKUFRNSTKFQUGHUHUSESTFJPVDMNZCMZUQVFUIURUQ - UKVEUIZBCGVABVECUQAULUJVGUTBCBAUMUNUOUP $. + copk vex dfop2lem1 rexbii bitri eqabi ) ADZBDZUAUBEZFUCZBCGZAHIZHJZVBKUDL + LZMNJVBIVCIHOOJFPVDLLMUEVDMQRSTKUFRNSTKFQUGHUHUSESTFJPVDMNZCMZUQVFUIURUQU + KVEUIZBCGVABVECUQAULUJVGUTBCBAUMUNUOUP $. $( Express the ordered pair via the set construction functors. (Contributed by SF, 2-Jan-2015.) $) @@ -42513,10 +42511,10 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \ ( vx vy cv wceq wrex cab csn cun cssetk cins3k cins2k cin cpw1 cimak csik ccompl cvv cxpk cop cphi c0c c1c csymdif cdif cnnc cidk ccnvk ccomk df-op wcel copk vex elimak dfphi2 eqeq2i opkelimagek bitr4i rexbii bicomi bitri - cimagek abbi2i dfop2lem2 uneq12i eqtr4i ) ABUACEZDEZUBZFZDAGZCHZVHVJUCIZJ - FDBGCHZJKLZKMZNUDOOZPRLVQMVPMKQQLJUEVROOPUFVRPVCUGSTNUHUGRSTNJZVCZAPZVQVT - UIKUJVNISTJLUEVRPRBPZJCDABUKWAVMWBVOVLCWAVHWAULVIVHUMVTULZDAGZVLDVTAVHCUN - ZUOVLWDVKWCDAVKVHVSVIPZFWCVJWFVHVIUPUQVIVHVSDUNWEURUSUTVAVBVDCDBVEVFVG $. + cimagek eqabi dfop2lem2 uneq12i eqtr4i ) ABUACEZDEZUBZFZDAGZCHZVHVJUCIZJF + DBGCHZJKLZKMZNUDOOZPRLVQMVPMKQQLJUEVROOPUFVRPVCUGSTNUHUGRSTNJZVCZAPZVQVTU + IKUJVNISTJLUEVRPRBPZJCDABUKWAVMWBVOVLCWAVHWAULVIVHUMVTULZDAGZVLDVTAVHCUNZ + UOVLWDVKWCDAVKVHVSVIPZFWCVJWFVHVIUPUQVIVHVSDUNWEURUSUTVAVBVDCDBVEVFVG $. $} ${ @@ -42535,11 +42533,11 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \ ) ) ) "_k A ) $= ( vx vy wcel cssetk cins3k cins2k cin cpw1 cimak ccompl csik cimagek cnnc cv cun cvv cxpk wceq wrex cproj1 cphi cab c1c csymdif cdif ccnvk df-proj1 - cidk copk opkelimagek opkelcnvk dfphi2 eqeq2i rexbii risset elimak abbi2i - vex 3bitr4ri eqtr4i ) AUABOZUBZADZBUCEFZEGZHUDIIZJKFVFGVEGELLFPUEVGIIJUFV - GJMNQRHUINKQRHPZMZUGZAJZBAUHVDBVKCOZVCSZCATVLVBUJVJDZCATVDVBVKDVMVNCAVBVL - UJVIDVLVHVBJZSVNVMVBVLVHBUSZCUSZUKVLVBVIVQVPULVCVOVLVBUMUNUTUOCVCAUPCVJAV - BVPUQUTURVA $. + cidk vex opkelimagek opkelcnvk dfphi2 eqeq2i 3bitr4ri rexbii risset eqabi + copk elimak eqtr4i ) AUABOZUBZADZBUCEFZEGZHUDIIZJKFVFGVEGELLFPUEVGIIJUFVG + JMNQRHUINKQRHPZMZUGZAJZBAUHVDBVKCOZVCSZCATVLVBUSVJDZCATVDVBVKDVMVNCAVBVLU + SVIDVLVHVBJZSVNVMVBVLVHBUJZCUJZUKVLVBVIVQVPULVCVOVLVBUMUNUOUPCVCAUQCVJAVB + VPUTUOURVA $. $( Express the second projection operator via the set construction functors. (Contributed by SF, 2-Jan-2015.) $) @@ -42557,10 +42555,10 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \ ( vx vy cv csn cun wcel cssetk cins2k cins3k cin cpw1 ccompl csik csymdif cimak cimagek cnnc cvv cxpk cproj2 cphi c0c cab c1c cdif cidk ccnvk ccomk df-proj2 copk wrex opkelcnvk dfop2lem1 bitri rexbii elimak risset 3bitr4i - wceq vex abbi2i eqtr4i ) AUABDZUBUCEZFZAGZBUDHIZHJZVHKUELLZPMJVHIVIIHNNJF - OVJLLPUFVJPQRSTKUGRMSTKFQUHHUIVEESTFJOVJPMZUHZAPZBAUJVGBVMCDZVDUKVLGZCAUL - VNVFUTZCAULVDVMGVGVOVPCAVOVDVNUKVKGVPVNVDVKCVABVAZUMBCUNUOUPCVLAVDVQUQCVF - AURUSVBVC $. + wceq vex eqabi eqtr4i ) AUABDZUBUCEZFZAGZBUDHIZHJZVHKUELLZPMJVHIVIIHNNJFO + VJLLPUFVJPQRSTKUGRMSTKFQUHHUIVEESTFJOVJPMZUHZAPZBAUJVGBVMCDZVDUKVLGZCAULV + NVFUTZCAULVDVMGVGVOVPCAVOVDVNUKVKGVPVNVDVKCVABVAZUMBCUNUOUPCVLAVDVQUQCVFA + URUSVBVC $. $} $( Equality theorem for ordered pairs. (Contributed by SF, 2-Jan-2015.) $) @@ -42978,23 +42976,23 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \ ( vz vy vw cnnc c0c wcel wa cv c1c wceq wrex cab eqeq2d cssetk cpw1 cimak cins3k wss wn cplc crab cphi wex weq wo eleq1 biimpcd con3d adantll ssel2 impcom adantlr nnc0suc sylib orel1 anidm anbi2d syl5bbr rexbidv syl5ibcom - sylc eqeq1 eqtr3 rexlimivw impbid1 rexbidva risset rexcom 3bitr4g abbi2dv - cif df-phi addceq1 rexrab r19.41v syl6bbr rexbiia bitri abbii syl6eqr cin - iftrue eqtri cvv dfrab2 cins2k ccompl csik cun csymdif cdif cimagek ccnvk - copk elimak opkelimagek opkelcnvk dfaddc2 eqeq2i 3bitr4i rexbii addcexlem - vex abbi2i 1cex pw1ex imakex imagekex cnvkex eqeltrri nncex eqeltri phieq - inex spcev syl ) BGUAZHBIZUBZJZBDKZEKZLUCZMZDBNZEGUDZUEZMZBAKZUEZMZAUFYCB - YDYLLUCZMZFKZYOMZJZDBNZAGNZFOZYJYCUUAFBYCDFUGZDBNYSAGNZDBNYQBIUUAYCUUCUUD - DBYCYDBIZJZUUCUUDUUFYPAGNZUUCUUDUUFYDHMZUBZUUHUUGUHZUUGYBUUEUUIXTUUEYBUUI - UUEUUHYAUUHUUEYAYDHBUIUJUKUNULUUFYDGIZUUJXTUUEUUKYBBGYDUMUOAYDUPUQUUHUUGU - RVDUUCYPYSAGYPYPYPJUUCYSYPUSUUCYPYRYPYDYQYOVEUTVAVBVCYSUUCAGYDYQYOVFVGVHV - IDYQBVJYSADGBVKVLVMYJYQYLGIZYOYLVNZMZAYINZFOUUBAFYIVOUUOUUAFUUOYPDBNZUUNJ - ZAGNUUAYHUUPUUNAEGEAUGZYGYPDBUURYFYOYDYEYLLVPPVBVQUUQYTAGUULUUQUUPYRJYTUU - LUUNYRUUPUULUUMYOYQUULYOYLWEPUTYPYRDBVRVSVTWAWBWFWCYNYKAYIYIYHEOZGWDWGYHE - GWHUUSGQTZQWIZWDLRZRZSWJTUVAWIUUTWIQWKWKTWLWMUVCRRSWNZUVCSZWOZWPZBSZUUSWG - YHEUVHYEUVHIYDYEWQUVGIZDBNYHDUVGBYEEXFZWRUVIYGDBYEYDWQUVFIYDUVEYESZMUVIYG - YEYDUVEUVJDXFZWSYDYEUVFUVLUVJWTYFUVKYDYELXAXBXCXDWAXGUVGBUVFUVEUVDUVCXEUV - BLXHXIXIXJXKXLCXJXMXNXQXOYLYIMYMYJBYLYIXPPXRXS $. + sylc eqeq1 rexlimivw impbid1 rexbidva risset rexcom 3bitr4g eqabdv df-phi + eqtr3 cif addceq1 rexrab iftrue r19.41v syl6bbr rexbiia bitri abbii eqtri + syl6eqr cin cvv dfrab2 cins2k ccompl csik csymdif cdif cimagek ccnvk copk + cun elimak opkelimagek opkelcnvk dfaddc2 eqeq2i 3bitr4i rexbii eqabi 1cex + addcexlem pw1ex imakex imagekex cnvkex eqeltrri nncex eqeltri phieq spcev + vex inex syl ) BGUAZHBIZUBZJZBDKZEKZLUCZMZDBNZEGUDZUEZMZBAKZUEZMZAUFYCBYD + YLLUCZMZFKZYOMZJZDBNZAGNZFOZYJYCUUAFBYCDFUGZDBNYSAGNZDBNYQBIUUAYCUUCUUDDB + YCYDBIZJZUUCUUDUUFYPAGNZUUCUUDUUFYDHMZUBZUUHUUGUHZUUGYBUUEUUIXTUUEYBUUIUU + EUUHYAUUHUUEYAYDHBUIUJUKUNULUUFYDGIZUUJXTUUEUUKYBBGYDUMUOAYDUPUQUUHUUGURV + DUUCYPYSAGYPYPYPJUUCYSYPUSUUCYPYRYPYDYQYOVEUTVAVBVCYSUUCAGYDYQYOVNVFVGVHD + YQBVIYSADGBVJVKVLYJYQYLGIZYOYLVOZMZAYINZFOUUBAFYIVMUUOUUAFUUOYPDBNZUUNJZA + GNUUAYHUUPUUNAEGEAUGZYGYPDBUURYFYOYDYEYLLVPPVBVQUUQYTAGUULUUQUUPYRJYTUULU + UNYRUUPUULUUMYOYQUULYOYLVRPUTYPYRDBVSVTWAWBWCWDWEYNYKAYIYIYHEOZGWFWGYHEGW + HUUSGQTZQWIZWFLRZRZSWJTUVAWIUUTWIQWKWKTWQWLUVCRRSWMZUVCSZWNZWOZBSZUUSWGYH + EUVHYEUVHIYDYEWPUVGIZDBNYHDUVGBYEEXQZWRUVIYGDBYEYDWPUVFIYDUVEYESZMUVIYGYE + YDUVEUVJDXQZWSYDYEUVFUVLUVJWTYFUVKYDYELXAXBXCXDWBXEUVGBUVFUVEUVDUVCXGUVBL + XFXHXHXIXJXKCXIXLXMXRXNYLYIMYMYJBYLYIXOPXPXS $. $( Lemma for ~ phiall . Any set without ` 0c ` is equal to the ` Phi ` of a set. (Contributed by Scott Fenton, 8-Apr-2021.) $) @@ -44323,18 +44321,18 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \ ccomk csn elimak df-rex elpw12 anbi1i r19.41v bitr4i exbii rexcom4 opkeq1 vex snex eleq1d ceqsexv 3bitr2i bitri exancom opkelxpk mpbiran elvvk elin rexbii 19.41vv 3bitr4i exrot3 anass 19.42v opkeq2 setconslem3 syl6bb opex - anbi1d exbidv eleq1 pm5.32i 3bitri 2exbii abbi2i ) CGZAGZBGZHZUAZWNWOUBZD - IZJZBKAKZCLLLMZMZNUCUCOZNNOZNPZTUDQQZRUEOXFPXEPXDUFUGXGQQZRUHXGRUIUJLMTUK - UJUELMTUFUIULZUCUNOXFXFXINUNUMUOUOLMUFOUGXGRUEUCOTXGRPUFPUGXHRUEZTZDQQZRZ - WMXMIEGZWMHZXKIZEXLSZFGZUOZUOZWMHZXKIZFDSZXAEXKXLWMCVEZUPXQXNXLIZXPJZEKZY - CXPEXLUQYGXNXTUAZXPJZFDSZEKYIEKZFDSYCYFYJEYFYHFDSZXPJYJYEYLXPFXNDURUSYHXP - FDUTVAVBYIFEDVCYKYBFDXPYBEXTXSVFZYHXOYAXKXNXTWMVDVGVHVPVIVJYCXRDIZYBJFKYB - YNJZFKZXAYBFDUQYNYBFVKYPWQYAXJIZJZYNJZFKZBKAKZXAYPYRBKAKZYNJZFKYSBKAKZFKU - UAYOUUCFYBUUBYNYAXCIZYQJWQBKAKZYQJYBUUBUUEUUFYQUUEWMXBIZUUFUUEXTLIUUGYMXT - WMLXBYMYDVLVMABWMVNVJUSYAXCXJVOWQYQABVQVRUSVBUUDUUCFYRYNABVQVBYSFABVSVIYT - WTABYTWQYQYNJZJZFKWQUUHFKZJWTYSUUIFWQYQYNVTVBWQUUHFWAWQUUJWSWQUUJXRWRUAZY - NJZFKWSWQUUHUULFWQYQUUKYNWQYQXTWPHZXJIUUKWQYAUUMXJWMWPXTWBVGXRWNWOFVEAVEZ - BVEZWCWDWFWGYNWSFWRWNWOUUNUUOWEXRWRDWHVHWDWIWJWKVJWJWJWL $. + anbi1d exbidv eleq1 pm5.32i 3bitri 2exbii eqabi ) CGZAGZBGZHZUAZWNWOUBZDI + ZJZBKAKZCLLLMZMZNUCUCOZNNOZNPZTUDQQZRUEOXFPXEPXDUFUGXGQQZRUHXGRUIUJLMTUKU + JUELMTUFUIULZUCUNOXFXFXINUNUMUOUOLMUFOUGXGRUEUCOTXGRPUFPUGXHRUEZTZDQQZRZW + MXMIEGZWMHZXKIZEXLSZFGZUOZUOZWMHZXKIZFDSZXAEXKXLWMCVEZUPXQXNXLIZXPJZEKZYC + XPEXLUQYGXNXTUAZXPJZFDSZEKYIEKZFDSYCYFYJEYFYHFDSZXPJYJYEYLXPFXNDURUSYHXPF + DUTVAVBYIFEDVCYKYBFDXPYBEXTXSVFZYHXOYAXKXNXTWMVDVGVHVPVIVJYCXRDIZYBJFKYBY + NJZFKZXAYBFDUQYNYBFVKYPWQYAXJIZJZYNJZFKZBKAKZXAYPYRBKAKZYNJZFKYSBKAKZFKUU + AYOUUCFYBUUBYNYAXCIZYQJWQBKAKZYQJYBUUBUUEUUFYQUUEWMXBIZUUFUUEXTLIUUGYMXTW + MLXBYMYDVLVMABWMVNVJUSYAXCXJVOWQYQABVQVRUSVBUUDUUCFYRYNABVQVBYSFABVSVIYTW + TABYTWQYQYNJZJZFKWQUUHFKZJWTYSUUIFWQYQYNVTVBWQUUHFWAWQUUJWSWQUUJXRWRUAZYN + JZFKWSWQUUHUULFWQYQUUKYNWQYQXTWPHZXJIUUKWQYAUUMXJWMWPXTWBVGXRWNWOFVEAVEZB + VEZWCWDWFWGYNWSFWRWNWOUUNUUOWEXRWRDWHVHWDWIWJWKVJWJWJWL $. $} ${ @@ -44718,11 +44716,11 @@ Ins2_k Ins3_k ( ( Ins2_k _S_k i^i ( vx vy vw vt cv cvv cxpk cssetk csik cins3k cins2k cin cpw1 cimak ccompl cun csymdif wcel cima wbr wrex cab c1c cdif cimagek cnnc cidk ccnvk ccomk vz c0c csn df-ima copk vex elimak cop setconslem6 opeq1 eleq1d opkelopkab - weq opeq2 df-br bitr4i rexbii bitri abbi2i eqtr4i ) ABUACGZDGZAUBZCBUCZDU - DHHHIIJKKLZJJLZJMZNUEOOZPQLVRMVQMVPRSVSOOZPUFVSPUGUHHINUIUHQHINRUGUJZKUKL - VRVRWAJUKUMUNUNHIRLSVSPQKLNVSPMRMSVTPQNAOOPZBPZDCABUOVODWCVMWCTVLVMUPWBTZ - CBUCVOCWBBVMDUQZURWDVNCBWDVLVMUSZATZVNEGZFGZUSZATVLWIUSZATWGULEFWBVLVMEFU - LAUTECVDWJWKAWHVLWIVAVBFDVDWKWFAWIVMVLVEVBCUQWEVCVLVMAVFVGVHVIVJVK $. + weq opeq2 df-br bitr4i rexbii bitri eqabi eqtr4i ) ABUACGZDGZAUBZCBUCZDUD + HHHIIJKKLZJJLZJMZNUEOOZPQLVRMVQMVPRSVSOOZPUFVSPUGUHHINUIUHQHINRUGUJZKUKLV + RVRWAJUKUMUNUNHIRLSVSPQKLNVSPMRMSVTPQNAOOPZBPZDCABUOVODWCVMWCTVLVMUPWBTZC + BUCVOCWBBVMDUQZURWDVNCBWDVLVMUSZATZVNEGZFGZUSZATVLWIUSZATWGULEFWBVLVMEFUL + AUTECVDWJWKAWHVLWIVAVBFDVDWKWFAWIVMVLVEVBCUQWEVCVLVMAVFVGVHVIVJVK $. $} $( The image of a set under a set is a set. (Contributed by SF, @@ -45979,7 +45977,7 @@ We use their notation ("onto" under the arrow). For alternate $d x y A $. opabbi2i.1 $e |- ( <. x , y >. e. A <-> ph ) $. $( Equality of a class variable and an ordered pair abstractions (inference - rule). Compare ~ abbi2i . (Contributed by Scott Fenton, + rule). Compare ~ eqabi . (Contributed by Scott Fenton, 18-Apr-2021.) $) opabbi2i $p |- A = { <. x , y >. | ph } $= ( cv cop wcel copab opabid2 opabbii eqtr3i ) BFCFGDHZBCIDABCIBCDJMABCEKL @@ -45990,7 +45988,7 @@ We use their notation ("onto" under the arrow). For alternate $d x y A $. $d x y ph $. opabbi2dv.1 $e |- ( ph -> ( <. x , y >. e. A <-> ps ) ) $. $( Deduce equality of a relation and an ordered-pair class builder. - Compare ~ abbi2dv . (Contributed by NM, 24-Feb-2014.) $) + Compare ~ eqabdv . (Contributed by NM, 24-Feb-2014.) $) opabbi2dv $p |- ( ph -> A = { <. x , y >. | ps } ) $= ( cv cop wcel copab opabid2 opabbidv syl5eqr ) AECGDGHEIZCDJBCDJCDEKANBCD FLM $. @@ -46268,17 +46266,17 @@ We use their notation ("onto" under the arrow). For alternate $( Alternate definition of domain. (Contributed by set.mm contributors, 5-Feb-2015.) $) dfdm2 $p |- dom A = { x | E. y x A y } $= - ( cv wbr wex cdm eldm abbi2i ) ADZBDCEBFACGBJCHI $. + ( cv wbr wex cdm eldm eqabi ) ADZBDCEBFACGBJCHI $. $( Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by set.mm contributors, 28-Dec-1996.) $) dfdm3 $p |- dom A = { x | E. y <. x , y >. e. A } $= - ( cv cop wcel wex cdm eldm2 abbi2i ) ADZBDECFBGACHBKCIJ $. + ( cv cop wcel wex cdm eldm2 eqabi ) ADZBDECFBGACHBKCIJ $. $( Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by set.mm contributors, 27-Dec-1996.) $) dfrn2 $p |- ran A = { y | E. x x A y } $= - ( cv wbr wex crn elrn abbi2i ) ADBDZCEAFBCGAJCHI $. + ( cv wbr wex crn elrn eqabi ) ADBDZCEAFBCGAJCHI $. $( Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by set.mm contributors, 28-Dec-1996.) $) @@ -46361,10 +46359,10 @@ We use their notation ("onto" under the arrow). For alternate (Revised by set.mm contributors, 27-Aug-2011.) $) dmun $p |- dom ( A u. B ) = ( dom A u. dom B ) $= ( vx vy cun cdm cv cop wcel wex cab dfdm3 wo eldm orbi12i elun df-br brun - wbr bitr3i exbii 19.43 bitri 3bitr4i abbi2i eqtr4i ) ABEZFCGZDGZHUGIZDJZC - KAFZBFZEZCDUGLUKCUNUHULIZUHUMIZMUHUIASZDJZUHUIBSZDJZMZUHUNIUKUOURUPUTDUHA - NDUHBNOUHULUMPUKUQUSMZDJVAUJVBDUJUHUIUGSVBUHUIUGQUHUIABRTUAUQUSDUBUCUDUEU - F $. + wbr bitr3i exbii 19.43 bitri 3bitr4i eqabi eqtr4i ) ABEZFCGZDGZHUGIZDJZCK + AFZBFZEZCDUGLUKCUNUHULIZUHUMIZMUHUIASZDJZUHUIBSZDJZMZUHUNIUKUOURUPUTDUHAN + DUHBNOUHULUMPUKUQUSMZDJVAUJVBDUJUHUIUGSVBUHUIUGQUHUIABRTUAUQUSDUBUCUDUEUF + $. $( The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by set.mm contributors, @@ -50062,12 +50060,12 @@ singleton of the first member (with no sethood assumptions on ` B ` ). ( vz cv wcel wbr wa wex weu cfv wceq wb wal elfv wi bi2 breq2 sylib alimi vex ceqsalv anim2i eximi elequ2 anbi12d cbvexv 19.40 simprd df-eu jca nfv sylibr nfeu1 nfa1 nfan nfex nfim bi1 ax-14 syl6 com23 imp3a anc2ri eximdv - sps com12 syl5bi exlimi imp impbii bitri abbi2i ) AFZBFZGZCVPDHZIZBJZVRBK - ZIZACDLZVOWCGVOEFZGZVRVPWDMZNZBOZIZEJZWBEBVOCDPWJWBWJVTWAWJWECWDDHZIZEJVT - WIWLEWHWKWEWHWFVRQZBOWKWGWMBVRWFRUAVRWKBWDEUBVPWDCDSUCTUDUEWLVSEBWDVPMWEV - QWKVREBAUFWDVPCDSUGUHTWJWHEJZWAWJWEEJWNWEWHEUIUJVRBEUKZUNULVTWAWJVSWAWJQB - WAWJBVRBUOWIBEWEWHBWEBUMWGBUPUQURUSWAWNVSWJWOVSWHWIEWHVSWIWHVSWEWGVSWEQBW - GVQVRWEWGVRVQWEWGVRWFVQWEQVRWFUTBEAVAVBVCVDVGVEVHVFVIVJVKVLVMVN $. + sps com12 syl5bi exlimi imp impbii bitri eqabi ) AFZBFZGZCVPDHZIZBJZVRBKZ + IZACDLZVOWCGVOEFZGZVRVPWDMZNZBOZIZEJZWBEBVOCDPWJWBWJVTWAWJWECWDDHZIZEJVTW + IWLEWHWKWEWHWFVRQZBOWKWGWMBVRWFRUAVRWKBWDEUBVPWDCDSUCTUDUEWLVSEBWDVPMWEVQ + WKVREBAUFWDVPCDSUGUHTWJWHEJZWAWJWEEJWNWEWHEUIUJVRBEUKZUNULVTWAWJVSWAWJQBW + AWJBVRBUOWIBEWEWHBWEBUMWGBUPUQURUSWAWNVSWJWOVSWHWIEWHVSWIWHVSWEWGVSWEQBWG + VQVRWEWGVRVQWEWGVRWFVQWEQVRWFUTBEAVAVBVCVDVGVEVHVFVIVJVKVLVMVN $. $} ${ @@ -51788,15 +51786,15 @@ H Isom ( R i^i ( A X. A ) ) , S ( A , B ) ) $= cv eliniseg anbi2i wceq crn wfo wf1o isof1o f1ofo forn eleq2d wfn fvelrnb f1ofn bitr3d anbi1d adantr anbi1i anass wi fnbrfvb adantrr bicomd anbi12d sylan isorel ancom breq1 pm5.32i 3bitr3g exp32 com23 imp pm5.32d rexbidv2 - syl5bb r19.41v syl6bb bitr4d abbi2dv df-ima syl6reqr ) ABDEFUAZCAIZJZBEKC - FLZMNZOZGUDZHUDZFPZGADKCMNZOZQZHUBFWPNWHWQHWKWMWKIZWMBIZWMWIEPZJZWHWQWRWS - WMWJIZJXAWMBWJUCXBWTWSEWIWMUEUFRWHXAWLFLZWMUGZGAQZWTJZWQWFXAXFSWGWFWSXEWT - WFWMFUHZIZWSXEWFXGBWMWFABFUIZXGBUGWFABFUJZXIABDEFUKZABFULTABFUMTUNWFFAUOZ - XHXESWFXJXLXKABFUQTZGAWMFUPTURUSUTWHWQXDWTJZGAQXFWHWNXNGWPAWLWPIZWNJZWLAI - ZWLCDPZWNJZJZWHXQXNJXPXQXRJZWNJXTXOYAWNXOXQWLWOIZJYAWLAWOUCYBXRXQDCWLUEUF - RVAXQXRWNVBRWHXQXSXNWFWGXQXSXNSZVCWFXQWGYCWFXQWGYCWFXQWGJJZWNXRJXDXCWIEPZ - JXSXNYDWNXDXRYEYDXDWNWFXQXDWNSZWGWFXLXQYFXMAWLWMFVDVHVEVFABWLCDEFVIVGWNXR - VJXDYEWTXCWMWIEVKVLVMVNVOVPVQVSVRXDWTGAVTWAWBVSWCHGFWPWDWE $. + syl5bb r19.41v syl6bb bitr4d eqabdv df-ima syl6reqr ) ABDEFUAZCAIZJZBEKCF + LZMNZOZGUDZHUDZFPZGADKCMNZOZQZHUBFWPNWHWQHWKWMWKIZWMBIZWMWIEPZJZWHWQWRWSW + MWJIZJXAWMBWJUCXBWTWSEWIWMUEUFRWHXAWLFLZWMUGZGAQZWTJZWQWFXAXFSWGWFWSXEWTW + FWMFUHZIZWSXEWFXGBWMWFABFUIZXGBUGWFABFUJZXIABDEFUKZABFULTABFUMTUNWFFAUOZX + HXESWFXJXLXKABFUQTZGAWMFUPTURUSUTWHWQXDWTJZGAQXFWHWNXNGWPAWLWPIZWNJZWLAIZ + WLCDPZWNJZJZWHXQXNJXPXQXRJZWNJXTXOYAWNXOXQWLWOIZJYAWLAWOUCYBXRXQDCWLUEUFR + VAXQXRWNVBRWHXQXSXNWFWGXQXSXNSZVCWFXQWGYCWFXQWGYCWFXQWGJJZWNXRJXDXCWIEPZJ + XSXNYDWNXDXRYEYDXDWNWFXQXDWNSZWGWFXLXQYFXMAWLWMFVDVHVEVFABWLCDEFVIVGWNXRV + JXDYEWTXCWMWIEVKVLVMVNVOVPVQVSVRXDWTGAVTWAWBVSWCHGFWPWDWE $. $} ${ @@ -55497,9 +55495,9 @@ result of an operator (deduction version). (Contributed by Paul 20-Feb-2015.) $) epprc $p |- -. _E e. _V $= ( vx cep cvv wcel cv wnel cab wn df-nel mpbi cfix ccompl wel wbr elfix epel - ru bitri notbii vex elcompl 3bitr4i abbi2i fixexg complexg syl syl5eqelr - mto ) BCDZAEZUJFZAGZCDZULCFUMHAQULCIJUIULBKZLZCUKAUOUJUNDZHAAMZHUJUODUKUPUQ - UPUJUJBNUQUJBOAAPRSUJUNATUAUJUJIUBUCUIUNCDUOCDBCUDUNCUEUFUGUH $. + ru bitri notbii vex elcompl 3bitr4i eqabi fixexg complexg syl syl5eqelr mto + ) BCDZAEZUJFZAGZCDZULCFUMHAQULCIJUIULBKZLZCUKAUOUJUNDZHAAMZHUJUODUKUPUQUPUJ + UJBNUQUJBOAAPRSUJUNATUAUJUJIUBUCUIUNCDUOCDBCUDUNCUEUFUGUH $. ${ $d f x y z p q $. @@ -55512,28 +55510,28 @@ result of an operator (deduction version). 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(Contributed by Paul ( vx vy vz va vb c1c cpw1fn wceq cv cfv cpw1 wss wex cvv wa vex exbii csn wcel el1c cpw wf1o wfn crn wi wral fnpw1fn cuni wrex df-pw1fn rnmpt sspw1 weq cab df1c2 sseq2i ssv biantrur 3bitr4i elpw df-rex anbi1i 19.41v excom - bitr4i unieq unisn syl6eq pw1eq eqeq2d ceqsexv bitri 3bitri abbi2i eqtr4i - snex syl anbi12i eeanv pw111 biimpi fveq2 pw1fnval eqeqan12d eqeq12 sneqb - a1i syl6bb 3imtr4d exlimivv sylbi rgen2a dff1o6 mpbir3an ) FFUAZGUBGFUCGU - DZWOHAIZGJZBIZGJZHZABUMZUEZBFUFAFUFUGWPWSWQUHZKZHZAFUIZBUNWOABFXEGAUJUKXG - BWOWSFLZWSCIZKZHZCMZWSWOSXGWSNKZLXINLZXKOZCMXHXLCWSNBPZULFXMWSUOUPXKXOCXN - XKXIUQURQUSWSFXPUTXGWQFSZXFOZAMWQXIRZHZXFOZCMZAMZXLXFAFVAXRYBAXRXTCMZXFOY - BXQYDXFCWQTVBXTXFCVCVEQYCYAAMZCMXLYAACVDYEXKCXFXKAXSXIVPXTXEXJWSXTXDXIHXE - XJHXTXDXSUHXIWQXSVFXICPVGVHXDXIVIVQVJVKQVLVMUSVNVOXCABFXQWSFSZOZWQDIZRZHZ - WSEIZRZHZOZEMDMZXCYGYJDMZYMEMZOYOXQYPYFYQDWQTEWSTVRYJYMDEVSVEYNXCDEYNYHKZ - YKKZHZDEUMZXAXBYTUUAUEYNYTUUAYHYKVTWAWGYJYMWRYRWTYSYJWRYIGJYRWQYIGWBYHDPZ - WCVHYMWTYLGJYSWSYLGWBYKEPWCVHWDYNXBYIYLHUUAWQYIWSYLWEYHYKUUBWFWHWIWJWKWLA - BFWOGWMWN $. + bitr4i snex unieq unisn syl6eq pw1eq syl eqeq2d bitri 3bitri eqabi eqtr4i + ceqsexv anbi12i eeanv pw111 biimpi fveq2 pw1fnval eqeqan12d eqeq12 syl6bb + a1i sneqb 3imtr4d exlimivv sylbi rgen2a dff1o6 mpbir3an ) FFUAZGUBGFUCGUD + ZWOHAIZGJZBIZGJZHZABUMZUEZBFUFAFUFUGWPWSWQUHZKZHZAFUIZBUNWOABFXEGAUJUKXGB + WOWSFLZWSCIZKZHZCMZWSWOSXGWSNKZLXINLZXKOZCMXHXLCWSNBPZULFXMWSUOUPXKXOCXNX + KXIUQURQUSWSFXPUTXGWQFSZXFOZAMWQXIRZHZXFOZCMZAMZXLXFAFVAXRYBAXRXTCMZXFOYB + XQYDXFCWQTVBXTXFCVCVEQYCYAAMZCMXLYAACVDYEXKCXFXKAXSXIVFXTXEXJWSXTXDXIHXEX + JHXTXDXSUHXIWQXSVGXICPVHVIXDXIVJVKVLVQQVMVNUSVOVPXCABFXQWSFSZOZWQDIZRZHZW + SEIZRZHZOZEMDMZXCYGYJDMZYMEMZOYOXQYPYFYQDWQTEWSTVRYJYMDEVSVEYNXCDEYNYHKZY + KKZHZDEUMZXAXBYTUUAUEYNYTUUAYHYKVTWAWGYJYMWRYRWTYSYJWRYIGJYRWQYIGWBYHDPZW + CVIYMWTYLGJYSWSYLGWBYKEPWCVIWDYNXBYIYLHUUAWQYIWSYLWEYHYKUUBWHWFWIWJWKWLAB + FWOGWMWN $. $} ${ @@ -55951,9 +55949,9 @@ result of an operator (deduction version). (Contributed by Paul { x | E! y x F y } $= ( vz cv wbr weu cid ccom ccompl cdif cdm wcel wex weq wi wal wa exbii nfv eldm fnfullfunlem1 wsb eu1 breq2 sbie equcom imbi12i anbi2i bitr2i 3bitri - albii abbi2i ) AEZBEZCFZBGZAHCIHJCIKZLZUNUSMUNUOURFZBNUPUNDEZCFZDBOZPZDQZ - RZBNZUQBUNURUAUTVFBDUNUOCUBSUQUPUPBDUCZBDOZPZDQZRZBNVGUPBDUPDTUDVLVFBVKVE - UPVJVDDVHVBVIVCUPVBBDVBBTUOVAUNCUEUFBDUGUHULUISUJUKUM $. + albii eqabi ) AEZBEZCFZBGZAHCIHJCIKZLZUNUSMUNUOURFZBNUPUNDEZCFZDBOZPZDQZR + ZBNZUQBUNURUAUTVFBDUNUOCUBSUQUPUPBDUCZBDOZPZDQZRZBNVGUPBDUPDTUDVLVFBVKVEU + PVJVDDVHVBVIVCUPVBBDVBBTUOVAUNCUEUFBDUGUHULUISUJUKUM $. $} ${ @@ -56121,13 +56119,13 @@ result of an operator (deduction version). (Contributed by Paul clos1ex $p |- Clos1 ( S , R ) e. _V $= ( va vb cv wss cima cvv csset wcel wbr vex brsset wex imaex bitri ssetex wa cclos1 cab cint df-clos1 csn cimage ccom cfix cin elin elimasn 3bitr2i - df-br elfix brco brimage anbi1i exbii breq1 syl6bb ceqsexv anbi12i abbi2i - cop wceq snex imageex coex fixex inex eqeltrri intex eqeltri ) BAUABEGZHZ - AVNIZVNHZTZEUBZUCJABEUDVSKBUEZIZKAUFZUGZUHZUIZVSJVREWEVNWELVNWALZVNWDLZTV - RVNWAWDUJWFVOWGVQWFBVNVDKLBVNKMVOKBVNUKBVNKUMBVNCENZOULWGVNVNWCMZVQVNWCUN - WIVNFGZWBMZWJVNKMZTZFPZVQFVNVNKWBUOWNWJVPVEZWLTZFPVQWMWPFWKWOWLVNWJAWHFNU - PUQURWLVQFVPAVNDWHQZWOWLVPVNKMVQWJVPVNKUSVPVNWQWHOUTVARRRVBRVCWAWDKVTSBVF - QWCKWBSADVGVHVIVJVKVLVM $. + cop df-br elfix brco wceq brimage anbi1i exbii breq1 syl6bb ceqsexv eqabi + anbi12i snex imageex coex fixex inex eqeltrri intex eqeltri ) BAUABEGZHZA + VNIZVNHZTZEUBZUCJABEUDVSKBUEZIZKAUFZUGZUHZUIZVSJVREWEVNWELVNWALZVNWDLZTVR + VNWAWDUJWFVOWGVQWFBVNUMKLBVNKMVOKBVNUKBVNKUNBVNCENZOULWGVNVNWCMZVQVNWCUOW + IVNFGZWBMZWJVNKMZTZFPZVQFVNVNKWBUPWNWJVPUQZWLTZFPVQWMWPFWKWOWLVNWJAWHFNUR + USUTWLVQFVPAVNDWHQZWOWLVPVNKMVQWJVPVNKVAVPVNWQWHOVBVCRRRVERVDWAWDKVTSBVFQ + WCKWBSADVGVHVIVJVKVLVM $. $} ${ @@ -57463,16 +57461,16 @@ result of an operator (deduction version). (Contributed by Paul cins2 ccnv csi cins3 csymdif cpw1 cec wceq cab df-qs elimapw1 wel wal wex ccompl elima1c elsymdif snex otelins2 opelssetsn otelins3 wbr df-br brcnv bitr3i opsnelsi elec 3bitr4i bibi12i notbii exbii opex alex dfcleq bitr4i - elcompl rexbii abbi2i eqtr4i ssetex ins2ex cnvexg siexg ins3exg symdifexg - 3syl sylancr 1cex imaexg sylancl complexg syl pw1exg syl2an syl5eqel ) BC - HZADHZUAABUBZIUCZBUDZUEZUFZUGZJKZUQZAUHZKZLWTEMZFMZBUIZUJZFANZEUKXIFEABUL - XNEXIXJXIHXKOZXJPZXGHZFANXNFXJXGAUMXQXMFAXQGEUNZGMZXLHZQZGUOZXMXPXFHZRYAR - ZGUPZRXQYBYCYEYCXSOZXPPZXEHZGUPYEGXPXEURYHYDGYHYGXAHZYGXDHZQZRYDYGXAXDUSY - KYAYIXRYJXTYIYFXJPIHXRYFXOXJIXKUTZVAXSXJGSZESZVBTYJYFXOPXCHZXTYFXOXJXCYNV - CXSXKPXBHZXKXSBVDZYOXTYPXSXKXBVDYQXSXKXBVEXSXKBVFVGXSXKXBYMFSVHXSXKBVIVJT - VKVLTVMTVLXPXFXOXJYLYNVNVRYAGVOVJGXJXLVPVQVSTVTWAWRXGLHZXHLHXILHWSWRXFLHZ - YRWRXELHZJLHYSWRXALHXDLHZYTIWBWCWRXBLHXCLHUUABCWDXBLWEXCLWFWHXAXDLLWGWIWJ - XEJLLWKWLXFLWMWNADWOXGXHLLWKWPWQ $. + elcompl rexbii eqabi eqtr4i ssetex ins2ex siexg ins3exg symdifexg sylancr + cnvexg 3syl 1cex imaexg sylancl complexg syl pw1exg syl2an syl5eqel ) BCH + ZADHZUAABUBZIUCZBUDZUEZUFZUGZJKZUQZAUHZKZLWTEMZFMZBUIZUJZFANZEUKXIFEABULX + NEXIXJXIHXKOZXJPZXGHZFANXNFXJXGAUMXQXMFAXQGEUNZGMZXLHZQZGUOZXMXPXFHZRYARZ + GUPZRXQYBYCYEYCXSOZXPPZXEHZGUPYEGXPXEURYHYDGYHYGXAHZYGXDHZQZRYDYGXAXDUSYK + YAYIXRYJXTYIYFXJPIHXRYFXOXJIXKUTZVAXSXJGSZESZVBTYJYFXOPXCHZXTYFXOXJXCYNVC + XSXKPXBHZXKXSBVDZYOXTYPXSXKXBVDYQXSXKXBVEXSXKBVFVGXSXKXBYMFSVHXSXKBVIVJTV + KVLTVMTVLXPXFXOXJYLYNVNVRYAGVOVJGXJXLVPVQVSTVTWAWRXGLHZXHLHXILHWSWRXFLHZY + RWRXELHZJLHYSWRXALHXDLHZYTIWBWCWRXBLHXCLHUUABCWHXBLWDXCLWEWIXAXDLLWFWGWJX + EJLLWKWLXFLWMWNADWOXGXHLLWKWPWQ $. $} ${ @@ -57736,14 +57734,14 @@ the first case of his notation (simple exponentiation) and subscript it ( vx cfuns c1st cimage ccnv cima cin c2nd cv wcel wceq wbr bitri 3bitr4i wa csn cpw wf cab cvv wfun cdm crn wss vex elfuns cop elimasn df-br brcnv elin brimage dfdm4 eqeq2i eqcom 3bitr2i anbi12i dfrn5 rexbii elima risset - wrex rnex elpw wfn df-f df-fn anbi1i abbi2i funsex 1stex cnvex snex imaex - imageex inex 2ndex pwex eqeltrri ) GHIZJZAUAZKZLZMIZJZBUBZKZLZABCNZUCZCUD - UEWPCWNWOWIOZWOWMOZTWOUFZWOUGZAPZTZWOUHZBUIZTZWOWNOWPWQXBWRXDWQWOGOZWOWHO - ZTXBWOGWHUPXFWSXGXAWOCUJZUKXGAWOULWFOAWOWFQZXAWFAWOUMAWOWFUNXIWOAWEQZXAAW - OWEUOXJAHWOKZPAWTPXAWOAHXHDUQWTXKAWOURUSAWTUTVARVAVBRWRXCWLOZXDFNZWOWKQZF - WLVGXMXCPZFWLVGWRXLXNXOFWLWOXMWJQXMMWOKZPXNXOWOXMMXHFUJUQXMWOWJUOXCXPXMWO - VCUSSVDFWOWKWLVEFXCWLVFSXCBWOXHVHVIRVBWOWIWMUPWPWOAVJZXDTXEABWOVKXQXBXDWO - AVLVMRSVNWIWMGWHVOWFWGWEHVPVTVQAVRVSWAWKWLWJMWBVTVQBEWCVSWAWD $. + wrex rnex elpw wfn df-f df-fn anbi1i eqabi 1stex imageex cnvex snex imaex + funsex inex 2ndex pwex eqeltrri ) GHIZJZAUAZKZLZMIZJZBUBZKZLZABCNZUCZCUDU + EWPCWNWOWIOZWOWMOZTWOUFZWOUGZAPZTZWOUHZBUIZTZWOWNOWPWQXBWRXDWQWOGOZWOWHOZ + TXBWOGWHUPXFWSXGXAWOCUJZUKXGAWOULWFOAWOWFQZXAWFAWOUMAWOWFUNXIWOAWEQZXAAWO + WEUOXJAHWOKZPAWTPXAWOAHXHDUQWTXKAWOURUSAWTUTVARVAVBRWRXCWLOZXDFNZWOWKQZFW + LVGXMXCPZFWLVGWRXLXNXOFWLWOXMWJQXMMWOKZPXNXOWOXMMXHFUJUQXMWOWJUOXCXPXMWOV + CUSSVDFWOWKWLVEFXCWLVFSXCBWOXHVHVIRVBWOWIWMUPWPWOAVJZXDTXEABWOVKXQXBXDWOA + VLVMRSVNWIWMGWHVTWFWGWEHVOVPVQAVRVSWAWKWLWJMWBVPVQBEWCVSWAWD $. $} ${ @@ -57788,10 +57786,10 @@ the first case of his notation (simple exponentiation) and subscript it 19-Apr-2019.) $) fnpm $p |- ^pm Fn _V $= ( vx vy vf cpm cvv cxp wfn cv wfun cpw crab df-pm cfuns cin wcel cab elin - wa vex mpbi abbi2i df-rab wral wceq elfuns rgenw 3eqtr2i xpex pwex funsex - wb rabbi inex eqeltrri fnmpt2i xpvv fneq2i ) DEEFZGDEGABEECHZIZCBHZAHZFZJ - ZKZDABCLVDMNZVEEVFUSVDOUSMOZRZCPVGCVDKZVEVHCVFUSVDMQUAVGCVDUBVGUTUKZCVDUC - VIVEUDVJCVDUSCSUEUFVGUTCVDULTUGVDMVCVAVBBSASUHUIUJUMUNUOUREDUPUQT $. + wa vex mpbi eqabi df-rab wral wceq elfuns rgenw rabbi 3eqtr2i xpex funsex + wb pwex inex eqeltrri fnmpt2i xpvv fneq2i ) DEEFZGDEGABEECHZIZCBHZAHZFZJZ + KZDABCLVDMNZVEEVFUSVDOUSMOZRZCPVGCVDKZVEVHCVFUSVDMQUAVGCVDUBVGUTUKZCVDUCV + IVEUDVJCVDUSCSUEUFVGUTCVDUGTUHVDMVCVAVBBSASUIULUJUMUNUOUREDUPUQT $. $} ${ @@ -58464,7 +58462,7 @@ the first case of his notation (simple exponentiation) and subscript it crn eqtr3i syl5eqr dff1o4 vex siex f1oen exlimiv sylbi csn wmo weq wi w3a copab fununiq sneqb 3expib alrimivv sneq mo4 alrimiv funopab cab wel eldm dmopab brelrn eleq2d wrex elpw1 spcev syl6bi rexlimivw com23 mpdi exlimdv - syl5bi breldm impbid1 bitr3d snelpw1 syl6bb abbi1dv brcnv 3imtr3g cnvopab + syl5bi breldm impbid1 bitr3d snelpw1 syl6bb eqabcdv brcnv 3imtr3g cnvopab f1ocnv elrn fneq1i enpw1lem1 impbii vtocl2g pm5.21nii ) ABGHZAIJZBIJZUEZA KZBKZGHZABGUFUUAYSIJZYTIJZUEYRYSYTGUFUUBYPUUCYQAUGBUGUHUIUASZUBSZGHZUUDKZ UUEKZGHZVGAUUEGHZYSUUHGHZVGYOUUAVGUAUBABIIUUDALZUUFUUJUUIUUKUUDAUUEGUJUUL @@ -58719,14 +58717,14 @@ the first case of his notation (simple exponentiation) and subscript it ( vy vx vz cpw cpw1 cpw1fn wf1o wbr c1c wss ax-mp wceq cv wrex wa wex vex wcel cres cen cima wf1 pw1fnf1o f1of1 pw1ss1c f1ores mp2an wb df-ima elpw cab sspw1 df-rex df-pw eqabri anbi1i exbii bitr2i 3bitri csn elpw1 bitr4i - r19.41v rexcom4 snex breq1 ceqsexv bitri rexbii bitr3i abbi2i eqtr4i mpbi - brpw1fn f1oeq3 pw1fnex pwex pw1ex resex f1oen ) AFZGZAGZFZHWDUAZIZWDWFUBJ - WDHWDUCZWGIZWHKKFZHUDZWDKLWJKWKHIWLUEKWKHUFMWCUGKWKWDHUHUIWIWFNWJWHUJWICO - ZDOZHJZCWDPZDUMWFDCHWDUKWPDWFWNWFTZWNEOZGNZEWCPZWPWQWNWELWRALZWSQZERZWTWN - WEDSZULEWNAXDUNWTWRWCTZWSQZERXCWSEWCUOXFXBEXEXAWSXAEWCEAUPUQURUSUTVAWPWMW - DTZWOQZCRWMWRVBZNZWOQZEWCPZCRZWTWOCWDUOXHXLCXHXJEWCPZWOQXLXGXNWOEWMWCVCUR - XJWOEWCVEVDUSXMXKCRZEWCPWTXKECWCVFXOWSEWCXOXIWNHJZWSWOXPCXIWRVGWMXIWNHVHV - IWRWNESVPVJVKVLVAVDVMVNWIWFWDWGVQMVOWDWFWGHWDVRWCABVSVTWAWBM $. + r19.41v rexcom4 snex breq1 ceqsexv brpw1fn bitri rexbii bitr3i eqabi mpbi + eqtr4i f1oeq3 pw1fnex pwex pw1ex resex f1oen ) AFZGZAGZFZHWDUAZIZWDWFUBJW + DHWDUCZWGIZWHKKFZHUDZWDKLWJKWKHIWLUEKWKHUFMWCUGKWKWDHUHUIWIWFNWJWHUJWICOZ + DOZHJZCWDPZDUMWFDCHWDUKWPDWFWNWFTZWNEOZGNZEWCPZWPWQWNWELWRALZWSQZERZWTWNW + EDSZULEWNAXDUNWTWRWCTZWSQZERXCWSEWCUOXFXBEXEXAWSXAEWCEAUPUQURUSUTVAWPWMWD + TZWOQZCRWMWRVBZNZWOQZEWCPZCRZWTWOCWDUOXHXLCXHXJEWCPZWOQXLXGXNWOEWMWCVCURX + JWOEWCVEVDUSXMXKCRZEWCPWTXKECWCVFXOWSEWCXOXIWNHJZWSWOXPCXIWRVGWMXIWNHVHVI + WRWNESVJVKVLVMVAVDVNVPWIWFWDWGVQMVOWDWFWGHWDVRWCABVSVTWAWBM $. $} ${ @@ -58926,12 +58924,12 @@ the first case of his notation (simple exponentiation) and subscript it ( vy wcel cv csn cfv wn cin cvv csset vex wceq wa wex bitri crab cfullfun cab dfrab2 eqtri cdm cuni1 ccompl elcompl wel cop elin wbr snex brfullfun eldm2 df-br eqcom 3bitr3i opelssetsn anbi12i exbii eluni1 3bitr4i xchbinx - fvex clel3 abbi2i fullfunex ssetex inex dmex uni1ex complex eqeltrri mpan - inexg syl5eqel ) BDHZCAIZVTJZEIZKZHZLZAUCZBMZNCWEABUAWGFWEABUDUEWFNHVSWGN - HWBUBZOMZUFZUGZUHZWFNWEAWLVTWLHVTWKHZWDVTWKAPZUIWAWJHZGIZWCQZAGUJZRZGSZWM - WDWOWAWPUKZWIHZGSWTGWAWIUPXBWSGXBXAWHHZXAOHZRWSXAWHOULXCWQXDWRWAWPWHUMWCW - PQXCWQWAWPWBVTUNUOWAWPWHUQWCWPURUSVTWPWNGPUTVATVBTVTWJWNVCGVTWCWAWBVFVGVD - VEVHWKWJWIWHOWBEPVIVJVKVLVMVNVOWFBNDVQVPVR $. + fvex clel3 eqabi fullfunex ssetex inex dmex uni1ex complex eqeltrri inexg + mpan syl5eqel ) BDHZCAIZVTJZEIZKZHZLZAUCZBMZNCWEABUAWGFWEABUDUEWFNHVSWGNH + WBUBZOMZUFZUGZUHZWFNWEAWLVTWLHVTWKHZWDVTWKAPZUIWAWJHZGIZWCQZAGUJZRZGSZWMW + DWOWAWPUKZWIHZGSWTGWAWIUPXBWSGXBXAWHHZXAOHZRWSXAWHOULXCWQXDWRWAWPWHUMWCWP + QXCWQWAWPWBVTUNUOWAWPWHUQWCWPURUSVTWPWNGPUTVATVBTVTWJWNVCGVTWCWAWBVFVGVDV + EVHWKWJWIWHOWBEPVIVJVKVLVMVNVOWFBNDVPVQVR $. $} ${ @@ -59293,20 +59291,20 @@ the first case of his notation (simple exponentiation) and subscript it vn cxp cab cmuc ctxp crn cins4 ccnv cins2 cin cima elima df-br elrn2 elin co wex oqelins4 elrn trtxp ancom op1st2nd anbi2i exbii opex breq1 ceqsexv vex brcross otelins2 brcnv bitr3i anbi12i bitri xpex breq2 rexbii crossex - abbi2i 2ndex 1stex txpex rnex ins4ex enex cnvex ins2ex inex imaexg ancoms - mpan sylan syl5eqelr rexeq abbidv rexbidv df-muc ovmpt2g mpd3an3 ) BGHZCG - HZDIZEIZAIZUCZJKZACLZEBLZDUDZMHBCUEUQXJNXAXBOXJPQRUFZUFZUGZUHZJUIZUJZUJZU - KZUGZCULZBULZMXIDYAXCYAHXDXCXTKZEBLXIEXCXTBUMYBXHEBYBXDXCSZXTHXEYCXSKZACL - XHXDXCXTUNAYCXSCUMYDXGACYDXEYCSZXSHZXGXEYCXSUNYFFIZYESZXRHZFURYGXFNZXCYGJ - KZOZFURXGFYEXRUOYIYLFYIYHXNHZYHXQHZOYLYHXNXQUPYMYJYNYKYMYGXEXDSZSZXMHZXDX - ESZYGPKZYJYGXEXDXCXMDVIUSYQXCYPXLKZDURXCYRNZXCYGPKZOZDURYSDYPXLUTYTUUCDYT - UUBXCYOXKKZOUUBUUAOUUCXCYGYOPXKVAUUDUUAUUBUUDXCXEQKZXCXDRKZOUUFUUEOUUAXCX - EXDQRVAUUEUUFVBXDXEXCEVIZAVIZVCTVDUUBUUAVBTVEUUBYSDYRXDXEUUGUUHVFXCYRYGPV - GVHTXDXEYGUUGUUHVJTYNYGYCSXPHYGXCSXOHZYKYGXEYCXPUUHVKYGXDXCXOUUGVKUUIYGXC - XOKYKYGXCXOUNYGXCJVLVMTVNVOVEYKXGFXFXDXEUUGUUHVPYGXFXCJVQVHTVOVRTVRVOVTXB - XAYAMHZXBXTMHZXAUUJXSMHXBUUKXRXNXQXMXLPXKVSQRWAWBWCWCWDWEXPXOJWFWGWHWHWIW - DXSCMGWJWLXTBMGWJWMWKWNUAUBBCGGXGAUBIZLZEUAIZLZDUDXJUEUUMEBLZDUDMUUNBNUUO - UUPDUUMEUUNBWOWPUULCNZUUPXIDUUQUUMXHEBXGAUULCWOWQWPAUAUBDEWRWSWT $. + eqabi 2ndex 1stex txpex rnex ins4ex enex cnvex ins2ex imaexg sylan ancoms + inex mpan syl5eqelr rexeq abbidv rexbidv df-muc ovmpt2g mpd3an3 ) BGHZCGH + ZDIZEIZAIZUCZJKZACLZEBLZDUDZMHBCUEUQXJNXAXBOXJPQRUFZUFZUGZUHZJUIZUJZUJZUK + ZUGZCULZBULZMXIDYAXCYAHXDXCXTKZEBLXIEXCXTBUMYBXHEBYBXDXCSZXTHXEYCXSKZACLX + HXDXCXTUNAYCXSCUMYDXGACYDXEYCSZXSHZXGXEYCXSUNYFFIZYESZXRHZFURYGXFNZXCYGJK + ZOZFURXGFYEXRUOYIYLFYIYHXNHZYHXQHZOYLYHXNXQUPYMYJYNYKYMYGXEXDSZSZXMHZXDXE + SZYGPKZYJYGXEXDXCXMDVIUSYQXCYPXLKZDURXCYRNZXCYGPKZOZDURYSDYPXLUTYTUUCDYTU + UBXCYOXKKZOUUBUUAOUUCXCYGYOPXKVAUUDUUAUUBUUDXCXEQKZXCXDRKZOUUFUUEOUUAXCXE + XDQRVAUUEUUFVBXDXEXCEVIZAVIZVCTVDUUBUUAVBTVEUUBYSDYRXDXEUUGUUHVFXCYRYGPVG + VHTXDXEYGUUGUUHVJTYNYGYCSXPHYGXCSXOHZYKYGXEYCXPUUHVKYGXDXCXOUUGVKUUIYGXCX + OKYKYGXCXOUNYGXCJVLVMTVNVOVEYKXGFXFXDXEUUGUUHVPYGXFXCJVQVHTVOVRTVRVOVTXBX + AYAMHZXBXTMHZXAUUJXSMHXBUUKXRXNXQXMXLPXKVSQRWAWBWCWCWDWEXPXOJWFWGWHWHWLWD + XSCMGWIWMXTBMGWIWJWKWNUAUBBCGGXGAUBIZLZEUAIZLZDUDXJUEUUMEBLZDUDMUUNBNUUOU + UPDUUMEUUNBWOWPUULCNZUUPXIDUUQUUMXHEBXGAUULCWOWQWPAUAUBDEWRWSWT $. $} ${ @@ -59432,7 +59430,7 @@ the first case of his notation (simple exponentiation) and subscript it [Rosser] p. 372. (Contributed by SF, 24-Feb-2015.) $) df0c2 $p |- 0c = Nc (/) $= ( vx c0 cen cec cv wbr cab cnc c0c dfec2 df-nc wceq wcel en0 ensym 3bitr4ri - el0c abbi2i 3eqtr4ri ) BCDBAEZCFZAGBHIABCJBKUAAITBCFTBLUATIMTNBTOTQPRS $. + el0c eqabi 3eqtr4ri ) BCDBAEZCFZAGBHIABCJBKUAAITBCFTBLUATIMTNBTOTQPRS $. $( Cardinal zero is a cardinal number. Corollary 1 to theorem XI.2.7 of [Rosser] p. 373. (Contributed by SF, 24-Feb-2015.) $) @@ -59447,14 +59445,14 @@ the first case of his notation (simple exponentiation) and subscript it ( vx vy vz vf c1c wcel cv cnc wceq wex csn cen wbr cvv vex exlimiv eqeq2d crn spcev sylbi cncs cec cab dfec2 df-nc el1c en2sn mp2an breq2 wf1o bren mpbiri wf f1of wfo f1ofo forn syl cfv cop wa wi fsn2 rnsnop syl6eq eqeq1d - rneq fvex sneq eqcoms syl6bi adantl sylc impbii abbi2i 3eqtr4ri snex nceq - bitri ax-mp elncs mpbir ) EUAFEAGZHZIZAJZEBGZKZHZIZWFWHLUBWHCGZLMZCUCWIEC - WHLUDWHUEWLCEWKEFWKWCKZIZAJZWLAWKUFWOWLWNWLAWNWLWHWMLMZWGNFWCNFWPBOZAOWGW - CNNUGUHWKWMWHLUIULPWLWHWKDGZUJZDJWOWHWKDUKWSWODWSWHWKWRUMZWRRZWKIZWOWHWKW - RUNWSWHWKWRUOXBWHWKWRUPWHWKWRUQURWTWGWRUSZWKFZWRWGXCUTKZIZVAXBWOVBZWGWKWR - WQVCXFXGXDXFXBXCKZWKIWOXFXAXHWKXFXAXERXHWRXEVGWGXCWQVDVEVFWOWKXHWNWKXHIAX - CWGWRVHWCXCIWMXHWKWCXCVIQSVJVKVLTVMPTVNVSVOVPWEWJAWHWGVQWCWHIWDWIEWCWHVRQ - SVTAEWAWB $. + rneq fvex sneq eqcoms syl6bi adantl sylc impbii bitri eqabi 3eqtr4ri snex + nceq ax-mp elncs mpbir ) EUAFEAGZHZIZAJZEBGZKZHZIZWFWHLUBWHCGZLMZCUCWIECW + HLUDWHUEWLCEWKEFWKWCKZIZAJZWLAWKUFWOWLWNWLAWNWLWHWMLMZWGNFWCNFWPBOZAOWGWC + NNUGUHWKWMWHLUIULPWLWHWKDGZUJZDJWOWHWKDUKWSWODWSWHWKWRUMZWRRZWKIZWOWHWKWR + UNWSWHWKWRUOXBWHWKWRUPWHWKWRUQURWTWGWRUSZWKFZWRWGXCUTKZIZVAXBWOVBZWGWKWRW + QVCXFXGXDXFXBXCKZWKIWOXFXAXHWKXFXAXERXHWRXEVGWGXCWQVDVEVFWOWKXHWNWKXHIAXC + WGWRVHWCXCIWMXHWKWCXCVIQSVJVKVLTVMPTVNVOVPVQWEWJAWHWGVRWCWHIWDWIEWCWHVSQS + VTAEWAWB $. $} ${ @@ -59807,9 +59805,9 @@ the first case of his notation (simple exponentiation) and subscript it wf wb wn snex opex elcompl elsymdif opelssetsn otelins2 otsnelsi3 wfn wss df-br brfns bitr3i opelco brimage dfrn5 eqeq2i bitr4i brsset rnex ceqsexv exbii sseq1 oteltxp bibi12i xchbinx exnal 3bitrri con1bii oqelins4 mapval - df-f eqabb ovex breq2 df-3an abbi2i cnvex imaexg mpan xpexg syl2an cnvexg - pw1fnex ins3exg syl ssetex ins3ex fnsex 2ndex coex ins2ex 1cex mpan2 3syl - imageex txpex si3ex symdifex imaex complex ins4ex enex inexg syl5eqelr + df-f eqabb ovex breq2 df-3an eqabi pw1fnex cnvex imaexg mpan xpexg syl2an + cnvexg ins3exg syl ssetex ins3ex fnsex 2ndex imageex coex 1cex mpan2 3syl + ins2ex txpex si3ex symdifex imaex complex ins4ex enex inexg syl5eqelr inex ) CDJZBEJZKZFUAZUBZCJZGUAZUBZBJZAUAZUUNUUQUEUCZLMZUDZGNZFNZAUFOUGZCU HZUVFBUHZUIZUGZUJZPUJZUKPULUMZUNZUOZUPZUQZUSZQUHZURZUTZLUGZUQZUQZVAZVBZVA ZQUHZQUHZRUVEAUWIUUTUWIJUUNVCZUUTSZUWHJZFNUVEFUUTUWHVDUWLUVDFUWLUUQVCZUWK @@ -59833,10 +59831,10 @@ the first case of his notation (simple exponentiation) and subscript it UWCJUXDUUTSUWBJZUXNUXDUWMUWKUWCUYGWKUXDUWJUUTUWBUYHWKVUKUXDUUTUWBMUXNUXDU UTUWBWOUXDUUTLVJWQVRVQTXFUXNUVBHUVAUUNUUQUEXRUXDUVAUUTLXSXEVRVQUWNUVKUWFV TUUPUUSUVBXTVPXFTXFTYAUUMUVIRJZUVKRJZUWIRJZUUKUVGRJZUVHRJZVULUULUVFRJZUUK - VUOOYHYBZUVFCRDYCYDVUQUULVUPVURUVFBREYCYDUVGUVHRRYEYFVULUVJRJVUMUVIRYGUVJ + VUOOYBYCZUVFCRDYDYEVUQUULVUPVURUVFBREYDYEUVGUVHRRYFYGVULUVJRJVUMUVIRYHUVJ RYIYJVUMUWGRJZUWHRJZVUNVUMUWFRJVUSUWEUWAUWDUVTUVSUVRQUVLUVQPYKYLUVPUVOUKU - VNYMPUVMYKULYNYTYOUUAUUBYPUUCYQUUDUUEUUFUWCUWBLUUGYBYPYPUUJXDUVKUWFRRUUHY - RVUSQRJZVUTYQUWGQRRYCYRVUTVVAVUNYQUWHQRRYCYRYSYSUUI $. + VNYMPUVMYKULYNYOYPUUAUUBYTUUCYQUUDUUEUUFUWCUWBLUUGYCYTYTUUJXDUVKUWFRRUUHY + RVUSQRJZVUTYQUWGQRRYDYRVUTVVAVUNYQUWHQRRYDYRYSYSUUI $. $} ${ @@ -60022,17 +60020,17 @@ the first case of his notation (simple exponentiation) and subscript it ( vm vn vt cv c0c cce co wne c1c c1st c2nd csn cima wcel wbr oveq1 neeq1d c0 wceq cplc ccnv cres cfullfun ccompl cab cvv wn vex elcompl cop wrex wa brres eliniseg anbi2i 0cex 3bitri rexbii elima risset 3bitr4i brfullfunop - necon3bbii bitri abbi2i 1stex 2ndex cnvex snex imaex resex ceex fullfunex - op1st2nd complex eqeltrri cncs cpw1 0cnc pw10 nulel0c ce0nnuli mp2an cnnc - weq eqeltri wi nnnc 1cnc 0ex pw1sn snel1c jctr wb ce0addcnnul syl5ibr syl - mpan2 finds ) BEZFGHZSIZFFGHZSIZCEZFGHZSIZXFJUAZFGHZSIZAFGHZSIBCAKLUBZFMZ - NZUCZGUDZUBZSMZNZNZUEZXCBUFUGXCBYBXAYBOXAYAOZUHXCXAYABUIZUJYCXBSYCXAFUKZX - TOZYESXQPXBSTDEZXAXPPZDXTULYGYETZDXTULYCYFYHYIDXTYHYGXAKPZYGXOOZUMYJYGFLP - ZUMYIYGXAKXOUNYKYLYJLFYGUOUPXAFYGYDUQVOURUSDXAXPXTUTDYEXTVAVBXQSYEUOXAFSG - YDUQVCURVDVEVFYAXPXTKXOVGXMXNLVHVIFVJVKVLXRXSXQGVMVNVISVJZVKVKVPVQXAFTXBX - DSXAFFGQRBCWFXBXGSXAXFFGQRXAXITXBXJSXAXIFGQRXAATXBXLSXAAFGQRFVROSVSZFOXEV - TYNSFWAWBWGSFWCWDXFWEOXFVROZXHXKWHXFWIXHXKYOXHJFGHSIZUMZXHYPJVROZXSVSZJOY - PWJYSXSMJSWKWLXSYMWMWGXSJWCWDWNYOYRXKYQWOWJXFJWPWSWQWRWT $. + op1st2nd necon3bbii bitri eqabi 1stex 2ndex cnvex snex imaex ceex complex + resex fullfunex eqeltrri weq cncs cpw1 0cnc pw10 nulel0c eqeltri ce0nnuli + mp2an cnnc wi nnnc 1cnc 0ex pw1sn snel1c wb ce0addcnnul mpan2 syl5ibr syl + jctr finds ) BEZFGHZSIZFFGHZSIZCEZFGHZSIZXFJUAZFGHZSIZAFGHZSIBCAKLUBZFMZN + ZUCZGUDZUBZSMZNZNZUEZXCBUFUGXCBYBXAYBOXAYAOZUHXCXAYABUIZUJYCXBSYCXAFUKZXT + OZYESXQPXBSTDEZXAXPPZDXTULYGYETZDXTULYCYFYHYIDXTYHYGXAKPZYGXOOZUMYJYGFLPZ + UMYIYGXAKXOUNYKYLYJLFYGUOUPXAFYGYDUQVDURUSDXAXPXTUTDYEXTVAVBXQSYEUOXAFSGY + DUQVCURVEVFVGYAXPXTKXOVHXMXNLVIVJFVKVLVOXRXSXQGVMVPVJSVKZVLVLVNVQXAFTXBXD + SXAFFGQRBCVRXBXGSXAXFFGQRXAXITXBXJSXAXIFGQRXAATXBXLSXAAFGQRFVSOSVTZFOXEWA + YNSFWBWCWDSFWEWFXFWGOXFVSOZXHXKWHXFWIXHXKYOXHJFGHSIZUMZXHYPJVSOZXSVTZJOYP + WJYSXSMJSWKWLXSYMWMWDXSJWEWFWSYOYRXKYQWNWJXFJWOWPWQWRWT $. $} ${ @@ -60553,30 +60551,30 @@ the first case of his notation (simple exponentiation) and subscript it leconnnc $p |- ( ( A e. Nn /\ B e. Nn ) -> ( A <_c B \/ B <_c A ) ) $= ( va vp cnnc wcel clec wbr wo cv wceq breq2 breq1 orbi12d imbi2d c0c cncs wi c1c wa vn vm cplc wn cab ccnv csn cima cun cvv elun eliniseg cop df-br - elimasn bitr4i orbi12i bitri abbi2i uneq2i eqtri abbii eqtr4i abexv lecex - unab imor cnvex snex imaex unex eqeltrri weq nnnc le0nc syl dflec2 nc0le1 - orc wrex 1cnc ax-mp mpbiri orim1i a1i orcomd adantl simpll simplr leaddc2 - simpr syl31anc ex orim12d biimprd com12 rexlimdva adantr sylbid addlecncs - mpd mpan2 peano2nc lectr mpd3an3 mpan2d ancoms olc syl6 jaod syl2an finds - a2d vtoclga imp ) AEFZBEFZABGHZBAGHZIZXQXPXTXPAUAJZGHZYAAGHZIZRXPXTRUABEY - ABKZYDXTXPYEYBXRYCXSYABAGLYABAGMNOXPYAEFZYDYFCJZYAGHZYAYGGHZIZRZYFPYAGHZY - APGHZIZRYFUBJZYAGHZYAYOGHZIZRYFYOSUCZYAGHZYAYSGHZIZRYFYDRCUBAYFUDZCUEZGUF - ZYAUGZUHZGUUFUHZUIZUIZYKCUEZUJUUJUUCYJIZCUEZUUKUUJUUDYJCUEZUIUUMUUIUUNUUD - YJCUUIYGUUIFYGUUGFZYGUUHFZIYJYGUUGUUHUKUUOYHUUPYIGYAYGULUUPYAYGUMGFYIGYAY - GUOYAYGGUNUPUQURUSUTUUCYJCVFVAYKUULCYFYJVGVBVCUUDUUIUUCCVDUUGUUHUUEUUFGVE - VHYAVIZVJGUUFVEUUQVJVKVKVLYGPKZYJYNYFUURYHYLYIYMYGPYAGMYGPYAGLNOCUBVMZYJY - RYFUUSYHYPYIYQYGYOYAGMYGYOYAGLNOYGYSKZYJUUBYFUUTYHYTYIUUAYGYSYAGMYGYSYAGL - NOYGAKZYJYDYFUVAYHYBYIYCYGAYAGMYGAYAGLNOYFYLYNYFYAQFZYLYAVNZYAVOVPYLYMVSV - PYOEFZYFYRUUBUVDYFYRUUBRZUVDYOQFZUVBUVEYFYOVNUVCUVFUVBTZYPUUBYQUVGYPYAYOD - JZUCZKZDQVTZUUBYOYADVQUVFUVKUUBRUVBUVFUVJUUBDQUVFUVHQFZTZYSUVIGHZUVIYSGHZ - IZUVJUUBRUVMSUVHGHZUVHSGHZIZUVPUVLUVSUVFUVLUVRUVQUVLUVHPKZUVQIZUVRUVQIZUV - HVRUWAUWBRUVLUVTUVRUVQUVTUVRPSGHZSQFZUWCWASVOWBUVHPSGMWCWDWEXAWFWGUVMUVQU - VNUVRUVOUVMUVQUVNUVMUVQTZUVFUWDUVLUVQUVNUVFUVLUVQWHUWDUWEWAWEUVFUVLUVQWIU - VMUVQWKUVHYOSWJWLWMUVMUVRUVOUVMUVRTZUVFUVLUWDUVRUVOUVFUVLUVRWHUVFUVLUVRWI - UWDUWFWAWEUVMUVRWKSYOUVHWJWLWMWNXAUVJUVPUUBUVJUUBUVPUVJYTUVNUUAUVOYAUVIYS - GLYAUVIYSGMNWOWPVPWQWRWSUVGYQUUAUUBUVBUVFYQUUARUVBUVFTYQYOYSGHZUUAUVFUWGU - VBUVFUWDUWGWAYOSWTXBWGUVBUVFYSQFZYQUWGTUUARUVFUWHUVBYOXCWGYAYOYSXDXEXFXGU - UAYTXHXIXJXKWMXMXLWPXNWPXO $. + elimasn bitr4i bitri eqabi uneq2i unab eqtri imor abbii abexv lecex cnvex + orbi12i eqtr4i snex imaex unex eqeltrri weq nnnc le0nc wrex dflec2 nc0le1 + syl orc ax-mp mpbiri orim1i a1i orcomd adantl simpll simplr simpr leaddc2 + 1cnc mpd syl31anc orim12d biimprd com12 rexlimdva adantr sylbid addlecncs + mpan2 peano2nc lectr mpd3an3 mpan2d ancoms olc syl6 jaod syl2an a2d finds + ex vtoclga imp ) AEFZBEFZABGHZBAGHZIZXQXPXTXPAUAJZGHZYAAGHZIZRXPXTRUABEYA + BKZYDXTXPYEYBXRYCXSYABAGLYABAGMNOXPYAEFZYDYFCJZYAGHZYAYGGHZIZRZYFPYAGHZYA + PGHZIZRYFUBJZYAGHZYAYOGHZIZRYFYOSUCZYAGHZYAYSGHZIZRYFYDRCUBAYFUDZCUEZGUFZ + YAUGZUHZGUUFUHZUIZUIZYKCUEZUJUUJUUCYJIZCUEZUUKUUJUUDYJCUEZUIUUMUUIUUNUUDY + JCUUIYGUUIFYGUUGFZYGUUHFZIYJYGUUGUUHUKUUOYHUUPYIGYAYGULUUPYAYGUMGFYIGYAYG + UOYAYGGUNUPVGUQURUSUUCYJCUTVAYKUULCYFYJVBVCVHUUDUUIUUCCVDUUGUUHUUEUUFGVEV + FYAVIZVJGUUFVEUUQVJVKVKVLYGPKZYJYNYFUURYHYLYIYMYGPYAGMYGPYAGLNOCUBVMZYJYR + YFUUSYHYPYIYQYGYOYAGMYGYOYAGLNOYGYSKZYJUUBYFUUTYHYTYIUUAYGYSYAGMYGYSYAGLN + OYGAKZYJYDYFUVAYHYBYIYCYGAYAGMYGAYAGLNOYFYLYNYFYAQFZYLYAVNZYAVOVSYLYMVTVS + YOEFZYFYRUUBUVDYFYRUUBRZUVDYOQFZUVBUVEYFYOVNUVCUVFUVBTZYPUUBYQUVGYPYAYODJ + ZUCZKZDQVPZUUBYOYADVQUVFUVKUUBRUVBUVFUVJUUBDQUVFUVHQFZTZYSUVIGHZUVIYSGHZI + ZUVJUUBRUVMSUVHGHZUVHSGHZIZUVPUVLUVSUVFUVLUVRUVQUVLUVHPKZUVQIZUVRUVQIZUVH + VRUWAUWBRUVLUVTUVRUVQUVTUVRPSGHZSQFZUWCWKSVOWAUVHPSGMWBWCWDWLWEWFUVMUVQUV + NUVRUVOUVMUVQUVNUVMUVQTZUVFUWDUVLUVQUVNUVFUVLUVQWGUWDUWEWKWDUVFUVLUVQWHUV + MUVQWIUVHYOSWJWMXMUVMUVRUVOUVMUVRTZUVFUVLUWDUVRUVOUVFUVLUVRWGUVFUVLUVRWHU + WDUWFWKWDUVMUVRWISYOUVHWJWMXMWNWLUVJUVPUUBUVJUUBUVPUVJYTUVNUUAUVOYAUVIYSG + LYAUVIYSGMNWOWPVSWQWRWSUVGYQUUAUUBUVBUVFYQUUARUVBUVFTYQYOYSGHZUUAUVFUWGUV + BUVFUWDUWGWKYOSWTXAWFUVBUVFYSQFZYQUWGTUUARUVFUWHUVBYOXBWFYAYOYSXCXDXEXFUU + AYTXGXHXIXJXMXKXLWPXNWPXO $. $} ${ @@ -61041,11 +61039,11 @@ the first case of his notation (simple exponentiation) and subscript it by SF, 19-Mar-2015.) $) nclennlem1 $p |- { x | A. n e. NC ( n <_c x -> n e. Nn ) } e. _V $= ( clec cnnc ccompl cres cncs cima cv wbr wcel wi wral wn vex elcompl wrex - wa bitri complex cab cvv elima brres anbi2i rexbii 3bitrri con1bii abbi2i - rexanali lecex nncex resex ncsex imaex eqeltrri ) CDEZFZGHZEZBIZAIZCJZVAD - KZLBGMZAUAUBVEAUTVBUTKVBUSKZNVEVBUSAOPVEVFVFVAVBURJZBGQVCVDNZRZBGQVENBVBU - RGUCVGVIBGVGVCVAUQKZRVIVAVBCUQUDVJVHVCVADBOPUESUFVCVDBGUJUGUHSUIUSURGCUQU - KDULTUMUNUOTUP $. + wa bitri complex cab cvv elima brres anbi2i rexbii rexanali 3bitrri eqabi + con1bii lecex nncex resex ncsex imaex eqeltrri ) CDEZFZGHZEZBIZAIZCJZVADK + ZLBGMZAUAUBVEAUTVBUTKVBUSKZNVEVBUSAOPVEVFVFVAVBURJZBGQVCVDNZRZBGQVENBVBUR + GUCVGVIBGVGVCVAUQKZRVIVAVBCUQUDVJVHVCVADBOPUESUFVCVDBGUGUHUJSUIUSURGCUQUK + DULTUMUNUOTUP $. $} ${ @@ -61233,10 +61231,10 @@ the first case of his notation (simple exponentiation) and subscript it nnltp1clem1 $p |- { x | x { x } ) = A $= - ( cv csn cvv wcel crab cab cmpt cdm df-rab eqid dmmpt snex biantru abbi2i - wa 3eqtr4i ) ACZDZEFZABGSBFZUAQZAHABTIZJBUAABKABTUDUDLMUCABUAUBSNOPR $. + ( cv csn cvv wcel crab wa cab cmpt df-rab eqid dmmpt snex biantru 3eqtr4i + cdm eqabi ) ACZDZEFZABGSBFZUAHZAIABTJZQBUAABKABTUDUDLMUCABUAUBSNORP $. $} ${ diff --git a/set.mm b/set.mm index 73a11350b..19e717c9a 100644 --- a/set.mm +++ b/set.mm @@ -26582,31 +26582,31 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim ${ $d x y A $. $d ph x y $. $d ps y $. - abbi2dv.1 $e |- ( ph -> ( x e. A <-> ps ) ) $. + eqabdv.1 $e |- ( ph -> ( x e. A <-> ps ) ) $. $( Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) Avoid ~ ax-11 . (Revised by Wolf Lammen, 6-May-2023.) $) - abbi2dv $p |- ( ph -> A = { x | ps } ) $= + eqabdv $p |- ( ph -> A = { x | ps } ) $= ( vy cab cv wcel wsb sbbidv clelsb1 bicomi df-clab 3bitr4g eqrdv ) AFDBCG ZACHDIZCFJZBCFJFHZDIZTQIARBCFEKSUACFDLMBFCNOP $. $} ${ $d x A $. $d ph x $. - abbi1dv.1 $e |- ( ph -> ( ps <-> x e. A ) ) $. + eqabcdv.1 $e |- ( ph -> ( ps <-> x e. A ) ) $. $( Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) $) - abbi1dv $p |- ( ph -> { x | ps } = A ) $= - ( cab cv wcel bicomd abbi2dv eqcomd ) ADBCFABCDABCGDHEIJK $. + eqabcdv $p |- ( ph -> { x | ps } = A ) $= + ( cab cv wcel bicomd eqabdv eqcomd ) ADBCFABCDABCGDHEIJK $. $} ${ $d x A $. - abbi2i.1 $e |- ( x e. A <-> ph ) $. + eqabi.1 $e |- ( x e. A <-> ph ) $. $( Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 26-May-1993.) Avoid ~ ax-11 . (Revised by Wolf Lammen, 6-May-2023.) $) - abbi2i $p |- A = { x | ph } $= - ( cab wceq wtru cv wcel wb a1i abbi2dv mptru ) CABEFGABCBHCIAJGDKLM $. + eqabi $p |- A = { x | ph } $= + ( cab wceq wtru cv wcel wb a1i eqabdv mptru ) CABEFGABCBHCIAJGDKLM $. $} ${ @@ -26647,7 +26647,7 @@ generally appear in a single form (either definitional, but more often (Contributed by NM, 26-Dec-1993.) (Revised by BJ, 10-Nov-2020.) $) abid1 $p |- A = { x | x e. A } $= - ( cv wcel biid abbi2i ) ACBDZABGEF $. + ( cv wcel biid eqabi ) ACBDZABGEF $. $} ${ @@ -26780,7 +26780,7 @@ generally appear in a single form (either definitional, but more often proper substitution of ` y ` for ` x ` into a class variable. (Contributed by NM, 14-Sep-2003.) $) sbab $p |- ( x = y -> A = { z | [ y / x ] z e. A } ) $= - ( weq cv wcel wsb sbequ12 abbi2dv ) ABECFDGZABHCDKABIJ $. + ( weq cv wcel wsb sbequ12 eqabdv ) ABECFDGZABHCDKABIJ $. $} @@ -31733,8 +31733,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is is equal to the restricting class. Deduction form of ~ rabeqc . (Contributed by Steven Nguyen, 7-Jun-2023.) $) rabeqcda $p |- ( ph -> { x e. A | ps } = A ) $= - ( crab cv wcel wa cab df-rab ex pm4.71d abbi2dv eqtr4id ) ABCDFCGDHZBIZCJ - DBCDKAQCDAPBAPBELMNO $. + ( crab cv wcel wa cab df-rab ex pm4.71d eqabdv eqtr4id ) ABCDFCGDHZBIZCJD + BCDKAQCDAPBAPBELMNO $. $} ${ @@ -34553,7 +34553,7 @@ general is seen either by substitution (when the variable ` V ` has no hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.) $) abidnf $p |- ( F/_ x A -> { z | A. x z e. A } = A ) $= - ( wnfc cv wcel wal sp nfcr nf5rd impbid2 abbi1dv ) ACDZBECFZAGZBCMONNAHMN + ( wnfc cv wcel wal sp nfcr nf5rd impbid2 eqabcdv ) ACDZBECFZAGZBCMONNAHMN AABCIJKL $. $} @@ -37042,7 +37042,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $( Substitution doesn't affect a constant ` B ` (in which ` x ` is not free). (Contributed by Mario Carneiro, 14-Oct-2016.) $) csbtt $p |- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B ) $= - ( vy wcel wnfc wa csb cv wsbc cab df-csb wnf wb nfcr sbctt sylan2 abbi1dv + ( vy wcel wnfc wa csb cv wsbc cab df-csb wnf wb nfcr sbctt sylan2 eqabcdv eqtrid ) BDFZACGZHZABCIEJCFZABKZELCAEBCMUCUEECUBUAUDANUEUDOAECPUDABDQRST $. $} @@ -37349,7 +37349,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use ` x , D ` by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.) $) csbie2g $p |- ( A e. V -> [_ A / x ]_ B = D ) $= - ( vz wcel csb cv wsbc cab df-csb wceq eleq2d sbcie2g abbi1dv eqtrid ) CGK + ( vz wcel csb cv wsbc cab df-csb wceq eleq2d sbcie2g eqabcdv eqtrid ) CGK ZACDLJMZDKZACNZJOFAJCDPUBUEJFUDUCEKUCFKABCGAMBMZQDEUCHRUFCQEFUCIRSTUA $. $} @@ -37392,13 +37392,13 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use cbvralcsf $p |- ( A. x e. A ph <-> A. y e. B ps ) $= ( vz vv cv wcel wi wal wsbc nfcri wral csb nfv nfcsb1v nfsbc1v id csbeq1a nfim weq eleq12d sbceq1a imbi12d cbvalv1 nfcv nfcsb csbeq1 cab df-csb wsb - nfsbc eleq2d sbsbc bitr3i abbi2i eqtr4i eqtrdi dfsbcq bitrdi bitri df-ral - sbie 3bitr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVPMOZCWAEUBZPZACWASZQZMR - VTVOWECMVOMUCWCWDCCMWBCWAEUDTACWAUEUHCMUIZVNWCAWDWFVMWAEWBWFUFCWAEUGUJACW - AUKULUMWEVSMDWCWDDDMWBDCWAEDWAUNZGUOTADCWAWGIUTUHVSMUCMDUIZWCVRWDBWHWAVQW - BFWHUFWHWBCVQEUBZFCWAVQEUPWINOZEPZCVQSZNUQFCNVQEURWLNFWJFPZWKCDUSWLWKWMCD - CNFHTCDUIEFWJKVAVKWKCDVBVCVDVEVFUJWHWDACVQSZBACWAVQVGWNACDUSBACDVBABCDJLV - KVCVHULUMVIACEVJBDFVJVL $. + nfsbc eleq2d sbsbc bitr3i eqabi eqtr4i eqtrdi dfsbcq bitrdi bitri 3bitr4i + sbie df-ral ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVPMOZCWAEUBZPZACWASZQZMRV + TVOWECMVOMUCWCWDCCMWBCWAEUDTACWAUEUHCMUIZVNWCAWDWFVMWAEWBWFUFCWAEUGUJACWA + UKULUMWEVSMDWCWDDDMWBDCWAEDWAUNZGUOTADCWAWGIUTUHVSMUCMDUIZWCVRWDBWHWAVQWB + FWHUFWHWBCVQEUBZFCWAVQEUPWINOZEPZCVQSZNUQFCNVQEURWLNFWJFPZWKCDUSWLWKWMCDC + NFHTCDUIEFWJKVAVKWKCDVBVCVDVEVFUJWHWDACVQSZBACWAVQVGWNACDUSBACDVBABCDJLVK + VCVHULUMVIACEVLBDFVLVJ $. $( A more general version of ~ cbvrexf that has no distinct variable restrictions. Changes bound variables using implicit substitution. @@ -37417,13 +37417,13 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use cbvreucsf $p |- ( E! x e. A ph <-> E! y e. B ps ) $= ( vz vv cv wcel wa weu wsb nfcri wreu csb nfv nfcsb1v nfan weq id csbeq1a nfs1v eleq12d sbequ12 anbi12d cbveu nfcv nfcsb nfsb csbeq1 cab wsbc sbsbc - abbii eleq2d sbie bicomi abbi2i df-csb 3eqtr4ri eqtrdi sbequ bitrdi bitri - df-reu 3bitr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACMSZQZM - RWAVPWFCMVPMUCWDWECCMWCCWBEUDTACMUIUECMUFZVOWDAWEWGVNWBEWCWGUGCWBEUHUJACM - UKULUMWFVTMDWDWEDDMWCDCWBEDWBUNGUOTACMDIUPUEVTMUCMDUFZWDVSWEBWHWBVRWCFWHU - GWHWCCVREUBZFCWBVREUQNOZEPZCDSZNURWKCVRUSZNURFWIWLWMNWKCDUTVAWLNFWLWJFPZW - KWNCDCNFHTCDUFEFWJKVBVCVDVECNVREVFVGVHUJWHWEACDSBAMDCVIABCDJLVCVJULUMVKAC - EVLBDFVLVM $. + abbii eleq2d sbie bicomi eqabi df-csb 3eqtr4ri eqtrdi sbequ bitrdi df-reu + bitri 3bitr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACMSZQZMR + WAVPWFCMVPMUCWDWECCMWCCWBEUDTACMUIUECMUFZVOWDAWEWGVNWBEWCWGUGCWBEUHUJACMU + KULUMWFVTMDWDWEDDMWCDCWBEDWBUNGUOTACMDIUPUEVTMUCMDUFZWDVSWEBWHWBVRWCFWHUG + WHWCCVREUBZFCWBVREUQNOZEPZCDSZNURWKCVRUSZNURFWIWLWMNWKCDUTVAWLNFWLWJFPZWK + WNCDCNFHTCDUFEFWJKVBVCVDVECNVREVFVGVHUJWHWEACDSBAMDCVIABCDJLVCVJULUMVLACE + VKBDFVKVM $. $( A more general version of ~ cbvrab with no distinct variable restrictions. Usage of this theorem is discouraged because it depends @@ -37432,7 +37432,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use cbvrabcsf $p |- { x e. A | ph } = { y e. B | ps } $= ( vz vv cv wcel wa cab wsb nfcri crab csb nfv nfcsb1v nfan weq id csbeq1a nfs1v eleq12d sbequ12 anbi12d cbvab nfcv nfcsb csbeq1 df-csb eleq2d sbsbc - nfsb wsbc bitr3i abbi2i eqtr4i eqtrdi sbequ bitrdi eqtri df-rab 3eqtr4i + nfsb wsbc bitr3i eqabi eqtr4i eqtrdi sbequ bitrdi eqtri df-rab 3eqtr4i sbie ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVOMOZCVTEUBZPZACMSZQZMRVSVNWDCMV NMUCWBWCCCMWACVTEUDTACMUIUECMUFZVMWBAWCWEVLVTEWAWEUGCVTEUHUJACMUKULUMWDVR MDWBWCDDMWADCVTEDVTUNGUOTACMDIUTUEVRMUCMDUFZWBVQWCBWFVTVPWAFWFUGWFWACVPEU @@ -39875,7 +39875,7 @@ technically classes despite morally (and provably) being sets, like ` 1 ` $( Alternate definition of the symmetric difference. (Contributed by NM, 17-Aug-2004.) (Revised by AV, 17-Aug-2022.) $) dfsymdif4 $p |- ( A /_\ B ) = { x | -. ( x e. A <-> x e. B ) } $= - ( cv wcel wb wn csymdif elsymdif abbi2i ) ADZBEKCEFGABCHKBCIJ $. + ( cv wcel wb wn csymdif elsymdif eqabi ) ADZBEKCEFGABCHKBCIJ $. $} $( Membership in a symmetric difference is an exclusive-or relationship. @@ -39890,7 +39890,7 @@ technically classes despite morally (and provably) being sets, like ` 1 ` $( Alternate definition of the symmetric difference. (Contributed by BJ, 30-Apr-2020.) $) dfsymdif2 $p |- ( A /_\ B ) = { x | ( x e. A \/_ x e. B ) } $= - ( cv wcel wxo csymdif elsymdifxor abbi2i ) ADZBEJCEFABCGJBCHI $. + ( cv wcel wxo csymdif elsymdifxor eqabi ) ADZBEJCEFABCGJBCHI $. $} ${ @@ -41470,7 +41470,7 @@ among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets, $( The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.) $) csbvarg $p |- ( A e. V -> [_ A / x ]_ x = A ) $= - ( vz vy wcel cvv cv csb wceq elex wsbc cab df-csb sbcel2gv abbi1dv eqtrid + ( vz vy wcel cvv cv csb wceq elex wsbc cab df-csb sbcel2gv eqabcdv eqtrid elv csbeq2i csbcow 3eqtr3i syl ) BCFBGFZABAHZIZBJBCKUCUEDHZEHZFEBLZDMZBEB AUGUDIZIEBUGIUEUIEBUJUGUJUGJEUGGFZUJUFUDFAUGLZDMUGADUGUDNUKULDUGAUFUGGOPQ RSAEBUDTEDBUGNUAUCUHDBEUFBGOPQUB $. @@ -42412,8 +42412,8 @@ fewer connectives (but probably less intuitive to understand). (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $) iftrue $p |- ( ph -> if ( ph , A , B ) = A ) $= - ( vx cif cv wcel wi wa cab dfif2 dedlem0a abbi2dv eqtr4id ) AABCEDFZCGZAH - OBGZAIHZDJBADBCKARDBAQPLMN $. + ( vx cif cv wcel wi wa cab dfif2 dedlem0a eqabdv eqtr4id ) AABCEDFZCGZAHO + BGZAIHZDJBADBCKARDBAQPLMN $. $} ${ @@ -42437,8 +42437,8 @@ fewer connectives (but probably less intuitive to understand). $( Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) $) iffalse $p |- ( -. ph -> if ( ph , A , B ) = B ) $= - ( vx wn cif cv wcel wa wo cab df-if dedlemb abbi2dv eqtr4id ) AEZABCFDGZB - HZAIQCHZPIJZDKCADBCLPTDCARSMNO $. + ( vx wn cif cv wcel wa wo cab df-if dedlemb eqabdv eqtr4id ) AEZABCFDGZBH + ZAIQCHZPIJZDKCADBCLPTDCARSMNO $. $} ${ @@ -43560,9 +43560,9 @@ will do (e.g., ` O = (/) ` and ` T = { (/) } ` , see ~ 0nep0 ). $( Alternate definition of a pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) $) dfpr2 $p |- { A , B } = { x | ( x = A \/ x = B ) } $= - ( cpr csn cun cv wceq wo df-pr wcel elun velsn orbi12i bitri abbi2i eqtri - cab ) BCDBEZCEZFZAGZBHZUBCHZIZARBCJUEAUAUBUAKUBSKZUBTKZIUEUBSTLUFUCUGUDAB - MACMNOPQ $. + ( cpr csn cun cv wceq cab df-pr wcel elun velsn orbi12i bitri eqabi eqtri + wo ) BCDBEZCEZFZAGZBHZUBCHZRZAIBCJUEAUAUBUAKUBSKZUBTKZRUEUBSTLUFUCUGUDABM + ACMNOPQ $. $} ${ @@ -43959,7 +43959,7 @@ will do (e.g., ` O = (/) ` and ` T = { (/) } ` , see ~ 0nep0 ). Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.) $) dftp2 $p |- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) } $= - ( cv wceq w3o ctp vex eltp abbi2i ) AEZBFLCFLDFGABCDHLBCDAIJK $. + ( cv wceq w3o ctp vex eltp eqabi ) AEZBFLCFLDFGABCDHLBCDAIJK $. $} ${ @@ -45531,10 +45531,10 @@ will do (e.g., ` O = (/) ` and ` T = { (/) } ` , see ~ 0nep0 ). dfopif $p |- <. A , B >. = if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) ) $= ( vx cop cvv wcel cv csn cpr w3a cab wa c0 df-op df-3an abbii wceq iftrue - cif ibar abbi2dv eqtr2d wss pm2.21 adantrd abssdv ss0 syl iffalse pm2.61i - wn eqtr4d 3eqtri ) ABDAEFZBEFZCGZAHABIIZFZJZCKUNUOLZURLZCKZUTUQMSZCABNUSV - ACUNUOUROPUTVBVCQUTVCUQVBUTUQMRUTVACUQUTURTUAUBUTUKZVBMVCVDVBMUCVBMQVDVAC - MVDUTUPMFZURUTVEUDUEUFVBUGUHUTUQMUIULUJUM $. + cif ibar eqabdv eqtr2d wn wss pm2.21 adantrd abssdv ss0 syl eqtr4d 3eqtri + iffalse pm2.61i ) ABDAEFZBEFZCGZAHABIIZFZJZCKUNUOLZURLZCKZUTUQMSZCABNUSVA + CUNUOUROPUTVBVCQUTVCUQVBUTUQMRUTVACUQUTURTUAUBUTUCZVBMVCVDVBMUDVBMQVDVACM + VDUTUPMFZURUTVEUEUFUGVBUHUIUTUQMULUJUMUK $. $( $j usage 'dfopif' avoids 'ax-10' 'ax-11' 'ax-12'; $) $} @@ -47383,8 +47383,8 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and $( An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) (Proof shortened by SN, 15-Jan-2025.) $) iunid $p |- U_ x e. A { x } = A $= - ( vy cv csn ciun wcel wrex df-iun clel5 velsn rexbii bitr4i abbi2i eqtr4i - cab weq ) ABADZEZFCDZSGZABHZCPBACBSIUBCBTBGCAQZABHUBABTJUAUCABCRKLMNO $. + ( vy cv csn ciun wcel wrex cab df-iun weq clel5 velsn rexbii bitr4i eqabi + eqtr4i ) ABADZEZFCDZSGZABHZCIBACBSJUBCBTBGCAKZABHUBABTLUAUCABCRMNOPQ $. $( $j usage 'iunid' avoids 'ax-10' 'ax-11' 'ax-12'; $) $( Obsolete version of ~ iunid as of 15-Jan-2025. (Contributed by NM, @@ -47413,8 +47413,8 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and $( An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.) $) 0iin $p |- |^|_ x e. (/) A = _V $= - ( vy c0 ciin cv wcel wral cab cvv df-iin vex ral0 2th abbi2i eqtr4i ) ADB - ECFZBGZADHZCIJACDBKSCJQJGSCLRAMNOP $. + ( vy c0 ciin cv wcel wral cab cvv df-iin vex ral0 2th eqabi eqtr4i ) ADBE + CFZBGZADHZCIJACDBKSCJQJGSCLRAMNOP $. $( Indexed intersection with a universal index class. When ` A ` doesn't depend on ` x ` , this evaluates to ` A ` by ~ 19.3 and ~ abid2 . When @@ -47635,7 +47635,7 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) $) iinxsng $p |- ( A e. V -> |^|_ x e. { A } B = C ) $= - ( vy wcel csn ciin cv wral cab df-iin wceq eleq2d ralsng abbi1dv eqtrid ) + ( vy wcel csn ciin cv wral cab df-iin wceq eleq2d ralsng eqabcdv eqtrid ) BEHZABIZCJGKZCHZAUALZGMDAGUACNTUDGDUCUBDHABEAKBOCDUBFPQRS $. $} @@ -47800,8 +47800,8 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.) $) pwpwab $p |- ~P ~P A = { x | U. x C_ A } $= - ( cv cuni wss cpw wcel cvv vex elpwpw mpbiran abbi2i ) ACZDBEZABFFZMOGMHG - NAIMBJKL $. + ( cv cuni wss cpw wcel cvv vex elpwpw mpbiran eqabi ) ACZDBEZABFFZMOGMHGN + AIMBJKL $. $} ${ @@ -54064,9 +54064,9 @@ dummy variables (but note that their equivalence uses ~ ax-sep ). relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.) $) epse $p |- _E Se A $= - ( vy vx cep wse cv wbr crab cvv wcel wral cab epel bicomi abbi2i eqeltrri - vex rabssab ssexi rgenw df-se mpbir ) ADEBFZCFZDGZBAHZIJZCAKUGCAUFUEBLZUD - UHIUEBUDUEUCUDJCUCMNOCQPUEBARSTCBADUAUB $. + ( vy vx cep wse wbr crab cvv wcel wral cab epel bicomi eqabi vex eqeltrri + cv rabssab ssexi rgenw df-se mpbir ) ADEBQZCQZDFZBAGZHIZCAJUGCAUFUEBKZUDU + HHUEBUDUEUCUDICUCLMNCOPUEBARSTCBADUAUB $. $} $( Similar to Theorem 7.2 of [TakeutiZaring] p. 35, except that the Axiom of @@ -54621,7 +54621,7 @@ of the restricted converse (see ~ cnvrescnv ). (Contributed by NM, of all sets which are binary relations. (Contributed by BJ, 21-Dec-2023.) $) pwvabrel $p |- ~P ( _V X. _V ) = { x | Rel x } $= - ( cv wrel cvv cxp cpw wcel wb pwvrel elv abbi2i ) ABZCZADDEFZLNGMHALDIJK $. + ( cv wrel cvv cxp cpw wcel wb pwvrel elv eqabi ) ABZCZADDEFZLNGMHALDIJK $. $( Two classes related by a binary relation are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) $) @@ -55836,7 +55836,7 @@ of the restricted converse (see ~ cnvrescnv ). (Contributed by NM, opabbi2dv.1 $e |- Rel A $. opabbi2dv.3 $e |- ( ph -> ( <. x , y >. e. A <-> ps ) ) $. $( Deduce equality of a relation and an ordered-pair class abstraction. - Compare ~ abbi2dv . (Contributed by NM, 24-Feb-2014.) $) + Compare ~ eqabdv . (Contributed by NM, 24-Feb-2014.) $) opabbi2dv $p |- ( ph -> A = { <. x , y >. | ps } ) $= ( cv cop wcel copab wrel wceq opabid2 ax-mp opabbidv eqtr3id ) AECHDHIEJZ CDKZBCDKELSEMFCDENOARBCDGPQ $. @@ -57897,8 +57897,8 @@ the restriction (of the relation) to the singleton containing this example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) $) iniseg $p |- ( B e. V -> ( `' A " { B } ) = { x | x A B } ) $= - ( wcel cvv ccnv csn cima cv wbr cab wceq elex vex eliniseg abbi2dv syl ) - CDECFEZBGCHIZAJZCBKZALMCDNSUBATBCUAFAOPQR $. + ( wcel cvv ccnv csn cima cv wbr cab wceq elex vex eliniseg eqabdv syl ) C + DECFEZBGCHIZAJZCBKZALMCDNSUBATBCUAFAOPQR $. $} ${ @@ -60057,9 +60057,9 @@ singleton of the first member (with no sethood assumptions on ` B ` ). of ` A ` exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) $) setlikespec $p |- ( ( X e. A /\ R Se A ) -> Pred ( R , A , X ) e. _V ) $= - ( vx wcel wse wa cv wbr crab cpred cvv wceq cab df-rab vex elpred abbi2dv - eqtr4id adantr seex ancoms eqeltrrd ) CAEZABFZGDHZCBIZDAJZABCKZLUDUHUIMUE - UDUHUFAEUGGZDNUIUGDAOUDUJDUIAABCUFDPQRSTUEUDUHLEDACBUAUBUC $. + ( vx wcel wse wa cv wbr crab cpred cvv wceq cab df-rab vex elpred eqtr4id + eqabdv adantr seex ancoms eqeltrrd ) CAEZABFZGDHZCBIZDAJZABCKZLUDUHUIMUEU + DUHUFAEUGGZDNUIUGDAOUDUJDUIAABCUFDPQSRTUEUDUHLEDACBUAUBUC $. $} $( Idempotent law for the predecessor class. (Contributed by Scott Fenton, @@ -60096,7 +60096,7 @@ singleton of the first member (with no sethood assumptions on ` B ` ). shortened by Andrew Salmon, 27-Aug-2011.) $) predep $p |- ( X e. B -> Pred ( _E , A , X ) = ( A i^i X ) ) $= ( vy wcel cep cpred ccnv csn cima cin df-pred cv wbr cab wrel wceq relcnv - relimasn eqtrid ax-mp wb cvv brcnvg elvd epelg bitrd abbi1dv ineq2d ) CBE + relimasn eqtrid ax-mp wb cvv brcnvg elvd epelg bitrd eqabcdv ineq2d ) CBE ZAFCGAFHZCIJZKACKAFCLUJULCAUJULCDMZUKNZDOZCUKPULUOQFRDCUKSUAUJUNDCUJUNUMC FNZUMCEUJUNUPUBDCUMBUCFUDUEUMCBUFUGUHTUIT $. $} @@ -65212,12 +65212,12 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). ( vz cv wcel wbr wa wex weu cfv weq wb wal elfv wi biimpr breq2 sylib eu6 alimi equsalvw anim2i eximi elequ2 anbi12d cbvexvw exsimpr sylibr jca nfv nfeu1 nfa1 nfan nfex nfim biimp syl6 impcomd anc2ri com12 eximdv biimtrid - ax9 sps exlimi imp impbii bitri abbi2i ) AFZBFZGZCVMDHZIZBJZVOBKZIZACDLZV - LVTGVLEFZGZVOBEMZNZBOZIZEJZVSEBVLCDPWGVSWGVQVRWGWBCWADHZIZEJVQWFWIEWEWHWB - WEWCVOQZBOWHWDWJBVOWCRUBVOWHBEVMWACDSUCTUDUEWIVPEBEBMWBVNWHVOEBAUFWAVMCDS - UGUHTWGWEEJZVRWBWEEUIVOBEUAZUJUKVQVRWGVPVRWGQBVRWGBVOBUMWFBEWBWEBWBBULWDB - UNUOUPUQVRWKVPWGWLVPWEWFEWEVPWFWEVPWBWDVPWBQBWDVOVNWBWDVOWCVNWBQVOWCURBEA - VEUSUTVFVAVBVCVDVGVHVIVJVK $. + ax9 sps exlimi imp impbii bitri eqabi ) AFZBFZGZCVMDHZIZBJZVOBKZIZACDLZVL + VTGVLEFZGZVOBEMZNZBOZIZEJZVSEBVLCDPWGVSWGVQVRWGWBCWADHZIZEJVQWFWIEWEWHWBW + EWCVOQZBOWHWDWJBVOWCRUBVOWHBEVMWACDSUCTUDUEWIVPEBEBMWBVNWHVOEBAUFWAVMCDSU + GUHTWGWEEJZVRWBWEEUIVOBEUAZUJUKVQVRWGVPVRWGQBVRWGBVOBUMWFBEWBWEBWBBULWDBU + NUOUPUQVRWKVPWGWLVPWEWFEWEVPWFWEVPWBWDVPWBQBWDVOVNWBWDVOWCVNWBQVOWCURBEAV + EUSUTVFVAVBVCVDVGVHVIVJVK $. $} ${ @@ -65763,7 +65763,7 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). shortened by Scott Fenton, 8-Aug-2024.) $) fnsnfv $p |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) ) $= ( vy wfn wcel wa csn cima cv wbr cab wceq imasng adantl velsn eqcom bitri - cfv fnbrfvb bitr2id abbi1dv eqtr2d ) CAEZBAFZGZCBHIZBDJZCKZDLZBCSZHZUEUGU + cfv fnbrfvb bitr2id eqabcdv eqtr2d ) CAEZBAFZGZCBHIZBDJZCKZDLZBCSZHZUEUGU JMUDDBACNOUFUIDULUHULFZUKUHMZUFUIUMUHUKMUNDUKPUHUKQRABUHCTUAUBUC $. $( Obsolete version of ~ fnsnfv as of 8-Aug-2024. (Contributed by NM, @@ -66863,7 +66863,7 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). $( Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.) $) fncnvima2 $p |- ( F Fn A -> ( `' F " B ) = { x e. A | ( F ` x ) e. B } ) $= - ( wfn ccnv cima cv wcel cfv wa cab crab elpreima abbi2dv df-rab eqtr4di ) + ( wfn ccnv cima cv wcel cfv wa cab crab elpreima eqabdv df-rab eqtr4di ) DBEZDFCGZAHZBITDJCIZKZALUAABMRUBASBTCDNOUAABPQ $. $( Inverse point images under functions expressed as abstractions. @@ -70362,16 +70362,16 @@ H Isom ( R i^i ( A X. A ) ) , S ( A , B ) ) $= dfima2 wceq crn wf1o wfo isof1o f1ofo forn eleq2d 3syl wfn fvelrnb bitr3d f1ofn cvv fvex vex eliniseg mp1i adantr anbi2d bitrid anbi1d anass adantl anbi12d isorel syl fnbrfvb bicomd sylan adantrr ancom breq1 pm5.32i bitri - wi exp32 com23 imp pm5.32d bitrd rexbidv2 r19.41v bitr4d abbi2dv eqtr4id - ) ABDEFUAZCAIZJZFADKCLMZNZMGOZHOZFPZGWOQZHUBBEKCFRZLMZNZGHFWOUDWMWSHXBWQX - BIWQBIZWQXAIZJZWMWSWQBXAUCWMXEWPFRZWQUEZGAQZWQWTEPZJZWSWKXEXJSWLWKXCXHXDX - IWKWQFUFZIZXCXHWKABFUGZABFUHZXLXCSABDEFUIZABFUJXNXKBWQABFUKULUMWKXMFAUNZX - LXHSXOABFUQZGAWQFUOUMUPWTURIXDXISWKCFUSEWTWQURHUTVAVBVIVCWMWSXGXIJZGAQXJW - MWRXRGWOAWMWPWOIZWRJZWPAIZWPCDPZWRJZJZYAXRJWLXTYDSWKWLXTYAYBJZWRJYDWLXSYE - WRXSYAWPWNIZJWLYEWPAWNUCWLYFYBYADCWPAGUTVAVDVEVFYAYBWRVGTVHWMYAYCXRWKWLYA - YCXRSZVTWKYAWLYGWKYAWLYGWKYAWLJJZYCXFWTEPZXGJZXRYHYBYIWRXGABWPCDEFVJWKYAW - RXGSZWLWKXPYAYKWKXMXPXOXQVKXPYAJXGWRAWPWQFVLVMVNVOVIYJXGYIJXRYIXGVPXGYIXI - XFWQWTEVQVRVSTWAWBWCWDWEWFXGXIGAWGTWHVEWIWJ $. + wi exp32 com23 imp pm5.32d bitrd rexbidv2 r19.41v bitr4d eqabdv eqtr4id ) + ABDEFUAZCAIZJZFADKCLMZNZMGOZHOZFPZGWOQZHUBBEKCFRZLMZNZGHFWOUDWMWSHXBWQXBI + WQBIZWQXAIZJZWMWSWQBXAUCWMXEWPFRZWQUEZGAQZWQWTEPZJZWSWKXEXJSWLWKXCXHXDXIW + KWQFUFZIZXCXHWKABFUGZABFUHZXLXCSABDEFUIZABFUJXNXKBWQABFUKULUMWKXMFAUNZXLX + HSXOABFUQZGAWQFUOUMUPWTURIXDXISWKCFUSEWTWQURHUTVAVBVIVCWMWSXGXIJZGAQXJWMW + RXRGWOAWMWPWOIZWRJZWPAIZWPCDPZWRJZJZYAXRJWLXTYDSWKWLXTYAYBJZWRJYDWLXSYEWR + XSYAWPWNIZJWLYEWPAWNUCWLYFYBYADCWPAGUTVAVDVEVFYAYBWRVGTVHWMYAYCXRWKWLYAYC + XRSZVTWKYAWLYGWKYAWLYGWKYAWLJJZYCXFWTEPZXGJZXRYHYBYIWRXGABWPCDEFVJWKYAWRX + GSZWLWKXPYAYKWKXMXPXOXQVKXPYAJXGWRAWPWQFVLVMVNVOVIYJXGYIJXRYIXGVPXGYIXIXF + WQWTEVQVRVSTWAWBWCWDWEWFXGXIGAWGTWHVEWIWJ $. $} ${ @@ -78252,19 +78252,19 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) fo1st $p |- 1st : _V -onto-> _V $= ( vx vy cvv c1st wfo wfn crn wceq cv csn cdm cuni vsnex dmex uniex df-1st fnmpti wrex cab wcel rnmpt vex cop opex op1sta eqcomi sneq dmeqd rspceeqv - unieqd mp2an 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAIZJZKZLZDU - TUSAMNOAPZQUQBIZVAHACRZBSCABCVADVBUAVDBCVCCTVDBUBZVCVCUCZCTVCVFJZKZLZHVDV - CVCUDVIVCVCVCVEVEUEUFAVFCVAVIVCURVFHZUTVHVJUSVGURVFUGUHUJUIUKULUMUNCCDUOU - P $. + unieqd mp2an 2th eqabi eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAIZJZKZLZDUT + USAMNOAPZQUQBIZVAHACRZBSCABCVADVBUAVDBCVCCTVDBUBZVCVCUCZCTVCVFJZKZLZHVDVC + VCUDVIVCVCVCVEVEUEUFAVFCVAVIVCURVFHZUTVHVJUSVGURVFUGUHUJUIUKULUMUNCCDUOUP + $. $( The ` 2nd ` function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) $) fo2nd $p |- 2nd : _V -onto-> _V $= ( vx vy cvv c2nd wfo wfn crn wceq csn cuni vsnex rnex uniex df-2nd fnmpti cv wrex cab rnmpt wcel vex cop opex op2nda eqcomi sneq rneqd unieqd mp2an - rspceeqv 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAPZIZGZJZDUSURA - KLMANZOUPBPZUTHACQZBRCABCUTDVASVCBCVBCTVCBUAZVBVBUBZCTVBVEIZGZJZHVCVBVBUC - VHVBVBVBVDVDUDUEAVECUTVHVBUQVEHZUSVGVIURVFUQVEUFUGUHUJUIUKULUMCCDUNUO $. + rspceeqv 2th eqabi eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAPZIZGZJZDUSURAK + LMANZOUPBPZUTHACQZBRCABCUTDVASVCBCVBCTVCBUAZVBVBUBZCTVBVEIZGZJZHVCVBVBUCV + HVBVBVBVDVDUDUEAVECUTVHVBUQVEHZUSVGVIURVFUQVEUFUGUHUJUIUKULUMCCDUNUO $. $} $( Uniqueness condition for the binary relation ` 1st ` . (Contributed by @@ -78436,8 +78436,8 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) functions. (Contributed by NM, 16-Sep-2006.) $) xp2 $p |- ( A X. B ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) } $= - ( cxp cv cvv wcel c1st cfv c2nd wa cab crab elxp7 abbi2i df-rab eqtr4i ) - BCDZAEZFFDZGSHIBGSJICGKZKZALUAATMUBARSBCNOUAATPQ $. + ( cxp cv cvv wcel c1st cfv c2nd wa cab crab elxp7 eqabi df-rab eqtr4i ) B + CDZAEZFFDZGSHIBGSJICGKZKZALUAATMUBARSBCNOUAATPQ $. $} ${ @@ -82552,11 +82552,11 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( vw cdm wrel ctpos cv cop wceq wbr wa wex cab wcel cvv cxp bitr3i bitrdi coprab wss relcnv dmtpos releqd mpbiri reltpos jctil relrelss sylib sseld ccnv elvvv syl6ib pm4.71rd 19.41vvv eleq1 df-br brtpos elv pm5.32i 3exbii - wb abbi2dv df-oprab eqtr4di ) DFZGZDHZEIZAIZBIZJZCIZJZKZVLVKJVNDLZMZCNBNA - NZEOVQABCUAVHVSEVIVHVJVIPZVPCNBNANZVTMZVSVHVTWAVHVTVJQQRQRZPWAVHVIWCVJVHV - IGZVIFZGZMVIWCUBVHWFWDVHWFVGULZGVGUCVHWEWGDUDUEUFDUGUHVIUIUJUKABCVJUMUNUO - WBVPVTMZCNBNANVSVPVTABCUPWHVRABCVPVTVQVPVTVOVIPZVQVJVOVIUQWIVMVNVILZVQVMV - NVIURWJVQVCCVKVLVNDQUSUTSTVAVBSTVDVQABCEVEVF $. + wb eqabdv df-oprab eqtr4di ) DFZGZDHZEIZAIZBIZJZCIZJZKZVLVKJVNDLZMZCNBNAN + ZEOVQABCUAVHVSEVIVHVJVIPZVPCNBNANZVTMZVSVHVTWAVHVTVJQQRQRZPWAVHVIWCVJVHVI + GZVIFZGZMVIWCUBVHWFWDVHWFVGULZGVGUCVHWEWGDUDUEUFDUGUHVIUIUJUKABCVJUMUNUOW + BVPVTMZCNBNANVSVPVTABCUPWHVRABCVPVTVQVPVTVOVIPZVQVJVOVIUQWIVMVNVILZVQVMVN + VIURWJVQVCCVKVLVNDQUSUTSTVAVBSTVDVQABCEVEVF $. $( Alternate definition of ` tpos ` . (Contributed by Mario Carneiro, 4-Oct-2015.) $) @@ -84846,8 +84846,8 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $= variables in ` A ` for later use. (Contributed by NM, 9-Apr-1995.) $) tfrlem3 $p |- A = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) } $= - ( cv wfn cfv cres wceq wral wa con0 wrex vex tfrlem3a abbi2i ) GJZCJZKDJZ - UBLUBUDMHLNDUCOPCQRGEABCDEFHUBIGSTUA $. + ( cv wfn cfv cres wceq wral wa con0 wrex vex tfrlem3a eqabi ) GJZCJZKDJZU + BLUBUDMHLNDUCOPCQRGEABCDEFHUBIGSTUA $. $} ${ @@ -91594,17 +91594,17 @@ the first case of his notation (simple exponentiation) and subscript it frel cdm imaeq2d imadmrn eqtr3di eqtr3d eqeq1d exbidv bitrid adantl sseld fdm frn vsnid eleq2 mpbiri impel adantll ffrn feq3 syl5ibcom imp ad2antrr vex fsng sylancl jca ex eximdv mpd df-rex sylibr w3a f1osng adantr f1oeq1 - wf1o bicomd f1of 3adant2 wss snssi 3ad2ant2 fssd rexlimdv3a bitrd abbi2dv - impbid ) AEJZDBJZUAKZLZBCUBZECDKZUCUDZAXIXOMXNCXINZXMACXNXIFOHXNOMADUGUEU - FAXPXMAXPXMAXPPZXJCMZXLPZBQZXMXQXIUHZXJKZLZBQZXTXQDXJXIUIZBUJZYDXPXIXNUKD - XNMZYFAXNCXIULADGMZYGIDGUMRBXNDXIUNUOXPYFYDSAYFYEBUPZYBLZBQXPYDYEBUQXPYJY - CBXPYIYAYBXPXIXNURZYIYAXPXIUSYKYILXNCXIVABDXIUTRXPXIXIVBZURYKYAXPYLXNXIXN - CXIVLVCXIVDVEVFVGVHVIVJTXQYCXSBXQYCXSXQYCPZXRXLXPYCXRAXPXJYAMZXRYCXPYACXJ - XNCXIVMVKYCYNXJYBMBVNYAYBXJVOVPVQVRYMXNYBXINZXLXPYCYOAXPYCYOXPXNYAXINYCYO - XNCXIVSYAYBXNXIVTWAWBVRYMYHXJOMZYOXLSAYHXPYCIWCBWDZDXJGOXIWEWFTWGWHWIWJXL - BCWKWLWHAXLXPBCAXRXLWMXNYBCXIAXLYOXRAXLPZXNYBXIWQZYOYRXNYBXKWQZYSAYTXLAYH - YPYTIYQDXJGOWNWFWOXLYTYSSAXLYSYTXNYBXIXKWPWRVJTXNYBXIWSRWTXRAYBCXAXLXJCXB - XCXDXEXHXFXG $. + wf1o bicomd f1of 3adant2 wss snssi 3ad2ant2 fssd rexlimdv3a impbid eqabdv + bitrd ) AEJZDBJZUAKZLZBCUBZECDKZUCUDZAXIXOMXNCXINZXMACXNXIFOHXNOMADUGUEUF + AXPXMAXPXMAXPPZXJCMZXLPZBQZXMXQXIUHZXJKZLZBQZXTXQDXJXIUIZBUJZYDXPXIXNUKDX + NMZYFAXNCXIULADGMZYGIDGUMRBXNDXIUNUOXPYFYDSAYFYEBUPZYBLZBQXPYDYEBUQXPYJYC + BXPYIYAYBXPXIXNURZYIYAXPXIUSYKYILXNCXIVABDXIUTRXPXIXIVBZURYKYAXPYLXNXIXNC + XIVLVCXIVDVEVFVGVHVIVJTXQYCXSBXQYCXSXQYCPZXRXLXPYCXRAXPXJYAMZXRYCXPYACXJX + NCXIVMVKYCYNXJYBMBVNYAYBXJVOVPVQVRYMXNYBXINZXLXPYCYOAXPYCYOXPXNYAXINYCYOX + NCXIVSYAYBXNXIVTWAWBVRYMYHXJOMZYOXLSAYHXPYCIWCBWDZDXJGOXIWEWFTWGWHWIWJXLB + CWKWLWHAXLXPBCAXRXLWMXNYBCXIAXLYOXRAXLPZXNYBXIWQZYOYRXNYBXKWQZYSAYTXLAYHY + PYTIYQDXJGOWNWFWOXLYTYSSAXLYSYTXNYBXIXKWPWRVJTXNYBXIWSRWTXRAYBCXAXLXJCXBX + CXDXEXFXHXG $. $} ${ @@ -91836,9 +91836,9 @@ the first case of his notation (simple exponentiation) and subscript it $( Infinite Cartesian product of a constant ` B ` . (Contributed by Mario Carneiro, 11-Jan-2015.) $) ixpconstg $p |- ( ( A e. V /\ B e. W ) -> X_ x e. A B = ( B ^m A ) ) $= - ( vf wcel cixp cmap co wa cv wf cab vex elixpconst abbi2i mapvalg eqtr4id - wceq ancoms ) CEGZBDGZABCHZCBIJZTUBUCKUDBCFLZMZFNUEUGFUDABCUFFOPQCBEDFRSU - A $. + ( vf wcel cixp cmap co wceq wa cv wf cab elixpconst eqabi mapvalg eqtr4id + vex ancoms ) CEGZBDGZABCHZCBIJZKUBUCLUDBCFMZNZFOUEUGFUDABCUFFTPQCBEDFRSUA + $. ixpconst.1 $e |- A e. _V $. ixpconst.2 $e |- B e. _V $. @@ -92045,8 +92045,8 @@ the first case of his notation (simple exponentiation) and subscript it NM, 21-Sep-2007.) $) ixp0x $p |- X_ x e. (/) A = { (/) } $= ( vf c0 cixp cv wfn cfv wcel wral wa cab csn dfixp wceq velsn fn0 biantru - ral0 3bitr2i abbi2i eqtr4i ) ADBECFZDGZAFUCHBIZADJZKZCLDMZADBCNUGCUHUCUHI - UCDOUDUGCDPUCQUFUDUEASRTUAUB $. + ral0 3bitr2i eqabi eqtr4i ) ADBECFZDGZAFUCHBIZADJZKZCLDMZADBCNUGCUHUCUHIU + CDOUDUGCDPUCQUFUDUEASRTUAUB $. $} ${ @@ -100692,39 +100692,39 @@ all reals (greatest real number) for which all positive reals are greater. eqsstrrid eqssd sseq1d reseq2d sqxpeqd difeq1 difun2 ineq1i 3eqtr3i weeq2 3anbi123d eqtrdi bitrd oieq1 oieq2 eqtrd dmeqd eqeq2d spcev syl21anc dmex brdomi simpr oion eqeltrdi simplr sylancr eqbrtrd simpll1 endomtr syl2an2 - oien ssdomg jca impbid2 bitrid abbi2dv eqtr4id 3pm3.2i ) IUAZFFUBZUCZOIUD - ZFJUEZIUFZAUGZFUHUIZAUJUKZPVPUWMLUGZUAZFOZQUXCULZUXBOZUXBUXCUXCUBZOZUMZUX - CUXBQUNZUOZRZBUGZUXCUXJUPZUAZPZRZBUQZLURZUWOUWMUXQLBUSZUAUXSIUXTMUTUXQLBV - AVBUXRLUWOUXQUXBUWOUEZBUXIUYAUXKUXPUXIUXBUWNOUYAUXIUXBUXGUWNUXDUXFUXHVCUX - IUXDUXDUXGUWNOUXDUXFUXHVDZUYBUXCFUXCFVEVFVGLUWNVHVIVJVKVLVMUWPUXTUDUXLLBU - 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(Contributed by Mario Carneiro, 14-Jan-2013.) $) hartogslem2 $p |- ( A e. V -> { x e. On | x ~<_ A } e. _V ) $= @@ -101643,8 +101643,8 @@ is that it denies the existence of a set containing itself ( ~ elirrv ). $( The Russell class is equal to the universe ` _V ` . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) $) ruv $p |- { x | x e/ x } = _V $= - ( cvv cv wnel cab wcel vex elirr nelir 2th abbi2i eqcomi ) BACZMDZAENABMBFN - AGMMMHIJKL $. + ( cvv cv wnel cab wcel vex elirr nelir 2th eqabi eqcomi ) BACZMDZAENABMBFNA + GMMMHIJKL $. $( Alternate proof of ~ ru , simplified using (indirectly) the Axiom of Regularity ~ ax-reg . (Contributed by Alan Sare, 4-Oct-2008.) @@ -102314,7 +102314,7 @@ of all inductive sets (which is the smallest inductive set, since $( A simplification of ~ df-om assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.) $) dfom4 $p |- _om = { x | A. y ( Lim y -> x e. y ) } $= - ( cv wlim wel wi wal com elom3 abbi2i ) BCDABEFBGAHBACIJ $. + ( cv wlim wel wi wal com elom3 eqabi ) BCDABEFBGAHBACIJ $. $} ${ @@ -105979,7 +105979,7 @@ of the previous layer (and the union of previous layers when the terms of rank. Definition 15.19 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) $) r1val2 $p |- ( A e. On -> ( R1 ` A ) = { x | ( rank ` x ) e. A } ) $= - ( con0 wcel cv crnk cfv cr1 vex rankr1a abbi2dv ) BCDAEZFGBDABHGLBAIJK $. + ( con0 wcel cv crnk cfv cr1 vex rankr1a eqabdv ) BCDAEZFGBDABHGLBAIJK $. $( The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113. (Contributed by NM, @@ -107857,9 +107857,9 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). See ~ cfval occur in the literature. (Contributed by NM, 7-Nov-2003.) $) cardval2 $p |- ( A e. dom card -> ( card ` A ) = { x e. On | x ~< A } ) $= ( ccrd cdm wcel cfv cv con0 csdm wbr wa cab crab cardon oneli pm4.71ri wb - cardsdomel ancoms pm5.32da bitr4id abbi2dv df-rab eqtr4di ) BCDEZBCFZAGZH - EZUGBIJZKZALUIAHMUEUJAUFUEUGUFEZUHUKKUJUKUHUFUGBNOPUEUHUIUKUHUEUIUKQUGBRS - TUAUBUIAHUCUD $. + cardsdomel ancoms pm5.32da bitr4id eqabdv df-rab eqtr4di ) BCDEZBCFZAGZHE + ZUGBIJZKZALUIAHMUEUJAUFUEUGUFEZUHUKKUJUKUHUFUGBNOPUEUHUIUKUHUEUIUKQUGBRST + UAUBUIAHUCUD $. $} ${ @@ -109528,7 +109528,7 @@ strictly dominates its argument (or the supremum of all ordinals that the finite ordinals in ` _om ` plus the transfinite alephs. (Contributed by NM, 10-Sep-2004.) $) cardnum $p |- { x | ( card ` x ) = x } = ( _om u. ran aleph ) $= - ( com cale crn cun cv ccrd cfv wceq cab wcel iscard3 bicomi abbi2i eqcomi ) + ( com cale crn cun cv ccrd cfv wceq cab wcel iscard3 bicomi eqabi eqcomi ) BCDEZAFZGHQIZAJRAPRQPKQLMNO $. ${ @@ -109588,12 +109588,12 @@ strictly dominates its argument (or the supremum of all ordinals that alephiso $p |- aleph Isom _E , _E ( On , { x | ( _om C_ x /\ ( card ` x ) = x ) } ) $= ( vy vz con0 com cv wss ccrd cfv wceq cab cep cale wiso wbr wral mpbir2an - wa wcel rgen2 wf1o wb wf1 wfo wf wi wfn alephfnon isinfcard bicomi abbi2i - crn df-fo fof ax-mp aleph11 biimpd dff13 df-f1o alephord2 epel fvex epeli - 3bitr4g df-isom ) DEAFZGVFHIVFJRZAKZLLMNDVHMUAZBFZCFZLOZVJMIZVKMIZLOZUBZC - DPBDPVIDVHMUCZDVHMUDZVQDVHMUEZVMVNJZVJVKJZUFZCDPBDPVRVSVRMDUGMULZVHJUHVGA - WCVGVFWCSVFUIUJUKDVHMUMQZDVHMUNUOWBBCDDVJDSVKDSRZVTWAVJVKUPUQTBCDVHMURQWD - DVHMUSQVPBCDDWEVJVKSVMVNSVLVOVJVKUTCVJVAVMVNVKMVBVCVDTBCDVHLLMVEQ $. + wa wcel rgen2 wf1o wb wf1 wfo wf wfn crn alephfnon isinfcard bicomi eqabi + wi df-fo fof ax-mp aleph11 biimpd dff13 alephord2 epel fvex epeli 3bitr4g + df-f1o df-isom ) DEAFZGVFHIVFJRZAKZLLMNDVHMUAZBFZCFZLOZVJMIZVKMIZLOZUBZCD + PBDPVIDVHMUCZDVHMUDZVQDVHMUEZVMVNJZVJVKJZULZCDPBDPVRVSVRMDUFMUGZVHJUHVGAW + CVGVFWCSVFUIUJUKDVHMUMQZDVHMUNUOWBBCDDVJDSVKDSRZVTWAVJVKUPUQTBCDVHMURQWDD + VHMVDQVPBCDDWEVJVKSVMVNSVLVOVJVKUSCVJUTVMVNVKMVAVBVCTBCDVHLLMVEQ $. $} $( The class of all transfinite cardinal numbers (the range of the aleph @@ -109956,19 +109956,19 @@ is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The wi df-xp sseqtrri ssexi sseq2 dmeq fneq2d anbi12d exbidv spcv csn cima wb fndm dmopab eleq2i weq elequ1 eleq2 elab 19.42v n0 anbi2i 3bitrri bitr4id bitr4i syl adantl fnfun funfvima3 sylan2 sylbid imp wbr cvv imasng anbi2d - wfun elv eqid brab abbii eqtri ibar abbi2dv eqtr4id eleq2d ad2antrl mpbid - exp32 ralrimiv eximi alrimiv cmpt aceq3lem impbii bitri ) UACHZDHZIZXMXNJ - ZKZLZCMZDNZBHZUCUBZYAXMOZYAPZUPZBAHZUDZCMZANZDCUEXTYIXTYHAXTXMEAQZFEQZLZE - FUFZIZXMYMJZKZLZCMZYHXSYRDYMYMYFYFUGZUHZYFYSARAUIUJYMYJFHZYSPZLZEFUFYTYLU - UCEFYLYJUUBYJYKUKYKYJUUBUUAEHZYFULUMUNUOEFYFYSUQURUSXNYMSZXRYQCUUEXOYNXQY - PXNYMXMUTUUEXPYOXMXNYMVAVBVCVDVEYQYGCYQYEBYFYQBAQZYBYDYQUUFYBLZLYCYMYAVFV - GZPZYDYQUUGUUIYQUUGYAXMJZPZUUIYPUUGUUKVHZYNYPUUJYOSZUULYOXMVIUUMUUGYAYOPZ - UUKUUNYAYLFMZETZPUUFFBQZLZFMZUUGYOUUPYAYLEFVJVKUUOUUSEYABRZEBVLZYLUURFUVA - YJUUFYKUUQEBAVMUUDYAUUAVNVCZVDVOUUSUUFUUQFMZLUUGUUFUUQFVPYBUVCUUFFYAVQVRW - AVSUUJYOYAVNVTWBWCYPYNXMWMZUUKUUIUPZYOXMWDUVDYNUVEYAXMYMWEUMWFWGWHUUFUUIY - DVHYQYBUUFUUHYAYCUUFUUHUUFGBQZLZGTZYAUUHYAGHZYMWIZGTZUVHUUHUVKSBGYAWJYMWK - WNUVJUVGGYLUURUVGEFYAUVIYMUUTGRUVBFGVLUUQUVFUUFFGBVMWLYMWOWPWQWRUUFUVGGYA - UUFUVFWSWTXAXBXCXDXEXFXGWBXHYIXSDADBEGCEXPUUDUVIXNWIGTXMOXIZUVLWOXJXHXKXL + wfun elv eqid brab abbii eqtri eqabdv eqtr4id eleq2d ad2antrl mpbid exp32 + ibar ralrimiv eximi alrimiv cmpt aceq3lem impbii bitri ) UACHZDHZIZXMXNJZ + KZLZCMZDNZBHZUCUBZYAXMOZYAPZUPZBAHZUDZCMZANZDCUEXTYIXTYHAXTXMEAQZFEQZLZEF + UFZIZXMYMJZKZLZCMZYHXSYRDYMYMYFYFUGZUHZYFYSARAUIUJYMYJFHZYSPZLZEFUFYTYLUU + CEFYLYJUUBYJYKUKYKYJUUBUUAEHZYFULUMUNUOEFYFYSUQURUSXNYMSZXRYQCUUEXOYNXQYP + XNYMXMUTUUEXPYOXMXNYMVAVBVCVDVEYQYGCYQYEBYFYQBAQZYBYDYQUUFYBLZLYCYMYAVFVG + ZPZYDYQUUGUUIYQUUGYAXMJZPZUUIYPUUGUUKVHZYNYPUUJYOSZUULYOXMVIUUMUUGYAYOPZU + UKUUNYAYLFMZETZPUUFFBQZLZFMZUUGYOUUPYAYLEFVJVKUUOUUSEYABRZEBVLZYLUURFUVAY + JUUFYKUUQEBAVMUUDYAUUAVNVCZVDVOUUSUUFUUQFMZLUUGUUFUUQFVPYBUVCUUFFYAVQVRWA + VSUUJYOYAVNVTWBWCYPYNXMWMZUUKUUIUPZYOXMWDUVDYNUVEYAXMYMWEUMWFWGWHUUFUUIYD + VHYQYBUUFUUHYAYCUUFUUHUUFGBQZLZGTZYAUUHYAGHZYMWIZGTZUVHUUHUVKSBGYAWJYMWKW + NUVJUVGGYLUURUVGEFYAUVIYMUUTGRUVBFGVLUUQUVFUUFFGBVMWLYMWOWPWQWRUUFUVGGYAU + UFUVFXEWSWTXAXBXCXDXFXGWBXHYIXSDADBEGCEXPUUDUVIXNWIGTXMOXIZUVLWOXJXHXKXL $. $} @@ -113268,7 +113268,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality cab imauni vex imaex dfiun2 eqtri weq rexrab eleq1 biimparc rexlimivw cdm ccnv cnvimass f1odm ad3antrrr sseqtrid cnvex elpw sylibr wfo f1ofo simprl sselda elpwid foimacnv syl2anc eqcomd simpr eqeq2d anbi12d rspcev impbid2 - eqeltrrd ex bitrid abbi1dv unieqd eqtrid simplr ssrab2 a1i sylib exlimddv + eqeltrrd ex bitrid eqabcdv unieqd eqtrid simplr ssrab2 a1i sylib exlimddv simprrl n0 rabn0 wo elrab anbi12i simprrr adantr simprlr sorpssi wb f1of1 wf1 simprll f1imass orbi12d mpbid sylan2b ralrimivva sorpss fin2i cbvrabv syl22anc elrab2 simprbi syl expr sylan2 ralrimiva exlimiv sylbi cvv relen @@ -114694,7 +114694,7 @@ version of the pigeonhole principle (for aleph-null pigeons and 2 holes) elssuni simprr adantr domentr syl2anc wb simpllr simprl sorpssi fincssdom sseldd simplr syl12anc mpbid biimtrid ralrimiv unissb eqssd breq2 rabbidv ex unieqd rspceeqv velsn peano1 wi dom0 biimpi 3imtr4g ssrdv uni0b eqcomd - a1i sylancr eqeq1 rexbidv jaod impbid elun bitr4di abbi1dv eqtrid ) AEUAZ + a1i sylancr eqeq1 rexbidv jaod impbid elun bitr4di eqabcdv eqtrid ) AEUAZ AFGZHZBICJZBJZKLZCAMZNZUBZUCDJZYPOZBIUDZDUEAPUFZUGZBDIYPYQYQUHUIYKYTDUUBY KYTYRAQZYRUUAQZTZYRUUBQYKYTUUEYKYSUUEBIYKYMIQZHZUUEYSYPAQZYPUUAQZTZUUGUUJ YOPUUGYOPOZHZUUIUUHUULYPPOZUUIUUKUUMUUGUUKYPPNPYOPUJUOUKULYPPUMUNRUPUUGYO @@ -118262,7 +118262,7 @@ it contains AC as a simple corollary (letting ` m ( i ) = (/) ` , this @( An alternate definition of the class of all cardinal numbers. @) cardnum2 @p |- Card = { x | ( card ` x ) = x } @= ( ccdn com cale crn cun cv ccrd cfv wceq cab df-cardn wcel iscard3 bicomi - abbi2i eqcomi eqtr4i ) BCDEFZAGZHITJZAKZLSUBUAASUATSMTNOPQR @. + eqabi eqcomi eqtr4i ) BCDEFZAGZHITJZAKZLSUBUAASUATSMTNOPQR @. @( [25-Jan-2005] @) @( [23-Sep-2004] @) @( The class of cardinal numbers is a proper class. Exercise 5(G)(b) of @@ -138302,14 +138302,14 @@ distinguish between finite and infinite sets (and therefore if the set size $( Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) $) nnzrab $p |- NN = { x e. ZZ | 1 <_ x } $= - ( cn cv cz wcel c1 cle wbr wa cab crab elnnz1 abbi2i df-rab eqtr4i ) BACZDE - FPGHZIZAJQADKRABPLMQADNO $. + ( cn cv cz wcel c1 cle wbr wa cab crab elnnz1 eqabi df-rab eqtr4i ) BACZDEF + PGHZIZAJQADKRABPLMQADNO $. $( Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) $) nn0zrab $p |- NN0 = { x e. ZZ | 0 <_ x } $= - ( cn0 cv cz wcel cc0 cle wbr wa cab crab elnn0z abbi2i df-rab eqtr4i ) BACZ - DEFPGHZIZAJQADKRABPLMQADNO $. + ( cn0 cv cz wcel cc0 cle wbr wa cab crab elnn0z eqabi df-rab eqtr4i ) BACZD + EFPGHZIZAJQADKRABPLMQADNO $. $( One is an integer. (Contributed by NM, 10-May-2004.) $) 1z $p |- 1 e. ZZ $= @@ -159132,7 +159132,7 @@ are used instead of sets because their representation is shorter (and more ( ( ! ` ( # ` A ) ) x. ( ( # ` B ) _C ( # ` A ) ) ) ) $= ( wcel chash cfv cfa cbc co cmul wceq wi c1 cc0 c0 oveq2d cn0 adantr cdiv syl vx vy vz cfn cv wf1 cab csn cun f1eq2 wfn f1fn fn0 sylib f1eq1 mpbiri - f10 impbii velsn bitr4i bitrdi abbi1dv fveq2d cvv 0ex hashsng ax-mp fveq2 + f10 impbii velsn bitr4i bitrdi eqabcdv fveq2d cvv 0ex hashsng ax-mp fveq2 eqtrdi hash0 fac0 oveq12d eqeq12d imbi2d abbidv 2fveq3 bcn0 1t1e1 eqtr2di hashcl wn caddc clt wbr wne wex abn0 cdom f1domg cle hashunsng elv adantl wa breq1d simprl snfi unfi sylancl simpl hashdom syl2anc ad2antrl nn0p1nn @@ -160301,9 +160301,9 @@ as finite sequences of _symbols_ (or _characters_) being elements of the 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) $) wrdval $p |- ( S e. V -> Word S = U_ l e. NN0 ( S ^m ( 0 ..^ l ) ) ) $= ( vw wcel cword cc0 cv cfzo co wf cn0 wrex cab cmap ciun df-word eliun wb - cvv ovex elmapg mpan2 rexbidv bitrid abbi2dv eqtr4id ) ABEZAFGCHZIJZADHZK - ZCLMZDNCLAUJOJZPZDACQUHUMDUOUKUOEUKUNEZCLMUHUMCUKLUNRUHUPULCLUHUJTEUPULSG - UIIUAAUJUKBTUBUCUDUEUFUG $. + cvv ovex elmapg mpan2 rexbidv bitrid eqabdv eqtr4id ) ABEZAFGCHZIJZADHZKZ + CLMZDNCLAUJOJZPZDACQUHUMDUOUKUOEUKUNEZCLMUHUMCUKLUNRUHUPULCLUHUJTEUPULSGU + IIUAAUJUKBTUBUCUDUEUFUG $. $} ${ @@ -160525,10 +160525,10 @@ of an open range of nonnegative integers (of length equal to the length of -> { w e. Word V | ( # ` w ) = N } = ( V ^m ( 0 ..^ N ) ) ) $= ( wcel cn0 wa cv chash cfv wceq cword crab cab cc0 cfzo co cmap wf cvv wb df-rab ovexd elmapg syldan iswrdi adantl fnfzo0hash adantll ex wrdf oveq2 - jca feq2d syl5ibcom imp impbid1 bitrd abbi2dv eqtr4id ) CDEZBFEZGZAHZIJZB - KZACLZMVDVGEZVFGZANCOBPQZRQZVFAVGUBVCVIAVKVCVDVKEZVJCVDSZVIVAVBVJTEVLVMUA - VCOBPUCCVJVDDTUDUEVCVMVIVCVMVIVCVMGVHVFVMVHVCCBVDUFUGVBVMVFVACVDBUHUIUMUJ - VHVFVMVHOVEPQZCVDSVFVMCVDUKVFVNVJCVDVEBOPULUNUOUPUQURUSUT $. + jca feq2d syl5ibcom imp impbid1 bitrd eqabdv eqtr4id ) CDEZBFEZGZAHZIJZBK + ZACLZMVDVGEZVFGZANCOBPQZRQZVFAVGUBVCVIAVKVCVDVKEZVJCVDSZVIVAVBVJTEVLVMUAV + COBPUCCVJVDDTUDUEVCVMVIVCVMVIVCVMGVHVFVMVHVCCBVDUFUGVBVMVFVACVDBUHUIUMUJV + HVFVMVHOVEPQZCVDSVFVMCVDUKVFVNVJCVDVEBOPULUNUOUPUQURUSUT $. $( Words as a mapping. (Contributed by Thierry Arnoux, 4-Mar-2020.) $) wrdmap $p |- ( ( V e. X /\ N e. NN0 ) @@ -167899,7 +167899,7 @@ a function (ordinarily on a subset of ` CC ` ) and produces a new { x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = ( ZZ>= ` ( B + A ) ) ) $= ( cz wcel wa cv cmin co cuz cfv cc crab cab caddc df-rab w3a simp2 ancoms wi zcn 3ad2ant1 npcand eluzadd 3adant2 eqeltrrd 3expib adantr a1i eluzsub - eluzelcn 3expia jcad impbid abbi1dv eqtrid ) BDEZCDEZFZAGZBHIZCJKEZALMUTL + eluzelcn 3expia jcad impbid eqabcdv eqtrid ) BDEZCDEZFZAGZBHIZCJKEZALMUTL EZVBFZANCBOIZJKZVBALPUSVDAVFUSVDUTVFEZUQVDVGTURUQVCVBVGUQVCVBQZVABOIZUTVF VHUTBUQVCVBRUQVCBLEVBBUAUBUCUQVBVIVFEZVCVBUQVJBCVAUDSUEUFUGUHUSVGVCVBVGVC TUSVEUTUKUIURUQVGVBTURUQVGVBBCUTUJULSUMUNUOUP $. @@ -186484,13 +186484,13 @@ The circle constant (tau = 2 pi) qnnen $p |- QQ ~~ NN $= ( vx vy vz cq cn cdom wbr cen cz cxp wcel cv cdiv com mp2an wceq wrex mp2 znnen cvv ccrd cdm co cmpo wfo omelon nnenom ensymi isnumi wb ennum ax-mp - con0 mpbir xpnum wfn crn eqid fnmpoi cab rnmpo elq abbi2i eqtr4i mpbir2an - ovex df-fo fodomnum nnex enref xpen xpnnen entri domentr wss nnssq ssdomg - qex sbth ) DEFGZEDFGZDEHGDIEJZFGZWBEHGVTWBUAUBZKZWBDABIEALZBLZMUCZUDZUEZW - CIWDKZEWDKZWEWKWLNUMKNEHGWLUFENUGUHNEUIOZIEHGZWKWLUJSIEUKULUNWMIEUOOWJWIW - BUPWIUQZDPABIEWHWIWIURZWFWGMVFUSWOCLZWHPBEQAIQZCUTDABCIEWHWIWPVAWRCDABWQV - BVCVDWBDWIVGVEWBDWIVHRWBEEJZEWNEEHGWBWSHGSEVIVJIEEEVKOVLVMDWBEVNODTKEDVOW - AVRVPEDTVQRDEVSO $. + con0 mpbir xpnum wfn crn eqid ovex fnmpoi cab rnmpo eqabi eqtr4i mpbir2an + elq df-fo fodomnum nnex enref xpen xpnnen entri domentr nnssq ssdomg sbth + wss qex ) DEFGZEDFGZDEHGDIEJZFGZWBEHGVTWBUAUBZKZWBDABIEALZBLZMUCZUDZUEZWC + IWDKZEWDKZWEWKWLNUMKNEHGWLUFENUGUHNEUIOZIEHGZWKWLUJSIEUKULUNWMIEUOOWJWIWB + UPWIUQZDPABIEWHWIWIURZWFWGMUSUTWOCLZWHPBEQAIQZCVADABCIEWHWIWPVBWRCDABWQVF + VCVDWBDWIVGVEWBDWIVHRWBEEJZEWNEEHGWBWSHGSEVIVJIEEEVKOVLVMDWBEVNODTKEDVRWA + VSVOEDTVPRDEVQO $. $} @@ -229975,14 +229975,14 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is O'Rear, 22-Aug-2015.) $) submacs $p |- ( G e. Mnd -> ( SubMnd ` G ) e. ( ACS ` B ) ) $= ( vs vx vy cmnd wcel csubmnd cfv c0g cv cplusg wral crab eqid cvv sylancr - co wa cpw cacs cab wss issubm anbi1i 3anass bitr4i bitr4di abbi2dv df-rab - w3a velpw eqtr4di cin inrab cmre cbs fvexi mreacs mp1i acsfn0 mndcl 3expb - mndidcl ralrimivva acsfn2 mreincl syl3anc eqeltrrid eqeltrd ) BGHZBIJZBKJ - ZDLZHZELZFLZBMJZSZVOHFVONEVONZTZDAUAZOZAUBJZVLVMVOWCHZWBTZDUCWDVLWGDVMVLV - OVMHVOAUDZVPWAULZWGEFAVSVOBVNCVNPZVSPZUEWGWHWBTWIWFWHWBDAUMUFWHVPWAUGUHUI - UJWBDWCUKUNVLWDVPDWCOZWADWCOZUOZWEVPWADWCUPVLWEWCUQJHZWLWEHZWMWEHZWNWEHAQ - HZWOVLABURCUSZQAUTVAVLWRVNAHWPWSABVNCWJVEVNQADVBRVLWRVTAHZFANEANWQWSVLWTE - FAAVLVQAHVRAHWTAVSBVQVRCWKVCVDVFVTQADEFVGRWLWMWEWCVHVIVJVK $. + co wa cpw cab wss issubm velpw anbi1i 3anass bitr4i bitr4di eqabdv df-rab + cacs w3a eqtr4di cin inrab cmre cbs fvexi mreacs mp1i mndidcl mndcl 3expb + acsfn0 ralrimivva acsfn2 mreincl syl3anc eqeltrrid eqeltrd ) BGHZBIJZBKJZ + DLZHZELZFLZBMJZSZVOHFVONEVONZTZDAUAZOZAULJZVLVMVOWCHZWBTZDUBWDVLWGDVMVLVO + VMHVOAUCZVPWAUMZWGEFAVSVOBVNCVNPZVSPZUDWGWHWBTWIWFWHWBDAUEUFWHVPWAUGUHUIU + JWBDWCUKUNVLWDVPDWCOZWADWCOZUOZWEVPWADWCUPVLWEWCUQJHZWLWEHZWMWEHZWNWEHAQH + ZWOVLABURCUSZQAUTVAVLWRVNAHWPWSABVNCWJVBVNQADVERVLWRVTAHZFANEANWQWSVLWTEF + AAVLVQAHVRAHWTAVSBVQVRCWKVCVDVFVTQADEFVGRWLWMWEWCVHVIVJVK $. $} ${ @@ -252846,16 +252846,16 @@ elements and the subgroup containing only the identity ( ~ simpgnsgbid ). ( 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 eqtr4di cycsubggenodd - ex rnmpt c2 ablsimpgfindlem2 ablsimpgfindlem1 pm2.61dane adantrr eqnetrrd - rexlimddv pm2.21ddne efald ) ABUCJZAWGUPZKZUAWGBUBLZMUDZMWIWGWJMAWHUEUFAW - KMNZWHAGOZCUGLZNZWLGBAWMWNUHUPZGBPWOGBPAGBCWNDWNQZFUIWPWOGBWMWNUJUKULAWMB - JZWOKZKZWMCUMLZLZWKMWTWMBBCUNLZHCXADXCQZXAQZACUOJZWSACFUQZRAWRWOURZWTBIOZ - HOZWMXCVGZUHZHSPZIUSHSXKUTZVAWTXMIBWTXIBJZXMWTXOXMWTXOKZWMBXIXCHCWNDWQXDA - CVBJWSXOETACVCJWSXOFTWTWRXOXHRXPWMWNAWRWOXOVDVEWTXOUEVFVPWTXLXOHSWTXJSJZX - LKZKZXIXKBWTXQXLVHXSBXCCXJWMDXDAXFWSXRXGTWTXQXLURWTWRXRXHRVIVJVKVLVMHISXK - XNXNQVQVNVOAWRXBMNZWOAWRKXTVRWMXCVGWNAGBXCCXAWNDWQXDXEEFVSAGBXCCXAWNDWQXD - XEEFVTWAWBWCWDRWEWF $. + co ex simprr mulgcld eqeltrd rexlimdvaa impbid eqabdv rnmpt cycsubggenodd + eqtr4di c2 ablsimpgfindlem2 ablsimpgfindlem1 pm2.61dane adantrr rexlimddv + eqnetrrd pm2.21ddne efald ) ABUCJZAWGUPZKZUAWGBUBLZMUDZMWIWGWJMAWHUEUFAWK + MNZWHAGOZCUGLZNZWLGBAWMWNUHUPZGBPWOGBPAGBCWNDWNQZFUIWPWOGBWMWNUJUKULAWMBJ + ZWOKZKZWMCUMLZLZWKMWTWMBBCUNLZHCXADXCQZXAQZACUOJZWSACFUQZRAWRWOURZWTBIOZH + OZWMXCVGZUHZHSPZIUSHSXKUTZVAWTXMIBWTXIBJZXMWTXOXMWTXOKZWMBXIXCHCWNDWQXDAC + VBJWSXOETACVCJWSXOFTWTWRXOXHRXPWMWNAWRWOXOVDVEWTXOUEVFVHWTXLXOHSWTXJSJZXL + KZKZXIXKBWTXQXLVIXSBXCCXJWMDXDAXFWSXRXGTWTXQXLURWTWRXRXHRVJVKVLVMVNHISXKX + NXNQVOVQVPAWRXBMNZWOAWRKXTVRWMXCVGWNAGBXCCXAWNDWQXDXEEFVSAGBXCCXAWNDWQXDX + EEFVTWAWBWDWCRWEWF $. $} ${ @@ -256706,22 +256706,22 @@ irreducible elements (see ~ df-irred ). (Contributed by Mario Carneiro, ( vv vw vf vx vy crg cv cbs cfv wceq co wa wral cmap crab wcel cab eqid crh cur cplusg cmulr csb cmpo cghm cmgp cmhm cin df-rnghom ancom r19.26-2 anbi1i anass 3bitri rabbii fvex oveq12 ancoms wb raleqbi1dv adantr anbi2d - raleq rabeqbidv csbie2 inrab 3eqtr4i wf cgrp ringgrp isghm3 syl2an df-rab - abbi2dv elmap abbii eqtri eqtr4di w3a cmnd ringmgp mgpbas ringidval ismhm - mgpplusg baib 3anass bitr4i ineq12d eqtr4id mpoeq3ia ) UABAHHCBIZJKZDAIZJ - KZWNUBKZEIZKWPUBKZLZFIZGIZWNUCKZMWSKXBWSKZXCWSKZWPUCKZMLZXBXCWNUDKZMWSKXE - XFWPUDKZMLZNZGCIZOZFXMOZNZEDIZXMPMZQZUEUEZUFBAHHWNWPUGMZWNUHKZWPUHKZUIMZU - JZUFFGDCEABUKBAHHXTYEWNHRZWPHRZNZXTXHGWOOFWOOZEWQWOPMZQZXKGWOOFWOOZXANZEY - JQZUJZYEXAXLGWOOZFWOOZNZEYJQZYIYMNZEYJQXTYOYRYTEYJYRYQXANYIYLNZXANYTXAYQU - LYQUUAXAXHXKFGWOWOUMUNYIYLXAUOUPUQCDWOWQXSYSWNJURZWPJURZXMWOLZXQWQLZNZXPY - REXRYJUUEUUDXRYJLXQWQXMWOPUSUTUUFXOYQXAUUDXOYQVAUUEXNYPFXMWOXLGXMWOVEVBVC - VDVFVGYIYMEYJVHVIYHYAYKYDYNYHYAWOWQWSVJZYINZESZYKYHUUHEYAYFWNVKRWPVKRWSYA - RUUHVAYGWNVLWPVLGFXDXGWNWPWSWOWQWOTZWQTZXDTXGTVMVNVPYKWSYJRZYINZESUUIYIEY - JVOUUMUUHEUULUUGYIWQWOWSUUCUUBVQZUNVRVSVTYHYDUUGYLXAWAZESZYNYHUUOEYDYFYBW - BRZYCWBRZWSYDRZUUOVAYGWNYBYBTZWCWPYCYCTZWCUUSUUQUURNUUOFGWOWQXIXJYBYCWSWT - WRWOWNYBUUTUUJWDWQWPYCUVAUUKWDWNXIYBUUTXITWGWPXJYCUVAXJTWGWNWRYBUUTWRTWEW - PWTYCUVAWTTWEWFWHVNVPYNUULYMNZESUUPYMEYJVOUVBUUOEUVBUUGYMNUUOUULUUGYMUUNU - NUUGYLXAWIWJVRVSVTWKWLWMVS $. + raleq rabeqbidv csbie2 inrab 3eqtr4i wf cgrp ringgrp isghm3 syl2an eqabdv + df-rab elmap abbii eqtri eqtr4di w3a cmnd mgpbas mgpplusg ringidval ismhm + ringmgp baib 3anass bitr4i ineq12d eqtr4id mpoeq3ia ) UABAHHCBIZJKZDAIZJK + ZWNUBKZEIZKWPUBKZLZFIZGIZWNUCKZMWSKXBWSKZXCWSKZWPUCKZMLZXBXCWNUDKZMWSKXEX + FWPUDKZMLZNZGCIZOZFXMOZNZEDIZXMPMZQZUEUEZUFBAHHWNWPUGMZWNUHKZWPUHKZUIMZUJ + ZUFFGDCEABUKBAHHXTYEWNHRZWPHRZNZXTXHGWOOFWOOZEWQWOPMZQZXKGWOOFWOOZXANZEYJ + QZUJZYEXAXLGWOOZFWOOZNZEYJQZYIYMNZEYJQXTYOYRYTEYJYRYQXANYIYLNZXANYTXAYQUL + YQUUAXAXHXKFGWOWOUMUNYIYLXAUOUPUQCDWOWQXSYSWNJURZWPJURZXMWOLZXQWQLZNZXPYR + EXRYJUUEUUDXRYJLXQWQXMWOPUSUTUUFXOYQXAUUDXOYQVAUUEXNYPFXMWOXLGXMWOVEVBVCV + DVFVGYIYMEYJVHVIYHYAYKYDYNYHYAWOWQWSVJZYINZESZYKYHUUHEYAYFWNVKRWPVKRWSYAR + UUHVAYGWNVLWPVLGFXDXGWNWPWSWOWQWOTZWQTZXDTXGTVMVNVOYKWSYJRZYINZESUUIYIEYJ + VPUUMUUHEUULUUGYIWQWOWSUUCUUBVQZUNVRVSVTYHYDUUGYLXAWAZESZYNYHUUOEYDYFYBWB + RZYCWBRZWSYDRZUUOVAYGWNYBYBTZWGWPYCYCTZWGUUSUUQUURNUUOFGWOWQXIXJYBYCWSWTW + RWOWNYBUUTUUJWCWQWPYCUVAUUKWCWNXIYBUUTXITWDWPXJYCUVAXJTWDWNWRYBUUTWRTWEWP + WTYCUVAWTTWEWFWHVNVOYNUULYMNZESUUPYMEYJVPUVBUUOEUVBUUGYMNUUOUULUUGYMUUNUN + UUGYLXAWIWJVRVSVTWKWLWMVS $. $} $( Define the ring isomorphism relation, analogous to ~ df-gic : Two @@ -263685,12 +263685,12 @@ Absolute value (abstract algebra) lsppr $p |- ( ph -> ( N ` { X , Y } ) = { v | E. k e. K E. l e. K v = ( ( k .x. X ) .+ ( l .x. Y ) ) } ) $= ( cpr cfv csn cun cv wceq wrex cab df-pr fveq2i clsm clmod wcel wss snssd - co lspun syl3anc clss eqid lspsncl syl2anc lsmspsn abbi2dv 3eqtr2d eqtrid - lsmsp ) AKLUCZHUDKUEZLUEZUFZHUDZBUGZEUGKDURMUGLDURCURUHMGUIEGUIZBUJZVJVMH - KLUKULAVNVKHUDZVLHUDZUFHUDZVRVSJUMUDZURZVQAJUNUOZVKIUPVLIUPVNVTUHTAKIUAUQ - ALIUBUQVKVLHIJNSUSUTAWCVRJVAUDZUOZVSWDUOZWBVTUHTAWCKIUOWETUAWDHIJKNWDVBZS - VCVDAWCLIUOWFTUBWDHIJLNWGSVCVDWAWDVRVSHJWGSWAVBZVIUTAVPBWBACWADVOEMFGHIJK - LNOPQRWHSTUAUBVEVFVGVH $. + co lspun syl3anc clss lspsncl syl2anc lsmsp lsmspsn eqabdv 3eqtr2d eqtrid + eqid ) AKLUCZHUDKUEZLUEZUFZHUDZBUGZEUGKDURMUGLDURCURUHMGUIEGUIZBUJZVJVMHK + LUKULAVNVKHUDZVLHUDZUFHUDZVRVSJUMUDZURZVQAJUNUOZVKIUPVLIUPVNVTUHTAKIUAUQA + LIUBUQVKVLHIJNSUSUTAWCVRJVAUDZUOZVSWDUOZWBVTUHTAWCKIUOWETUAWDHIJKNWDVIZSV + BVCAWCLIUOWFTUBWDHIJLNWGSVBVCWAWDVRVSHJWGSWAVIZVDUTAVPBWBACWADVOEMFGHIJKL + NOPQRWHSTUAUBVEVFVGVH $. $d k l v Z $. $d v .+ $. $d v .x. $. $d v K $. $( Member of the span of a pair of vectors. (Contributed by NM, @@ -266537,10 +266537,10 @@ left ideal which is also a right ideal (or a left ideal over the opposite ( K ` { G } ) = { x | G .|| x } ) $= ( va crg wcel wa cv cfv wceq wrex wb cbs eqtri wbr csn cmulr co eqcom a1i rexbidv crglmod rlmlmod cid rlmsca2 cnx baseid strfvi rlmbas rlmvsca crsp - clmod clspn rspval lspsnel sylan eqid dvdsr2 adantl 3bitr4d abbi2dv ) DKL - ZEBLZMZEANZCUAZAEUBFOZVJVKJNEDUCOZUDZPZJBQZVOVKPZJBQZVKVMLZVLVJVPVRJBVPVR - RVJVKVOUEUFUGVHDUHOZURLVIVTVQRDUIVNVKJDUJOBFBWAEDUKDSULSOBUMGUNBDSOWASOGD - UOTDUPFDUQOWAUSOHDUTTVAVBVIVLVSRVHJBCDVNEVKGIVNVCVDVEVFVG $. + clmod clspn rspval lspsnel sylan eqid dvdsr2 adantl 3bitr4d eqabdv ) DKLZ + EBLZMZEANZCUAZAEUBFOZVJVKJNEDUCOZUDZPZJBQZVOVKPZJBQZVKVMLZVLVJVPVRJBVPVRR + VJVKVOUEUFUGVHDUHOZURLVIVTVQRDUIVNVKJDUJOBFBWAEDUKDSULSOBUMGUNBDSOWASOGDU + OTDUPFDUQOWAUSOHDUTTVAVBVIVLVSRVHJBCDVNEVKGIVNVCVDVEVFVG $. $} ${ @@ -269390,16 +269390,16 @@ According to Wikipedia ("Integer", 25-May-2019, ( vx wcel cfv co wceq cc0 chash wa wf1o wb cz eqid bitrd 3syl cvv cn cphi cv cgcd c1 cfzo crab czrh cres ccnv cima dfphi2 cab cbs wfn cif cn0 nnnn0 znf1o syl wne nnne0 ifnefalse reseq2 f1oeq1d f1oeq2 mpbid elpreima adantl - f1ofn fvres eleq1d elfzoelz znunit syl2an pm5.32da abbi2dv df-rab eqtr4di - fveq2d cen wbr wf1 wss f1ocnv f1of1 ovexd unitss fvexi f1imaen2g syl22anc - a1i cui hasheni 3eqtr2rd ) BUAGZBUBHFUCZBUDIUEJZFKBUFIZUGZLHCUHHZWSUIZUJZ - AUKZLHZALHZFBULWPXDWTLWPXDWQWSGZWRMZFUMWTWPXHFXDWPWQXDGZXGWQXBHZAGZMZXHWP - WSCUNHZXBNZXBWSUOXIXLOWPBKJPWSUPZXMXAXOUIZNZXNWPBUQGZXQBURZXMXPBXOCDXMQZX - PQXOQUSUTWPBKVAXOWSJZXQXNOBVBBKPWSVCYAXQXOXMXBNXNYAXOXMXPXBXOWSXAVDVEXOWS - XMXBVFRSVGZWSXMXBVJWSWQAXBVHSWPXGXKWRWPXGMZXKWQXAHZAGZWRYCXJYDAXGXJYDJWPW - QWSXAVKVIVLWPXRWQPGYEWROXGXSWQKBVMWQAXABCDEXAQVNVORVPRVQWRFWSVRVSVTWPXDAW - AWBZXEXFJWPXMWSXCWCZWSTGAXMWDZATGZYFWPXNXMWSXCNYGYBWSXMXBWEXMWSXCWFSWPKBU - FWGYHWPXMCAXTEWHWLYIWPACWMEWIWLXMWSAXCTWJWKXDAWNUTWO $. + f1ofn eleq1d elfzoelz znunit syl2an pm5.32da eqabdv df-rab eqtr4di fveq2d + fvres cen wbr wf1 wss f1ocnv f1of1 ovexd a1i cui fvexi f1imaen2g syl22anc + unitss hasheni 3eqtr2rd ) BUAGZBUBHFUCZBUDIUEJZFKBUFIZUGZLHCUHHZWSUIZUJZA + UKZLHZALHZFBULWPXDWTLWPXDWQWSGZWRMZFUMWTWPXHFXDWPWQXDGZXGWQXBHZAGZMZXHWPW + SCUNHZXBNZXBWSUOXIXLOWPBKJPWSUPZXMXAXOUIZNZXNWPBUQGZXQBURZXMXPBXOCDXMQZXP + QXOQUSUTWPBKVAXOWSJZXQXNOBVBBKPWSVCYAXQXOXMXBNXNYAXOXMXPXBXOWSXAVDVEXOWSX + MXBVFRSVGZWSXMXBVJWSWQAXBVHSWPXGXKWRWPXGMZXKWQXAHZAGZWRYCXJYDAXGXJYDJWPWQ + WSXAVTVIVKWPXRWQPGYEWROXGXSWQKBVLWQAXABCDEXAQVMVNRVORVPWRFWSVQVRVSWPXDAWA + WBZXEXFJWPXMWSXCWCZWSTGAXMWDZATGZYFWPXNXMWSXCNYGYBWSXMXBWEXMWSXCWFSWPKBUF + WGYHWPXMCAXTEWMWHYIWPACWIEWJWHXMWSAXCTWKWLXDAWNUTWO $. znrrg.e $e |- E = ( RLReg ` Y ) $. $( The regular elements of ` Z/nZ ` are exactly the units. (This theorem @@ -275599,7 +275599,7 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, wi vex wb wral wfn ffn adantl elixp baib syl cz ffvelcdm adantll eleqtrdi cuz nn0uz adantr ffvelcdmda nn0zd elfz5 syl2anc ralbidva ffnd simpl inidm eqidd ofrfvalg bitr4d psrbaglecl 3expia pm4.71rd 3bitrrd pm5.21ndd eqtrid - ex abbi1dv ccnv cn cima cmap wceq cnveq imaeq1d eleq1d simprbi fzfid cdif + ex eqabcdv ccnv cn cima cmap wceq cnveq imaeq1d eleq1d simprbi fzfid cdif elrab2 csn csupp fcdmnn0suppg mpdan eqimss id c0ex suppssrg oveq2d eqtrdi fz0sn ixpfi2 eqeltrd ) DBHZAIZDJUAUBZABUCZGEKGIZDLZMNZUDZOXQXTXRBHZXSPZAU EYDXSABUFXQYFAYDXQEQXRUGZYFXRYDHZXQYEYGXSYEYGUPXQBCXREFUHRUIYHYGUPXQYHXRG @@ -275623,7 +275623,7 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, wf mprg sseli vex elixpconst sylib wral wfn ffn adantl elixp baib syl cuz wi cz ffvelcdm adantll nn0uz eleqtrdi psrbagfOLD ffvelcdmda nn0zd syl2anc elfz5 ralbidva ffnd simpll inidm eqidd ofrfval bitr4d psrbagleclOLD 3exp2 - imp31 pm4.71rd 3bitrrd ex pm5.21ndd abbi1dv cmap simpr wceq cnveq imaeq1d + imp31 pm4.71rd 3bitrrd ex pm5.21ndd eqabcdv cmap simpr wceq cnveq imaeq1d eqtrid eleq1d elrab2 simprd fzfid cdif csn cvv csupp jca fcdmnn0supp 3syl eqimss c0ex suppssr oveq2d fz0sn eqtrdi ixpfi2 eqeltrd ) EFIZDBIZJZAKZDLU AUBZABUCZHEMHKZDNZUDOZUFZPYEYHYFBIZYGJZAUGYLYGABUEYEYNAYLYEEQYFURZYNYFYLI @@ -297883,8 +297883,8 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.) $) tgval3 $p |- ( B e. V -> ( topGen ` B ) = { x | E. y ( y C_ B /\ x = U. y ) } ) $= - ( wcel cv wss cuni wceq wa wex ctg cfv eltg3 abbi2dv ) CDEBFZCGAFZPHIJBKA - CLMBQCDNO $. + ( wcel cv wss cuni wceq wa wex ctg cfv eltg3 eqabdv ) CDEBFZCGAFZPHIJBKAC + LMBQCDNO $. $( Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.) $) @@ -308436,18 +308436,18 @@ arbitrary neighborhoods (such as "locally compact", which is actually cmpo eqid txval adantr oveq1d txbasex tgrest syl2an cab cin wb elrest vex xpexg inex1 ad2ant2r xpeq1 eqeq2d rexbidv ad2ant2l xpeq2 adantl sylan9bbr rexxfr2d wral xpex rgen2w ineq1 inxp eqtrdi rexrnmpo ax-mp bitr4di bitr4d - a1i abbi2dv rnmpo eqtr4di fveq2d 3eqtr2d ovex mp2an ) CEMZDFMZNZAGMZBHMZN - ZNZCDUCOZABUDZPOZIJCAPOZDBPOZIQZJQZUDZUHZUEZUFUGZWTXAUCOZWPWSKLCDKQZLQZUD - ZUHZUEZUFUGZWRPOZXMWRPOZUFUGZXGWPWQXNWRPWLWQXNRWOKLXMCDEFXMUIZUJUKULWLXMS - MZWRSMZXQXORWOKLXMCDEFXRUMZABGHVAZWRXMSSUNUOWPXPXFUFWPXPUAQZXDRZJXATZIWTT - ZUAUPXFWPYFUAXPWPYCXPMZYCUBQZWRUQZRZUBXMTZYFWLXSXTYGYKURWOYAYBUBYCWRXMSSU - SUOWPYFYCXIAUQZXJBUQZUDZRZLDTZKCTZYKWPYEYPIKYLWTCSYLSMWPXICMNXIAKUTZVBWBW - JWMXBWTMXBYLRZKCTURWKWNKXBACEGUSVCYSYEYCYLXCUDZRZJXATWPYPYSYDUUAJXAYSXDYT - YCXBYLXCVDVEVFWPUUAYOJLYMXADSYMSMWPXJDMNXJBLUTZVBWBWKWNXCXAMXCYMRZLDTURWJ - WMLXCBDFHUSVGUUCUUAYOURWPUUCYTYNYCXCYMYLVHVEVIVKVJVKXKSMZLDVLKCVLYKYQURUU - 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ciun syl12anc simprbda mpbid impbida abbi2dv eqtr4id df-fn sylanbrc rneqd - wb adantr rniun wral vex snnz rnxp ax-mp biimpa ctopon toptopon simpr w3a - trnei syl31anc flfelbas ex ssrdv syl3anc syldan eqsstrid ralrimiva sylibr - cfil iunss eqsstrd df-f ) AFGHUAUKRZEUBZXPUCZDSEDXPULAXPUDZXPUEZEUFXQAGUG - TZHUHTZCDFULZCESZXSKLMNCDEFGHIJUIUJAXTBUMZQUMZUNZXPTZQUOZBUPEBQXPUQAYIBEA - YEETZYIAYJURZAYECGUSRRZTZYFFHYEVGZGUTRRCVAUKZVBUKRZTZQUOZYIAYJVCAYMYJAYLE - YEOVDZVEZYKYPVFVHYRPQYPVIVJAYIYMYRURZAYIYMYQURZQUOUUAAYHUUBQAYHYGBYLYNYPV - KZVSZTUUBAXPUUDYGAYAHUGTZYCYDXPUUDUFKAYBUUELHVLVMZMNBCDFGHEIJVNUJZVDBYLYP - YFVOVPVQYMYQQVRVPZVEVTAYIURYMYJAYIYMYRUUHWAAYMYJWIYIYSWJWBWCWDWEXPEWFWGAX - RUUDUCZDAXPUUDUUGWHAUUIBYLUUCUCZVSZDBYLUUCWKAUUJDSZBYLWLUUKDSAUULBYLAYMUR - UUJYPDYNVFVHUUJYPUFYEBWMWNYNYPWOWPAYMYJYPDSZAYMYJYSWQYKHDWRRTZYOCXLRTZYCU - UMAUUNYJAUUEUUNUUFHDJWSVJWJYKGEWRRTZYDYJYMUUOAUUPYJAYAUUPKGEIWSVJWJAYDYJN - WJAYJWTYTUUPYDYJXAYMUUOCYEGEXBWQXCAYCYJMWJUUNUUOYCXAZQYPDUUQYQYFDTYFFHYOD - CXDXEXFXGXHXIXJBYLUUJDXMXKXIXNEDXPXOWG $. + ciun syl12anc simprbda adantr mpbid impbida eqabdv eqtr4id df-fn sylanbrc + wb rneqd rniun wral vex snnz rnxp ax-mp biimpa ctopon cfil toptopon simpr + w3a trnei syl31anc flfelbas ssrdv syl3anc syldan eqsstrid ralrimiva iunss + ex sylibr eqsstrd df-f ) AFGHUAUKRZEUBZXPUCZDSEDXPULAXPUDZXPUEZEUFXQAGUGT + ZHUHTZCDFULZCESZXSKLMNCDEFGHIJUIUJAXTBUMZQUMZUNZXPTZQUOZBUPEBQXPUQAYIBEAY + EETZYIAYJURZAYECGUSRRZTZYFFHYEVGZGUTRRCVAUKZVBUKRZTZQUOZYIAYJVCAYMYJAYLEY + EOVDZVEZYKYPVFVHYRPQYPVIVJAYIYMYRURZAYIYMYQURZQUOUUAAYHUUBQAYHYGBYLYNYPVK + ZVSZTUUBAXPUUDYGAYAHUGTZYCYDXPUUDUFKAYBUUELHVLVMZMNBCDFGHEIJVNUJZVDBYLYPY + FVOVPVQYMYQQVRVPZVEVTAYIURYMYJAYIYMYRUUHWAAYMYJWIYIYSWBWCWDWEWFXPEWGWHAXR + UUDUCZDAXPUUDUUGWJAUUIBYLUUCUCZVSZDBYLUUCWKAUUJDSZBYLWLUUKDSAUULBYLAYMURU + UJYPDYNVFVHUUJYPUFYEBWMWNYNYPWOWPAYMYJYPDSZAYMYJYSWQYKHDWRRTZYOCWSRTZYCUU + MAUUNYJAUUEUUNUUFHDJWTVJWBYKGEWRRTZYDYJYMUUOAUUPYJAYAUUPKGEIWTVJWBAYDYJNW + BAYJXAYTUUPYDYJXBYMUUOCYEGEXCWQXDAYCYJMWBUUNUUOYCXBZQYPDUUQYQYFDTYFFHYODC + XEXLXFXGXHXIXJBYLUUJDXKXMXIXNEDXPXOWH $. $d b d u v x y z A $. $d b u v w x y B $. $d b c d u v w x y C $. $d b c d u v w x y z F $. $d b c d u v w x y z J $. @@ -329070,7 +329070,7 @@ Normed space homomorphisms (bounded linear operators) ( cxr wcel cabs cc0 co cc cle wbr wa cfv cr wb adantl wceq cmin eqtrd cab ccnv cicc cima cv crab wfn absf ffn elpreima mp2b w3a df-3an abscl absge0 rexrd jca biantrurd bitr4id 0xr simpl elicc1 sylancr 0cn cnmetdval abssub - mpan subid1 fveq2d breq1d 3bitr4d pm5.32da bitrid abbi2dv df-rab eqtr4di + mpan subid1 fveq2d breq1d 3bitr4d pm5.32da bitrid eqabdv df-rab eqtr4di wf ) CEFZGUBHCUCIZUDZAUEZJFZHWABIZCKLZMZAUAWDAJUFVRWEAVTWAVTFZWBWAGNZVSFZ MZVRWEJOGVQGJUGWFWIPUHJOGUIJWAVSGUJUKVRWBWHWDVRWBMZWGEFZHWGKLZWGCKLZULZWM WHWDWJWNWKWLMZWMMWMWKWLWMUMWJWOWMWBWOVRWBWKWLWBWGWAUNUPWAUOUQQURUSWJHEFVR @@ -343531,11 +343531,11 @@ Hilbert space (in the algebraic sense, meaning that all algebraically shft2rab $p |- ( ph -> A = { y e. RR | ( y - -u C ) e. B } ) $= ( cv cr wcel cmin co wa crab cc wceq syl2anr eleq1d cneg sseld caddc recn cab pm4.71rd recnd subneg adantr eleq12d wb id readdcl oveq1 elrab3 pncan - syl 3bitrd pm5.32da bitr4d abbi2dv df-rab eqtr4di ) ADCJZKLZVDFUAMNZELZOZ - CUEVGCKPAVHCDAVDDLZVEVIOVHAVIVEADKVDGUBUFAVEVGVIAVEOZVGVDFUCNZBJZFMNZDLZB - KPZLZVKFMNZDLZVIVJVFVKEVOVEVDQLZFQLZVFVKRAVDUDZAFHUGZVDFUHSAEVORVEIUIUJVJ - VKKLZVPVRUKVEVEFKLWCAVEULHVDFUMSVNVRBVKKVLVKRVMVQDVLVKFMUNTUOUQVJVQVDDVEV - SVTVQVDRAWAWBVDFUPSTURUSUTVAVGCKVBVC $. + syl 3bitrd pm5.32da bitr4d eqabdv df-rab eqtr4di ) ADCJZKLZVDFUAMNZELZOZC + UEVGCKPAVHCDAVDDLZVEVIOVHAVIVEADKVDGUBUFAVEVGVIAVEOZVGVDFUCNZBJZFMNZDLZBK + PZLZVKFMNZDLZVIVJVFVKEVOVEVDQLZFQLZVFVKRAVDUDZAFHUGZVDFUHSAEVORVEIUIUJVJV + KKLZVPVRUKVEVEFKLWCAVEULHVDFUMSVNVRBVKKVLVKRVMVQDVLVKFMUNTUOUQVJVQVDDVEVS + VTVQVDRAWAWBVDFUPSTURUSUTVAVGCKVBVC $. ${ ovolshft.4 $e |- M = { y e. RR* | @@ -343625,12 +343625,12 @@ Hilbert space (in the algebraic sense, meaning that all algebraically sca2rab $p |- ( ph -> A = { y e. RR | ( ( 1 / C ) x. y ) e. B } ) $= ( cv cr wcel cdiv co cmul wa crab wceq adantr eleq1d c1 sseld pm4.71rd wb cab eleq2d crp rprecred remulcl oveq2 elrab3 syl simpr recnd rpcnd rpne0d - sylancom divrec2d oveq2d divcan2d eqtr3d 3bitrd pm5.32da abbi2dv eqtr4di - bitr4d df-rab ) ADCJZKLZUAFMNZVHONZELZPZCUEVLCKQAVMCDAVHDLZVIVNPVMAVNVIAD + sylancom divrec2d oveq2d divcan2d eqtr3d 3bitrd pm5.32da bitr4d eqtr4di + eqabdv df-rab ) ADCJZKLZUAFMNZVHONZELZPZCUEVLCKQAVMCDAVHDLZVIVNPVMAVNVIAD KVHGUBUCAVIVLVNAVIPZVLVKFBJZONZDLZBKQZLZFVKONZDLZVNVOEVSVKAEVSRVIISUFVOVK KLZVTWBUDAVIVJKLWCVOFAFUGLVIHSZUHVJVHUIUQVRWBBVKKVPVKRVQWADVPVKFOUJTUKULV OWAVHDVOFVHFMNZONWAVHVOWEVKFOVOVHFVOVHAVIUMUNZVOFWDUOZVOFWDUPZURUSVOVHFWF - WGWHUTVATVBVCVFVDVLCKVGVE $. + WGWHUTVATVBVCVDVFVLCKVGVE $. ovolsca.4 $e |- ( ph -> ( vol* ` A ) e. RR ) $. ${ @@ -383789,7 +383789,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 c1 adantr eqtrd co cvma cv crab cif cn0 prmnn nnnn0 nnexpcl syl2an vmaval syl csn cab df-rab wi w3a prmdvdsexpb biimpd 3coml 3expa expimpd simpl cz eqid prmz iddvdsexp sylan jca eleq1 breq1 anbi12d syl5ibrcom impbid velsn - bitr4di abbi1dv eqtrid hashsng iftrued unieqd unisng 3eqtrd ) ADEZBFEZGZA + bitr4di eqabcdv eqtrid hashsng iftrued unieqd unisng 3eqtrd ) ADEZBFEZGZA BHUAZUBIZCUCZWGJKZCDUDZLIZRMZWKNZOIZPUEZWOAOIWFWGFEZWHWPMWDAFEBUFEWQWEAUG BUHABUIUJWGWKCWKVEUKULWFWMWOPWFWLAUMZLIZRWFWKWRLWFWKWIDEZWJGZCUNWRWJCDUOW FXACWRWFXAWIAMZWIWREWFXAXBWFWTWJXBWDWEWTWJXBUPZWTWDWEXCWTWDWEUQWJXBWIABUR @@ -402104,11 +402104,11 @@ a multiplicative function (but not completely multiplicative). bdayfo $p |- bday : No -onto-> On $= ( vx vy csur con0 cbday wfo wfn crn wceq cdm wcel wral dmexg rgen df-bday cv cvv mptfng mpbi c1o wrex cab rnmpt csn cxp noxp1o 1oex snnz dmxp ax-mp - wne eqcomi dmeq rspceeqv sylancl nodmon eleq1a syl rexlimiv impbii abbi2i - c0 wi eqtr4i df-fo mpbir2an ) CDEFECGZEHZDIAPZJZQKZACLVGVKACVICMNACVJEAOZ - RSVHBPZVJIZACUAZBUBDABCVJEVLUCVOBDVMDKZVOVPVMTUDZUEZCKVMVRJZIVOVMUFVSVMVQ - VBUKVSVMITUGUHVMVQUIUJULAVRCVJVSVMVIVRUMUNUOVNVPACVICKVJDKVNVPVCVIUPVJDVM - UQURUSUTVAVDCDEVEVF $. + c0 wne eqcomi rspceeqv sylancl wi nodmon eleq1a syl rexlimiv impbii eqabi + dmeq eqtr4i df-fo mpbir2an ) CDEFECGZEHZDIAPZJZQKZACLVGVKACVICMNACVJEAOZR + SVHBPZVJIZACUAZBUBDABCVJEVLUCVOBDVMDKZVOVPVMTUDZUEZCKVMVRJZIVOVMUFVSVMVQU + KULVSVMITUGUHVMVQUIUJUMAVRCVJVSVMVIVRVCUNUOVNVPACVICKVJDKVNVPUPVIUQVJDVMU + RUSUTVAVBVDCDEVEVF $. $} @@ -405548,15 +405548,15 @@ property of surreals and will be used (via surreal cuts) to prove many wbr cab con0 cuni cpw cxp crab madeval scutcl biimpd mpan9 pm4.71ri abbii co cop breq1 anbi12d rexxp cin imaindm dmscut ineq2i imaeq2i eqtri eleq2i cdm vex elima anbi1i anass bitri rexbii2 3bitri df-br df-ov eqeq1i wfn wb - elin wf scutf ffn ax-mp fnbrfvb mpan bitrid pm5.32i 2rexbii abbi2i df-rab - 3bitr4i 3eqtr4i eqtrdi ) BUAFBGHIGBJUBUCZWLUDZJZCKZDKZLSZWOWPIULZAKZMZNZD - WLOZCWLOZAPUEZBUFXCATWSPFZXCNZATWNXDXCXFAXCXEXBXECWLXAXEDWLWQWRPFZWTXEWOW - PUGWTXGXEWRWSPQUHUIRRUJUKXCAWNEKZLFZXHWSISZNZEWMOZWOWPUMZLFZXMWSISZNZDWLO - CWLOWSWNFZXCXKXPECDWLWLXHXMMXIXNXJXOXHXMLQXHXMWSIUNUOUPXQWSIWMLUQZJZFXJEX - ROXLWNXSWSWNIWMIVDZUQZJXSWMIURYAXRIXTLWMUSUTVAVBVCEWSIXRAVEVFXJXKEXRWMXHX - RFZXJNXHWMFZXINZXJNYCXKNYBYDXJXHWMLVQVGYCXIXJVHVIVJVKXAXPCDWLWLXAXNWTNXPW - QXNWTWOWPLVLVGXNWTXOWTXMIHZWSMZXNXOWRYEWSWOWPIVMVNILVOZXNYFXOVPLPIVRYGVSL - PIVTWALXMWSIWBWCWDWEVIWFWIWGXCAPWHWJWK $. + elin wf scutf ffn ax-mp fnbrfvb mpan bitrid pm5.32i 2rexbii 3bitr4i eqabi + df-rab 3eqtr4i eqtrdi ) BUAFBGHIGBJUBUCZWLUDZJZCKZDKZLSZWOWPIULZAKZMZNZDW + LOZCWLOZAPUEZBUFXCATWSPFZXCNZATWNXDXCXFAXCXEXBXECWLXAXEDWLWQWRPFZWTXEWOWP + UGWTXGXEWRWSPQUHUIRRUJUKXCAWNEKZLFZXHWSISZNZEWMOZWOWPUMZLFZXMWSISZNZDWLOC + WLOWSWNFZXCXKXPECDWLWLXHXMMXIXNXJXOXHXMLQXHXMWSIUNUOUPXQWSIWMLUQZJZFXJEXR + OXLWNXSWSWNIWMIVDZUQZJXSWMIURYAXRIXTLWMUSUTVAVBVCEWSIXRAVEVFXJXKEXRWMXHXR + FZXJNXHWMFZXINZXJNYCXKNYBYDXJXHWMLVQVGYCXIXJVHVIVJVKXAXPCDWLWLXAXNWTNXPWQ + XNWTWOWPLVLVGXNWTXOWTXMIHZWSMZXNXOWRYEWSWOWPIVMVNILVOZXNYFXOVPLPIVRYGVSLP + IVTWALXMWSIWBWCWDWEVIWFWGWHXCAPWIWJWK $. $} ${ @@ -406934,7 +406934,7 @@ property of surreals and will be used (via surreal cuts) to prove many addsrid $p |- ( A e. No -> ( A +s 0s ) = A ) $= ( vb vx vy vz vw cv c0s cadds co wceq weq oveq1 eqeq12d wcel cfv cun wrex id c0 va csur cleft cright wral cab cscut 0sno addsval mpan2 adantr elun1 - simpr rspcva syl2anr eqeq2d equcom bitrdi rexbidva risset bitr4di abbi1dv + simpr rspcva syl2anr eqeq2d equcom bitrdi rexbidva risset bitr4di eqabcdv rex0 left0s rexeqi mtbir abf a1i uneq12d un0 eqtrdi elun2 right0s oveq12d wa lrcut 3eqtrd ex noinds ) UAGZHIJZVTKZBGZHIJZWCKZAHIJZAKUABAUABLZWAWDVT WCVTWCHIMWGSNVTAKZWAWFVTAVTAHIMWHSNVTUBOZWEBVTUCPZVTUDPZQZUEZWBWIWMVOZWAC @@ -408784,7 +408784,7 @@ property of surreals and will be used (via surreal cuts) to prove many cleft wral cab cscut 1sno mulsval mpan2 adantr elun1 rspcva sylan adantll wa ancoms muls01 leftssno sseli adantl addsridd syl subsid1 eqeq2d equcom eqtrd bitrdi rexbidva left1s rexeqi elexi oveq2 oveq2d rexsn bitri rexbii - csn 0sno risset 3bitr4g abbi1dv wn rex0 right1s mtbir a1i abf uneq12d un0 + csn 0sno risset 3bitr4g eqabcdv wn rex0 right1s mtbir a1i abf uneq12d un0 nrex eqtrdi elun2 rightssno bitr4di 0un lrcut 3eqtrd ex noinds ) UAGZHIJZ XMKZBGZHIJZXPKZAHIJZAKUABAUABLZXNXQXMXPXMXPHIUJXTUKULXMAKZXNXSXMAXMAHIUJY AUKULXMMNZXRBXMUPUMZXMUNUMZUOZUQZXOYBYFVHZXNCGZDGZHIJZXMUBGZIJZOJZYIYKIJZ @@ -408817,7 +408817,7 @@ property of surreals and will be used (via surreal cuts) to prove many cleft wral cab cscut 1sno mulsval adantr muls02 elun1 rspcva sylan ancoms wa mpan adantll leftssno sseli adantl addslid eqtrd subsid1 eqeq2d equcom syl bitrdi rexbidva csn left1s rexeqi 0sno elexi oveq1 oveq1d rexsn bitri - rexbidv risset 3bitr4g abbi1dv right1s mtbir abf a1i uneq12d eqtrdi elun2 + rexbidv risset 3bitr4g eqabcdv right1s mtbir abf a1i uneq12d eqtrdi elun2 rex0 un0 rightssno lrcut 3eqtrd ex noinds ) GUAHZIJZXIKZGBHZIJZXLKZGAIJZA KUABAUABLZXJXMXIXLXIXLGIUJXPUKULXIAKZXJXOXIAXIAGIUJXQUKULXIMNZXNBXIUPUMZX IUNUMZUOZUQZXKXRYBVHZXJCHZUBHZXIIJZGDHZIJZOJZYEYGIJZPJZKZDXSQZUBGUPUMZQZC @@ -451453,7 +451453,7 @@ Additional material on group theory (deprecated) ( vx vy cgr wcel wfn crn wceq cv cfv wral eqid grpoinvcl grpo2inv fveq2 wa wi wf1o co cgi crio cmpt riotaex fnmpti grpoinvfval fneq1d mpbiri wrex cab fnrnfv syl eqcomd rspceeqv syl2anc ex simpr adantr eqeltrd rexlimdva2 - impbid abbi2dv eqtr4d wb eqeqan12d anandis imbitrid ralrimivva syl3anbrc + impbid eqabdv eqtr4d wb eqeqan12d anandis imbitrid ralrimivva syl3anbrc dff1o6 ) AHIZBCJZBKZCLFMZBNZGMZBNZLZVQVSLZUAZGCOFCOCCBUBVNVOFCVSVQAUCAUDN ZLZGCUEZUFZCJFCWFWGWEGCUGWGPUHVNCBWGFGWDABCDWDPEUIUJUKZVNVPVSVRLZFCULZGUM ZCVNVOVPWKLWHFGCBUNUOVNWJGCVNVSCIZWJVNWLWJVNWLTZVTCIVSVTBNZLWJVSABCDEQWMW @@ -465292,13 +465292,13 @@ space independently from the axioms (see comments in ~ ax-hilex ). Note cmap crab cab ccauold ccau cin df-rab df-hcau elin ancom wf wb hlex elmap nnex c1 nnuz cxmet imsxmet mp1i 1zzd eqidd id iscauf wceq ffvelcdm adantr eluznn sylan2 anassrs h2hmetdval syl2anc breq1d ralbidva rexbidva ralbidv - cnv bitrd sylbi pm5.32i 3bitri abbi2i 3eqtr4i ) GKZHKZLZIKZWJLZUAMUBLZJKZ - NUCZIWIUDLZOZGPUEZJQOZHRPUFMZUGWJXASZWTTZHUHUIAUJLZXAUKZWTHXAULJGIHUMXCHX - EWJXESWJXDSZXBTXBXFTXCWJXDXAUNXFXBUOXBXFWTXBPRWJUPZXFWTUQRPWJBREURUTUSXGX - FWKWMAMZWONUCZIWQOZGPUEZJQOWTXGJWMWKAGIWJVARPVBBWBSARVCLSXGDABREFVDVEXGVF - XGWLPSZTWMVGXGWIPSZTZWKVGXGVHVIXGXKWSJQXGXJWRGPXNXIWPIWQXNWLWQSZTZXHWNWON - XPWKRSZWMRSZXHWNVJXNXQXOPRWIWJVKVLXGXMXOXRXMXOTXGXLXRWLWIVMPRWLWJVKVNVOWK - WMABCDEFVPVQVRVSVTWAWCWDWEWFWGWH $. + cnv bitrd sylbi pm5.32i 3bitri eqabi 3eqtr4i ) GKZHKZLZIKZWJLZUAMUBLZJKZN + UCZIWIUDLZOZGPUEZJQOZHRPUFMZUGWJXASZWTTZHUHUIAUJLZXAUKZWTHXAULJGIHUMXCHXE + WJXESWJXDSZXBTXBXFTXCWJXDXAUNXFXBUOXBXFWTXBPRWJUPZXFWTUQRPWJBREURUTUSXGXF + WKWMAMZWONUCZIWQOZGPUEZJQOWTXGJWMWKAGIWJVARPVBBWBSARVCLSXGDABREFVDVEXGVFX + GWLPSZTWMVGXGWIPSZTZWKVGXGVHVIXGXKWSJQXGXJWRGPXNXIWPIWQXNWLWQSZTZXHWNWONX + PWKRSZWMRSZXHWNVJXNXQXOPRWIWJVKVLXGXMXOXRXMXOTXGXLXRWLWIVMPRWLWJVKVNVOWKW + MABCDEFVPVQVRVSVTWAWCWDWEWFWGWH $. $} ${ @@ -469884,9 +469884,9 @@ to a member of the subspace (Definition of complete subspace in [Beran] (New usage is discouraged.) $) dfch2 $p |- CH = { x e. ~P ~H | ( _|_ ` ( _|_ ` x ) ) = x } $= ( cch cv chba cpw wcel cort cfv wceq cab crab wss chss ococ occl 3syl eleq1 - wa jca imbitrid impcom impbii velpw anbi1i bitr4i abbi2i df-rab eqtr4i ) BA - CZDEZFZUIGHZGHZUIIZRZAJUNAUJKUOABUIBFZUIDLZUNRZUOUPURUPUQUNUIMUINSUNUQUPUQU - MBFZUNUPUQULBFULDLUSUIOULMULOPUMUIBQTUAUBUKUQUNADUCUDUEUFUNAUJUGUH $. + wa jca imbitrid impcom impbii velpw anbi1i bitr4i eqabi df-rab eqtr4i ) BAC + ZDEZFZUIGHZGHZUIIZRZAJUNAUJKUOABUIBFZUIDLZUNRZUOUPURUPUQUNUIMUINSUNUQUPUQUM + BFZUNUPUQULBFULDLUSUIOULMULOPUMUIBQTUAUBUKUQUNADUCUDUEUFUNAUJUGUH $. ${ $d x y A $. $d x B $. @@ -475254,14 +475254,14 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. ( vw vx vz vt vy cv cmv co cno cfv clt chba wral crp wcel wa wi wrex cmap wbr crab cab ccop df-rab df-cnop wf wceq hilmetdval normsub eqtrd adantll ccn breq1d ffvelcdm anim12dan syl anassrs imbi12d rexbidv ralbidv pm5.32i - ralbidva cxmet hilxmet metcn mp2an ax-hilex anbi1i 3bitr4i abbi2i 3eqtr4i - wb elmap ) EJZFJZKLMNZGJZOUDZVRHJZNZVSWCNZKLMNZIJZOUDZUAZEPQZGRUBZIRQZFPQ - ZHPPUCLZUEWCWNSZWMTZHUFUGBBUPLZWMHWNUHFIGEHUIWPHWQPPWCUJZVSVRALZWAOUDZWEW - DALZWGOUDZUAZEPQZGRUBZIRQZFPQZTZWRWMTWCWQSZWPWRXGWMWRXFWLFPWRVSPSZTZXEWKI - RXKXDWJGRXKXCWIEPXKVRPSZTZWTWBXBWHXMWSVTWAOXJXLWSVTUKWRXJXLTZWSVSVRKLMNVT - VSVRACULVSVRUMUNUOUQXMXAWFWGOWRXJXLXAWFUKZWRXNTWEPSZWDPSZTZXOWRXJXPXLXQPP - VSWCURPPVRWCURUSXRXAWEWDKLMNWFWEWDACULWEWDUMUNUTVAUQVBVFVCVDVFVEAPVGNSZXS - XIXHVPACVHZXTFIGEAAWCBBPPDDVIVJWOWRWMPPWCVKVKVQVLVMVNVO $. + ralbidva cxmet wb hilxmet metcn mp2an ax-hilex elmap anbi1i 3bitr4i eqabi + 3eqtr4i ) EJZFJZKLMNZGJZOUDZVRHJZNZVSWCNZKLMNZIJZOUDZUAZEPQZGRUBZIRQZFPQZ + HPPUCLZUEWCWNSZWMTZHUFUGBBUPLZWMHWNUHFIGEHUIWPHWQPPWCUJZVSVRALZWAOUDZWEWD + ALZWGOUDZUAZEPQZGRUBZIRQZFPQZTZWRWMTWCWQSZWPWRXGWMWRXFWLFPWRVSPSZTZXEWKIR + XKXDWJGRXKXCWIEPXKVRPSZTZWTWBXBWHXMWSVTWAOXJXLWSVTUKWRXJXLTZWSVSVRKLMNVTV + SVRACULVSVRUMUNUOUQXMXAWFWGOWRXJXLXAWFUKZWRXNTWEPSZWDPSZTZXOWRXJXPXLXQPPV + SWCURPPVRWCURUSXRXAWEWDKLMNWFWEWDACULWEWDUMUNUTVAUQVBVFVCVDVFVEAPVGNSZXSX + IXHVHACVIZXTFIGEAAWCBBPPDDVJVKWOWRWMPPWCVLVLVMVNVOVPVQ $. hhcn.4 $e |- K = ( TopOpen ` CCfld ) $. $( The continuous functionals of Hilbert space. (Contributed by Mario @@ -475271,7 +475271,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. wbr wi wrex cmap crab cab ccnfn df-rab df-cnfn wf ccom hilmetdval normsub ccn wceq eqtrd adantll breq1d ffvelcdm anim12dan cnmetdval abssub anassrs eqid syl imbi12d ralbidva rexbidv ralbidv pm5.32i cxmet wb hilxmet cnxmet - cnfldtopn metcn mp2an cnex ax-hilex elmap anbi1i 3bitr4i abbi2i 3eqtr4i ) + cnfldtopn metcn mp2an cnex ax-hilex elmap anbi1i 3bitr4i eqabi 3eqtr4i ) GUAZHUAZUBLUCMZIUAZNUFZWJJUAZMZWKWOMZUDLUEMZKUAZNUFZUGZGOPZIQUHZKQPZHOPZJ ROUILZUJWOXFSZXETZJUKULBCUSLZXEJXFUMHKIGJUNXHJXIORWOUOZWKWJALZWMNUFZWQWPU EUDUPZLZWSNUFZUGZGOPZIQUHZKQPZHOPZTZXJXETWOXISZXHXJXTXEXJXSXDHOXJWKOSZTZX @@ -479263,8 +479263,8 @@ Positive operators (cont.) by NM, 24-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) $) pjhmopidm $p |- ran projh = { t e. HrmOp | ( t o. t ) = t } $= - ( cpjh crn cv cho wcel ccom wceq wa cab crab dfpjop abbi2i df-rab eqtr4i - ) BCZADZEFQQGQHZIZAJRAEKSAPQLMRAENO $. + ( cpjh crn cv cho wcel ccom wceq wa cab crab dfpjop eqabi df-rab eqtr4i ) + BCZADZEFQQGQHZIZAJRAEKSAPQLMRAENO $. $} $( A projection operator is idempotent. Part of Theorem 26.1 of [Halmos] @@ -479312,12 +479312,12 @@ Positive operators (cont.) ( vy cpjh crn wcel cv chba cfv wceq wa cab crab cch wss elpjch simpld syl sylan syl5ibcom chss sseld wrex wfn wb cho wf elpjhmop hmopf ffnd fvelrnb ccom fvco3 elpjidm adantr fveq1d eqtr3d fveq2 id eqeq12d rexlimdva sylbid - jcad fnfvelrn eleq1 expimpd impbid abbi2dv df-rab eqtr4di ) BDEFZBEZAGZHF - ZVMBIZVMJZKZALVPAHMVKVQAVLVKVMVLFZVQVKVRVNVPVKVLHVMVKVLNFZVLHOVKVSBVLDIJB - PQVLUARUBVKVRCGZBIZVMJZCHUCZVPVKBHUDZVRWCUEVKHHBVKBUFFHHBUGZBUHBUIRZUJZCH - VMBUKRVKWBVPCHVKVTHFZKZWABIZWAJWBVPWIVTBBULZIZWJWAVKWEWHWLWJJWFHHVTBBUMSW - IVTWKBVKWKBJWHBUNUOUPUQWBWJVOWAVMWAVMBURWBUSUTTVAVBVCVKVNVPVRVKVNKVOVLFZV - PVRVKWDVNWMWGHVMBVDSVOVMVLVETVFVGVHVPAHVIVJ $. + jcad fnfvelrn eleq1 expimpd impbid eqabdv df-rab eqtr4di ) BDEFZBEZAGZHFZ + VMBIZVMJZKZALVPAHMVKVQAVLVKVMVLFZVQVKVRVNVPVKVLHVMVKVLNFZVLHOVKVSBVLDIJBP + QVLUARUBVKVRCGZBIZVMJZCHUCZVPVKBHUDZVRWCUEVKHHBVKBUFFHHBUGZBUHBUIRZUJZCHV + MBUKRVKWBVPCHVKVTHFZKZWABIZWAJWBVPWIVTBBULZIZWJWAVKWEWHWLWJJWFHHVTBBUMSWI + VTWKBVKWKBJWHBUNUOUPUQWBWJVOWAVMWAVMBURWBUSUTTVAVBVCVKVNVPVRVKVNKVOVLFZVP + VRVKWDVNWMWGHVMBVDSVOVMVLVETVFVGVHVPAHVIVJ $. $} ${ @@ -485613,13 +485613,13 @@ Class abstractions (a.k.a. class builders) ofrn2 $p |- ( ph -> ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) ) $= ( vz va vx vy cv co crn wcel cfv wceq wrex cab cof cxp cima wa wfn simprl ffnd fnfvelrn syl2an2r simprr rspceov syl3anc rexlimdvaa cmpt inidm eqidd - ss2abdv offval rneqd eqid rnmpt eqtrdi wss wb frnd xpss12 syl2anc abbi2dv - ovelimab 3sstr4d ) AMQZNQZFUAZVPGUAZERZUBZNBUCZMUDZVOOQPQERUBPGSZUCOFSZUC - ZMUDFGEUERZSZEWDWCUFZUGZAWAWEMAVTWENBAVPBTZVTUHZUHVQWDTZVRWCTZVTWEAFBUIWK - WJWLABCFIUKZAWJVTUJZBVPFULUMAGBUIWKWJWMABCGJUKZWOBVPGULUMAWJVTUNOPWDWCVQV - RVOEUOUPUQVAAWGNBVSURZSWBAWFWQANBBVQVREBFGHHWNWPLLBUSAWJUHZVQUTWRVRUTVBVC - NMBVSWQWQVDVEVFAWEMWIAECCUFZUIWHWSVGZVOWITWEVHAWSDEKUKAWDCVGWCCVGWTABCFIV - IABCGJVIWDCWCCVJVKOPWSWDWCVOEVMVKVLVN $. + ss2abdv offval rneqd eqid rnmpt eqtrdi wss wb frnd xpss12 ovelimab eqabdv + syl2anc 3sstr4d ) AMQZNQZFUAZVPGUAZERZUBZNBUCZMUDZVOOQPQERUBPGSZUCOFSZUCZ + MUDFGEUERZSZEWDWCUFZUGZAWAWEMAVTWENBAVPBTZVTUHZUHVQWDTZVRWCTZVTWEAFBUIWKW + JWLABCFIUKZAWJVTUJZBVPFULUMAGBUIWKWJWMABCGJUKZWOBVPGULUMAWJVTUNOPWDWCVQVR + VOEUOUPUQVAAWGNBVSURZSWBAWFWQANBBVQVREBFGHHWNWPLLBUSAWJUHZVQUTWRVRUTVBVCN + MBVSWQWQVDVEVFAWEMWIAECCUFZUIWHWSVGZVOWITWEVHAWSDEKUKAWDCVGWCCVGWTABCFIVI + ABCGJVIWDCWCCVJVMOPWSWDWCVOEVKVMVLVN $. $} ${ @@ -547182,9 +547182,9 @@ Set induction (or epsilon induction) fobigcup $p |- Bigcup : _V -onto-> _V $= ( vx vy cvv cbigcup wfo wfn crn wceq cuni wcel wral uniexg rgen dfbigcup2 cv mptfng mpbi wrex cab rnmpt vex csn vsnex unisnv eqcomi unieq mp2an 2th - rspceeqv abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFZDGZCHAOZIZCJZACKUNURACUPC - LMACUQDANZPQUOBOZUQHACRZBSCABCUQDUSTVABCUTCJVABUAUTUBZCJUTVBIZHVABUCVCUTB - UDUEAVBCUQVCUTUPVBUFUIUGUHUJUKCCDULUM $. + rspceeqv eqabi eqtr4i df-fo mpbir2an ) CCDEDCFZDGZCHAOZIZCJZACKUNURACUPCL + MACUQDANZPQUOBOZUQHACRZBSCABCUQDUSTVABCUTCJVABUAUTUBZCJUTVBIZHVABUCVCUTBU + DUEAVBCUQVCUTUPVBUFUIUGUHUJUKCCDULUM $. $} $( ` Bigcup ` is a function over the universal class. (Contributed by Scott @@ -547427,7 +547427,7 @@ Set induction (or epsilon induction) Scott Fenton, 20-Nov-2013.) 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See ~ brofs and A. x e. X ( ( nei ` j ) ` { x } ) = { n e. ~P X | ( ( F ` x ) i^i ~P n ) =/= (/) } ) $= ( wcel vz cv ctopon cfv csn cnei cpw cin c0 wne crab wceq wral wa wreu wi - weu wal neibastop1 cab wss neibastop2 velpw anbi1i bitr4di abbi2dv df-rab - eqtr4di ralrimiva sneq fveq2d fveq2 ineq1d neeq1d rabbidv cbvralvw sylibr - eqeq12d cuni toponuni eqimss2 sspwuni ad2antlr sseqin2 sylib wrex ctop wb + weu wal neibastop1 cab wss neibastop2 velpw anbi1i bitr4di eqabdv eqtr4di + df-rab ralrimiva sneq fveq2d fveq2 ineq1d neeq1d rabbidv eqeq12d cbvralvw + sylibr cuni toponuni eqimss2 syl sspwuni ad2antlr sseqin2 sylib wrex ctop topontop ad3antlr eltop2 ssralv adantl simprr eleq2d sseq2d biimpa sylan2 elpwi sselda adantrr adantr eqid isneip baibd syl21anc pweq ineq2d elrab3 - syl 3bitr3d expr ralimdva syld an32s ralbi bitrd rabbi2dva eqtr3d alrimiv - imp expl eleq1 fveq1d eqeq1d ralbidv anbi12d eqeu syl121anc df-reu ) AGUB - ZMUCUDZTZBUBZUEZYKUFUDZUDZYNJUDZHUBZUGZUHZUIUJZHMUGZUKZULZBMUMZUNZGUQZUUF - GYLUOAKYLTZUUIYOKUFUDZUDZUUDULZBMUMZUUGYKKULZUPZGURUUHABDEIJKLMNOPQUSZUUP - AUAUBZUEZUUJUDZUUQJUDZYTUHZUIUJZHUUCUKZULZUAMUMUUMAUVDUAMAUUQMTUNZUUSYSUU - CTZUVBUNZHUTUVCUVEUVGHUUSUVEYSUUSTYSMVAZUVBUNUVGABCDEFUUQIJKYSLMNOPQRSVBU - VFUVHUVBHMVCVDVEVFUVBHUUCVGVHVIUULUVDBUAMYNUUQULZUUKUUSUUDUVCUVIYOUURUUJY - NUUQVJVKUVIUUBUVBHUUCUVIUUAUVAUIUVIYRUUTYTYNUUQJVLVMVNVOVRVPVQAUUOGAYMUUF - UUNAYMUNZUUFUNZUUCYKUHZYKKUVKYKUUCVAZUVLYKULYMUVMAUUFYMYKVSZMVAZUVMYMMUVN - ULZUVOMYKVTZUVNMWAXJYKMWBVQWCYKUUCWDWEUVKUVLYRIUBZUGZUHZUIUJZBUVRUMZIUUCU - KKUVKUWBIUUCYKUVKUVRUUCTZUNZUVRYKTZYNUUQTUUQUVRVAUNUAYKWFZBUVRUMZUWBUWDYK - WGTZUWEUWGWHYMUWHAUUFUWCMYKWIZWJBUAUVRYKWKXJUWDUWFUWAWHZBUVRUMZUWGUWBWHUV - JUWCUUFUWKUVJUWCUNZUUFUWKUWLUUFUUEBUVRUMZUWKUWCUUFUWMUPZUVJUWCUVRMVAZUWNU - VRMWSZUUEBUVRMWLXJWMUWLUUEUWJBUVRUWLYNUVRTZUUEUWJUWLUWQUUEUNZUNZUVRYQTZUV - RUUDTZUWFUWAUWSYQUUDUVRUWLUWQUUEWNWOUWSUWHYNUVNTZUVRUVNVAZUWTUWFWHYMUWHAU - WCUWRUWIWJUWLUWQUXBUUEUWLUVRUVNYNUWCUVJUWOUXCUWPUVJUWOUXCUVJMUVNUVRYMUVPA - UVQWMWPWQWRZWTXAUWLUXCUWRUXDXBUWHUXBUNUWTUXCUWFYNUAYKUVRUVNUVNXCXDXEXFUWC - UXAUWAWHUVJUWRUUBUWAHUVRUUCYSUVRULZUUAUVTUIUXEYTUVSYRYSUVRXGXHVNXIWCXKXLX - MXNYAXOUWFUWABUVRXPXJXQXRQVHXSYBXTUUGUUIUUMUNGKYLUUNYMUUIUUFUUMYKKYLYCUUN - UUEUULBMUUNYQUUKUUDUUNYOYPUUJYKKUFVLYDYEYFYGYHYIUUFGYLYJVQ $. + wb 3bitr3d expr ralimdva syld imp an32s ralbi bitrd rabbi2dva eqtr3d expl + alrimiv eleq1 fveq1d eqeq1d ralbidv anbi12d eqeu syl121anc df-reu ) AGUBZ + MUCUDZTZBUBZUEZYKUFUDZUDZYNJUDZHUBZUGZUHZUIUJZHMUGZUKZULZBMUMZUNZGUQZUUFG + YLUOAKYLTZUUIYOKUFUDZUDZUUDULZBMUMZUUGYKKULZUPZGURUUHABDEIJKLMNOPQUSZUUPA + UAUBZUEZUUJUDZUUQJUDZYTUHZUIUJZHUUCUKZULZUAMUMUUMAUVDUAMAUUQMTUNZUUSYSUUC + TZUVBUNZHUTUVCUVEUVGHUUSUVEYSUUSTYSMVAZUVBUNUVGABCDEFUUQIJKYSLMNOPQRSVBUV + FUVHUVBHMVCVDVEVFUVBHUUCVHVGVIUULUVDBUAMYNUUQULZUUKUUSUUDUVCUVIYOUURUUJYN + UUQVJVKUVIUUBUVBHUUCUVIUUAUVAUIUVIYRUUTYTYNUUQJVLVMVNVOVPVQVRAUUOGAYMUUFU + UNAYMUNZUUFUNZUUCYKUHZYKKUVKYKUUCVAZUVLYKULYMUVMAUUFYMYKVSZMVAZUVMYMMUVNU + LZUVOMYKVTZUVNMWAWBYKMWCVRWDYKUUCWEWFUVKUVLYRIUBZUGZUHZUIUJZBUVRUMZIUUCUK + KUVKUWBIUUCYKUVKUVRUUCTZUNZUVRYKTZYNUUQTUUQUVRVAUNUAYKWGZBUVRUMZUWBUWDYKW + HTZUWEUWGXJYMUWHAUUFUWCMYKWIZWJBUAUVRYKWKWBUWDUWFUWAXJZBUVRUMZUWGUWBXJUVJ + UWCUUFUWKUVJUWCUNZUUFUWKUWLUUFUUEBUVRUMZUWKUWCUUFUWMUPZUVJUWCUVRMVAZUWNUV + RMWSZUUEBUVRMWLWBWMUWLUUEUWJBUVRUWLYNUVRTZUUEUWJUWLUWQUUEUNZUNZUVRYQTZUVR + UUDTZUWFUWAUWSYQUUDUVRUWLUWQUUEWNWOUWSUWHYNUVNTZUVRUVNVAZUWTUWFXJYMUWHAUW + CUWRUWIWJUWLUWQUXBUUEUWLUVRUVNYNUWCUVJUWOUXCUWPUVJUWOUXCUVJMUVNUVRYMUVPAU + VQWMWPWQWRZWTXAUWLUXCUWRUXDXBUWHUXBUNUWTUXCUWFYNUAYKUVRUVNUVNXCXDXEXFUWCU + XAUWAXJUVJUWRUUBUWAHUVRUUCYSUVRULZUUAUVTUIUXEYTUVSYRYSUVRXGXHVNXIWDXKXLXM + XNXOXPUWFUWABUVRXQWBXRXSQVGXTYAYBUUGUUIUUMUNGKYLUUNYMUUIUUFUUMYKKYLYCUUNU + UEUULBMUUNYQUUKUUDUUNYOYPUUJYKKUFVLYDYEYFYGYHYIUUFGYLYJVR $. $} @@ -561408,7 +561408,7 @@ FOL part ( ~ bj-ru0 ) and then two versions ( ~ bj-ru1 and ~ bj-ru ). $d x A $. $( Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.) $) bj-snglinv $p |- A = { x | { x } e. sngl A } $= - ( cv csn bj-csngl wcel bj-snglc abbi2i ) ACZDBEFABIBGH $. + ( cv csn bj-csngl wcel bj-snglc eqabi ) ACZDBEFABIBGH $. $} ${ @@ -587940,8 +587940,8 @@ A collection of theorems for commuting equalities (or $( Restricted coset of ` B ` . (Contributed by Peter Mazsa, 9-Dec-2018.) $) ecres $p |- [ B ] ( R |` A ) = { x | ( B e. A /\ B R x ) } $= - ( wcel cv wbr wa cres cec wb cvv elecres elv abbi2i ) CBECAFZDGHZACDBIJZP - REQKABCPDLMNO $. + ( wcel cv wbr wa cres cec wb cvv elecres elv eqabi ) CBECAFZDGHZACDBIJZPR + EQKABCPDLMNO $. $} ${ @@ -587949,8 +587949,8 @@ A collection of theorems for commuting equalities (or $( The restricted coset of ` B ` when ` B ` is an element of the restriction. (Contributed by Peter Mazsa, 16-Oct-2018.) $) ecres2 $p |- ( B e. A -> [ B ] ( R |` A ) = [ B ] R ) $= - ( vy wcel cres cec cv wbr cab wa wb cvv elecres baib abbi2dv dfec2 eqtr4d - elv ) BAEZBCAFGZBDHZCIZDJBCGTUCDUAUBUAEZTUCUDTUCKLDABUBCMNSOPDBCAQR $. + ( vy wcel cres cec cv wbr cab wa cvv elecres elv baib eqabdv dfec2 eqtr4d + wb ) BAEZBCAFGZBDHZCIZDJBCGTUCDUAUBUAEZTUCUDTUCKSDABUBCLMNOPDBCAQR $. $} ${ @@ -588160,7 +588160,7 @@ A collection of theorems for commuting equalities (or $( Alternate definition of domain. (Contributed by Peter Mazsa, 2-Mar-2018.) $) dfdm6 $p |- dom R = { x | [ x ] R =/= (/) } $= - ( cv cec c0 wne cdm ecdmn0 abbi2i ) ACZBDEFABGJBHI $. + ( cv cec c0 wne cdm ecdmn0 eqabi ) ACZBDEFABGJBHI $. $} ${ @@ -597595,9 +597595,9 @@ Functionals and kernels of a left vector space (or module) 15-Apr-2014.) $) lkrval2 $p |- ( ( W e. X /\ G e. F ) -> ( K ` G ) = { x e. V | ( G ` x ) = .0. } ) $= - ( wcel cvv cfv cv wceq wa crab elex ellkr abbi2dv df-rab eqtr4di sylan - cab ) GHOGPOZDCOZDEQZARZDQISZAFUAZSGHUBUIUJTZUKULFOUMTZAUHUNUOUPAUKBCDEFG - ULPIJKLMNUCUDUMAFUEUFUG $. + ( wcel cvv cfv cv wceq wa crab elex cab ellkr eqabdv df-rab eqtr4di sylan + ) GHOGPOZDCOZDEQZARZDQISZAFUAZSGHUBUIUJTZUKULFOUMTZAUCUNUOUPAUKBCDEFGULPI + JKLMNUDUEUMAFUFUGUH $. ellkr2.w $e |- ( ph -> W e. Y ) $. ellkr2.g $e |- ( ph -> G e. F ) $. @@ -598300,14 +598300,14 @@ Functionals and kernels of a left vector space (or module) eleq1 adantlr simplll simplr lkrshp3 mpbid ex wi eqimss2 eqimss eqssd a1i jcad reximdva mpd clss cun clspn w3a lkrshp 3adant3r eqid islshp 3ad2ant1 neeq1 uneq1 fveqeq2d rexbidv 3anbi123d adantl 3ad2ant3 rexlimdv3a sylibrd - wb mpbird impbid abbi2dv ) GUAQZBUBZFHUCUDUEZIUBZWHERZSZTZBCUFZIDWGWJDQZW - NWGWOWNWGWOTZWKWJSZBCUFWNWJBCDEGMNOUGWPWQWMBCWPWHCQZTZWQWIWLWSWQWIWSWQTZW - KDQZWIWPWQXAWRWOWQXAWGWQXAWOWKWJDUJUHUIUKWTACWHDEFGHJKLMNOWGWOWRWQULWPWRW - QUMUNUOUPWQWLUQWSWQWJWKWJWKURWKWJUSUTVAVBVCVDUPWGWNWJGVERZQZWJFUEZWJPUBUC - ZVFZGVGRZRFSZPFUFZVHZWOWGWMXJBCWGWRWMVHZXJWKXBQZWKFUEZWKXEVFZXGRFSZPFUFZV - HZXKXAXQWGWRWIXAWLACWHDEFGHJKLMNOVIVJWGWRXAXQWCWMPXBWKDXGFGUAJXGVKZXBVKZM - VLVMUOWMWGXJXQWCZWRWLXTWIWLXCXLXDXMXIXPWJWKXBUJWJWKFVNWLXHXOPFWLXFXNFXGWJ - WKXEVOVPVQVRVSVTWDWAPXBWJDXGFGUAJXRXSMVLWBWEWF $. + wb mpbird impbid eqabdv ) GUAQZBUBZFHUCUDUEZIUBZWHERZSZTZBCUFZIDWGWJDQZWN + WGWOWNWGWOTZWKWJSZBCUFWNWJBCDEGMNOUGWPWQWMBCWPWHCQZTZWQWIWLWSWQWIWSWQTZWK + DQZWIWPWQXAWRWOWQXAWGWQXAWOWKWJDUJUHUIUKWTACWHDEFGHJKLMNOWGWOWRWQULWPWRWQ + UMUNUOUPWQWLUQWSWQWJWKWJWKURWKWJUSUTVAVBVCVDUPWGWNWJGVERZQZWJFUEZWJPUBUCZ + VFZGVGRZRFSZPFUFZVHZWOWGWMXJBCWGWRWMVHZXJWKXBQZWKFUEZWKXEVFZXGRFSZPFUFZVH + ZXKXAXQWGWRWIXAWLACWHDEFGHJKLMNOVIVJWGWRXAXQWCWMPXBWKDXGFGUAJXGVKZXBVKZMV + LVMUOWMWGXJXQWCZWRWLXTWIWLXCXLXDXMXIXPWJWKXBUJWJWKFVNWLXHXOPFWLXFXNFXGWJW + KXEVOVPVQVRVSVTWDWAPXBWJDXGFGUAJXRXSMVLWBWEWF $. $d s F $. $d s K $. $d g s U $. $d s V $. $d s .0. $. $( The predicate "is a hyperplane" (of a left module or left vector space). @@ -651171,7 +651171,7 @@ first fundamental theorem of projective geometry (see ~ mapdpg ). @) sneqd eqeq12d fveq12d riotaeqbidv syld3an3 bitrid sbc3ieSBC oveq12d eqidd oveq123d - ifbieq12d oveq1d anbi2d mpteq12dv syld3an2 bitrdi abbi1dv mptex sylan9eq + ifbieq12d oveq1d anbi2d mpteq12dv syld3an2 bitrdi eqabcdv mptex sylan9eq eqid fvmpt ) ARCVIZUBPVIZVJSBUABVKZMVLZTVMZVIZUUJNVMZNGVNZUUJVLZTVMZFVMZL VKZVLZQVMZVOZMUUJIVNZVLZTVMZFVMZNUUSHVNZVLZQVMZVOZVJZLOVPZVQZVRZVOUSUUHUU ISUBUTPVAVKZBVBVKZVSVMZUUJVCVKZVLZVDVKZVMZVIZUUJVEVKZVMZUWCVFVKZVTVMZVNZU @@ -651403,7 +651403,7 @@ first fundamental theorem of projective geometry (see ~ mapdpg ). @) wb fveq2d oveqd eqeq2d ralbidv riotabidv mpteq2dv eleq2d sbcbidvSBC cvv fvex bitrd eqeltri w3a simp2 simp1 eqtrd simp3 fveq12d eqidd eqeq12d raleqbidv - oveq123d riotaeqbidv mpteq12dv syld3an2 sbc3ieSBC bitrdi abbi1dv mptex + oveq123d riotaeqbidv mpteq12dv syld3an2 sbc3ieSBC bitrdi eqabcdv mptex fvmpt eqid sylan9eq ) AMQUNZPKUNZUOLBEBUPZDUPZIUQZNURZCUPZXONURZHUQZUSZDOUTZCEV AZVBZUSUHXLXMLPUIKUJUPZBUKUPZXNXOULUPZVGURZUQZUMUPZURZXRXOYJURZUIUPZMVCUR @@ -652097,7 +652097,7 @@ first fundamental theorem of projective geometry (see ~ mapdpg ). @) simp2 fveq2d eqtrd eqtr4di simp3 id fveq1 eqeq1d anbi12d riotabidv ifeq2d w3a mpteq2dv eleq2d syl sbcie xpeq2 xpeq1d simp1 eqeq2d oveqd riotaeqbidv fveq12d ifeq12d mpteq12dv bitrid syl3anc sbc3ie xpeq12d fveqeq2d ifbieq2d - sneqd bitrdi abbi1dv eqid fvexi xpex mptex fvmpt sylan9eq ) AMCVCZRJVCZVD + sneqd bitrdi eqabcdv eqid fvexi xpex mptex fvmpt sylan9eq ) AMCVCZRJVCZVD KBQEVEZQVEZBVFZVGVHZSVIZFUUCVJZPVHZNVHIVFZVJZLVHZVIZUUBVKVHZVKVHZUUCOVLZV JZPVHZNVHUUKVGVHZUUGGVLZVJZLVHZVIZVDZIEVNZVMZVOZVIUMYRYSKRUNJUOVFZBUPVFZU QVFZVEZUVFVEZUUCURVFZVPVHZVIZUSVFZVPVHZUUEUTVFZVHZVAVFZVHZUUHVBVFZVHZVIZU @@ -652945,7 +652945,7 @@ first fundamental theorem of projective geometry (see ~ mapdpg ). @) imbi2d riotaeqbidv mpteq2dv eleq2d sbceqbid sbcbidv opex fvex w3a eqtr4di wb simp1 simp2 simp3 eqtrd id fveq1 riotabidv syl sbcie fveq2i eqtr2i a1i sneqd fveq12d uneq12d notbid oteq1d fveq1i raleqbidv bitrid sbc3ie bitrdi - imbi12d mpteq12dv syl3anc abbi1dv eqid mptfvmpt sylan9eq ) ANEUOZQKUOZUPH + imbi12d mpteq12dv syl3anc eqabcdv eqid mptfvmpt sylan9eq ) ANEUOZQKUOZUPH DPCUQZJURZOUSZDUQZURZOUSZUTZUOZVAZBUQZYNJJMUSZYNVBZLUSZYQVBZLUSZVCZVDZCPV EZBGVFZVGZVCUHYLYMHQUIKUJUQZDUKUQZYNULUQZURZUMUQZVHUSZUSZYRUUSUSZUTZUOZVA ZUUCYNUUPUUPUIUQZNVIUSZUSZUSZYNVBZUNUQZUSZYQVBZUVJUSZVCZVDZCUUOVEZBUVENVJ @@ -654294,7 +654294,7 @@ fixed reference functional determined by this vector (corresponding to cbs csca cdvh cab chg hgmapffval fveq1d eqtrid fveq2 eqtr4di 2fveq3 oveqd co eqeq2d ralbidv riotabidv mpteq2dv eleq2d sbceqbid sbcbidv fvexi wb w3a simp2 simp1 fveq2d eqtrd simp3 fveq12d eqidd oveq123d eqeq12d riotaeqbidv - fvex raleqbidv mpteq12dv syld3an2 sbc3ie bitrdi abbi1dv mptfvmpt sylan9eq + fvex raleqbidv mpteq12dv syld3an2 sbc3ie bitrdi eqabcdv mptfvmpt sylan9eq eqid syl ) AMQUNZPKUNZUOLBEBUPZDUPZIVRZNUQZCUPZXMNUQZHVRZURZDOUSZCEUTZVAZ URUHXJXKLPUIKUJUPZBUKUPZXLXMULUPZVBUQZVRZUMUPZUQZXPXMYHUQZUIUPZMVCUQZUQVB UQZVRZURZDYEVFUQZUSZCYDUTZVAZUNZUMYKMVDUQZUQZVEZUKYEVGUQZVFUQZVEZULYKMVHU @@ -655493,17 +655493,17 @@ fixed reference functional determined by this vector (corresponding to ( cfv cv wcel wceq wral wa cab crab chlt wss dochssv syl2anc pm4.71rd csn sseld clspn clss eqid clmod dvhlmod adantr dochlss lspsnel5 cdih dihlsprn simpr crn dochcl dochord snssd dochocsp sseq2d bitr4d 3bitrd dfss3 bitrdi - dochsscl ad2antrr sselda simplr dochsncom bitrd ralbidva pm5.32da abbi2dv - hdmapellkr df-rab eqtr4di ) ALIUCZBUDZJUEZWLCUDZEUCUCMUFZCLUGZUHZBUIWPBJU - JAWQBWKAWLWKUEZWMWRUHWQAWRWMAWKJWLAHUKUEKGUEUHZLJULZWKJULUAUBFGHIJKLNOPSU - MUNUQUOAWMWRWPAWMUHZWRWNWLUPZIUCZUEZCLUGZWPXAWRLXCULZXEXAWRXBFURUCZUCZWKU - LWKIUCZXHIUCZULZXFXAFUSUCZWKXGJFWLPXLUTZXGUTZAFVAUEWMAFGHKNOUAVBVCAWKXLUE - ZWMAWSWTXOUAUBXLFGHIJKLNOPXMSVDUNVCAWMVHZVEXAGKHVFUCUCZHIKXHWKNXQUTZSAWSW - MUAVCZXAWSWMXHXQVIZUEXSXPFGXQHXGJKWLNOPXNXRVGUNAWKXTUEZWMAWSWTYAUAUBFGXQH - IJKLNXROPSVJUNVCVKXAXKXIXCULXFXAXJXCXIXAFGHXGIJKXBNOSPXNXSXAWLJXPVLZVMVNX - AFGXQHIJKLXCNOPXRSXSAWTWMUBVCZXAWSXBJULXCXTUEXSYBFGXQHIJKXBNXROPSVJUNVSVO - VPCLXCVQVRXAWOXDCLXAWNLUEZUHZWOWLWNUPIUCUEXDYEDEFGHIJKWNWLMNSOPQRTAWSWMYD - UAVTZXALJWNYCWAZAWMYDWBZWHYEFGHIJKWLWNNSOPYFYHYGWCWDWEVOWFWDWGWPBJWIWJ $. + dochsscl ad2antrr sselda simplr hdmapellkr bitrd ralbidva pm5.32da eqabdv + dochsncom df-rab eqtr4di ) ALIUCZBUDZJUEZWLCUDZEUCUCMUFZCLUGZUHZBUIWPBJUJ + AWQBWKAWLWKUEZWMWRUHWQAWRWMAWKJWLAHUKUEKGUEUHZLJULZWKJULUAUBFGHIJKLNOPSUM + UNUQUOAWMWRWPAWMUHZWRWNWLUPZIUCZUEZCLUGZWPXAWRLXCULZXEXAWRXBFURUCZUCZWKUL + WKIUCZXHIUCZULZXFXAFUSUCZWKXGJFWLPXLUTZXGUTZAFVAUEWMAFGHKNOUAVBVCAWKXLUEZ + WMAWSWTXOUAUBXLFGHIJKLNOPXMSVDUNVCAWMVHZVEXAGKHVFUCUCZHIKXHWKNXQUTZSAWSWM + UAVCZXAWSWMXHXQVIZUEXSXPFGXQHXGJKWLNOPXNXRVGUNAWKXTUEZWMAWSWTYAUAUBFGXQHI + JKLNXROPSVJUNVCVKXAXKXIXCULXFXAXJXCXIXAFGHXGIJKXBNOSPXNXSXAWLJXPVLZVMVNXA + FGXQHIJKLXCNOPXRSXSAWTWMUBVCZXAWSXBJULXCXTUEXSYBFGXQHIJKXBNXROPSVJUNVSVOV + PCLXCVQVRXAWOXDCLXAWNLUEZUHZWOWLWNUPIUCUEXDYEDEFGHIJKWNWLMNSOPQRTAWSWMYDU + AVTZXALJWNYCWAZAWMYDWBZWCYEFGHIJKWLWNNSOPYFYHYGWHWDWEVOWFWDWGWPBJWIWJ $. $} $c HLHil $. @@ -662037,8 +662037,8 @@ D Fn ( I ... ( M - 1 ) ) /\ 1-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.) $) ruvALT $p |- { x | x e/ x } = _V $= - ( cvv cv wnel cab wcel vex elirrv nelir 2th abbi2i eqcomi ) BACZMDZAENABMBF - NAGMMAHIJKL $. + ( cvv cv wnel cab wcel vex elirrv nelir 2th eqabi eqcomi ) BACZMDZAENABMBFN + AGMMAHIJKL $. $( Alternative to ~ wcdeq and ~ df-cdeq . This flattens the syntax representation ( wi ( weq vx vy ) wph ) to ( sn-wcdeq vx vy wph ), @@ -683487,7 +683487,7 @@ B C_ ( A ^o B ) ) $= $d A a x $. $( Two ways to express a class. (Contributed by RP, 13-Feb-2025.) $) rp-abid $p |- A = { x | E. a e. A x = a } $= - ( weq wrex cv clel5 abbi2i ) ACDCBEABCBAFGH $. + ( weq wrex cv clel5 eqabi ) ACDCBEABCBAFGH $. $} ${ @@ -697469,10 +697469,10 @@ collection and union ( ~ mnuop3d ), from which closure under pairing dfuniv2 $p |- Univ = { y | A. z e. y A. f e. ~P y E. w e. y ( ~P z C_ ( y i^i w ) /\ ( z i^i U. f ) C_ U. ( f i^i ~P ~P w ) ) } $= ( vi vv vu vk vm vn vr cv cpw cin wss cuni wa wrex wral wel vq vp vl cgru - wcel wi wal wb cvv grumnueq ismnu elv ismnushort ralbii bitr4i abbi2i ) B - LZMZALZCLZNOUQDLZPNVAUTMMNPOQCUSRDUSMSZBUSSZAUDZUSVDUEZURUSOURUTOEFTFDTQF - USREGTGLPUTOQGVARUFEUQSQCUSRDUGQZBUSSZVCVEVGUHABCFGUSDEHIJVDUIKUAUBUCHIJK - UAUBUCUJUKULVBVFBUSBCFGUSDEUMUNUOUP $. + wcel wi wal wb cvv grumnueq ismnu elv ismnushort ralbii bitr4i eqabi ) BL + ZMZALZCLZNOUQDLZPNVAUTMMNPOQCUSRDUSMSZBUSSZAUDZUSVDUEZURUSOURUTOEFTFDTQFU + SREGTGLPUTOQGVARUFEUQSQCUSRDUGQZBUSSZVCVEVGUHABCFGUSDEHIJVDUIKUAUBUCHIJKU + AUBUCUJUKULVBVFBUSBCFGUSDEUMUNUOUP $. $} ${ @@ -700094,7 +700094,7 @@ notation for set theory (it is not defined in the predicate calculus $( Equality between two ways of saying "the complement of ` A ` ". (Contributed by Andrew Salmon, 15-Jul-2011.) $) compeq $p |- ( _V \ A ) = { x | -. x e. A } $= - ( cv wcel wn cvv cdif velcomp abbi2i ) ACBDEAFBGABHI $. + ( cv wcel wn cvv cdif velcomp eqabi ) ACBDEAFBGABHI $. $} $( The complement of ` A ` is not equal to ` A ` . (Contributed by Andrew @@ -783343,11 +783343,11 @@ their group (addition) operations are equal for all pairs of elements of 27-Feb-2020.) $) submgmacs $p |- ( G e. Mgm -> ( SubMgm ` G ) e. ( ACS ` B ) ) $= ( vx vy vs cmgm wcel csubmgm cfv cv cplusg co wral cpw crab cacs cab cvv - wa wss eqid issubmgm velpw anbi1i bitr4di abbi2dv eqtr4di cbs fvexi mgmcl - df-rab 3expb ralrimivva acsfn2 sylancr eqeltrd ) BGHZBIJZDKZEKZBLJZMZFKZH - EVDNDVDNZFAOZPZAQJZURUSVDVFHZVETZFRVGURVJFUSURVDUSHVDAUAZVETVJDEAVBVDBCVB - UBZUCVIVKVEFAUDUEUFUGVEFVFULUHURASHVCAHZEANDANVGVHHABUICUJURVMDEAAURUTAHV - AAHVMABUTVAVBCVLUKUMUNVCSAFDEUOUPUQ $. + wa wss eqid issubmgm velpw anbi1i bitr4di eqabdv df-rab eqtr4di cbs fvexi + mgmcl 3expb ralrimivva acsfn2 sylancr eqeltrd ) BGHZBIJZDKZEKZBLJZMZFKZHE + VDNDVDNZFAOZPZAQJZURUSVDVFHZVETZFRVGURVJFUSURVDUSHVDAUAZVETVJDEAVBVDBCVBU + BZUCVIVKVEFAUDUEUFUGVEFVFUHUIURASHVCAHZEANDANVGVHHABUJCUKURVMDEAAURUTAHVA + AHVMABUTVAVBCVLULUMUNVCSAFDEUOUPUQ $. $} ${ @@ -798745,8 +798745,8 @@ separated by (open) neighborhoods. See ~ sepnsepo for the equivalence ( ( c i^i d ) = (/) -> E. x e. j E. y e. j ( c C_ x /\ d C_ y /\ ( x i^i y ) = (/) ) ) } $= ( cnrm cv ctop wcel cin c0 wceq wss w3a wrex wi ccld cfv wral wa cab crab - isnrm3 abbi2i df-rab eqtr4i ) FCGZHIDGZEGZJKLUHAGZMUIBGZMUJUKJKLNBUGOAUGO - PEUGQRZSDULSZTZCUAUMCHUBUNCFABUGDEUCUDUMCHUEUF $. + isnrm3 eqabi df-rab eqtr4i ) FCGZHIDGZEGZJKLUHAGZMUIBGZMUJUKJKLNBUGOAUGOP + EUGQRZSDULSZTZCUAUMCHUBUNCFABUGDEUCUDUMCHUEUF $. $( A topological space is normal if any disjoint closed sets can be separated by neighborhoods. An alternate definition of ~ df-nrm . @@ -798755,8 +798755,8 @@ separated by (open) neighborhoods. See ~ sepnsepo for the equivalence ( ( c i^i d ) = (/) -> E. x e. ( ( nei ` j ) ` c ) E. y e. ( ( nei ` j ) ` d ) ( x i^i y ) = (/) ) } $= ( cnrm cv ctop wcel cin c0 wceq cnei cfv wrex wi ccld wral wa cab isnrm4 - crab abbi2i df-rab eqtr4i ) FCGZHIDGZEGZJKLAGBGJKLBUHUFMNZNOAUGUINOPEUFQN - ZRDUJRZSZCTUKCHUBULCFABUFDEUAUCUKCHUDUE $. + crab eqabi df-rab eqtr4i ) FCGZHIDGZEGZJKLAGBGJKLBUHUFMNZNOAUGUINOPEUFQNZ + RDUJRZSZCTUKCHUBULCFABUFDEUAUCUKCHUDUE $. $} ${