mmdefinitions - Metamath Proof Explorer

List of Syntax, Axioms (ax-) and Definitions (df-)

Ref	Expression (see link for any distinct variable requirements)
wn 3	wff -. ph
wi 4	wff ( ph -> ps )
ax-mp 5	|- ph   &    |- ( ph -> ps )   =>    |- ps
ax-1 6	|- ( ph -> ( ps -> ph ) )
ax-2 7	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
ax-3 8	|- ( ( -. ph -> -. ps ) -> ( ps -> ph ) )
wb 196	wff ( ph <-> ps )
df-bi 197	|- -. ( ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) -> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) )
wo 383	wff ( ph \/ ps )
wa 384	wff ( ph /\ ps )
df-or 385	|- ( ( ph \/ ps ) <-> ( -. ph -> ps ) )
df-an 386	|- ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) )
wif 1012	wff if- ( ph , ps , ch )
df-ifp 1013	|- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
w3o 1036	wff ( ph \/ ps \/ ch )
w3a 1037	wff ( ph /\ ps /\ ch )
df-3or 1038	|- ( ( ph \/ ps \/ ch ) <-> ( ( ph \/ ps ) \/ ch ) )
df-3an 1039	|- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
wnan 1447	wff ( ph -/\ ps )
df-nan 1448	|- ( ( ph -/\ ps ) <-> -. ( ph /\ ps ) )
wxo 1464	wff ( ph \/_ ps )
df-xor 1465	|- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) )
wal 1481	wff A. x ph
cv 1482	class x
wceq 1483	wff A = B
wtru 1484	wff T.
df-tru 1486	|- ( T. <-> ( A. x x = x -> A. x x = x ) )
wfal 1488	wff F.
df-fal 1489	|- ( F. <-> -. T. )
whad 1532	wff hadd ( ph , ps , ch )
df-had 1533	|- (hadd ( ph , ps , ch ) <-> ( ( ph \/_ ps ) \/_ ch ) )
wcad 1545	wff cadd ( ph , ps , ch )
df-cad 1546	|- (cadd ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) )
wex 1704	wff E. x ph
df-ex 1705	|- ( E. x ph <-> -. A. x -. ph )
wnf 1708	wff F/ x ph
wnfOLD 1709	wff F/ x ph
df-nf 1710	|- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
df-nfOLD 1721	|- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
ax-gen 1722	|- ph   =>    |- A. x ph
ax-4 1737	|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
ax-5 1839	|- ( ph -> A. x ph )
wsb 1880	wff [ y / x ] ph
df-sb 1881	|- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
ax-6 1888	|- -. A. x -. x = y
ax-7 1935	|- ( x = y -> ( x = z -> y = z ) )
wcel 1990	wff A e. B
ax-8 1992	|- ( x = y -> ( x e. z -> y e. z ) )
ax-9 1999	|- ( x = y -> ( z e. x -> z e. y ) )
ax-10 2019	|- ( -. A. x ph -> A. x -. A. x ph )
ax-11 2034	|- ( A. x A. y ph -> A. y A. x ph )
ax-12 2047	|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
ax-13 2246	|- ( -. x = y -> ( y = z -> A. x y = z ) )
weu 2470	wff E! x ph
wmo 2471	wff E* x ph
df-eu 2474	|- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
df-mo 2475	|- ( E* x ph <-> ( E. x ph -> E! x ph ) )
ax-ext 2602	|- ( A. z ( z e. x <-> z e. y ) -> x = y )
cab 2608	class { x | ph }
df-clab 2609	|- ( x e. { y | ph } <-> [ x / y ] ph )
df-cleq 2615	|- ( A. x ( x e. y <-> x e. z ) -> y = z )   =>    |- ( A = B <-> A. x ( x e. A <-> x e. B ) )
df-clel 2618	|- ( A e. B <-> E. x ( x = A /\ x e. B ) )
wnfc 2751	wff F/_ x A
df-nfc 2753	|- ( F/_ x A <-> A. y F/ x y e. A )
wne 2794	wff A =/= B
df-ne 2795	|- ( A =/= B <-> -. A = B )
wnel 2897	wff A e/ B
df-nel 2898	|- ( A e/ B <-> -. A e. B )
wral 2912	wff A. x e. A ph
wrex 2913	wff E. x e. A ph
wreu 2914	wff E! x e. A ph
wrmo 2915	wff E* x e. A ph
crab 2916	class { x e. A | ph }
df-ral 2917	|- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
df-rex 2918	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
df-reu 2919	|- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
df-rmo 2920	|- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
df-rab 2921	|- { x e. A | ph } = { x | ( x e. A /\ ph ) }
cvv 3200	class _V
df-v 3202	|- _V = { x | x = x }
wcdeq 3418	wff CondEq ( x = y -> ph )
df-cdeq 3419	|- (CondEq ( x = y -> ph ) <-> ( x = y -> ph ) )
wsbc 3435	wff [. A / x ]. ph
df-sbc 3436	|- ( [. A / x ]. ph <-> A e. { x | ph } )
csb 3533	class [_ A / x ]_ B
df-csb 3534	|- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
cdif 3571	class ( A \ B )
cun 3572	class ( A u. B )
cin 3573	class ( A i^i B )
wss 3574	wff A C_ B
wpss 3575	wff A C. B
df-dif 3577	|- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
df-un 3579	|- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
df-in 3581	|- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
df-ss 3588	|- ( A C_ B <-> ( A i^i B ) = A )
df-pss 3590	|- ( A C. B <-> ( A C_ B /\ A =/= B ) )
csymdif 3843	class ( A /_\ B )
df-symdif 3844	|- ( A /_\ B ) = ( ( A \ B ) u. ( B \ A ) )
c0 3915	class (/)
df-nul 3916	|- (/) = ( _V \ _V )
cif 4086	class if ( ph , A , B )
df-if 4087	|- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
cpw 4158	class ~P A
df-pw 4160	|- ~P A = { x | x C_ A }
csn 4177	class { A }
df-sn 4178	|- { A } = { x | x = A }
cpr 4179	class { A , B }
df-pr 4180	|- { A , B } = ( { A } u. { B } )
ctp 4181	class { A , B , C }
df-tp 4182	|- { A , B , C } = ( { A , B } u. { C } )
cop 4183	class <. A , B >.
df-op 4184	|- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
cotp 4185	class <. A , B , C >.
df-ot 4186	|- <. A , B , C >. = <. <. A , B >. , C >.
cuni 4436	class U. A
df-uni 4437	|- U. A = { x | E. y ( x e. y /\ y e. A ) }
cint 4475	class |^| A
df-int 4476	|- |^| A = { x | A. y ( y e. A -> x e. y ) }
ciun 4520	class U_ x e. A B
ciin 4521	class |^|_ x e. A B
df-iun 4522	|- U_ x e. A B = { y | E. x e. A y e. B }
df-iin 4523	|- |^|_ x e. A B = { y | A. x e. A y e. B }
wdisj 4620	wff Disj x e. A B
df-disj 4621	|- (Disj x e. A B <-> A. y E* x e. A y e. B )
wbr 4653	wff A R B
df-br 4654	|- ( A R B <-> <. A , B >. e. R )
copab 4712	class { <. x , y >. | ph }
df-opab 4713	|- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
cmpt 4729	class ( x e. A |-> B )
df-mpt 4730	|- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
wtr 4752	wff Tr A
df-tr 4753	|- ( Tr A <-> U. A C_ A )
ax-rep 4771	|- ( A. w E. y A. z ( A. y ph -> z = y ) -> E. y A. z ( z e. y <-> E. w ( w e. x /\ A. y ph ) ) )
ax-sep 4781	|- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
ax-nul 4789	|- E. x A. y -. y e. x
ax-pow 4843	|- E. y A. z ( A. w ( w e. z -> w e. x ) -> z e. y )
ax-pr 4906	|- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
cid 5023	class _I
df-id 5024	|- _I = { <. x , y >. | x = y }
cep 5028	class _E
df-eprel 5029	|- _E = { <. x , y >. | x e. y }
wpo 5033	wff R Po A
wor 5034	wff R Or A
df-po 5035	|- ( R Po A <-> A. x e. A A. y e. A A. z e. A ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) )
df-so 5036	|- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) )
wfr 5070	wff R Fr A
wse 5071	wff R Se A
wwe 5072	wff R We A
df-fr 5073	|- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) )
df-se 5074	|- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
df-we 5075	|- ( R We A <-> ( R Fr A /\ R Or A ) )
cxp 5112	class ( A X. B )
ccnv 5113	class `' A
cdm 5114	class dom A
crn 5115	class ran A
cres 5116	class ( A |` B )
cima 5117	class ( A " B )
ccom 5118	class ( A o. B )
wrel 5119	wff Rel A
df-xp 5120	|- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
df-rel 5121	|- ( Rel A <-> A C_ ( _V X. _V ) )
df-cnv 5122	|- `' A = { <. x , y >. | y A x }
df-co 5123	|- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
df-dm 5124	|- dom A = { x | E. y x A y }
df-rn 5125	|- ran A = dom `' A
df-res 5126	|- ( A |` B ) = ( A i^i ( B X. _V ) )
df-ima 5127	|- ( A " B ) = ran ( A |` B )
cpred 5679	class Pred ( R , A , X )
df-pred 5680	|- Pred ( R , A , X ) = ( A i^i ( `' R " { X } ) )
word 5722	wff Ord A
con0 5723	class On
wlim 5724	wff Lim A
csuc 5725	class suc A
df-ord 5726	|- ( Ord A <-> ( Tr A /\ _E We A ) )
df-on 5727	|- On = { x | Ord x }
df-lim 5728	|- ( Lim A <-> ( Ord A /\ A =/= (/) /\ A = U. A ) )
df-suc 5729	|- suc A = ( A u. { A } )
cio 5849	class ( iota x ph )
df-iota 5851	|- ( iota x ph ) = U. { y | { x | ph } = { y } }
wfun 5882	wff Fun A
wfn 5883	wff A Fn B
wf 5884	wff F : A --> B
wf1 5885	wff F : A -1-1-> B
wfo 5886	wff F : A -onto-> B
wf1o 5887	wff F : A -1-1-onto-> B
cfv 5888	class ( F ` A )
wiso 5889	wff H Isom R , S ( A , B )
df-fun 5890	|- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) )
df-fn 5891	|- ( A Fn B <-> ( Fun A /\ dom A = B ) )
df-f 5892	|- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
df-f1 5893	|- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
df-fo 5894	|- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
df-f1o 5895	|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
df-fv 5896	|- ( F ` A ) = ( iota x A F x )
df-isom 5897	|- ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
crio 6610	class ( iota_ x e. A ph )
df-riota 6611	|- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
co 6650	class ( A F B )
coprab 6651	class { <. <. x , y >. , z >. | ph }
cmpt2 6652	class ( x e. A , y e. B |-> C )
df-ov 6653	|- ( A F B ) = ( F ` <. A , B >. )
df-oprab 6654	|- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
df-mpt2 6655	|- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
cof 6895	class oF R
cofr 6896	class oR R
df-of 6897	|- oF R = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
df-ofr 6898	|- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
crpss 6936	class [ C.]
df-rpss 6937	|- [ C.] = { <. x , y >. | x C. y }
ax-un 6949	|- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
com 7065	class om
df-om 7066	|- om = { x e. On | A. y ( Lim y -> x e. y ) }
c1st 7166	class 1st
c2nd 7167	class 2nd
df-1st 7168	|- 1st = ( x e. _V |-> U. dom { x } )
df-2nd 7169	|- 2nd = ( x e. _V |-> U. ran { x } )
csupp 7295	class supp
df-supp 7296	|- supp = ( x e. _V , z e. _V |-> { i e. dom x | ( x " { i } ) =/= { z } } )
ctpos 7351	class tpos F
df-tpos 7352	|- tpos F = ( F o. ( x e. ( `' dom F u. { (/) } ) |-> U. `' { x } ) )
ccur 7391	class curry A
cunc 7392	class uncurry A
df-cur 7393	|- curry F = ( x e. dom dom F |-> { <. y , z >. | <. x , y >. F z } )
df-unc 7394	|- uncurry F = { <. <. x , y >. , z >. | y ( F ` x ) z }
cund 7398	class Undef
df-undef 7399	|- Undef = ( s e. _V |-> ~P U. s )
cwrecs 7406	class wrecs ( R , A , F )
df-wrecs 7407	|- wrecs ( R , A , F ) = U. { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) }
wsmo 7442	wff Smo A
df-smo 7443	|- ( Smo A <-> ( A : dom A --> On /\ Ord dom A /\ A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) ) ) )
crecs 7467	class recs ( F )
df-recs 7468	|- recs ( F ) = wrecs ( _E , On , F )
crdg 7505	class rec ( F , I )
df-rdg 7506	|- rec ( F , I ) = recs ( ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) )
cseqom 7542	class seq𝜔 ( F , I )
df-seqom 7543	|- seq𝜔 ( F , I ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) " om )
c1o 7553	class 1o
c2o 7554	class 2o
c3o 7555	class 3o
c4o 7556	class 4o
coa 7557	class +o
comu 7558	class .o
coe 7559	class ^o
df-1o 7560	|- 1o = suc (/)
df-2o 7561	|- 2o = suc 1o
df-3o 7562	|- 3o = suc 2o
df-4o 7563	|- 4o = suc 3o
df-oadd 7564	|- +o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> suc z ) , x ) ` y ) )
df-omul 7565	|- .o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z +o x ) ) , (/) ) ` y ) )
df-oexp 7566	|- ^o = ( x e. On , y e. On |-> if ( x = (/) , ( 1o \ y ) , ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) ) )
wer 7739	wff R Er A
cec 7740	class [ A ] R
cqs 7741	class ( A /. R )
df-er 7742	|- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
df-ec 7744	|- [ A ] R = ( R " { A } )
df-qs 7748	|- ( A /. R ) = { y | E. x e. A y = [ x ] R }
cmap 7857	class ^m
cpm 7858	class ^pm
df-map 7859	|- ^m = ( x e. _V , y e. _V |-> { f | f : y --> x } )
df-pm 7860	|- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
cixp 7908	class X_ x e. A B
df-ixp 7909	|- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
cen 7952	class ~~
cdom 7953	class ~<_
csdm 7954	class ~<
cfn 7955	class Fin
df-en 7956	|- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
df-dom 7957	|- ~<_ = { <. x , y >. | E. f f : x -1-1-> y }
df-sdom 7958	|- ~< = ( ~<_ \ ~~ )
df-fin 7959	|- Fin = { x | E. y e. om x ~~ y }
cfsupp 8275	class finSupp
df-fsupp 8276	|- finSupp = { <. r , z >. | ( Fun r /\ ( r supp z ) e. Fin ) }
cfi 8316	class fi
df-fi 8317	|- fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )
csup 8346	class sup ( A , B , R )
cinf 8347	class inf ( A , B , R )
df-sup 8348	|- sup ( A , B , R ) = U. { x e. B | ( A. y e. A -. x R y /\ A. y e. B ( y R x -> E. z e. A y R z ) ) }
df-inf 8349	|- inf ( A , B , R ) = sup ( A , B , `' R )
coi 8414	class OrdIso ( R , A )
df-oi 8415	|- OrdIso ( R , A ) = if ( ( R We A /\ R Se A ) , (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) |` { x e. On | E. t e. A A. z e. (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) " x ) z R t } ) , (/) )
char 8461	class har
cwdom 8462	class ~<_*
df-har 8463	|- har = ( x e. _V |-> { y e. On | y ~<_ x } )
df-wdom 8464	|- ~<_* = { <. x , y >. | ( x = (/) \/ E. z z : y -onto-> x ) }
ax-reg 8497	|- ( E. y y e. x -> E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) )
ax-inf 8535	|- E. y ( x e. y /\ A. z ( z e. y -> E. w ( z e. w /\ w e. y ) ) )
ax-inf2 8538	|- E. x ( E. y ( y e. x /\ A. z -. z e. y ) /\ A. y ( y e. x -> E. z ( z e. x /\ A. w ( w e. z <-> ( w e. y \/ w = y ) ) ) ) )
ccnf 8558	class CNF
df-cnf 8559	|- CNF = ( x e. On , y e. On |-> ( f e. { g e. ( x ^m y ) | g finSupp (/) } |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) )
ctc 8612	class TC
df-tc 8613	|- TC = ( x e. _V |-> |^| { y | ( x C_ y /\ Tr y ) } )
cr1 8625	class R1
crnk 8626	class rank
df-r1 8627	|- R1 = rec ( ( x e. _V |-> ~P x ) , (/) )
df-rank 8628	|- rank = ( x e. _V |-> |^| { y e. On | x e. ( R1 ` suc y ) } )
ccrd 8761	class card
cale 8762	class aleph
ccf 8763	class cf
wacn 8764	class AC A
df-card 8765	|- card = ( x e. _V |-> |^| { y e. On | y ~~ x } )
df-aleph 8766	|- aleph = rec (har , om )
df-cf 8767	|- cf = ( x e. On |-> |^| { y | E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. v e. x E. u e. z v C_ u ) ) } )
df-acn 8768	|- AC A = { x | ( A e. _V /\ A. f e. ( ( ~P x \ { (/) } ) ^m A ) E. g A. y e. A ( g ` y ) e. ( f ` y ) ) }
wac 8938	wff CHOICE
df-ac 8939	|- (CHOICE <-> A. x E. f ( f C_ x /\ f Fn dom x ) )
ccda 8989	class +c
df-cda 8990	|- +c = ( x e. _V , y e. _V |-> ( ( x X. { (/) } ) u. ( y X. { 1o } ) ) )
cfin1a 9100	class FinIa
cfin2 9101	class FinII
cfin4 9102	class FinIV
cfin3 9103	class FinIII
cfin5 9104	class FinV
cfin6 9105	class FinVI
cfin7 9106	class FinVII
df-fin1a 9107	|- FinIa = { x | A. y e. ~P x ( y e. Fin \/ ( x \ y ) e. Fin ) }
df-fin2 9108	|- FinII = { x | A. y e. ~P ~P x ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) }
df-fin4 9109	|- FinIV = { x | -. E. y ( y C. x /\ y ~~ x ) }
df-fin3 9110	|- FinIII = { x | ~P x e. FinIV }
df-fin5 9111	|- FinV = { x | ( x = (/) \/ x ~< ( x +c x ) ) }
df-fin6 9112	|- FinVI = { x | ( x ~< 2o \/ x ~< ( x X. x ) ) }
df-fin7 9113	|- FinVII = { x | -. E. y e. ( On \ om ) x ~~ y }
ax-cc 9257	|- ( x ~~ om -> E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z ) )
ax-dc 9268	|- ( ( E. y E. z y x z /\ ran x C_ dom x ) -> E. f A. n e. om ( f ` n ) x ( f ` suc n ) )
ax-ac 9281	|- E. y A. z A. w ( ( z e. w /\ w e. x ) -> E. v A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v ) )
ax-ac2 9285	|- E. y A. z E. v A. u ( ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) ) \/ ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) ) )
cgch 9442	class GCH
df-gch 9443	|- GCH = ( Fin u. { x | A. y -. ( x ~< y /\ y ~< ~P x ) } )
cwina 9504	class InaccW
cina 9505	class Inacc
df-wina 9506	|- InaccW = { x | ( x =/= (/) /\ ( cf ` x ) = x /\ A. y e. x E. z e. x y ~< z ) }
df-ina 9507	|- Inacc = { x | ( x =/= (/) /\ ( cf ` x ) = x /\ A. y e. x ~P y ~< x ) }
cwun 9522	class WUni
cwunm 9523	class wUniCl
df-wun 9524	|- WUni = { u | ( Tr u /\ u =/= (/) /\ A. x e. u ( U. x e. u /\ ~P x e. u /\ A. y e. u { x , y } e. u ) ) }
df-wunc 9525	|- wUniCl = ( x e. _V |-> |^| { u e. WUni | x C_ u } )
ctsk 9570	class Tarski
df-tsk 9571	|- Tarski = { y | ( A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) /\ A. z e. ~P y ( z ~~ y \/ z e. y ) ) }
cgru 9612	class Univ
df-gru 9613	|- Univ = { u | ( Tr u /\ A. x e. u ( ~P x e. u /\ A. y e. u { x , y } e. u /\ A. y e. ( u ^m x ) U. ran y e. u ) ) }
ax-groth 9645	|- E. y ( x e. y /\ A. z e. y ( A. w ( w C_ z -> w e. y ) /\ E. w e. y A. v ( v C_ z -> v e. w ) ) /\ A. z ( z C_ y -> ( z ~~ y \/ z e. y ) ) )
ctskm 9659	class tarskiMap
df-tskm 9660	|- tarskiMap = ( x e. _V |-> |^| { y e. Tarski | x e. y } )
cnpi 9666	class N.
cpli 9667	class +N
cmi 9668	class .N
clti 9669	class <N
cplpq 9670	class +pQ
cmpq 9671	class .pQ
cltpq 9672	class <pQ
ceq 9673	class ~Q
cnq 9674	class Q.
c1q 9675	class 1Q
cerq 9676	class /Q
cplq 9677	class +Q
cmq 9678	class .Q
crq 9679	class *Q
cltq 9680	class <Q
cnp 9681	class P.
c1p 9682	class 1P
cpp 9683	class +P.
cmp 9684	class .P.
cltp 9685	class <P
cer 9686	class ~R
cnr 9687	class R.
c0r 9688	class 0R
c1r 9689	class 1R
cm1r 9690	class -1R
cplr 9691	class +R
cmr 9692	class .R
cltr 9693	class <R
df-ni 9694	|- N. = ( om \ { (/) } )
df-pli 9695	|- +N = ( +o |` ( N. X. N. ) )
df-mi 9696	|- .N = ( .o |` ( N. X. N. ) )
df-lti 9697	|- <N = ( _E i^i ( N. X. N. ) )
df-plpq 9730	|- +pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
df-mpq 9731	|- .pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
df-ltpq 9732	|- <pQ = { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) }
df-enq 9733	|- ~Q = { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) ) ) }
df-nq 9734	|- Q. = { x e. ( N. X. N. ) | A. y e. ( N. X. N. ) ( x ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` x ) ) }
df-erq 9735	|- /Q = ( ~Q i^i ( ( N. X. N. ) X. Q. ) )
df-plq 9736	|- +Q = ( ( /Q o. +pQ ) |` ( Q. X. Q. ) )
df-mq 9737	|- .Q = ( ( /Q o. .pQ ) |` ( Q. X. Q. ) )
df-1nq 9738	|- 1Q = <. 1o , 1o >.
df-rq 9739	|- *Q = ( `' .Q " { 1Q } )
df-ltnq 9740	|- <Q = ( <pQ i^i ( Q. X. Q. ) )
df-np 9803	|- P. = { x | ( ( (/) C. x /\ x C. Q. ) /\ A. y e. x ( A. z ( z <Q y -> z e. x ) /\ E. z e. x y <Q z ) ) }
df-1p 9804	|- 1P = { x | x <Q 1Q }
df-plp 9805	|- +P. = ( x e. P. , y e. P. |-> { w | E. v e. x E. u e. y w = ( v +Q u ) } )
df-mp 9806	|- .P. = ( x e. P. , y e. P. |-> { w | E. v e. x E. u e. y w = ( v .Q u ) } )
df-ltp 9807	|- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ x C. y ) }
df-enr 9877	|- ~R = { <. x , y >. | ( ( x e. ( P. X. P. ) /\ y e. ( P. X. P. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) ) }
df-nr 9878	|- R. = ( ( P. X. P. ) /. ~R )
df-plr 9879	|- +R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R ) ) }
df-mr 9880	|- .R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. f ) ) , ( ( w .P. f ) +P. ( v .P. u ) ) >. ] ~R ) ) }
df-ltr 9881	|- <R = { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\ ( z +P. u ) <P ( w +P. v ) ) ) }
df-0r 9882	|- 0R = [ <. 1P , 1P >. ] ~R
df-1r 9883	|- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
df-m1r 9884	|- -1R = [ <. 1P , ( 1P +P. 1P ) >. ] ~R
cc 9934	class CC
cr 9935	class RR
cc0 9936	class 0
c1 9937	class 1
ci 9938	class _i
caddc 9939	class +
cltrr 9940	class <RR
cmul 9941	class x.
df-c 9942	|- CC = ( R. X. R. )
df-0 9943	|- 0 = <. 0R , 0R >.
df-1 9944	|- 1 = <. 1R , 0R >.
df-i 9945	|- _i = <. 0R , 1R >.
df-r 9946	|- RR = ( R. X. { 0R } )
df-add 9947	|- + = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) }
df-mul 9948	|- x. = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) }
df-lt 9949	|- <RR = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ E. z E. w ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) /\ z <R w ) ) }
ax-cnex 9992	|- CC e. _V
ax-resscn 9993	|- RR C_ CC
ax-1cn 9994	|- 1 e. CC
ax-icn 9995	|- _i e. CC
ax-addcl 9996	|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
ax-addrcl 9997	|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
ax-mulcl 9998	|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
ax-mulrcl 9999	|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
ax-mulcom 10000	|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
ax-addass 10001	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
ax-mulass 10002	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
ax-distr 10003	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) )
ax-i2m1 10004	|- ( ( _i x. _i ) + 1 ) = 0
ax-1ne0 10005	|- 1 =/= 0
ax-1rid 10006	|- ( A e. RR -> ( A x. 1 ) = A )
ax-rnegex 10007	|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
ax-rrecex 10008	|- ( ( A e. RR /\ A =/= 0 ) -> E. x e. RR ( A x. x ) = 1 )
ax-cnre 10009	|- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
ax-pre-lttri 10010	|- ( ( A e. RR /\ B e. RR ) -> ( A <RR B <-> -. ( A = B \/ B <RR A ) ) )
ax-pre-lttrn 10011	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <RR B /\ B <RR C ) -> A <RR C ) )
ax-pre-ltadd 10012	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( C + A ) <RR ( C + B ) ) )
ax-pre-mulgt0 10013	|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )
ax-pre-sup 10014	|- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <RR x ) -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) )
ax-addf 10015	|- + : ( CC X. CC ) --> CC
ax-mulf 10016	|- x. : ( CC X. CC ) --> CC
cpnf 10071	class +oo
cmnf 10072	class -oo
cxr 10073	class RR*
clt 10074	class <
cle 10075	class <_
df-pnf 10076	|- +oo = ~P U. CC
df-mnf 10077	|- -oo = ~P +oo
df-xr 10078	|- RR* = ( RR u. { +oo , -oo } )
df-ltxr 10079	|- < = ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) )
df-le 10080	|- <_ = ( ( RR* X. RR* ) \ `' < )
cmin 10266	class -
cneg 10267	class -u A
df-sub 10268	|- - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
df-neg 10269	|- -u A = ( 0 - A )
cdiv 10684	class /
df-div 10685	|- / = ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) )
cn 11020	class NN
df-nn 11021	|- NN = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 1 ) " om )
c2 11070	class 2
c3 11071	class 3
c4 11072	class 4
c5 11073	class 5
c6 11074	class 6
c7 11075	class 7
c8 11076	class 8
c9 11077	class 9
c10 11078	class 10
df-2 11079	|- 2 = ( 1 + 1 )
df-3 11080	|- 3 = ( 2 + 1 )
df-4 11081	|- 4 = ( 3 + 1 )
df-5 11082	|- 5 = ( 4 + 1 )
df-6 11083	|- 6 = ( 5 + 1 )
df-7 11084	|- 7 = ( 6 + 1 )
df-8 11085	|- 8 = ( 7 + 1 )
df-9 11086	|- 9 = ( 8 + 1 )
df-10OLD 11087	|- 10 = ( 9 + 1 )
cn0 11292	class NN0
df-n0 11293	|- NN0 = ( NN u. { 0 } )
cxnn0 11363	class NN0*
df-xnn0 11364	|- NN0* = ( NN0 u. { +oo } )
cz 11377	class ZZ
df-z 11378	|- ZZ = { n e. RR | ( n = 0 \/ n e. NN \/ -u n e. NN ) }
cdc 11493	class ; A B
df-dec 11494	|- ; A B = ( ( ( 9 + 1 ) x. A ) + B )
cuz 11687	class ZZ>=
df-uz 11688	|- ZZ>= = ( j e. ZZ |-> { k e. ZZ | j <_ k } )
cq 11788	class QQ
df-q 11789	|- QQ = ( / " ( ZZ X. NN ) )
crp 11832	class RR+
df-rp 11833	|- RR+ = { x e. RR | 0 < x }
cxne 11943	class -e A
cxad 11944	class +e
cxmu 11945	class xe
df-xneg 11946	|- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
df-xadd 11947	|- +e = ( x e. RR* , y e. RR* |-> if ( x = +oo , if ( y = -oo , 0 , +oo ) , if ( x = -oo , if ( y = +oo , 0 , -oo ) , if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) ) ) )
df-xmul 11948	|- xe = ( x e. RR* , y e. RR* |-> if ( ( x = 0 \/ y = 0 ) , 0 , if ( ( ( ( 0 < y /\ x = +oo ) \/ ( y < 0 /\ x = -oo ) ) \/ ( ( 0 < x /\ y = +oo ) \/ ( x < 0 /\ y = -oo ) ) ) , +oo , if ( ( ( ( 0 < y /\ x = -oo ) \/ ( y < 0 /\ x = +oo ) ) \/ ( ( 0 < x /\ y = -oo ) \/ ( x < 0 /\ y = +oo ) ) ) , -oo , ( x x. y ) ) ) ) )
cioo 12175	class (,)
cioc 12176	class (,]
cico 12177	class [,)
cicc 12178	class [,]
df-ioo 12179	|- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
df-ioc 12180	|- (,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z <_ y ) } )
df-ico 12181	|- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
df-icc 12182	|- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
cfz 12326	class ...
df-fz 12327	|- ... = ( m e. ZZ , n e. ZZ |-> { k e. ZZ | ( m <_ k /\ k <_ n ) } )
cfzo 12465	class ..^
df-fzo 12466	|- ..^ = ( m e. ZZ , n e. ZZ |-> ( m ... ( n - 1 ) ) )
cfl 12591	class |_
cceil 12592	class ⌈
df-fl 12593	|- |_ = ( x e. RR |-> ( iota_ y e. ZZ ( y <_ x /\ x < ( y + 1 ) ) ) )
df-ceil 12594	|- ⌈ = ( x e. RR |-> -u ( |_ ` -u x ) )
cmo 12668	class mod
df-mod 12669	|- mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
cseq 12801	class seq M ( .+ , F )
df-seq 12802	|- seq M ( .+ , F ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om )
cexp 12860	class ^
df-exp 12861	|- ^ = ( x e. CC , y e. ZZ |-> if ( y = 0 , 1 , if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) ) ) )
cfa 13060	class !
df-fac 13061	|- ! = ( { <. 0 , 1 >. } u. seq 1 ( x. , _I ) )
cbc 13089	class _C
df-bc 13090	|- _C = ( n e. NN0 , k e. ZZ |-> if ( k e. ( 0 ... n ) , ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) , 0 ) )
chash 13117	class #
df-hash 13118	|- # = ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om ) o. card ) u. ( ( _V \ Fin ) X. { +oo } ) )
cword 13291	class Word S
clsw 13292	class lastS
cconcat 13293	class ++
cs1 13294	class <" A ">
csubstr 13295	class substr
csplice 13296	class splice
creverse 13297	class reverse
creps 13298	class repeatS
df-word 13299	|- Word S = { w | E. l e. NN0 w : ( 0..^ l ) --> S }
df-lsw 13300	|- lastS = ( w e. _V |-> ( w ` ( ( # ` w ) - 1 ) ) )
df-concat 13301	|- ++ = ( s e. _V , t e. _V |-> ( x e. ( 0..^ ( ( # ` s ) + ( # ` t ) ) ) |-> if ( x e. ( 0..^ ( # ` s ) ) , ( s ` x ) , ( t ` ( x - ( # ` s ) ) ) ) ) )
df-s1 13302	|- <" A "> = { <. 0 , ( _I ` A ) >. }
df-substr 13303	|- substr = ( s e. _V , b e. ( ZZ X. ZZ ) |-> if ( ( ( 1st ` b )..^ ( 2nd ` b ) ) C_ dom s , ( x e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( x + ( 1st ` b ) ) ) ) , (/) ) )
df-splice 13304	|- splice = ( s e. _V , b e. _V |-> ( ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) ++ ( 2nd ` b ) ) ++ ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) ) )
df-reverse 13305	|- reverse = ( s e. _V |-> ( x e. ( 0..^ ( # ` s ) ) |-> ( s ` ( ( ( # ` s ) - 1 ) - x ) ) ) )
df-reps 13306	|- repeatS = ( s e. _V , n e. NN0 |-> ( x e. ( 0..^ n ) |-> s ) )
ccsh 13534	class cyclShift
df-csh 13535	|- cyclShift = ( w e. { f | E. l e. NN0 f Fn ( 0..^ l ) } , n e. ZZ |-> if ( w = (/) , (/) , ( ( w substr <. ( n mod ( # ` w ) ) , ( # ` w ) >. ) ++ ( w substr <. 0 , ( n mod ( # ` w ) ) >. ) ) ) )
cs2 13586	class <" A B ">
cs3 13587	class <" A B C ">
cs4 13588	class <" A B C D ">
cs5 13589	class <" A B C D E ">
cs6 13590	class <" A B C D E F ">
cs7 13591	class <" A B C D E F G ">
cs8 13592	class <" A B C D E F G H ">
df-s2 13593	|- <" A B "> = ( <" A "> ++ <" B "> )
df-s3 13594	|- <" A B C "> = ( <" A B "> ++ <" C "> )
df-s4 13595	|- <" A B C D "> = ( <" A B C "> ++ <" D "> )
df-s5 13596	|- <" A B C D E "> = ( <" A B C D "> ++ <" E "> )
df-s6 13597	|- <" A B C D E F "> = ( <" A B C D E "> ++ <" F "> )
df-s7 13598	|- <" A B C D E F G "> = ( <" A B C D E F "> ++ <" G "> )
df-s8 13599	|- <" A B C D E F G H "> = ( <" A B C D E F G "> ++ <" H "> )
ctcl 13724	class t+
crtcl 13725	class t*
df-trcl 13726	|- t+ = ( x e. _V |-> |^| { z | ( x C_ z /\ ( z o. z ) C_ z ) } )
df-rtrcl 13727	|- t* = ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } )
crelexp 13760	class ^r
df-relexp 13761	|- ^r = ( r e. _V , n e. NN0 |-> if ( n = 0 , ( _I |` ( dom r u. ran r ) ) , ( seq 1 ( ( x e. _V , y e. _V |-> ( x o. r ) ) , ( z e. _V |-> r ) ) ` n ) ) )
crtrcl 13795	class t*rec
df-rtrclrec 13796	|- t*rec = ( r e. _V |-> U_ n e. NN0 ( r ^r n ) )
cshi 13806	class shift
df-shft 13807	|- shift = ( f e. _V , x e. CC |-> { <. y , z >. | ( y e. CC /\ ( y - x ) f z ) } )
csgn 13826	class sgn
df-sgn 13827	|- sgn = ( x e. RR* |-> if ( x = 0 , 0 , if ( x < 0 , -u 1 , 1 ) ) )
ccj 13836	class *
cre 13837	class Re
cim 13838	class Im
df-cj 13839	|- * = ( x e. CC |-> ( iota_ y e. CC ( ( x + y ) e. RR /\ ( _i x. ( x - y ) ) e. RR ) ) )
df-re 13840	|- Re = ( x e. CC |-> ( ( x + ( * ` x ) ) / 2 ) )
df-im 13841	|- Im = ( x e. CC |-> ( Re ` ( x / _i ) ) )
csqrt 13973	class sqr
cabs 13974	class abs
df-sqrt 13975	|- sqr = ( x e. CC |-> ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) ) )
df-abs 13976	|- abs = ( x e. CC |-> ( sqr ` ( x x. ( * ` x ) ) ) )
clsp 14201	class limsup
df-limsup 14202	|- limsup = ( x e. _V |-> inf ( ran ( k e. RR |-> sup ( ( ( x " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
cli 14215	class ~~>
crli 14216	class ~~> r
co1 14217	class O(1)
clo1 14218	class <_O(1)
df-clim 14219	|- ~~> = { <. f , y >. | ( y e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) ) }
df-rlim 14220	|- ~~> r = { <. f , x >. | ( ( f e. ( CC ^pm RR ) /\ x e. CC ) /\ A. y e. RR+ E. z e. RR A. w e. dom f ( z <_ w -> ( abs ` ( ( f ` w ) - x ) ) < y ) ) }
df-o1 14221	|- O(1) = { f e. ( CC ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m }
df-lo1 14222	|- <_O(1) = { f e. ( RR ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( f ` y ) <_ m }
csu 14416	class sum_ k e. A B
df-sum 14417	|- sum_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) )
cprod 14635	class prod_ k e. A B
df-prod 14636	|- prod_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y ) /\ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) )
cfallfac 14735	class FallFac
crisefac 14736	class RiseFac
df-risefac 14737	|- RiseFac = ( x e. CC , n e. NN0 |-> prod_ k e. ( 0 ... ( n - 1 ) ) ( x + k ) )
df-fallfac 14738	|- FallFac = ( x e. CC , n e. NN0 |-> prod_ k e. ( 0 ... ( n - 1 ) ) ( x - k ) )
cbp 14777	class BernPoly
df-bpoly 14778	|- BernPoly = ( m e. NN0 , x e. CC |-> (wrecs ( < , NN0 , ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) ) ` m ) )
ce 14792	class exp
ceu 14793	class _e
csin 14794	class sin
ccos 14795	class cos
ctan 14796	class tan
cpi 14797	class pi
df-ef 14798	|- exp = ( x e. CC |-> sum_ k e. NN0 ( ( x ^ k ) / ( ! ` k ) ) )
df-e 14799	|- _e = ( exp ` 1 )
df-sin 14800	|- sin = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) )
df-cos 14801	|- cos = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) )
df-tan 14802	|- tan = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) / ( cos ` x ) ) )
df-pi 14803	|- pi = inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < )
cdvds 14983	class ||
df-dvds 14984	|- || = { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. n e. ZZ ( n x. x ) = y ) }
cbits 15141	class bits
csad 15142	class sadd
csmu 15143	class smul
df-bits 15144	|- bits = ( n e. ZZ |-> { m e. NN0 | -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) ) } )
df-sad 15173	|- sadd = ( x e. ~P NN0 , y e. ~P NN0 |-> { k e. NN0 | hadd ( k e. x , k e. y , (/) e. ( seq 0 ( ( c e. 2o , m e. NN0 |-> if (cadd ( m e. x , m e. y , (/) e. c ) , 1o , (/) ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) ` k ) ) } )
df-smu 15198	|- smul = ( x e. ~P NN0 , y e. ~P NN0 |-> { k e. NN0 | k e. ( seq 0 ( ( p e. ~P NN0 , m e. NN0 |-> ( p sadd { n e. NN0 | ( m e. x /\ ( n - m ) e. y ) } ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) ` ( k + 1 ) ) } )
cgcd 15216	class gcd
df-gcd 15217	|- gcd = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) )
clcm 15301	class lcm
clcmf 15302	class lcm
df-lcm 15303	|- lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) )
df-lcmf 15304	|- lcm = ( z e. ~P ZZ |-> if ( 0 e. z , 0 , inf ( { n e. NN | A. m e. z m || n } , RR , < ) ) )
cprime 15385	class Prime
df-prm 15386	|- Prime = { p e. NN | { n e. NN | n || p } ~~ 2o }
cnumer 15441	class numer
cdenom 15442	class denom
df-numer 15443	|- numer = ( y e. QQ |-> ( 1st ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ y = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )
df-denom 15444	|- denom = ( y e. QQ |-> ( 2nd ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ y = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )
codz 15468	class odZ
cphi 15469	class phi
df-odz 15470	|- odZ = ( n e. NN |-> ( x e. { x e. ZZ | ( x gcd n ) = 1 } |-> inf ( { m e. NN | n || ( ( x ^ m ) - 1 ) } , RR , < ) ) )
df-phi 15471	|- phi = ( n e. NN |-> ( # ` { x e. ( 1 ... n ) | ( x gcd n ) = 1 } ) )
cpc 15541	class pCnt
df-pc 15542	|- pCnt = ( p e. Prime , r e. QQ |-> if ( r = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( r = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) ) ) ) )
cgz 15633	class ZZ[_i]
df-gz 15634	|- ZZ[_i] = { x e. CC | ( ( Re ` x ) e. ZZ /\ ( Im ` x ) e. ZZ ) }
cvdwa 15669	class AP
cvdwm 15670	class MonoAP
cvdwp 15671	class PolyAP
df-vdwap 15672	|- AP = ( k e. NN0 |-> ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( k - 1 ) ) |-> ( a + ( m x. d ) ) ) ) )
df-vdwmc 15673	|- MonoAP = { <. k , f >. | E. c ( ran (AP ` k ) i^i ~P ( `' f " { c } ) ) =/= (/) }
df-vdwpc 15674	|- PolyAP = { <. <. m , k >. , f >. | E. a e. NN E. d e. ( NN ^m ( 1 ... m ) ) ( A. i e. ( 1 ... m ) ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) C_ ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) ) = m ) }
cram 15703	class Ramsey
df-ram 15705	|- Ramsey = ( m e. NN0 , r e. _V |-> inf ( { n e. NN0 | A. s ( n <_ ( # ` s ) -> A. f e. ( dom r ^m { y e. ~P s | ( # ` y ) = m } ) E. c e. dom r E. x e. ~P s ( ( r ` c ) <_ ( # ` x ) /\ A. y e. ~P x ( ( # ` y ) = m -> ( f ` y ) = c ) ) ) } , RR* , < ) )
cprmo 15735	class #p
df-prmo 15736	|- #p = ( n e. NN0 |-> prod_ k e. ( 1 ... n ) if ( k e. Prime , k , 1 ) )
cstr 15853	class Struct
cnx 15854	class ndx
csts 15855	class sSet
cslot 15856	class Slot A
cbs 15857	class Base
cress 15858	class ↾s
df-struct 15859	|- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
df-ndx 15860	|- ndx = ( _I |` NN )
df-slot 15861	|- Slot A = ( x e. _V |-> ( x ` A ) )
df-base 15863	|- Base = Slot 1
df-sets 15864	|- sSet = ( s e. _V , e e. _V |-> ( ( s |` ( _V \ dom { e } ) ) u. { e } ) )
df-ress 15865	|- ↾s = ( w e. _V , x e. _V |-> if ( ( Base ` w ) C_ x , w , ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) ) )
cplusg 15941	class +g
cmulr 15942	class .r
cstv 15943	class *r
csca 15944	class Scalar
cvsca 15945	class .s
cip 15946	class .i
cts 15947	class TopSet
cple 15948	class le
coc 15949	class oc
cds 15950	class dist
cunif 15951	class UnifSet
chom 15952	class Hom
cco 15953	class comp
df-plusg 15954	|- +g = Slot 2
df-mulr 15955	|- .r = Slot 3
df-starv 15956	|- *r = Slot 4
df-sca 15957	|- Scalar = Slot 5
df-vsca 15958	|- .s = Slot 6
df-ip 15959	|- .i = Slot 8
df-tset 15960	|- TopSet = Slot 9
df-ple 15961	|- le = Slot ; 1 0
df-ocomp 15963	|- oc = Slot ; 1 1
df-ds 15964	|- dist = Slot ; 1 2
df-unif 15965	|- UnifSet = Slot ; 1 3
df-hom 15966	|- Hom = Slot ; 1 4
df-cco 15967	|- comp = Slot ; 1 5
crest 16081	class ↾t
ctopn 16082	class TopOpen
df-rest 16083	|- ↾t = ( j e. _V , x e. _V |-> ran ( y e. j |-> ( y i^i x ) ) )
df-topn 16084	|- TopOpen = ( w e. _V |-> ( (TopSet ` w ) ↾t ( Base ` w ) ) )
ctg 16098	class topGen
cpt 16099	class Xt_
c0g 16100	class 0g
cgsu 16101	class gsumg
df-0g 16102	|- 0g = ( g e. _V |-> ( iota e ( e e. ( Base ` g ) /\ A. x e. ( Base ` g ) ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) ) ) )
df-gsum 16103	|- gsumg = ( w e. _V , f e. _V |-> [_ { x e. ( Base ` w ) | A. y e. ( Base ` w ) ( ( x ( +g ` w ) y ) = y /\ ( y ( +g ` w ) x ) = y ) } / o ]_ if ( ran f C_ o , ( 0g ` w ) , if ( dom f e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) , ( iota x E. g [. ( `' f " ( _V \ o ) ) / y ]. ( g : ( 1 ... ( # ` y ) ) -1-1-onto-> y /\ x = ( seq 1 ( ( +g ` w ) , ( f o. g ) ) ` ( # ` y ) ) ) ) ) ) )
df-topgen 16104	|- topGen = ( x e. _V |-> { y | y C_ U. ( x i^i ~P y ) } )
df-pt 16105	|- Xt_ = ( f e. _V |-> ( topGen ` { x | E. g ( ( g Fn dom f /\ A. y e. dom f ( g ` y ) e. ( f ` y ) /\ E. z e. Fin A. y e. ( dom f \ z ) ( g ` y ) = U. ( f ` y ) ) /\ x = X_ y e. dom f ( g ` y ) ) } ) )
cprds 16106	class X_s
cpws 16107	class ^s
df-prds 16108	|- X_s = ( s e. _V , r e. _V |-> [_ X_ x e. dom r ( Base ` ( r ` x ) ) / v ]_ [_ ( f e. v , g e. v |-> X_ x e. dom r ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) ) / h ]_ ( ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( f e. v , g e. v |-> ( x e. dom r |-> ( ( f ` x ) ( +g ` ( r ` x ) ) ( g ` x ) ) ) ) >. , <. ( .r ` ndx ) , ( f e. v , g e. v |-> ( x e. dom r |-> ( ( f ` x ) ( .r ` ( r ` x ) ) ( g ` x ) ) ) ) >. } u. { <. (Scalar ` ndx ) , s >. , <. ( .s ` ndx ) , ( f e. ( Base ` s ) , g e. v |-> ( x e. dom r |-> ( f ( .s ` ( r ` x ) ) ( g ` x ) ) ) ) >. , <. ( .i ` ndx ) , ( f e. v , g e. v |-> ( s gsumg ( x e. dom r |-> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) ) ) ) >. } ) u. ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. r ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ v /\ A. x e. dom r ( f ` x ) ( le ` ( r ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. v , g e. v |-> sup ( ( ran ( x e. dom r |-> ( ( f ` x ) ( dist ` ( r ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } u. { <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( a e. ( v X. v ) , c e. v |-> ( d e. ( c h ( 2nd ` a ) ) , e e. ( h ` a ) |-> ( x e. dom r |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( r ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } ) ) )
df-pws 16110	|- ^s = ( r e. _V , i e. _V |-> ( (Scalar ` r ) X_s ( i X. { r } ) ) )
cordt 16159	class ordTop
cxrs 16160	class RR*s
df-ordt 16161	|- ordTop = ( r e. _V |-> ( topGen ` ( fi ` ( { dom r } u. ran ( ( x e. dom r |-> { y e. dom r | -. y r x } ) u. ( x e. dom r |-> { y e. dom r | -. x r y } ) ) ) ) ) )
df-xrs 16162	|- RR*s = ( { <. ( Base ` ndx ) , RR* >. , <. ( +g ` ndx ) , +e >. , <. ( .r ` ndx ) , xe >. } u. { <. (TopSet ` ndx ) , (ordTop ` <_ ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( x e. RR* , y e. RR* |-> if ( x <_ y , ( y +e -e x ) , ( x +e -e y ) ) ) >. } )
cqtop 16163	class qTop
cimas 16164	class "s
cqus 16165	class /.s
cxps 16166	class X.s
df-qtop 16167	|- qTop = ( j e. _V , f e. _V |-> { s e. ~P ( f " U. j ) | ( ( `' f " s ) i^i U. j ) e. j } )
df-imas 16168	|- "s = ( f e. _V , r e. _V |-> [_ ( Base ` r ) / v ]_ ( ( { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } u. { <. (Scalar ` ndx ) , (Scalar ` r ) >. , <. ( .s ` ndx ) , U_ q e. v ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } >. } ) u. { <. (TopSet ` ndx ) , ( ( TopOpen ` r ) qTop f ) >. , <. ( le ` ndx ) , ( ( f o. ( le ` r ) ) o. `' f ) >. , <. ( dist ` ndx ) , ( x e. ran f , y e. ran f |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) , RR* , < ) ) >. } ) )
df-qus 16169	|- /.s = ( r e. _V , e e. _V |-> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) )
df-xps 16170	|- X.s = ( r e. _V , s e. _V |-> ( `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> `' ( { x } +c { y } ) ) "s ( (Scalar ` r ) X_s `' ( { r } +c { s } ) ) ) )
cmre 16242	class Moore
cmrc 16243	class mrCls
cmri 16244	class mrInd
cacs 16245	class ACS
df-mre 16246	|- Moore = ( x e. _V |-> { c e. ~P ~P x | ( x e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) } )
df-mrc 16247	|- mrCls = ( c e. U. ran Moore |-> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) )
df-mri 16248	|- mrInd = ( c e. U. ran Moore |-> { s e. ~P U. c | A. x e. s -. x e. ( (mrCls ` c ) ` ( s \ { x } ) ) } )
df-acs 16249	|- ACS = ( x e. _V |-> { c e. (Moore ` x ) | E. f ( f : ~P x --> ~P x /\ A. s e. ~P x ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) } )
ccat 16325	class Cat
ccid 16326	class Id
chomf 16327	class Hom f
ccomf 16328	class compf
df-cat 16329	|- Cat = { c | [. ( Base ` c ) / b ]. [. ( Hom ` c ) / h ]. [. (comp ` c ) / o ]. A. x e. b ( E. g e. ( x h x ) A. y e. b ( A. f e. ( y h x ) ( g ( <. y , x >. o x ) f ) = f /\ A. f e. ( x h y ) ( f ( <. x , x >. o y ) g ) = f ) /\ A. y e. b A. z e. b A. f e. ( x h y ) A. g e. ( y h z ) ( ( g ( <. x , y >. o z ) f ) e. ( x h z ) /\ A. w e. b A. k e. ( z h w ) ( ( k ( <. y , z >. o w ) g ) ( <. x , y >. o w ) f ) = ( k ( <. x , z >. o w ) ( g ( <. x , y >. o z ) f ) ) ) ) }
df-cid 16330	|- Id = ( c e. Cat |-> [_ ( Base ` c ) / b ]_ [_ ( Hom ` c ) / h ]_ [_ (comp ` c ) / o ]_ ( x e. b |-> ( iota_ g e. ( x h x ) A. y e. b ( A. f e. ( y h x ) ( g ( <. y , x >. o x ) f ) = f /\ A. f e. ( x h y ) ( f ( <. x , x >. o y ) g ) = f ) ) ) )
df-homf 16331	|- Hom f = ( c e. _V |-> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( x ( Hom ` c ) y ) ) )
df-comf 16332	|- compf = ( c e. _V |-> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) , y e. ( Base ` c ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` c ) y ) , f e. ( ( Hom ` c ) ` x ) |-> ( g ( x (comp ` c ) y ) f ) ) ) )
coppc 16371	class oppCat
df-oppc 16372	|- oppCat = ( f e. _V |-> ( ( f sSet <. ( Hom ` ndx ) , tpos ( Hom ` f ) >. ) sSet <. (comp ` ndx ) , ( u e. ( ( Base ` f ) X. ( Base ` f ) ) , z e. ( Base ` f ) |-> tpos ( <. z , ( 2nd ` u ) >. (comp ` f ) ( 1st ` u ) ) ) >. ) )
cmon 16388	class Mono
cepi 16389	class Epi
df-mon 16390	|- Mono = ( c e. Cat |-> [_ ( Base ` c ) / b ]_ [_ ( Hom ` c ) / h ]_ ( x e. b , y e. b |-> { f e. ( x h y ) | A. z e. b Fun `' ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) } ) )
df-epi 16391	|- Epi = ( c e. Cat |-> tpos (Mono ` (oppCat ` c ) ) )
csect 16404	class Sect
cinv 16405	class Inv
ciso 16406	class Iso
df-sect 16407	|- Sect = ( c e. Cat |-> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> { <. f , g >. | [. ( Hom ` c ) / h ]. ( ( f e. ( x h y ) /\ g e. ( y h x ) ) /\ ( g ( <. x , y >. (comp ` c ) x ) f ) = ( ( Id ` c ) ` x ) ) } ) )
df-inv 16408	|- Inv = ( c e. Cat |-> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( ( x (Sect ` c ) y ) i^i `' ( y (Sect ` c ) x ) ) ) )
df-iso 16409	|- Iso = ( c e. Cat |-> ( ( x e. _V |-> dom x ) o. (Inv ` c ) ) )
ccic 16455	class ~=c𝑐
df-cic 16456	|- ~=c𝑐 = ( c e. Cat |-> ( ( Iso ` c ) supp (/) ) )
cssc 16467	class C_cat
cresc 16468	class |`cat
csubc 16469	class Subcat
df-ssc 16470	|- C_cat = { <. h , j >. | E. t ( j Fn ( t X. t ) /\ E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( j ` x ) ) }
df-resc 16471	|- |`cat = ( c e. _V , h e. _V |-> ( ( c ↾s dom dom h ) sSet <. ( Hom ` ndx ) , h >. ) )
df-subc 16472	|- Subcat = ( c e. Cat |-> { h | ( h C_cat ( Hom f ` c ) /\ [. dom dom h / s ]. A. x e. s ( ( ( Id ` c ) ` x ) e. ( x h x ) /\ A. y e. s A. z e. s A. f e. ( x h y ) A. g e. ( y h z ) ( g ( <. x , y >. (comp ` c ) z ) f ) e. ( x h z ) ) ) } )
cfunc 16514	class Func
cidfu 16515	class idfunc
ccofu 16516	class o.func
cresf 16517	class |`f
df-func 16518	|- Func = ( t e. Cat , u e. Cat |-> { <. f , g >. | [. ( Base ` t ) / b ]. ( f : b --> ( Base ` u ) /\ g e. X_ z e. ( b X. b ) ( ( ( f ` ( 1st ` z ) ) ( Hom ` u ) ( f ` ( 2nd ` z ) ) ) ^m ( ( Hom ` t ) ` z ) ) /\ A. x e. b ( ( ( x g x ) ` ( ( Id ` t ) ` x ) ) = ( ( Id ` u ) ` ( f ` x ) ) /\ A. y e. b A. z e. b A. m e. ( x ( Hom ` t ) y ) A. n e. ( y ( Hom ` t ) z ) ( ( x g z ) ` ( n ( <. x , y >. (comp ` t ) z ) m ) ) = ( ( ( y g z ) ` n ) ( <. ( f ` x ) , ( f ` y ) >. (comp ` u ) ( f ` z ) ) ( ( x g y ) ` m ) ) ) ) } )
df-idfu 16519	|- idfunc = ( t e. Cat |-> [_ ( Base ` t ) / b ]_ <. ( _I |` b ) , ( z e. ( b X. b ) |-> ( _I |` ( ( Hom ` t ) ` z ) ) ) >. )
df-cofu 16520	|- o.func = ( g e. _V , f e. _V |-> <. ( ( 1st ` g ) o. ( 1st ` f ) ) , ( x e. dom dom ( 2nd ` f ) , y e. dom dom ( 2nd ` f ) |-> ( ( ( ( 1st ` f ) ` x ) ( 2nd ` g ) ( ( 1st ` f ) ` y ) ) o. ( x ( 2nd ` f ) y ) ) ) >. )
df-resf 16521	|- |`f = ( f e. _V , h e. _V |-> <. ( ( 1st ` f ) |` dom dom h ) , ( x e. dom h |-> ( ( ( 2nd ` f ) ` x ) |` ( h ` x ) ) ) >. )
cful 16562	class Full
cfth 16563	class Faith
df-full 16564	|- Full = ( c e. Cat , d e. Cat |-> { <. f , g >. | ( f ( c Func d ) g /\ A. x e. ( Base ` c ) A. y e. ( Base ` c ) ran ( x g y ) = ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) ) } )
df-fth 16565	|- Faith = ( c e. Cat , d e. Cat |-> { <. f , g >. | ( f ( c Func d ) g /\ A. x e. ( Base ` c ) A. y e. ( Base ` c ) Fun `' ( x g y ) ) } )
cnat 16601	class Nat
cfuc 16602	class FuncCat
df-nat 16603	|- Nat = ( t e. Cat , u e. Cat |-> ( f e. ( t Func u ) , g e. ( t Func u ) |-> [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` t ) ( ( r ` x ) ( Hom ` u ) ( s ` x ) ) | A. x e. ( Base ` t ) A. y e. ( Base ` t ) A. h e. ( x ( Hom ` t ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` u ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` u ) ( s ` y ) ) ( a ` x ) ) } ) )
df-fuc 16604	|- FuncCat = ( t e. Cat , u e. Cat |-> { <. ( Base ` ndx ) , ( t Func u ) >. , <. ( Hom ` ndx ) , ( t Nat u ) >. , <. (comp ` ndx ) , ( v e. ( ( t Func u ) X. ( t Func u ) ) , h e. ( t Func u ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( t Nat u ) h ) , a e. ( f ( t Nat u ) g ) |-> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } )
cinito 16638	class InitO
ctermo 16639	class TermO
czeroo 16640	class ZeroO
df-inito 16641	|- InitO = ( c e. Cat |-> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } )
df-termo 16642	|- TermO = ( c e. Cat |-> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( b ( Hom ` c ) a ) } )
df-zeroo 16643	|- ZeroO = ( c e. Cat |-> ( (InitO ` c ) i^i (TermO ` c ) ) )
cdoma 16670	class domA
ccoda 16671	class coda
carw 16672	class Nat
choma 16673	class Homa
df-doma 16674	|- domA = ( 1st o. 1st )
df-coda 16675	|- coda = ( 2nd o. 1st )
df-homa 16676	|- Homa = ( c e. Cat |-> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) |-> ( { x } X. ( ( Hom ` c ) ` x ) ) ) )
df-arw 16677	|- Nat = ( c e. Cat |-> U. ran (Homa ` c ) )
cida 16703	class Ida
ccoa 16704	class compa
df-ida 16705	|- Ida = ( c e. Cat |-> ( x e. ( Base ` c ) |-> <. x , x , ( ( Id ` c ) ` x ) >. ) )
df-coa 16706	|- compa = ( c e. Cat |-> ( g e. (Nat ` c ) , f e. { h e. (Nat ` c ) | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) ( 2nd ` f ) ) >. ) )
csetc 16725	class SetCat
df-setc 16726	|- SetCat = ( u e. _V |-> { <. ( Base ` ndx ) , u >. , <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( y ^m x ) ) >. , <. (comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( z ^m ( 2nd ` v ) ) , f e. ( ( 2nd ` v ) ^m ( 1st ` v ) ) |-> ( g o. f ) ) ) >. } )
ccatc 16744	class CatCat
df-catc 16745	|- CatCat = ( u e. _V |-> [_ ( u i^i Cat ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. , <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } )
cestrc 16762	class ExtStrCat
df-estrc 16763	|- ExtStrCat = ( u e. _V |-> { <. ( Base ` ndx ) , u >. , <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. (comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } )
cxpc 16808	class X.c
c1stf 16809	class 1stF
c2ndf 16810	class 2ndF
cprf 16811	class ⟨,⟩F
df-xpc 16812	|- X.c = ( r e. _V , s e. _V |-> [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ [_ ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) / h ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } )
df-1stf 16813	|- 1stF = ( r e. Cat , s e. Cat |-> [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ <. ( 1st |` b ) , ( x e. b , y e. b |-> ( 1st |` ( x ( Hom ` ( r X.c s ) ) y ) ) ) >. )
df-2ndf 16814	|- 2ndF = ( r e. Cat , s e. Cat |-> [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ <. ( 2nd |` b ) , ( x e. b , y e. b |-> ( 2nd |` ( x ( Hom ` ( r X.c s ) ) y ) ) ) >. )
df-prf 16815	|- ⟨,⟩F = ( f e. _V , g e. _V |-> [_ dom ( 1st ` f ) / b ]_ <. ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) , ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) >. )
cevlf 16849	class evalF
ccurf 16850	class curryF
cuncf 16851	class uncurryF
cdiag 16852	class Δfunc
df-evlf 16853	|- evalF = ( c e. Cat , d e. Cat |-> <. ( f e. ( c Func d ) , x e. ( Base ` c ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( c Func d ) X. ( Base ` c ) ) , y e. ( ( c Func d ) X. ( Base ` c ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( c Nat d ) n ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` d ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. )
df-curf 16854	|- curryF = ( e e. _V , f e. _V |-> [_ ( 1st ` e ) / c ]_ [_ ( 2nd ` e ) / d ]_ <. ( x e. ( Base ` c ) |-> <. ( y e. ( Base ` d ) |-> ( x ( 1st ` f ) y ) ) , ( y e. ( Base ` d ) , z e. ( Base ` d ) |-> ( g e. ( y ( Hom ` d ) z ) |-> ( ( ( Id ` c ) ` x ) ( <. x , y >. ( 2nd ` f ) <. x , z >. ) g ) ) ) >. ) , ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( g e. ( x ( Hom ` c ) y ) |-> ( z e. ( Base ` d ) |-> ( g ( <. x , z >. ( 2nd ` f ) <. y , z >. ) ( ( Id ` d ) ` z ) ) ) ) ) >. )
df-uncf 16855	|- uncurryF = ( c e. _V , f e. _V |-> ( ( ( c ` 1 ) evalF ( c ` 2 ) ) o.func ( ( f o.func ( ( c ` 0 ) 1stF ( c ` 1 ) ) ) ⟨,⟩F ( ( c ` 0 ) 2ndF ( c ` 1 ) ) ) ) )
df-diag 16856	|- Δfunc = ( c e. Cat , d e. Cat |-> ( <. c , d >. curryF ( c 1stF d ) ) )
chof 16888	class HomF
cyon 16889	class Yon
df-hof 16890	|- HomF = ( c e. Cat |-> <. ( Hom f ` c ) , [_ ( Base ` c ) / b ]_ ( x e. ( b X. b ) , y e. ( b X. b ) |-> ( f e. ( ( 1st ` y ) ( Hom ` c ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` c ) ` x ) |-> ( ( g ( x (comp ` c ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` c ) ( 2nd ` y ) ) f ) ) ) ) >. )
df-yon 16891	|- Yon = ( c e. Cat |-> ( <. c , (oppCat ` c ) >. curryF (HomF ` (oppCat ` c ) ) ) )
cpreset 16926	class Preset
cdrs 16927	class Dirset
df-preset 16928	|- Preset = { f | [. ( Base ` f ) / b ]. [. ( le ` f ) / r ]. A. x e. b A. y e. b A. z e. b ( x r x /\ ( ( x r y /\ y r z ) -> x r z ) ) }
df-drs 16929	|- Dirset = { f e. Preset | [. ( Base ` f ) / b ]. [. ( le ` f ) / r ]. ( b =/= (/) /\ A. x e. b A. y e. b E. z e. b ( x r z /\ y r z ) ) }
cpo 16940	class Poset
cplt 16941	class lt
club 16942	class lub
cglb 16943	class glb
cjn 16944	class join
cmee 16945	class meet
df-poset 16946	|- Poset = { f | E. b E. r ( b = ( Base ` f ) /\ r = ( le ` f ) /\ A. x e. b A. y e. b A. z e. b ( x r x /\ ( ( x r y /\ y r x ) -> x = y ) /\ ( ( x r y /\ y r z ) -> x r z ) ) ) }
df-plt 16958	|- lt = ( p e. _V |-> ( ( le ` p ) \ _I ) )
df-lub 16974	|- lub = ( p e. _V |-> ( ( s e. ~P ( Base ` p ) |-> ( iota_ x e. ( Base ` p ) ( A. y e. s y ( le ` p ) x /\ A. z e. ( Base ` p ) ( A. y e. s y ( le ` p ) z -> x ( le ` p ) z ) ) ) ) |` { s | E! x e. ( Base ` p ) ( A. y e. s y ( le ` p ) x /\ A. z e. ( Base ` p ) ( A. y e. s y ( le ` p ) z -> x ( le ` p ) z ) ) } ) )
df-glb 16975	|- glb = ( p e. _V |-> ( ( s e. ~P ( Base ` p ) |-> ( iota_ x e. ( Base ` p ) ( A. y e. s x ( le ` p ) y /\ A. z e. ( Base ` p ) ( A. y e. s z ( le ` p ) y -> z ( le ` p ) x ) ) ) ) |` { s | E! x e. ( Base ` p ) ( A. y e. s x ( le ` p ) y /\ A. z e. ( Base ` p ) ( A. y e. s z ( le ` p ) y -> z ( le ` p ) x ) ) } ) )
df-join 16976	|- join = ( p e. _V |-> { <. <. x , y >. , z >. | { x , y } ( lub ` p ) z } )
df-meet 16977	|- meet = ( p e. _V |-> { <. <. x , y >. , z >. | { x , y } ( glb ` p ) z } )
ctos 17033	class Toset
df-toset 17034	|- Toset = { f e. Poset | [. ( Base ` f ) / b ]. [. ( le ` f ) / r ]. A. x e. b A. y e. b ( x r y \/ y r x ) }
cp0 17037	class 0.
cp1 17038	class 1.
df-p0 17039	|- 0. = ( p e. _V |-> ( ( glb ` p ) ` ( Base ` p ) ) )
df-p1 17040	|- 1. = ( p e. _V |-> ( ( lub ` p ) ` ( Base ` p ) ) )
clat 17045	class Lat
df-lat 17046	|- Lat = { p e. Poset | ( dom ( join ` p ) = ( ( Base ` p ) X. ( Base ` p ) ) /\ dom ( meet ` p ) = ( ( Base ` p ) X. ( Base ` p ) ) ) }
ccla 17107	class CLat
df-clat 17108	|- CLat = { p e. Poset | ( dom ( lub ` p ) = ~P ( Base ` p ) /\ dom ( glb ` p ) = ~P ( Base ` p ) ) }
codu 17128	class ODual
df-odu 17129	|- ODual = ( w e. _V |-> ( w sSet <. ( le ` ndx ) , `' ( le ` w ) >. ) )
cipo 17151	class toInc
df-ipo 17152	|- toInc = ( f e. _V |-> [_ { <. x , y >. | ( { x , y } C_ f /\ x C_ y ) } / o ]_ ( { <. ( Base ` ndx ) , f >. , <. (TopSet ` ndx ) , (ordTop ` o ) >. } u. { <. ( le ` ndx ) , o >. , <. ( oc ` ndx ) , ( x e. f |-> U. { y e. f | ( y i^i x ) = (/) } ) >. } ) )
cdlat 17191	class DLat
df-dlat 17192	|- DLat = { k e. Lat | [. ( Base ` k ) / b ]. [. ( join ` k ) / j ]. [. ( meet ` k ) / m ]. A. x e. b A. y e. b A. z e. b ( x m ( y j z ) ) = ( ( x m y ) j ( x m z ) ) }
cps 17198	class PosetRel
ctsr 17199	class TosetRel
df-ps 17200	|- PosetRel = { r | ( Rel r /\ ( r o. r ) C_ r /\ ( r i^i `' r ) = ( _I |` U. U. r ) ) }
df-tsr 17201	|- TosetRel = { r e. PosetRel | ( dom r X. dom r ) C_ ( r u. `' r ) }
cdir 17228	class DirRel
ctail 17229	class tail
df-dir 17230	|- DirRel = { r | ( ( Rel r /\ ( _I |` U. U. r ) C_ r ) /\ ( ( r o. r ) C_ r /\ ( U. U. r X. U. U. r ) C_ ( `' r o. r ) ) ) }
df-tail 17231	|- tail = ( r e. DirRel |-> ( x e. U. U. r |-> ( r " { x } ) ) )
cplusf 17239	class +f
cmgm 17240	class Mgm
df-plusf 17241	|- +f = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) y ) ) )
df-mgm 17242	|- Mgm = { g | [. ( Base ` g ) / b ]. [. ( +g ` g ) / o ]. A. x e. b A. y e. b ( x o y ) e. b }
csgrp 17283	class SGrp
df-sgrp 17284	|- SGrp = { g e. Mgm | [. ( Base ` g ) / b ]. [. ( +g ` g ) / o ]. A. x e. b A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) }
cmnd 17294	class Mnd
df-mnd 17295	|- Mnd = { g e. SGrp | [. ( Base ` g ) / b ]. [. ( +g ` g ) / p ]. E. e e. b A. x e. b ( ( e p x ) = x /\ ( x p e ) = x ) }
cmhm 17333	class MndHom
csubmnd 17334	class SubMnd
df-mhm 17335	|- MndHom = ( s e. Mnd , t e. Mnd |-> { f e. ( ( Base ` t ) ^m ( Base ` s ) ) | ( A. x e. ( Base ` s ) A. y e. ( Base ` s ) ( f ` ( x ( +g ` s ) y ) ) = ( ( f ` x ) ( +g ` t ) ( f ` y ) ) /\ ( f ` ( 0g ` s ) ) = ( 0g ` t ) ) } )
df-submnd 17336	|- SubMnd = ( s e. Mnd |-> { t e. ~P ( Base ` s ) | ( ( 0g ` s ) e. t /\ A. x e. t A. y e. t ( x ( +g ` s ) y ) e. t ) } )
cfrmd 17384	class freeMnd
cvrmd 17385	class varFMnd
df-frmd 17386	|- freeMnd = ( i e. _V |-> { <. ( Base ` ndx ) , Word i >. , <. ( +g ` ndx ) , ( ++ |` (Word i X. Word i ) ) >. } )
df-vrmd 17387	|- varFMnd = ( i e. _V |-> ( j e. i |-> <" j "> ) )
cgrp 17422	class Grp
cminusg 17423	class invg
csg 17424	class -g
df-grp 17425	|- Grp = { g e. Mnd | A. a e. ( Base ` g ) E. m e. ( Base ` g ) ( m ( +g ` g ) a ) = ( 0g ` g ) }
df-minusg 17426	|- invg = ( g e. _V |-> ( x e. ( Base ` g ) |-> ( iota_ w e. ( Base ` g ) ( w ( +g ` g ) x ) = ( 0g ` g ) ) ) )
df-sbg 17427	|- -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
cmg 17540	class .g
df-mulg 17541	|- .g = ( g e. _V |-> ( n e. ZZ , x e. ( Base ` g ) |-> if ( n = 0 , ( 0g ` g ) , [_ seq 1 ( ( +g ` g ) , ( NN X. { x } ) ) / s ]_ if ( 0 < n , ( s ` n ) , ( ( invg ` g ) ` ( s ` -u n ) ) ) ) ) )
csubg 17588	class SubGrp
cnsg 17589	class NrmSGrp
cqg 17590	class ~QG
df-subg 17591	|- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
df-nsg 17592	|- NrmSGrp = ( w e. Grp |-> { s e. (SubGrp ` w ) | [. ( Base ` w ) / b ]. [. ( +g ` w ) / p ]. A. x e. b A. y e. b ( ( x p y ) e. s <-> ( y p x ) e. s ) } )
df-eqg 17593	|- ~QG = ( r e. _V , i e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i ) } )
cghm 17657	class GrpHom
df-ghm 17658	|- GrpHom = ( s e. Grp , t e. Grp |-> { g | [. ( Base ` s ) / w ]. ( g : w --> ( Base ` t ) /\ A. x e. w A. y e. w ( g ` ( x ( +g ` s ) y ) ) = ( ( g ` x ) ( +g ` t ) ( g ` y ) ) ) } )
cgim 17699	class GrpIso
cgic 17700	class ~=g𝑔
df-gim 17701	|- GrpIso = ( s e. Grp , t e. Grp |-> { g e. ( s GrpHom t ) | g : ( Base ` s ) -1-1-onto-> ( Base ` t ) } )
df-gic 17702	|- ~=g𝑔 = ( `' GrpIso " ( _V \ 1o ) )
cga 17722	class GrpAct
df-ga 17723	|- GrpAct = ( g e. Grp , s e. _V |-> [_ ( Base ` g ) / b ]_ { m e. ( s ^m ( b X. s ) ) | A. x e. s ( ( ( 0g ` g ) m x ) = x /\ A. y e. b A. z e. b ( ( y ( +g ` g ) z ) m x ) = ( y m ( z m x ) ) ) } )
ccntz 17748	class Cntz
ccntr 17749	class Cntr
df-cntz 17750	|- Cntz = ( m e. _V |-> ( s e. ~P ( Base ` m ) |-> { x e. ( Base ` m ) | A. y e. s ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) } ) )
df-cntr 17751	|- Cntr = ( m e. _V |-> ( (Cntz ` m ) ` ( Base ` m ) ) )
coppg 17775	class oppg
df-oppg 17776	|- oppg = ( w e. _V |-> ( w sSet <. ( +g ` ndx ) , tpos ( +g ` w ) >. ) )
csymg 17797	class SymGrp
df-symg 17798	|- SymGrp = ( x e. _V |-> [_ { h | h : x -1-1-onto-> x } / b ]_ { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( f e. b , g e. b |-> ( f o. g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( x X. { ~P x } ) ) >. } )
cpmtr 17861	class pmTrsp
df-pmtr 17862	|- pmTrsp = ( d e. _V |-> ( p e. { y e. ~P d | y ~~ 2o } |-> ( z e. d |-> if ( z e. p , U. ( p \ { z } ) , z ) ) ) )
cpsgn 17909	class pmSgn
cevpm 17910	class pmEven
df-psgn 17911	|- pmSgn = ( d e. _V |-> ( x e. { p e. ( Base ` ( SymGrp ` d ) ) | dom ( p \ _I ) e. Fin } |-> ( iota s E. w e. Word ran (pmTrsp ` d ) ( x = ( ( SymGrp ` d ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) )
df-evpm 17912	|- pmEven = ( d e. _V |-> ( `' (pmSgn ` d ) " { 1 } ) )
cod 17944	class od
cgex 17945	class gEx
cpgp 17946	class pGrp
cslw 17947	class pSyl
df-od 17948	|- od = ( g e. _V |-> ( x e. ( Base ` g ) |-> [_ { n e. NN | ( n (.g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) ) )
df-gex 17949	|- gEx = ( g e. _V |-> [_ { n e. NN | A. x e. ( Base ` g ) ( n (.g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) )
df-pgp 17950	|- pGrp = { <. p , g >. | ( ( p e. Prime /\ g e. Grp ) /\ A. x e. ( Base ` g ) E. n e. NN0 ( ( od ` g ) ` x ) = ( p ^ n ) ) }
df-slw 17951	|- pSyl = ( p e. Prime , g e. Grp |-> { h e. (SubGrp ` g ) | A. k e. (SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g ↾s k ) ) <-> h = k ) } )
clsm 18049	class LSSum
cpj1 18050	class proj1
df-lsm 18051	|- LSSum = ( w e. _V |-> ( t e. ~P ( Base ` w ) , u e. ~P ( Base ` w ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` w ) y ) ) ) )
df-pj1 18052	|- proj1 = ( w e. _V |-> ( t e. ~P ( Base ` w ) , u e. ~P ( Base ` w ) |-> ( z e. ( t ( LSSum ` w ) u ) |-> ( iota_ x e. t E. y e. u z = ( x ( +g ` w ) y ) ) ) ) )
cefg 18119	class ~FG
cfrgp 18120	class freeGrp
cvrgp 18121	class varFGrp
df-efg 18122	|- ~FG = ( i e. _V |-> |^| { r | ( r Er Word ( i X. 2o ) /\ A. x e. Word ( i X. 2o ) A. n e. ( 0 ... ( # ` x ) ) A. y e. i A. z e. 2o x r ( x splice <. n , n , <" <. y , z >. <. y , ( 1o \ z ) >. "> >. ) ) } )
df-frgp 18123	|- freeGrp = ( i e. _V |-> ( (freeMnd ` ( i X. 2o ) ) /.s ( ~FG ` i ) ) )
df-vrgp 18124	|- varFGrp = ( i e. _V |-> ( j e. i |-> [ <" <. j , (/) >. "> ] ( ~FG ` i ) ) )
ccmn 18193	class CMnd
cabl 18194	class Abel
df-cmn 18195	|- CMnd = { g e. Mnd | A. a e. ( Base ` g ) A. b e. ( Base ` g ) ( a ( +g ` g ) b ) = ( b ( +g ` g ) a ) }
df-abl 18196	|- Abel = ( Grp i^i CMnd )
ccyg 18279	class CycGrp
df-cyg 18280	|- CycGrp = { g e. Grp | E. x e. ( Base ` g ) ran ( n e. ZZ |-> ( n (.g ` g ) x ) ) = ( Base ` g ) }
cdprd 18392	class DProd
cdpj 18393	class dProj
df-dprd 18394	|- DProd = ( g e. Grp , s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) } |-> ran ( f e. { h e. X_ x e. dom s ( s ` x ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) )
df-dpj 18395	|- dProj = ( g e. Grp , s e. ( dom DProd " { g } ) |-> ( i e. dom s |-> ( ( s ` i ) ( proj1 ` g ) ( g DProd ( s |` ( dom s \ { i } ) ) ) ) ) )
cmgp 18489	class mulGrp
df-mgp 18490	|- mulGrp = ( w e. _V |-> ( w sSet <. ( +g ` ndx ) , ( .r ` w ) >. ) )
cur 18501	class 1r
df-ur 18502	|- 1r = ( 0g o. mulGrp )
csrg 18505	class SRing
df-srg 18506	|- SRing = { f e. CMnd | ( (mulGrp ` f ) e. Mnd /\ [. ( Base ` f ) / r ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. [. ( 0g ` f ) / n ]. A. x e. r ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) ) ) }
crg 18547	class Ring
ccrg 18548	class CRing
df-ring 18549	|- Ring = { f e. Grp | ( (mulGrp ` f ) e. Mnd /\ [. ( Base ` f ) / r ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. A. x e. r A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) ) }
df-cring 18550	|- CRing = { f e. Ring | (mulGrp ` f ) e. CMnd }
coppr 18622	class oppr
df-oppr 18623	|- oppr = ( f e. _V |-> ( f sSet <. ( .r ` ndx ) , tpos ( .r ` f ) >. ) )
cdsr 18638	class ||r
cui 18639	class Unit
cir 18640	class Irred
df-dvdsr 18641	|- ||r = ( w e. _V |-> { <. x , y >. | ( x e. ( Base ` w ) /\ E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y ) } )
df-unit 18642	|- Unit = ( w e. _V |-> ( `' ( ( ||r ` w ) i^i ( ||r ` (oppr ` w ) ) ) " { ( 1r ` w ) } ) )
df-irred 18643	|- Irred = ( w e. _V |-> [_ ( ( Base ` w ) \ (Unit ` w ) ) / b ]_ { z e. b | A. x e. b A. y e. b ( x ( .r ` w ) y ) =/= z } )
cinvr 18671	class invr
df-invr 18672	|- invr = ( r e. _V |-> ( invg ` ( (mulGrp ` r ) ↾s (Unit ` r ) ) ) )
cdvr 18682	class /r
df-dvr 18683	|- /r = ( r e. _V |-> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) )
crh 18712	class RingHom
crs 18713	class RingIso
cric 18714	class ~=r
df-rnghom 18715	|- RingHom = ( r e. Ring , s e. Ring |-> [_ ( Base ` r ) / v ]_ [_ ( Base ` s ) / w ]_ { f e. ( w ^m v ) | ( ( f ` ( 1r ` r ) ) = ( 1r ` s ) /\ A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) ) } )
df-rngiso 18716	|- RingIso = ( r e. _V , s e. _V |-> { f e. ( r RingHom s ) | `' f e. ( s RingHom r ) } )
df-ric 18718	|- ~=r = ( `' RingIso " ( _V \ 1o ) )
cdr 18747	class DivRing
cfield 18748	class Field
df-drng 18749	|- DivRing = { r e. Ring | (Unit ` r ) = ( ( Base ` r ) \ { ( 0g ` r ) } ) }
df-field 18750	|- Field = ( DivRing i^i CRing )
csubrg 18776	class SubRing
crgspn 18777	class RingSpan
df-subrg 18778	|- SubRing = ( w e. Ring |-> { s e. ~P ( Base ` w ) | ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s ) } )
df-rgspn 18779	|- RingSpan = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. (SubRing ` w ) | s C_ t } ) )
cabv 18816	class AbsVal
df-abv 18817	|- AbsVal = ( r e. Ring |-> { f e. ( ( 0 [,) +oo ) ^m ( Base ` r ) ) | A. x e. ( Base ` r ) ( ( ( f ` x ) = 0 <-> x = ( 0g ` r ) ) /\ A. y e. ( Base ` r ) ( ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) x. ( f ` y ) ) /\ ( f ` ( x ( +g ` r ) y ) ) <_ ( ( f ` x ) + ( f ` y ) ) ) ) } )
cstf 18843	class *rf
csr 18844	class *Ring
df-staf 18845	|- *rf = ( f e. _V |-> ( x e. ( Base ` f ) |-> ( ( *r ` f ) ` x ) ) )
df-srng 18846	|- *Ring = { f | [. ( *rf ` f ) / i ]. ( i e. ( f RingHom (oppr ` f ) ) /\ i = `' i ) }
clmod 18863	class LMod
cscaf 18864	class .sf
df-lmod 18865	|- LMod = { g e. Grp | [. ( Base ` g ) / v ]. [. ( +g ` g ) / a ]. [. (Scalar ` g ) / f ]. [. ( .s ` g ) / s ]. [. ( Base ` f ) / k ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. ( f e. Ring /\ A. q e. k A. r e. k A. x e. v A. w e. v ( ( ( r s w ) e. v /\ ( r s ( w a x ) ) = ( ( r s w ) a ( r s x ) ) /\ ( ( q p r ) s w ) = ( ( q s w ) a ( r s w ) ) ) /\ ( ( ( q t r ) s w ) = ( q s ( r s w ) ) /\ ( ( 1r ` f ) s w ) = w ) ) ) }
df-scaf 18866	|- .sf = ( g e. _V |-> ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) ) )
clss 18932	class LSubSp
df-lss 18933	|- LSubSp = ( w e. _V |-> { s e. ( ~P ( Base ` w ) \ { (/) } ) | A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s } )
clspn 18971	class LSpan
df-lsp 18972	|- LSpan = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( LSubSp ` w ) | s C_ t } ) )
clmhm 19019	class LMHom
clmim 19020	class LMIso
clmic 19021	class ~=ph𝑚
df-lmhm 19022	|- LMHom = ( s e. LMod , t e. LMod |-> { f e. ( s GrpHom t ) | [. (Scalar ` s ) / w ]. ( (Scalar ` t ) = w /\ A. x e. ( Base ` w ) A. y e. ( Base ` s ) ( f ` ( x ( .s ` s ) y ) ) = ( x ( .s ` t ) ( f ` y ) ) ) } )
df-lmim 19023	|- LMIso = ( s e. LMod , t e. LMod |-> { g e. ( s LMHom t ) | g : ( Base ` s ) -1-1-onto-> ( Base ` t ) } )
df-lmic 19024	|- ~=ph𝑚 = ( `' LMIso " ( _V \ 1o ) )
clbs 19074	class LBasis
df-lbs 19075	|- LBasis = ( w e. _V |-> { b e. ~P ( Base ` w ) | [. ( LSpan ` w ) / n ]. [. (Scalar ` w ) / s ]. ( ( n ` b ) = ( Base ` w ) /\ A. x e. b A. y e. ( ( Base ` s ) \ { ( 0g ` s ) } ) -. ( y ( .s ` w ) x ) e. ( n ` ( b \ { x } ) ) ) } )
clvec 19102	class LVec
df-lvec 19103	|- LVec = { f e. LMod | (Scalar ` f ) e. DivRing }
csra 19168	class subringAlg
crglmod 19169	class ringLMod
clidl 19170	class LIdeal
crsp 19171	class RSpan
df-sra 19172	|- subringAlg = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> ( ( ( w sSet <. (Scalar ` ndx ) , ( w ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` w ) >. ) sSet <. ( .i ` ndx ) , ( .r ` w ) >. ) ) )
df-rgmod 19173	|- ringLMod = ( w e. _V |-> ( (subringAlg ` w ) ` ( Base ` w ) ) )
df-lidl 19174	|- LIdeal = ( LSubSp o. ringLMod )
df-rsp 19175	|- RSpan = ( LSpan o. ringLMod )
c2idl 19231	class 2Ideal
df-2idl 19232	|- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
clpidl 19241	class LPIdeal
clpir 19242	class LPIR
df-lpidl 19243	|- LPIdeal = ( w e. Ring |-> U_ g e. ( Base ` w ) { ( (RSpan ` w ) ` { g } ) } )
df-lpir 19244	|- LPIR = { w e. Ring | (LIdeal ` w ) = (LPIdeal ` w ) }
cnzr 19257	class NzRing
df-nzr 19258	|- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
crlreg 19279	class RLReg
cdomn 19280	class Domn
cidom 19281	class IDomn
cpid 19282	class PID
df-rlreg 19283	|- RLReg = ( r e. _V |-> { x e. ( Base ` r ) | A. y e. ( Base ` r ) ( ( x ( .r ` r ) y ) = ( 0g ` r ) -> y = ( 0g ` r ) ) } )
df-domn 19284	|- Domn = { r e. NzRing | [. ( Base ` r ) / b ]. [. ( 0g ` r ) / z ]. A. x e. b A. y e. b ( ( x ( .r ` r ) y ) = z -> ( x = z \/ y = z ) ) }
df-idom 19285	|- IDomn = ( CRing i^i Domn )
df-pid 19286	|- PID = (IDomn i^i LPIR )
casa 19309	class AssAlg
casp 19310	class AlgSpan
cascl 19311	class algSc
df-assa 19312	|- AssAlg = { w e. ( LMod i^i Ring ) | [. (Scalar ` w ) / f ]. ( f e. CRing /\ A. r e. ( Base ` f ) A. x e. ( Base ` w ) A. y e. ( Base ` w ) [. ( .s ` w ) / s ]. [. ( .r ` w ) / t ]. ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) ) }
df-asp 19313	|- AlgSpan = ( w e. AssAlg |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( (SubRing ` w ) i^i ( LSubSp ` w ) ) | s C_ t } ) )
df-ascl 19314	|- algSc = ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) |-> ( x ( .s ` w ) ( 1r ` w ) ) ) )
cmps 19351	class mPwSer
cmvr 19352	class mVar
cmpl 19353	class mPoly
cltb 19354	class <bag
copws 19355	class ordPwSer
df-psr 19356	|- mPwSer = ( i e. _V , r e. _V |-> [_ { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]_ [_ ( ( Base ` r ) ^m d ) / b ]_ ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( oF ( +g ` r ) |` ( b X. b ) ) >. , <. ( .r ` ndx ) , ( f e. b , g e. b |-> ( k e. d |-> ( r gsumg ( x e. { y e. d | y oR <_ k } |-> ( ( f ` x ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , r >. , <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. } ) )
df-mvr 19357	|- mVar = ( i e. _V , r e. _V |-> ( x e. i |-> ( f e. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } |-> if ( f = ( y e. i |-> if ( y = x , 1 , 0 ) ) , ( 1r ` r ) , ( 0g ` r ) ) ) ) )
df-mpl 19358	|- mPoly = ( i e. _V , r e. _V |-> [_ ( i mPwSer r ) / w ]_ ( w ↾s { f e. ( Base ` w ) | f finSupp ( 0g ` r ) } ) )
df-ltbag 19359	|- <bag = ( r e. _V , i e. _V |-> { <. x , y >. | ( { x , y } C_ { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } /\ E. z e. i ( ( x ` z ) < ( y ` z ) /\ A. w e. i ( z r w -> ( x ` w ) = ( y ` w ) ) ) ) } )
df-opsr 19360	|- ordPwSer = ( i e. _V , s e. _V |-> ( r e. ~P ( i X. i ) |-> [_ ( i mPwSer s ) / p ]_ ( p sSet <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ ( Base ` p ) /\ ( [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r <bag i ) z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. ) ) )
ces 19504	class evalSub
cevl 19505	class eval
df-evls 19506	|- evalSub = ( i e. _V , s e. CRing |-> [_ ( Base ` s ) / b ]_ ( r e. (SubRing ` s ) |-> [_ ( i mPoly ( s ↾s r ) ) / w ]_ ( iota_ f e. ( w RingHom ( s ^s ( b ^m i ) ) ) ( ( f o. (algSc ` w ) ) = ( x e. r |-> ( ( b ^m i ) X. { x } ) ) /\ ( f o. ( i mVar ( s ↾s r ) ) ) = ( x e. i |-> ( g e. ( b ^m i ) |-> ( g ` x ) ) ) ) ) ) )
df-evl 19507	|- eval = ( i e. _V , r e. _V |-> ( ( i evalSub r ) ` ( Base ` r ) ) )
cmhp 19537	class mHomP
cpsd 19538	class mPSDer
cslv 19539	class selectVars
cai 19540	class AlgInd
df-mhp 19541	|- mHomP = ( i e. _V , r e. _V |-> ( n e. NN0 |-> { f e. ( Base ` ( i mPoly r ) ) | ( f supp ( 0g ` r ) ) C_ { g e. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } | sum_ j e. NN0 ( g ` j ) = n } } ) )
df-psd 19542	|- mPSDer = ( i e. _V , r e. _V |-> ( x e. i |-> ( f e. ( Base ` ( i mPwSer r ) ) |-> ( k e. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } |-> ( ( ( k ` x ) + 1 ) (.g ` r ) ( f ` ( k oF + ( y e. i |-> if ( y = x , 1 , 0 ) ) ) ) ) ) ) ) )
df-selv 19543	|- selectVars = ( i e. _V , r e. _V |-> ( j e. ~P i |-> ( f e. ( i mPoly r ) |-> [_ ( ( i \ j ) mPoly r ) / s ]_ [_ ( x e. (Scalar ` s ) |-> ( x ( .s ` s ) ( 1r ` s ) ) ) / c ]_ ( ( ( ( i evalSub s ) ` ( c "s r ) ) ` ( c o. f ) ) ` ( x e. i |-> if ( x e. j , ( ( j mVar ( ( i \ j ) mPoly r ) ) ` x ) , ( c o. ( ( ( i \ j ) mVar r ) ` x ) ) ) ) ) ) ) )
df-algind 19544	|- AlgInd = ( w e. _V , k e. ~P ( Base ` w ) |-> { v e. ~P ( Base ` w ) | Fun `' ( f e. ( Base ` ( v mPoly ( w ↾s k ) ) ) |-> ( ( ( ( v evalSub w ) ` k ) ` f ) ` ( _I |` v ) ) ) } )
cps1 19545	class PwSer1
cv1 19546	class var1
cpl1 19547	class Poly1
cco1 19548	class coe1
ctp1 19549	class toPoly1
df-psr1 19550	|- PwSer1 = ( r e. _V |-> ( ( 1o ordPwSer r ) ` (/) ) )
df-vr1 19551	|- var1 = ( r e. _V |-> ( ( 1o mVar r ) ` (/) ) )
df-ply1 19552	|- Poly1 = ( r e. _V |-> ( (PwSer1 ` r ) ↾s ( Base ` ( 1o mPoly r ) ) ) )
df-coe1 19553	|- coe1 = ( f e. _V |-> ( n e. NN0 |-> ( f ` ( 1o X. { n } ) ) ) )
df-toply1 19554	|- toPoly1 = ( f e. _V |-> ( n e. ( NN0 ^m 1o ) |-> ( f ` ( n ` (/) ) ) ) )
ces1 19678	class evalSub1
ce1 19679	class eval1
df-evls1 19680	|- evalSub1 = ( s e. _V , r e. ~P ( Base ` s ) |-> [_ ( Base ` s ) / b ]_ ( ( x e. ( b ^m ( b ^m 1o ) ) |-> ( x o. ( y e. b |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub s ) ` r ) ) )
df-evl1 19681	|- eval1 = ( r e. _V |-> [_ ( Base ` r ) / b ]_ ( ( x e. ( b ^m ( b ^m 1o ) ) |-> ( x o. ( y e. b |-> ( 1o X. { y } ) ) ) ) o. ( 1o eval r ) ) )
cpsmet 19730	class PsMet
cxmt 19731	class *Met
cme 19732	class Met
cbl 19733	class ball
cfbas 19734	class fBas
cfg 19735	class filGen
cmopn 19736	class MetOpen
cmetu 19737	class metUnif
df-psmet 19738	|- PsMet = ( x e. _V |-> { d e. ( RR* ^m ( x X. x ) ) | A. y e. x ( ( y d y ) = 0 /\ A. z e. x A. w e. x ( y d z ) <_ ( ( w d y ) +e ( w d z ) ) ) } )
df-xmet 19739	|- *Met = ( x e. _V |-> { d e. ( RR* ^m ( x X. x ) ) | A. y e. x A. z e. x ( ( ( y d z ) = 0 <-> y = z ) /\ A. w e. x ( y d z ) <_ ( ( w d y ) +e ( w d z ) ) ) } )
df-met 19740	|- Met = ( x e. _V |-> { d e. ( RR ^m ( x X. x ) ) | A. y e. x A. z e. x ( ( ( y d z ) = 0 <-> y = z ) /\ A. w e. x ( y d z ) <_ ( ( w d y ) + ( w d z ) ) ) } )
df-bl 19741	|- ball = ( d e. _V |-> ( x e. dom dom d , z e. RR* |-> { y e. dom dom d | ( x d y ) < z } ) )
df-mopn 19742	|- MetOpen = ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
df-fbas 19743	|- fBas = ( w e. _V |-> { x e. ~P ~P w | ( x =/= (/) /\ (/) e/ x /\ A. y e. x A. z e. x ( x i^i ~P ( y i^i z ) ) =/= (/) ) } )
df-fg 19744	|- filGen = ( w e. _V , x e. ( fBas ` w ) |-> { y e. ~P w | ( x i^i ~P y ) =/= (/) } )
df-metu 19745	|- metUnif = ( d e. U. ran PsMet |-> ( ( dom dom d X. dom dom d ) filGen ran ( a e. RR+ |-> ( `' d " ( 0 [,) a ) ) ) ) )
ccnfld 19746	class ℂfld
df-cnfld 19747	|- ℂfld = ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
zring 19818	class ℤring
df-zring 19819	|- ℤring = (ℂfld ↾s ZZ )
czrh 19848	class ZRHom
czlm 19849	class ZMod
cchr 19850	class chr
czn 19851	class ℤ/nℤ
df-zrh 19852	|- ZRHom = ( r e. _V |-> U. (ℤring RingHom r ) )
df-zlm 19853	|- ZMod = ( g e. _V |-> ( ( g sSet <. (Scalar ` ndx ) , ℤring >. ) sSet <. ( .s ` ndx ) , (.g ` g ) >. ) )
df-chr 19854	|- chr = ( g e. _V |-> ( ( od ` g ) ` ( 1r ` g ) ) )
df-zn 19855	|- ℤ/nℤ = ( n e. NN0 |-> [_ℤring / z ]_ [_ ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) / s ]_ ( s sSet <. ( le ` ndx ) , [_ ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) / f ]_ ( ( f o. <_ ) o. `' f ) >. ) )
crefld 19950	class RRfld
df-refld 19951	|- RRfld = (ℂfld ↾s RR )
cphl 19969	class PreHil
cipf 19970	class .if
df-phl 19971	|- PreHil = { g e. LVec | [. ( Base ` g ) / v ]. [. ( .i ` g ) / h ]. [. (Scalar ` g ) / f ]. ( f e. *Ring /\ A. x e. v ( ( y e. v |-> ( y h x ) ) e. ( g LMHom (ringLMod ` f ) ) /\ ( ( x h x ) = ( 0g ` f ) -> x = ( 0g ` g ) ) /\ A. y e. v ( ( *r ` f ) ` ( x h y ) ) = ( y h x ) ) ) }
df-ipf 19972	|- .if = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( .i ` g ) y ) ) )
cocv 20004	class ocv
ccss 20005	class CSubSp
cthl 20006	class toHL
df-ocv 20007	|- ocv = ( h e. _V |-> ( s e. ~P ( Base ` h ) |-> { x e. ( Base ` h ) | A. y e. s ( x ( .i ` h ) y ) = ( 0g ` (Scalar ` h ) ) } ) )
df-css 20008	|- CSubSp = ( h e. _V |-> { s | s = ( ( ocv ` h ) ` ( ( ocv ` h ) ` s ) ) } )
df-thl 20009	|- toHL = ( h e. _V |-> ( (toInc ` ( CSubSp ` h ) ) sSet <. ( oc ` ndx ) , ( ocv ` h ) >. ) )
cpj 20044	class proj
chs 20045	class Hil
cobs 20046	class OBasis
df-pj 20047	|- proj = ( h e. _V |-> ( ( x e. ( LSubSp ` h ) |-> ( x ( proj1 ` h ) ( ( ocv ` h ) ` x ) ) ) i^i ( _V X. ( ( Base ` h ) ^m ( Base ` h ) ) ) ) )
df-hil 20048	|- Hil = { h e. PreHil | dom ( proj ` h ) = ( CSubSp ` h ) }
df-obs 20049	|- OBasis = ( h e. PreHil |-> { b e. ~P ( Base ` h ) | ( A. x e. b A. y e. b ( x ( .i ` h ) y ) = if ( x = y , ( 1r ` (Scalar ` h ) ) , ( 0g ` (Scalar ` h ) ) ) /\ ( ( ocv ` h ) ` b ) = { ( 0g ` h ) } ) } )
cdsmm 20075	class (+)m
df-dsmm 20076	|- (+)m = ( s e. _V , r e. _V |-> ( ( s X_s r ) ↾s { f e. X_ x e. dom r ( Base ` ( r ` x ) ) | { x e. dom r | ( f ` x ) =/= ( 0g ` ( r ` x ) ) } e. Fin } ) )
cfrlm 20090	class freeLMod
df-frlm 20091	|- freeLMod = ( r e. _V , i e. _V |-> ( r (+)m ( i X. { (ringLMod ` r ) } ) ) )
cuvc 20121	class unitVec
df-uvc 20122	|- unitVec = ( r e. _V , i e. _V |-> ( j e. i |-> ( k e. i |-> if ( k = j , ( 1r ` r ) , ( 0g ` r ) ) ) ) )
clindf 20143	class LIndF
clinds 20144	class LIndS
df-lindf 20145	|- LIndF = { <. f , w >. | ( f : dom f --> ( Base ` w ) /\ [. (Scalar ` w ) / s ]. A. x e. dom f A. k e. ( ( Base ` s ) \ { ( 0g ` s ) } ) -. ( k ( .s ` w ) ( f ` x ) ) e. ( ( LSpan ` w ) ` ( f " ( dom f \ { x } ) ) ) ) }
df-linds 20146	|- LIndS = ( w e. _V |-> { s e. ~P ( Base ` w ) | ( _I |` s ) LIndF w } )
cmmul 20189	class maMul
df-mamu 20190	|- maMul = ( r e. _V , o e. _V |-> [_ ( 1st ` ( 1st ` o ) ) / m ]_ [_ ( 2nd ` ( 1st ` o ) ) / n ]_ [_ ( 2nd ` o ) / p ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. p ) ) |-> ( i e. m , k e. p |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) )
cmat 20213	class Mat
df-mat 20214	|- Mat = ( n e. Fin , r e. _V |-> ( ( r freeLMod ( n X. n ) ) sSet <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. ) )
cdmat 20294	class DMat
cscmat 20295	class ScMat
df-dmat 20296	|- DMat = ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat r ) ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } )
df-scmat 20297	|- ScMat = ( n e. Fin , r e. _V |-> [_ ( n Mat r ) / a ]_ { m e. ( Base ` a ) | E. c e. ( Base ` r ) m = ( c ( .s ` a ) ( 1r ` a ) ) } )
cmvmul 20346	class maVecMul
df-mvmul 20347	|- maVecMul = ( r e. _V , o e. _V |-> [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) )
cmarrep 20362	class matRRep
cmatrepV 20363	class matRepV
df-marrep 20364	|- matRRep = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) , s e. ( Base ` r ) |-> ( k e. n , l e. n |-> ( i e. n , j e. n |-> if ( i = k , if ( j = l , s , ( 0g ` r ) ) , ( i m j ) ) ) ) ) )
df-marepv 20365	|- matRepV = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) , v e. ( ( Base ` r ) ^m n ) |-> ( k e. n |-> ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )
csubma 20382	class subMat
df-subma 20383	|- subMat = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( k e. n , l e. n |-> ( i e. ( n \ { k } ) , j e. ( n \ { l } ) |-> ( i m j ) ) ) ) )
cmdat 20390	class maDet
df-mdet 20391	|- maDet = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( r gsumg ( p e. ( Base ` ( SymGrp ` n ) ) |-> ( ( ( ( ZRHom ` r ) o. (pmSgn ` n ) ) ` p ) ( .r ` r ) ( (mulGrp ` r ) gsumg ( x e. n |-> ( ( p ` x ) m x ) ) ) ) ) ) ) )
cmadu 20438	class maAdju
cminmar1 20439	class minMatR1
df-madu 20440	|- maAdju = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( i e. n , j e. n |-> ( ( n maDet r ) ` ( k e. n , l e. n |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) ) ) )
df-minmar1 20441	|- minMatR1 = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( k e. n , l e. n |-> ( i e. n , j e. n |-> if ( i = k , if ( j = l , ( 1r ` r ) , ( 0g ` r ) ) , ( i m j ) ) ) ) ) )
ccpmat 20508	class ConstPolyMat
cmat2pmat 20509	class matToPolyMat
ccpmat2mat 20510	class cPolyMatToMat
df-cpmat 20511	|- ConstPolyMat = ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat (Poly1 ` r ) ) ) | A. i e. n A. j e. n A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) } )
df-mat2pmat 20512	|- matToPolyMat = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( x e. n , y e. n |-> ( (algSc ` (Poly1 ` r ) ) ` ( x m y ) ) ) ) )
df-cpmat2mat 20513	|- cPolyMatToMat = ( n e. Fin , r e. _V |-> ( m e. ( n ConstPolyMat r ) |-> ( x e. n , y e. n |-> ( (coe1 ` ( x m y ) ) ` 0 ) ) ) )
cdecpmat 20567	class decompPMat
df-decpmat 20568	|- decompPMat = ( m e. _V , k e. NN0 |-> ( i e. dom dom m , j e. dom dom m |-> ( (coe1 ` ( i m j ) ) ` k ) ) )
cpm2mp 20597	class pMatToMatPoly
df-pm2mp 20598	|- pMatToMatPoly = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat (Poly1 ` r ) ) ) |-> [_ ( n Mat r ) / a ]_ [_ (Poly1 ` a ) / q ]_ ( q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` q ) ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) ) ) ) ) )
cchpmat 20631	class CharPlyMat
df-chpmat 20632	|- CharPlyMat = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( ( n maDet (Poly1 ` r ) ) ` ( ( (var1 ` r ) ( .s ` ( n Mat (Poly1 ` r ) ) ) ( 1r ` ( n Mat (Poly1 ` r ) ) ) ) ( -g ` ( n Mat (Poly1 ` r ) ) ) ( ( n matToPolyMat r ) ` m ) ) ) ) )
ctop 20698	class Top
df-top 20699	|- Top = { x | ( A. y e. ~P x U. y e. x /\ A. y e. x A. z e. x ( y i^i z ) e. x ) }
ctopon 20715	class TopOn
df-topon 20716	|- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
ctps 20736	class TopSp
df-topsp 20737	|- TopSp = { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) }
ctb 20749	class TopBases
df-bases 20750	|- TopBases = { x | A. y e. x A. z e. x ( y i^i z ) C_ U. ( x i^i ~P ( y i^i z ) ) }
ccld 20820	class Clsd
cnt 20821	class int
ccl 20822	class cls
df-cld 20823	|- Clsd = ( j e. Top |-> { x e. ~P U. j | ( U. j \ x ) e. j } )
df-ntr 20824	|- int = ( j e. Top |-> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) )
df-cls 20825	|- cls = ( j e. Top |-> ( x e. ~P U. j |-> |^| { y e. ( Clsd ` j ) | x C_ y } ) )
cnei 20901	class nei
df-nei 20902	|- nei = ( j e. Top |-> ( x e. ~P U. j |-> { y e. ~P U. j | E. g e. j ( x C_ g /\ g C_ y ) } ) )
clp 20938	class limPt
cperf 20939	class Perf
df-lp 20940	|- limPt = ( j e. Top |-> ( x e. ~P U. j |-> { y | y e. ( ( cls ` j ) ` ( x \ { y } ) ) } ) )
df-perf 20941	|- Perf = { j e. Top | ( ( limPt ` j ) ` U. j ) = U. j }
ccn 21028	class Cn
ccnp 21029	class CnP
clm 21030	class ~~> t
df-cn 21031	|- Cn = ( j e. Top , k e. Top |-> { f e. ( U. k ^m U. j ) | A. y e. k ( `' f " y ) e. j } )
df-cnp 21032	|- CnP = ( j e. Top , k e. Top |-> ( x e. U. j |-> { f e. ( U. k ^m U. j ) | A. y e. k ( ( f ` x ) e. y -> E. g e. j ( x e. g /\ ( f " g ) C_ y ) ) } ) )
df-lm 21033	|- ~~> t = ( j e. Top |-> { <. f , x >. | ( f e. ( U. j ^pm CC ) /\ x e. U. j /\ A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
ct0 21110	class Kol2
ct1 21111	class Fre
cha 21112	class Haus
creg 21113	class Reg
cnrm 21114	class Nrm
ccnrm 21115	class CNrm
cpnrm 21116	class PNrm
df-t0 21117	|- Kol2 = { j e. Top | A. x e. U. j A. y e. U. j ( A. o e. j ( x e. o <-> y e. o ) -> x = y ) }
df-t1 21118	|- Fre = { x e. Top | A. a e. U. x { a } e. ( Clsd ` x ) }
df-haus 21119	|- Haus = { j e. Top | A. x e. U. j A. y e. U. j ( x =/= y -> E. n e. j E. m e. j ( x e. n /\ y e. m /\ ( n i^i m ) = (/) ) ) }
df-reg 21120	|- Reg = { j e. Top | A. x e. j A. y e. x E. z e. j ( y e. z /\ ( ( cls ` j ) ` z ) C_ x ) }
df-nrm 21121	|- Nrm = { j e. Top | A. x e. j A. y e. ( ( Clsd ` j ) i^i ~P x ) E. z e. j ( y C_ z /\ ( ( cls ` j ) ` z ) C_ x ) }
df-cnrm 21122	|- CNrm = { j e. Top | A. x e. ~P U. j ( j ↾t x ) e. Nrm }
df-pnrm 21123	|- PNrm = { j e. Nrm | ( Clsd ` j ) C_ ran ( f e. ( j ^m NN ) |-> |^| ran f ) }
ccmp 21189	class Comp
df-cmp 21190	|- Comp = { x e. Top | A. y e. ~P x ( U. x = U. y -> E. z e. ( ~P y i^i Fin ) U. x = U. z ) }
cconn 21214	class Conn
df-conn 21215	|- Conn = { j e. Top | ( j i^i ( Clsd ` j ) ) = { (/) , U. j } }
c1stc 21240	class 1stc
c2ndc 21241	class 2ndc
df-1stc 21242	|- 1stc = { j e. Top | A. x e. U. j E. y e. ~P j ( y ~<_ om /\ A. z e. j ( x e. z -> x e. U. ( y i^i ~P z ) ) ) }
df-2ndc 21243	|- 2ndc = { j | E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = j ) }
clly 21267	class Locally A
cnlly 21268	class 𝑛Locally A
df-lly 21269	|- Locally A = { j e. Top | A. x e. j A. y e. x E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. A ) }
df-nlly 21270	|- 𝑛Locally A = { j e. Top | A. x e. j A. y e. x E. u e. ( ( ( nei ` j ) ` { y } ) i^i ~P x ) ( j ↾t u ) e. A }
cref 21305	class Ref
cptfin 21306	class PtFin
clocfin 21307	class LocFin
df-ref 21308	|- Ref = { <. x , y >. | ( U. y = U. x /\ A. z e. x E. w e. y z C_ w ) }
df-ptfin 21309	|- PtFin = { x | A. y e. U. x { z e. x | y e. z } e. Fin }
df-locfin 21310	|- LocFin = ( x e. Top |-> { y | ( U. x = U. y /\ A. p e. U. x E. n e. x ( p e. n /\ { s e. y | ( s i^i n ) =/= (/) } e. Fin ) ) } )
ckgen 21336	class 𝑘Gen
df-kgen 21337	|- 𝑘Gen = ( j e. Top |-> { x e. ~P U. j | A. k e. ~P U. j ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) } )
ctx 21363	class tX
cxko 21364	class ^ko
df-tx 21365	|- tX = ( r e. _V , s e. _V |-> ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) )
df-xko 21366	|- ^ko = ( s e. Top , r e. Top |-> ( topGen ` ( fi ` ran ( k e. { x e. ~P U. r | ( r ↾t x ) e. Comp } , v e. s |-> { f e. ( r Cn s ) | ( f " k ) C_ v } ) ) ) )
ckq 21496	class KQ
df-kq 21497	|- KQ = ( j e. Top |-> ( j qTop ( x e. U. j |-> { y e. j | x e. y } ) ) )
chmeo 21556	class Homeo
chmph 21557	class ~=
df-hmeo 21558	|- Homeo = ( j e. Top , k e. Top |-> { f e. ( j Cn k ) | `' f e. ( k Cn j ) } )
df-hmph 21559	|- ~= = ( `' Homeo " ( _V \ 1o ) )
cfil 21649	class Fil
df-fil 21650	|- Fil = ( z e. _V |-> { f e. ( fBas ` z ) | A. x e. ~P z ( ( f i^i ~P x ) =/= (/) -> x e. f ) } )
cufil 21703	class UFil
cufl 21704	class UFL
df-ufil 21705	|- UFil = ( g e. _V |-> { f e. ( Fil ` g ) | A. x e. ~P g ( x e. f \/ ( g \ x ) e. f ) } )
df-ufl 21706	|- UFL = { x | A. f e. ( Fil ` x ) E. g e. ( UFil ` x ) f C_ g }
cfm 21737	class FilMap
cflim 21738	class fLim
cflf 21739	class fLimf
cfcls 21740	class fClus
cfcf 21741	class fClusf
df-fm 21742	|- FilMap = ( x e. _V , f e. _V |-> ( y e. ( fBas ` dom f ) |-> ( x filGen ran ( t e. y |-> ( f " t ) ) ) ) )
df-flim 21743	|- fLim = ( j e. Top , f e. U. ran Fil |-> { x e. U. j | ( ( ( nei ` j ) ` { x } ) C_ f /\ f C_ ~P U. j ) } )
df-flf 21744	|- fLimf = ( x e. Top , y e. U. ran Fil |-> ( f e. ( U. x ^m U. y ) |-> ( x fLim ( ( U. x FilMap f ) ` y ) ) ) )
df-fcls 21745	|- fClus = ( j e. Top , f e. U. ran Fil |-> if ( U. j = U. f , |^|_ x e. f ( ( cls ` j ) ` x ) , (/) ) )
df-fcf 21746	|- fClusf = ( j e. Top , f e. U. ran Fil |-> ( g e. ( U. j ^m U. f ) |-> ( j fClus ( ( U. j FilMap g ) ` f ) ) ) )
ccnext 21863	class CnExt
df-cnext 21864	|- CnExt = ( j e. Top , k e. Top |-> ( f e. ( U. k ^pm U. j ) |-> U_ x e. ( ( cls ` j ) ` dom f ) ( { x } X. ( ( k fLimf ( ( ( nei ` j ) ` { x } ) ↾t dom f ) ) ` f ) ) ) )
ctmd 21874	class TopMnd
ctgp 21875	class TopGrp
df-tmd 21876	|- TopMnd = { f e. ( Mnd i^i TopSp ) | [. ( TopOpen ` f ) / j ]. ( +f ` f ) e. ( ( j tX j ) Cn j ) }
df-tgp 21877	|- TopGrp = { f e. ( Grp i^i TopMnd ) | [. ( TopOpen ` f ) / j ]. ( invg ` f ) e. ( j Cn j ) }
ctsu 21929	class tsums
df-tsms 21930	|- tsums = ( w e. _V , f e. _V |-> [_ ( ~P dom f i^i Fin ) / s ]_ ( ( ( TopOpen ` w ) fLimf ( s filGen ran ( z e. s |-> { y e. s | z C_ y } ) ) ) ` ( y e. s |-> ( w gsumg ( f |` y ) ) ) ) )
ctrg 21959	class TopRing
ctdrg 21960	class TopDRing
ctlm 21961	class TopMod
ctvc 21962	class TopVec
df-trg 21963	|- TopRing = { r e. ( TopGrp i^i Ring ) | (mulGrp ` r ) e. TopMnd }
df-tdrg 21964	|- TopDRing = { r e. ( TopRing i^i DivRing ) | ( (mulGrp ` r ) ↾s (Unit ` r ) ) e. TopGrp }
df-tlm 21965	|- TopMod = { w e. (TopMnd i^i LMod ) | ( (Scalar ` w ) e. TopRing /\ ( .sf ` w ) e. ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) ) ) }
df-tvc 21966	|- TopVec = { w e. TopMod | (Scalar ` w ) e. TopDRing }
cust 22003	class UnifOn
df-ust 22004	|- UnifOn = ( x e. _V |-> { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) } )
cutop 22034	class unifTop
df-utop 22035	|- unifTop = ( u e. U. ran UnifOn |-> { a e. ~P dom U. u | A. x e. a E. v e. u ( v " { x } ) C_ a } )
cuss 22057	class UnifSt
cusp 22058	class UnifSp
ctus 22059	class toUnifSp
df-uss 22060	|- UnifSt = ( f e. _V |-> ( ( UnifSet ` f ) ↾t ( ( Base ` f ) X. ( Base ` f ) ) ) )
df-usp 22061	|- UnifSp = { f | ( (UnifSt ` f ) e. (UnifOn ` ( Base ` f ) ) /\ ( TopOpen ` f ) = (unifTop ` (UnifSt ` f ) ) ) }
df-tus 22062	|- toUnifSp = ( u e. U. ran UnifOn |-> ( { <. ( Base ` ndx ) , dom U. u >. , <. ( UnifSet ` ndx ) , u >. } sSet <. (TopSet ` ndx ) , (unifTop ` u ) >. ) )
cucn 22079	class Cnu
df-ucn 22080	|- Cnu = ( u e. U. ran UnifOn , v e. U. ran UnifOn |-> { f e. ( dom U. v ^m dom U. u ) | A. s e. v E. r e. u A. x e. dom U. u A. y e. dom U. u ( x r y -> ( f ` x ) s ( f ` y ) ) } )
ccfilu 22090	class CauFilu
df-cfilu 22091	|- CauFilu = ( u e. U. ran UnifOn |-> { f e. ( fBas ` dom U. u ) | A. v e. u E. a e. f ( a X. a ) C_ v } )
ccusp 22101	class CUnifSp
df-cusp 22102	|- CUnifSp = { w e. UnifSp | A. c e. ( Fil ` ( Base ` w ) ) ( c e. (CauFilu ` (UnifSt ` w ) ) -> ( ( TopOpen ` w ) fLim c ) =/= (/) ) }
cxme 22122	class *MetSp
cmt 22123	class MetSp
ctmt 22124	class toMetSp
df-xms 22125	|- *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
df-ms 22126	|- MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
df-tms 22127	|- toMetSp = ( d e. U. ran *Met |-> ( { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } sSet <. (TopSet ` ndx ) , ( MetOpen ` d ) >. ) )
cnm 22381	class norm
cngp 22382	class NrmGrp
ctng 22383	class toNrmGrp
cnrg 22384	class NrmRing
cnlm 22385	class NrmMod
cnvc 22386	class NrmVec
df-nm 22387	|- norm = ( w e. _V |-> ( x e. ( Base ` w ) |-> ( x ( dist ` w ) ( 0g ` w ) ) ) )
df-ngp 22388	|- NrmGrp = { g e. ( Grp i^i MetSp ) | ( ( norm ` g ) o. ( -g ` g ) ) C_ ( dist ` g ) }
df-tng 22389	|- toNrmGrp = ( g e. _V , f e. _V |-> ( ( g sSet <. ( dist ` ndx ) , ( f o. ( -g ` g ) ) >. ) sSet <. (TopSet ` ndx ) , ( MetOpen ` ( f o. ( -g ` g ) ) ) >. ) )
df-nrg 22390	|- NrmRing = { w e. NrmGrp | ( norm ` w ) e. (AbsVal ` w ) }
df-nlm 22391	|- NrmMod = { w e. (NrmGrp i^i LMod ) | [. (Scalar ` w ) / f ]. ( f e. NrmRing /\ A. x e. ( Base ` f ) A. y e. ( Base ` w ) ( ( norm ` w ) ` ( x ( .s ` w ) y ) ) = ( ( ( norm ` f ) ` x ) x. ( ( norm ` w ) ` y ) ) ) }
df-nvc 22392	|- NrmVec = (NrmMod i^i LVec )
cnmo 22509	class normOp
cnghm 22510	class NGHom
cnmhm 22511	class NMHom
df-nmo 22512	|- normOp = ( s e. NrmGrp , t e. NrmGrp |-> ( f e. ( s GrpHom t ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. ( Base ` s ) ( ( norm ` t ) ` ( f ` x ) ) <_ ( r x. ( ( norm ` s ) ` x ) ) } , RR* , < ) ) )
df-nghm 22513	|- NGHom = ( s e. NrmGrp , t e. NrmGrp |-> ( `' ( s normOp t ) " RR ) )
df-nmhm 22514	|- NMHom = ( s e. NrmMod , t e. NrmMod |-> ( ( s LMHom t ) i^i ( s NGHom t ) ) )
cii 22678	class II
ccncf 22679	class -cn->
df-ii 22680	|- II = ( MetOpen ` ( ( abs o. - ) |` ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) )
df-cncf 22681	|- -cn-> = ( a e. ~P CC , b e. ~P CC |-> { f e. ( b ^m a ) | A. x e. a A. e e. RR+ E. d e. RR+ A. y e. a ( ( abs ` ( x - y ) ) < d -> ( abs ` ( ( f ` x ) - ( f ` y ) ) ) < e ) } )
chtpy 22766	class Htpy
cphtpy 22767	class PHtpy
cphtpc 22768	class ~=ph
df-htpy 22769	|- Htpy = ( x e. Top , y e. Top |-> ( f e. ( x Cn y ) , g e. ( x Cn y ) |-> { h e. ( ( x tX II ) Cn y ) | A. s e. U. x ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) )
df-phtpy 22770	|- PHtpy = ( x e. Top |-> ( f e. ( II Cn x ) , g e. ( II Cn x ) |-> { h e. ( f ( II Htpy x ) g ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) } ) )
df-phtpc 22791	|- ~=ph = ( x e. Top |-> { <. f , g >. | ( { f , g } C_ ( II Cn x ) /\ ( f ( PHtpy ` x ) g ) =/= (/) ) } )
cpco 22800	class *p
comi 22801	class Om1
comn 22802	class OmN
cpi1 22803	class pi1
cpin 22804	class piN
df-pco 22805	|- *p = ( j e. Top |-> ( f e. ( II Cn j ) , g e. ( II Cn j ) |-> ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( f ` ( 2 x. x ) ) , ( g ` ( ( 2 x. x ) - 1 ) ) ) ) ) )
df-om1 22806	|- Om1 = ( j e. Top , y e. U. j |-> { <. ( Base ` ndx ) , { f e. ( II Cn j ) | ( ( f ` 0 ) = y /\ ( f ` 1 ) = y ) } >. , <. ( +g ` ndx ) , ( *p ` j ) >. , <. (TopSet ` ndx ) , ( j ^ko II ) >. } )
df-omn 22807	|- OmN = ( j e. Top , y e. U. j |-> seq 0 ( ( ( x e. _V , p e. _V |-> <. ( ( TopOpen ` ( 1st ` x ) ) Om1 ( 2nd ` x ) ) , ( ( 0 [,] 1 ) X. { ( 2nd ` x ) } ) >. ) o. 1st ) , <. { <. ( Base ` ndx ) , U. j >. , <. (TopSet ` ndx ) , j >. } , y >. ) )
df-pi1 22808	|- pi1 = ( j e. Top , y e. U. j |-> ( ( j Om1 y ) /.s ( ~=ph ` j ) ) )
df-pin 22809	|- piN = ( j e. Top , p e. U. j |-> ( n e. NN0 |-> ( ( 1st ` ( ( j OmN p ) ` n ) ) /.s if ( n = 0 , { <. x , y >. | E. f e. ( II Cn j ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) } , ( ~=ph ` ( TopOpen ` ( 1st ` ( ( j OmN p ) ` ( n - 1 ) ) ) ) ) ) ) ) )
cclm 22862	class CMod
df-clm 22863	|- CMod = { w e. LMod | [. (Scalar ` w ) / f ]. [. ( Base ` f ) / k ]. ( f = (ℂfld ↾s k ) /\ k e. (SubRing ` ℂfld ) ) }
ccvs 22923	class CVec
df-cvs 22924	|- CVec = (CMod i^i LVec )
ccph 22966	class CPreHil
ctch 22967	class toCHil
df-cph 22968	|- CPreHil = { w e. ( PreHil i^i NrmMod ) | [. (Scalar ` w ) / f ]. [. ( Base ` f ) / k ]. ( f = (ℂfld ↾s k ) /\ ( sqr " ( k i^i ( 0 [,) +oo ) ) ) C_ k /\ ( norm ` w ) = ( x e. ( Base ` w ) |-> ( sqr ` ( x ( .i ` w ) x ) ) ) ) }
df-tch 22969	|- toCHil = ( w e. _V |-> ( w toNrmGrp ( x e. ( Base ` w ) |-> ( sqr ` ( x ( .i ` w ) x ) ) ) ) )
ccfil 23050	class CauFil
cca 23051	class Cau
cms 23052	class CMet
df-cfil 23053	|- CauFil = ( d e. U. ran *Met |-> { f e. ( Fil ` dom dom d ) | A. x e. RR+ E. y e. f ( d " ( y X. y ) ) C_ ( 0 [,) x ) } )
df-cau 23054	|- Cau = ( d e. U. ran *Met |-> { f e. ( dom dom d ^pm CC ) | A. x e. RR+ E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( f ` j ) ( ball ` d ) x ) } )
df-cmet 23055	|- CMet = ( x e. _V |-> { d e. ( Met ` x ) | A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) } )
ccms 23129	class CMetSp
cbn 23130	class Ban
chl 23131	class CHil
df-cms 23132	|- CMetSp = { w e. MetSp | [. ( Base ` w ) / b ]. ( ( dist ` w ) |` ( b X. b ) ) e. ( CMet ` b ) }
df-bn 23133	|- Ban = { w e. (NrmVec i^i CMetSp ) | (Scalar ` w ) e. CMetSp }
df-hl 23134	|- CHil = (Ban i^i CPreHil )
crrx 23171	class ℝ^
cehl 23172	class 𝔼hil
df-rrx 23173	|- ℝ^ = ( i e. _V |-> (toCHil ` (RRfld freeLMod i ) ) )
df-ehl 23174	|- 𝔼hil = ( n e. NN0 |-> (ℝ^ ` ( 1 ... n ) ) )
covol 23231	class vol*
cvol 23232	class vol
df-ovol 23233	|- vol* = ( x e. ~P RR |-> inf ( { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } , RR* , < ) )
df-vol 23234	|- vol = ( vol* |` { x | A. y e. ( `' vol* " RR ) ( vol* ` y ) = ( ( vol* ` ( y i^i x ) ) + ( vol* ` ( y \ x ) ) ) } )
cmbf 23383	class MblFn
citg1 23384	class S.1
citg2 23385	class S.2
cibl 23386	class L^1
citg 23387	class S. A B _d x
df-mbf 23388	|- MblFn = { f e. ( CC ^pm RR ) | A. x e. ran (,) ( ( `' ( Re o. f ) " x ) e. dom vol /\ ( `' ( Im o. f ) " x ) e. dom vol ) }
df-itg1 23389	|- S.1 = ( f e. { g e. MblFn | ( g : RR --> RR /\ ran g e. Fin /\ ( vol ` ( `' g " ( RR \ { 0 } ) ) ) e. RR ) } |-> sum_ x e. ( ran f \ { 0 } ) ( x x. ( vol ` ( `' f " { x } ) ) ) )
df-itg2 23390	|- S.2 = ( f e. ( ( 0 [,] +oo ) ^m RR ) |-> sup ( { x | E. g e. dom S.1 ( g oR <_ f /\ x = ( S.1 ` g ) ) } , RR* , < ) )
df-ibl 23391	|- L^1 = { f e. MblFn | A. k e. ( 0 ... 3 ) ( S.2 ` ( x e. RR |-> [_ ( Re ` ( ( f ` x ) / ( _i ^ k ) ) ) / y ]_ if ( ( x e. dom f /\ 0 <_ y ) , y , 0 ) ) ) e. RR }
df-itg 23392	|- S. A B _d x = sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) ) )
c0p 23436	class 0p
df-0p 23437	|- 0p = ( CC X. { 0 } )
cdit 23610	class S__ [ A -> B ] C _d x
df-ditg 23611	|- S__ [ A -> B ] C _d x = if ( A <_ B , S. ( A (,) B ) C _d x , -u S. ( B (,) A ) C _d x )
climc 23626	class lim CC
cdv 23627	class _D
cdvn 23628	class Dn
ccpn 23629	class C^n
df-limc 23630	|- lim CC = ( f e. ( CC ^pm CC ) , x e. CC |-> { y | [. ( TopOpen ` ℂfld ) / j ]. ( z e. ( dom f u. { x } ) |-> if ( z = x , y , ( f ` z ) ) ) e. ( ( ( j ↾t ( dom f u. { x } ) ) CnP j ) ` x ) } )
df-dv 23631	|- _D = ( s e. ~P CC , f e. ( CC ^pm s ) |-> U_ x e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t s ) ) ` dom f ) ( { x } X. ( ( z e. ( dom f \ { x } ) |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) )
df-dvn 23632	|- Dn = ( s e. ~P CC , f e. ( CC ^pm s ) |-> seq 0 ( ( ( x e. _V |-> ( s _D x ) ) o. 1st ) , ( NN0 X. { f } ) ) )
df-cpn 23633	|- C^n = ( s e. ~P CC |-> ( x e. NN0 |-> { f e. ( CC ^pm s ) | ( ( s Dn f ) ` x ) e. ( dom f -cn-> CC ) } ) )
cmdg 23813	class mDeg
cdg1 23814	class deg1
df-mdeg 23815	|- mDeg = ( i e. _V , r e. _V |-> ( f e. ( Base ` ( i mPoly r ) ) |-> sup ( ran ( h e. ( f supp ( 0g ` r ) ) |-> (ℂfld gsumg h ) ) , RR* , < ) ) )
df-deg1 23816	|- deg1 = ( r e. _V |-> ( 1o mDeg r ) )
cmn1 23885	class Monic1p
cuc1p 23886	class Unic1p
cq1p 23887	class quot1p
cr1p 23888	class rem1p
cig1p 23889	class idlGen1p
df-mon1 23890	|- Monic1p = ( r e. _V |-> { f e. ( Base ` (Poly1 ` r ) ) | ( f =/= ( 0g ` (Poly1 ` r ) ) /\ ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) = ( 1r ` r ) ) } )
df-uc1p 23891	|- Unic1p = ( r e. _V |-> { f e. ( Base ` (Poly1 ` r ) ) | ( f =/= ( 0g ` (Poly1 ` r ) ) /\ ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) e. (Unit ` r ) ) } )
df-q1p 23892	|- quot1p = ( r e. _V |-> [_ (Poly1 ` r ) / p ]_ [_ ( Base ` p ) / b ]_ ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) )
df-r1p 23893	|- rem1p = ( r e. _V |-> [_ ( Base ` (Poly1 ` r ) ) / b ]_ ( f e. b , g e. b |-> ( f ( -g ` (Poly1 ` r ) ) ( ( f (quot1p ` r ) g ) ( .r ` (Poly1 ` r ) ) g ) ) ) )
df-ig1p 23894	|- idlGen1p = ( r e. _V |-> ( i e. (LIdeal ` (Poly1 ` r ) ) |-> if ( i = { ( 0g ` (Poly1 ` r ) ) } , ( 0g ` (Poly1 ` r ) ) , ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) ) ) ) )
cply 23940	class Poly
cidp 23941	class Xp
ccoe 23942	class coeff
cdgr 23943	class deg
df-ply 23944	|- Poly = ( x e. ~P CC |-> { f | E. n e. NN0 E. a e. ( ( x u. { 0 } ) ^m NN0 ) f = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) } )
df-idp 23945	|- Xp = ( _I |` CC )
df-coe 23946	|- coeff = ( f e. (Poly ` CC ) |-> ( iota_ a e. ( CC ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ f = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) ) )
df-dgr 23947	|- deg = ( f e. (Poly ` CC ) |-> sup ( ( `' (coeff ` f ) " ( CC \ { 0 } ) ) , NN0 , < ) )
cquot 24045	class quot
df-quot 24046	|- quot = ( f e. (Poly ` CC ) , g e. ( (Poly ` CC ) \ { 0p } ) |-> ( iota_ q e. (Poly ` CC ) [. ( f oF - ( g oF x. q ) ) / r ]. ( r = 0p \/ (deg ` r ) < (deg ` g ) ) ) )
caa 24069	class AA
df-aa 24070	|- AA = U_ f e. ( (Poly ` ZZ ) \ { 0p } ) ( `' f " { 0 } )
ctayl 24107	class Tayl
cana 24108	class Ana
df-tayl 24109	|- Tayl = ( s e. { RR , CC } , f e. ( CC ^pm s ) |-> ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( s Dn f ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( s Dn f ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) ) )
df-ana 24110	|- Ana = ( s e. { RR , CC } |-> { f e. ( CC ^pm s ) | A. x e. dom f x e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t s ) ) ` dom ( f i^i ( +oo ( s Tayl f ) x ) ) ) } )
culm 24130	class ~~> u
df-ulm 24131	|- ~~> u = ( s e. _V |-> { <. f , y >. | E. n e. ZZ ( f : ( ZZ>= ` n ) --> ( CC ^m s ) /\ y : s --> CC /\ A. x e. RR+ E. j e. ( ZZ>= ` n ) A. k e. ( ZZ>= ` j ) A. z e. s ( abs ` ( ( ( f ` k ) ` z ) - ( y ` z ) ) ) < x ) } )
clog 24301	class log
ccxp 24302	class ^c
df-log 24303	|- log = `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
df-cxp 24304	|- ^c = ( x e. CC , y e. CC |-> if ( x = 0 , if ( y = 0 , 1 , 0 ) , ( exp ` ( y x. ( log ` x ) ) ) ) )
clogb 24502	class logb
df-logb 24503	|- logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
casin 24589	class arcsin
cacos 24590	class arccos
catan 24591	class arctan
df-asin 24592	|- arcsin = ( x e. CC |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) ) ) )
df-acos 24593	|- arccos = ( x e. CC |-> ( ( pi / 2 ) - (arcsin ` x ) ) )
df-atan 24594	|- arctan = ( x e. ( CC \ { -u _i , _i } ) |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) )
carea 24682	class area
df-area 24683	|- area = ( s e. { t e. ~P ( RR X. RR ) | ( A. x e. RR ( t " { x } ) e. ( `' vol " RR ) /\ ( x e. RR |-> ( vol ` ( t " { x } ) ) ) e. L^1 ) } |-> S. RR ( vol ` ( s " { x } ) ) _d x )
cem 24718	class gamma
df-em 24719	|- gamma = sum_ k e. NN ( ( 1 / k ) - ( log ` ( 1 + ( 1 / k ) ) ) )
czeta 24739	class zeta
df-zeta 24740	|- zeta = ( iota_ f e. ( ( CC \ { 1 } ) -cn-> CC ) A. s e. ( CC \ { 1 } ) ( ( 1 - ( 2 ^c ( 1 - s ) ) ) x. ( f ` s ) ) = sum_ n e. NN0 ( sum_ k e. ( 0 ... n ) ( ( ( -u 1 ^ k ) x. ( n _C k ) ) x. ( ( k + 1 ) ^c s ) ) / ( 2 ^ ( n + 1 ) ) ) )
clgam 24742	class log _G
cgam 24743	class _G
cigam 24744	class 1/ _G
df-lgam 24745	|- log _G = ( z e. ( CC \ ( ZZ \ NN ) ) |-> ( sum_ m e. NN ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) - ( log ` z ) ) )
df-gam 24746	|- _G = ( exp o. log _G )
df-igam 24747	|- 1/ _G = ( x e. CC |-> if ( x e. ( ZZ \ NN ) , 0 , ( 1 / ( _G ` x ) ) ) )
ccht 24817	class theta
cvma 24818	class Λ
cchp 24819	class ψ
cppi 24820	class π
cmu 24821	class mmu
csgm 24822	class sigma
df-cht 24823	|- theta = ( x e. RR |-> sum_ p e. ( ( 0 [,] x ) i^i Prime ) ( log ` p ) )
df-vma 24824	|- Λ = ( x e. NN |-> [_ { p e. Prime | p || x } / s ]_ if ( ( # ` s ) = 1 , ( log ` U. s ) , 0 ) )
df-chp 24825	|- ψ = ( x e. RR |-> sum_ n e. ( 1 ... ( |_ ` x ) ) (Λ ` n ) )
df-ppi 24826	|- π = ( x e. RR |-> ( # ` ( ( 0 [,] x ) i^i Prime ) ) )
df-mu 24827	|- mmu = ( x e. NN |-> if ( E. p e. Prime ( p ^ 2 ) || x , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || x } ) ) ) )
df-sgm 24828	|- sigma = ( x e. CC , n e. NN |-> sum_ k e. { p e. NN | p || n } ( k ^c x ) )
cdchr 24957	class DChr
df-dchr 24958	|- DChr = ( n e. NN |-> [_ (ℤ/nℤ ` n ) / z ]_ [_ { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } / b ]_ { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( oF x. |` ( b X. b ) ) >. } )
clgs 25019	class /L
df-lgs 25020	|- /L = ( a e. ZZ , n e. ZZ |-> if ( n = 0 , if ( ( a ^ 2 ) = 1 , 1 , 0 ) , ( if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) ) ) ` ( abs ` n ) ) ) ) )
cstrkg 25329	class TarskiG
cstrkgc 25330	class TarskiGC
cstrkgb 25331	class TarskiGB
cstrkgcb 25332	class TarskiGCB
cstrkgld 25333	class DimTarskiG≥
cstrkge 25334	class TarskiGE
citv 25335	class Itv
clng 25336	class LineG
df-itv 25337	|- Itv = Slot ; 1 6
df-lng 25338	|- LineG = Slot ; 1 7
df-trkgc 25347	|- TarskiGC = { f | [. ( Base ` f ) / p ]. [. ( dist ` f ) / d ]. ( A. x e. p A. y e. p ( x d y ) = ( y d x ) /\ A. x e. p A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y ) ) }
df-trkgb 25348	|- TarskiGB = { f | [. ( Base ` f ) / p ]. [. (Itv ` f ) / i ]. ( A. x e. p A. y e. p ( y e. ( x i x ) -> x = y ) /\ A. x e. p A. y e. p A. z e. p A. u e. p A. v e. p ( ( u e. ( x i z ) /\ v e. ( y i z ) ) -> E. a e. p ( a e. ( u i y ) /\ a e. ( v i x ) ) ) /\ A. s e. ~P p A. t e. ~P p ( E. a e. p A. x e. s A. y e. t x e. ( a i y ) -> E. b e. p A. x e. s A. y e. t b e. ( x i y ) ) ) }
df-trkgcb 25349	|- TarskiGCB = { f | [. ( Base ` f ) / p ]. [. ( dist ` f ) / d ]. [. (Itv ` f ) / i ]. ( A. x e. p A. y e. p A. z e. p A. u e. p A. a e. p A. b e. p A. c e. p A. v e. p ( ( ( x =/= y /\ y e. ( x i z ) /\ b e. ( a i c ) ) /\ ( ( ( x d y ) = ( a d b ) /\ ( y d z ) = ( b d c ) ) /\ ( ( x d u ) = ( a d v ) /\ ( y d u ) = ( b d v ) ) ) ) -> ( z d u ) = ( c d v ) ) /\ A. x e. p A. y e. p A. a e. p A. b e. p E. z e. p ( y e. ( x i z ) /\ ( y d z ) = ( a d b ) ) ) }
df-trkge 25350	|- TarskiGE = { f | [. ( Base ` f ) / p ]. [. (Itv ` f ) / i ]. A. x e. p A. y e. p A. z e. p A. u e. p A. v e. p ( ( u e. ( x i v ) /\ u e. ( y i z ) /\ x =/= u ) -> E. a e. p E. b e. p ( y e. ( x i a ) /\ z e. ( x i b ) /\ v e. ( a i b ) ) ) }
df-trkgld 25351	|- DimTarskiG≥ = { <. g , n >. | [. ( Base ` g ) / p ]. [. ( dist ` g ) / d ]. [. (Itv ` g ) / i ]. E. f ( f : ( 1..^ n ) -1-1-> p /\ E. x e. p E. y e. p E. z e. p ( A. j e. ( 2..^ n ) ( ( ( f ` 1 ) d x ) = ( ( f ` j ) d x ) /\ ( ( f ` 1 ) d y ) = ( ( f ` j ) d y ) /\ ( ( f ` 1 ) d z ) = ( ( f ` j ) d z ) ) /\ -. ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) ) ) }
df-trkg 25352	|- TarskiG = ( (TarskiGC i^i TarskiGB ) i^i (TarskiGCB i^i { f | [. ( Base ` f ) / p ]. [. (Itv ` f ) / i ]. (LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) } ) )
ccgrg 25405	class cgrG
df-cgrg 25406	|- cgrG = ( g e. _V |-> { <. a , b >. | ( ( a e. ( ( Base ` g ) ^pm RR ) /\ b e. ( ( Base ` g ) ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) ( dist ` g ) ( a ` j ) ) = ( ( b ` i ) ( dist ` g ) ( b ` j ) ) ) ) } )
cismt 25427	class Ismt
df-ismt 25428	|- Ismt = ( g e. _V , h e. _V |-> { f | ( f : ( Base ` g ) -1-1-onto-> ( Base ` h ) /\ A. a e. ( Base ` g ) A. b e. ( Base ` g ) ( ( f ` a ) ( dist ` h ) ( f ` b ) ) = ( a ( dist ` g ) b ) ) } )
cleg 25477	class ≤G
df-leg 25478	|- ≤G = ( g e. _V |-> { <. e , f >. | [. ( Base ` g ) / p ]. [. ( dist ` g ) / d ]. [. (Itv ` g ) / i ]. E. x e. p E. y e. p ( f = ( x d y ) /\ E. z e. p ( z e. ( x i y ) /\ e = ( x d z ) ) ) } )
chlg 25495	class hlG
df-hlg 25496	|- hlG = ( g e. _V |-> ( c e. ( Base ` g ) |-> { <. a , b >. | ( ( a e. ( Base ` g ) /\ b e. ( Base ` g ) ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c (Itv ` g ) b ) \/ b e. ( c (Itv ` g ) a ) ) ) ) } ) )
cmir 25547	class pInvG
df-mir 25548	|- pInvG = ( g e. _V |-> ( m e. ( Base ` g ) |-> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( m ( dist ` g ) b ) = ( m ( dist ` g ) a ) /\ m e. ( b (Itv ` g ) a ) ) ) ) ) )
crag 25588	class ∟G
df-rag 25589	|- ∟G = ( g e. _V |-> { w e. Word ( Base ` g ) | ( ( # ` w ) = 3 /\ ( ( w ` 0 ) ( dist ` g ) ( w ` 2 ) ) = ( ( w ` 0 ) ( dist ` g ) ( ( (pInvG ` g ) ` ( w ` 1 ) ) ` ( w ` 2 ) ) ) ) } )
cperpg 25590	class ⟂G
df-perpg 25591	|- ⟂G = ( g e. _V |-> { <. a , b >. | ( ( a e. ran (LineG ` g ) /\ b e. ran (LineG ` g ) ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` g ) ) } )
chpg 25649	class hpG
df-hpg 25650	|- hpG = ( g e. _V |-> ( d e. ran (LineG ` g ) |-> { <. a , b >. | [. ( Base ` g ) / p ]. [. (Itv ` g ) / i ]. E. c e. p ( ( ( a e. ( p \ d ) /\ c e. ( p \ d ) ) /\ E. t e. d t e. ( a i c ) ) /\ ( ( b e. ( p \ d ) /\ c e. ( p \ d ) ) /\ E. t e. d t e. ( b i c ) ) ) } ) )
cmid 25664	class midG
clmi 25665	class lInvG
df-mid 25666	|- midG = ( g e. _V |-> ( a e. ( Base ` g ) , b e. ( Base ` g ) |-> ( iota_ m e. ( Base ` g ) b = ( ( (pInvG ` g ) ` m ) ` a ) ) ) )
df-lmi 25667	|- lInvG = ( g e. _V |-> ( m e. ran (LineG ` g ) |-> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( a (midG ` g ) b ) e. m /\ ( m (⟂G ` g ) ( a (LineG ` g ) b ) \/ a = b ) ) ) ) ) )
ccgra 25699	class cgrA
df-cgra 25700	|- cgrA = ( g e. _V |-> { <. a , b >. | [. ( Base ` g ) / p ]. [. (hlG ` g ) / k ]. ( ( a e. ( p ^m ( 0..^ 3 ) ) /\ b e. ( p ^m ( 0..^ 3 ) ) ) /\ E. x e. p E. y e. p ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( k ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( k ` ( b ` 1 ) ) ( b ` 2 ) ) ) } )
cinag 25726	class inA
cleag 25727	class ≤∠
df-inag 25728	|- inA = ( g e. _V |-> { <. p , t >. | ( ( p e. ( Base ` g ) /\ t e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. ( Base ` g ) ( x e. ( ( t ` 0 ) (Itv ` g ) ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( (hlG ` g ) ` ( t ` 1 ) ) p ) ) ) ) } )
df-leag 25732	|- ≤∠ = ( g e. _V |-> { <. a , b >. | ( ( a e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) /\ b e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) ) /\ E. x e. ( Base ` g ) ( x (inA ` g ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` g ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } )
ceqlg 25745	class eqltrG
df-eqlg 25746	|- eqltrG = ( g e. _V |-> { x e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) | x (cgrG ` g ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> } )
cttg 25753	class toTG
df-ttg 25754	|- toTG = ( w e. _V |-> [_ ( x e. ( Base ` w ) , y e. ( Base ` w ) |-> { z e. ( Base ` w ) | E. k e. ( 0 [,] 1 ) ( z ( -g ` w ) x ) = ( k ( .s ` w ) ( y ( -g ` w ) x ) ) } ) / i ]_ ( ( w sSet <. (Itv ` ndx ) , i >. ) sSet <. (LineG ` ndx ) , ( x e. ( Base ` w ) , y e. ( Base ` w ) |-> { z e. ( Base ` w ) | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) >. ) )
cee 25768	class EE
cbtwn 25769	class Btwn
ccgr 25770	class Cgr
df-ee 25771	|- EE = ( n e. NN |-> ( RR ^m ( 1 ... n ) ) )
df-btwn 25772	|- Btwn = `' { <. <. x , z >. , y >. | E. n e. NN ( ( x e. ( EE ` n ) /\ z e. ( EE ` n ) /\ y e. ( EE ` n ) ) /\ E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... n ) ( y ` i ) = ( ( ( 1 - t ) x. ( x ` i ) ) + ( t x. ( z ` i ) ) ) ) }
df-cgr 25773	|- Cgr = { <. x , y >. | E. n e. NN ( ( x e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` x ) ` i ) - ( ( 2nd ` x ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) ) }
ceeng 25857	class EEG
df-eeng 25858	|- EEG = ( n e. NN |-> ( { <. ( Base ` ndx ) , ( EE ` n ) >. , <. ( dist ` ndx ) , ( x e. ( EE ` n ) , y e. ( EE ` n ) |-> sum_ i e. ( 1 ... n ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. } u. { <. (Itv ` ndx ) , ( x e. ( EE ` n ) , y e. ( EE ` n ) |-> { z e. ( EE ` n ) | z Btwn <. x , y >. } ) >. , <. (LineG ` ndx ) , ( x e. ( EE ` n ) , y e. ( ( EE ` n ) \ { x } ) |-> { z e. ( EE ` n ) | ( z Btwn <. x , y >. \/ x Btwn <. z , y >. \/ y Btwn <. x , z >. ) } ) >. } ) )
cedgf 25867	class .ef
df-edgf 25868	|- .ef = Slot ; 1 8
cvtx 25874	class Vtx
ciedg 25875	class iEdg
df-vtx 25876	|- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
df-iedg 25877	|- iEdg = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 2nd ` g ) , (.ef ` g ) ) )
cedg 25939	class Edg
df-edg 25940	|- Edg = ( g e. _V |-> ran (iEdg ` g ) )
cuhgr 25951	class UHGraph
cushgr 25952	class USHGraph
df-uhgr 25953	|- UHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) }
df-ushgr 25954	|- USHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> ( ~P v \ { (/) } ) }
cupgr 25975	class UPGraph
cumgr 25976	class UMGraph
df-upgr 25977	|- UPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } }
df-umgr 25978	|- UMGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }
cuspgr 26043	class USPGraph
cusgr 26044	class USGraph
df-uspgr 26045	|- USPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } }
df-usgr 26046	|- USGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }
csubgr 26159	class SubGraph
df-subgr 26160	|- SubGraph = { <. s , g >. | ( (Vtx ` s ) C_ (Vtx ` g ) /\ (iEdg ` s ) = ( (iEdg ` g ) |` dom (iEdg ` s ) ) /\ (Edg ` s ) C_ ~P (Vtx ` s ) ) }
cfusgr 26208	class FinUSGraph
df-fusgr 26209	|- FinUSGraph = { g e. USGraph | (Vtx ` g ) e. Fin }
cnbgr 26224	class NeighbVtx
cuvtxa 26225	class UnivVtx
ccplgr 26226	class ComplGraph
ccusgr 26227	class ComplUSGraph
df-nbgr 26228	|- NeighbVtx = ( g e. _V , v e. (Vtx ` g ) |-> { n e. ( (Vtx ` g ) \ { v } ) | E. e e. (Edg ` g ) { v , n } C_ e } )
df-uvtxa 26230	|- UnivVtx = ( g e. _V |-> { v e. (Vtx ` g ) | A. n e. ( (Vtx ` g ) \ { v } ) n e. ( g NeighbVtx v ) } )
df-cplgr 26231	|- ComplGraph = { g | A. v e. (Vtx ` g ) v e. (UnivVtx ` g ) }
df-cusgr 26232	|- ComplUSGraph = { g e. USGraph | g e. ComplGraph }
cvtxdg 26361	class VtxDeg
df-vtxdg 26362	|- VtxDeg = ( g e. _V |-> [_ (Vtx ` g ) / v ]_ [_ (iEdg ` g ) / e ]_ ( u e. v |-> ( ( # ` { x e. dom e | u e. ( e ` x ) } ) +e ( # ` { x e. dom e | ( e ` x ) = { u } } ) ) ) )
crgr 26451	class RegGraph
crusgr 26452	class RegUSGraph
df-rgr 26453	|- RegGraph = { <. g , k >. | ( k e. NN0* /\ A. v e. (Vtx ` g ) ( (VtxDeg ` g ) ` v ) = k ) }
df-rusgr 26454	|- RegUSGraph = { <. g , k >. | ( g e. USGraph /\ g RegGraph k ) }
cewlks 26491	class EdgWalks
cwlks 26492	class Walks
cwlkson 26493	class WalksOn
df-ewlks 26494	|- EdgWalks = ( g e. _V , s e. NN0* |-> { f | [. (iEdg ` g ) / i ]. ( f e. Word dom i /\ A. k e. ( 1..^ ( # ` f ) ) s <_ ( # ` ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) ) ) } )
df-wlks 26495	|- Walks = ( g e. _V |-> { <. f , p >. | ( f e. Word dom (iEdg ` g ) /\ p : ( 0 ... ( # ` f ) ) --> (Vtx ` g ) /\ A. k e. ( 0..^ ( # ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( f ` k ) ) ) ) } )
df-wlkson 26496	|- WalksOn = ( g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( f (Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) } ) )
ctrls 26587	class Trails
ctrlson 26588	class TrailsOn
df-trls 26589	|- Trails = ( g e. _V |-> { <. f , p >. | ( f (Walks ` g ) p /\ Fun `' f ) } )
df-trlson 26590	|- TrailsOn = ( g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( f ( a (WalksOn ` g ) b ) p /\ f (Trails ` g ) p ) } ) )
cpths 26608	class Paths
cspths 26609	class SPaths
cpthson 26610	class PathsOn
cspthson 26611	class SPathsOn
df-pths 26612	|- Paths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ Fun `' ( p |` ( 1..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1..^ ( # ` f ) ) ) ) = (/) ) } )
df-spths 26613	|- SPaths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ Fun `' p ) } )
df-pthson 26614	|- PathsOn = ( g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( f ( a (TrailsOn ` g ) b ) p /\ f (Paths ` g ) p ) } ) )
df-spthson 26615	|- SPathsOn = ( g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( f ( a (TrailsOn ` g ) b ) p /\ f (SPaths ` g ) p ) } ) )
cclwlks 26666	class ClWalks
df-clwlks 26667	|- ClWalks = ( g e. _V |-> { <. f , p >. | ( f (Walks ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
ccrcts 26679	class Circuits
ccycls 26680	class Cycles
df-crcts 26681	|- Circuits = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
df-cycls 26682	|- Cycles = ( g e. _V |-> { <. f , p >. | ( f (Paths ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
cwwlks 26717	class WWalks
cwwlksn 26718	class WWalksN
cwwlksnon 26719	class WWalksNOn
cwwspthsn 26720	class WSPathsN
cwwspthsnon 26721	class WSPathsNOn
df-wwlks 26722	|- WWalks = ( g e. _V |-> { w e. Word (Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) ) } )
df-wwlksn 26723	|- WWalksN = ( n e. NN0 , g e. _V |-> { w e. (WWalks ` g ) | ( # ` w ) = ( n + 1 ) } )
df-wwlksnon 26724	|- WWalksNOn = ( n e. NN0 , g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { w e. ( n WWalksN g ) | ( ( w ` 0 ) = a /\ ( w ` n ) = b ) } ) )
df-wspthsn 26725	|- WSPathsN = ( n e. NN0 , g e. _V |-> { w e. ( n WWalksN g ) | E. f f (SPaths ` g ) w } )
df-wspthsnon 26726	|- WSPathsNOn = ( n e. NN0 , g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { w e. ( a ( n WWalksNOn g ) b ) | E. f f ( a (SPathsOn ` g ) b ) w } ) )
cclwwlks 26875	class ClWWalks
cclwwlksn 26876	class ClWWalksN
df-clwwlks 26877	|- ClWWalks = ( g e. _V |-> { w e. Word (Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { ( lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) } )
df-clwwlksn 26878	|- ClWWalksN = ( n e. NN , g e. _V |-> { w e. (ClWWalks ` g ) | ( # ` w ) = n } )
cconngr 27046	class ConnGraph
df-conngr 27047	|- ConnGraph = { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. v E. f E. p f ( k (PathsOn ` g ) n ) p }
ceupth 27057	class EulerPaths
df-eupth 27058	|- EulerPaths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ f : ( 0..^ ( # ` f ) ) -onto-> dom (iEdg ` g ) ) } )
cfrgr 27120	class FriendGraph
df-frgr 27121	|- FriendGraph = { g | ( g e. USGraph /\ [. (Vtx ` g ) / v ]. [. (Edg ` g ) / e ]. A. k e. v A. l e. ( v \ { k } ) E! x e. v { { x , k } , { x , l } } C_ e ) }
cplig 27326	class Plig
df-plig 27327	|- Plig = { x | ( A. a e. U. x A. b e. U. x ( a =/= b -> E! l e. x ( a e. l /\ b e. l ) ) /\ A. l e. x E. a e. U. x E. b e. U. x ( a =/= b /\ a e. l /\ b e. l ) /\ E. a e. U. x E. b e. U. x E. c e. U. x A. l e. x -. ( a e. l /\ b e. l /\ c e. l ) ) }
crpm 27339	class RPrime
df-rprm 27340	|- RPrime = ( w e. _V |-> [_ ( Base ` w ) / b ]_ { p e. ( b \ ( (Unit ` w ) u. { ( 0g ` w ) } ) ) | A. x e. b A. y e. b [. ( ||r ` w ) / d ]. ( p d ( x ( .r ` w ) y ) -> ( p d x \/ p d y ) ) } )
cgr 27343	class GrpOp
cgi 27344	class GId
cgn 27345	class inv
cgs 27346	class /g
df-grpo 27347	|- GrpOp = { g | E. t ( g : ( t X. t ) --> t /\ A. x e. t A. y e. t A. z e. t ( ( x g y ) g z ) = ( x g ( y g z ) ) /\ E. u e. t A. x e. t ( ( u g x ) = x /\ E. y e. t ( y g x ) = u ) ) }
df-gid 27348	|- GId = ( g e. _V |-> ( iota_ u e. ran g A. x e. ran g ( ( u g x ) = x /\ ( x g u ) = x ) ) )
df-ginv 27349	|- inv = ( g e. GrpOp |-> ( x e. ran g |-> ( iota_ z e. ran g ( z g x ) = (GId ` g ) ) ) )
df-gdiv 27350	|- /g = ( g e. GrpOp |-> ( x e. ran g , y e. ran g |-> ( x g ( ( inv ` g ) ` y ) ) ) )
cablo 27398	class AbelOp
df-ablo 27399	|- AbelOp = { g e. GrpOp | A. x e. ran g A. y e. ran g ( x g y ) = ( y g x ) }
cvc 27413	class CVecOLD
df-vc 27414	|- CVecOLD = { <. g , s >. | ( g e. AbelOp /\ s : ( CC X. ran g ) --> ran g /\ A. x e. ran g ( ( 1 s x ) = x /\ A. y e. CC ( A. z e. ran g ( y s ( x g z ) ) = ( ( y s x ) g ( y s z ) ) /\ A. z e. CC ( ( ( y + z ) s x ) = ( ( y s x ) g ( z s x ) ) /\ ( ( y x. z ) s x ) = ( y s ( z s x ) ) ) ) ) ) }
cnv 27439	class NrmCVec
cpv 27440	class +v
cba 27441	class BaseSet
cns 27442	class .sOLD
cn0v 27443	class 0vec
cnsb 27444	class -v
cnmcv 27445	class normCV
cims 27446	class IndMet
df-nv 27447	|- NrmCVec = { <. <. g , s >. , n >. | ( <. g , s >. e. CVecOLD /\ n : ran g --> RR /\ A. x e. ran g ( ( ( n ` x ) = 0 -> x = (GId ` g ) ) /\ A. y e. CC ( n ` ( y s x ) ) = ( ( abs ` y ) x. ( n ` x ) ) /\ A. y e. ran g ( n ` ( x g y ) ) <_ ( ( n ` x ) + ( n ` y ) ) ) ) }
df-va 27450	|- +v = ( 1st o. 1st )
df-ba 27451	|- BaseSet = ( x e. _V |-> ran ( +v ` x ) )
df-sm 27452	|- .sOLD = ( 2nd o. 1st )
df-0v 27453	|- 0vec = (GId o. +v )
df-vs 27454	|- -v = ( /g o. +v )
df-nmcv 27455	|- normCV = 2nd
df-ims 27456	|- IndMet = ( u e. NrmCVec |-> ( ( normCV ` u ) o. ( -v ` u ) ) )
cdip 27555	class .iOLD
df-dip 27556	|- .iOLD = ( u e. NrmCVec |-> ( x e. ( BaseSet ` u ) , y e. ( BaseSet ` u ) |-> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` u ) ` ( x ( +v ` u ) ( ( _i ^ k ) ( .sOLD ` u ) y ) ) ) ^ 2 ) ) / 4 ) ) )
css 27576	class SubSp
df-ssp 27577	|- SubSp = ( u e. NrmCVec |-> { w e. NrmCVec | ( ( +v ` w ) C_ ( +v ` u ) /\ ( .sOLD ` w ) C_ ( .sOLD ` u ) /\ ( normCV ` w ) C_ ( normCV ` u ) ) } )
clno 27595	class LnOp
cnmoo 27596	class normOpOLD
cblo 27597	class BLnOp
c0o 27598	class 0op
df-lno 27599	|- LnOp = ( u e. NrmCVec , w e. NrmCVec |-> { t e. ( ( BaseSet ` w ) ^m ( BaseSet ` u ) ) | A. x e. CC A. y e. ( BaseSet ` u ) A. z e. ( BaseSet ` u ) ( t ` ( ( x ( .sOLD ` u ) y ) ( +v ` u ) z ) ) = ( ( x ( .sOLD ` w ) ( t ` y ) ) ( +v ` w ) ( t ` z ) ) } )
df-nmoo 27600	|- normOpOLD = ( u e. NrmCVec , w e. NrmCVec |-> ( t e. ( ( BaseSet ` w ) ^m ( BaseSet ` u ) ) |-> sup ( { x | E. z e. ( BaseSet ` u ) ( ( ( normCV ` u ) ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } , RR* , < ) ) )
df-blo 27601	|- BLnOp = ( u e. NrmCVec , w e. NrmCVec |-> { t e. ( u LnOp w ) | ( ( u normOpOLD w ) ` t ) < +oo } )
df-0o 27602	|- 0op = ( u e. NrmCVec , w e. NrmCVec |-> ( ( BaseSet ` u ) X. { ( 0vec ` w ) } ) )
caj 27603	class adj
chmo 27604	class HmOp
df-aj 27605	|- adj = ( u e. NrmCVec , w e. NrmCVec |-> { <. t , s >. | ( t : ( BaseSet ` u ) --> ( BaseSet ` w ) /\ s : ( BaseSet ` w ) --> ( BaseSet ` u ) /\ A. x e. ( BaseSet ` u ) A. y e. ( BaseSet ` w ) ( ( t ` x ) ( .iOLD ` w ) y ) = ( x ( .iOLD ` u ) ( s ` y ) ) ) } )
df-hmo 27606	|- HmOp = ( u e. NrmCVec |-> { t e. dom ( u adj u ) | ( ( u adj u ) ` t ) = t } )
ccphlo 27667	class CPreHil OLD
df-ph 27668	|- CPreHil OLD = ( NrmCVec i^i { <. <. g , s >. , n >. | A. x e. ran g A. y e. ran g ( ( ( n ` ( x g y ) ) ^ 2 ) + ( ( n ` ( x g ( -u 1 s y ) ) ) ^ 2 ) ) = ( 2 x. ( ( ( n ` x ) ^ 2 ) + ( ( n ` y ) ^ 2 ) ) ) } )
ccbn 27718	class CBan
df-cbn 27719	|- CBan = { u e. NrmCVec | ( IndMet ` u ) e. ( CMet ` ( BaseSet ` u ) ) }
chlo 27741	class CHilOLD
df-hlo 27742	|- CHilOLD = ( CBan i^i CPreHil OLD )
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
chil 27776	class ~H
cva 27777	class +h
csm 27778	class .h
csp 27779	class .ih
cno 27780	class normh
c0v 27781	class 0h
cmv 27782	class -h
ccau 27783	class Cauchy
chli 27784	class ~~>v
csh 27785	class SH
cch 27786	class CH
cort 27787	class _|_
cph 27788	class +H
cspn 27789	class span
chj 27790	class vH
chsup 27791	class \/H
c0h 27792	class 0H
ccm 27793	class C_H
cpjh 27794	class proj h
chos 27795	class +op
chot 27796	class .op
chod 27797	class -op
chfs 27798	class +fn
chft 27799	class .fn
ch0o 27800	class 0hop
chio 27801	class Iop
cnop 27802	class normop
ccop 27803	class ContOp
clo 27804	class LinOp
cbo 27805	class BndLinOp
cuo 27806	class UniOp
cho 27807	class HrmOp
cnmf 27808	class normfn
cnl 27809	class null
ccnfn 27810	class ContFn
clf 27811	class LinFn
cado 27812	class adjh
cbr 27813	class bra
ck 27814	class ketbra
cleo 27815	class <_op
cei 27816	class eigvec
cel 27817	class eigval
cspc 27818	class Lambda
cst 27819	class States
chst 27820	class CHStates
ccv 27821	class <oH
cat 27822	class HAtoms
cmd 27823	class MH
cdmd 27824	class MH*
df-hnorm 27825	|- normh = ( x e. dom dom .ih |-> ( sqr ` ( x .ih x ) ) )
df-hba 27826	|- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. )
df-h0v 27827	|- 0h = ( 0vec ` <. <. +h , .h >. , normh >. )
df-hvsub 27828	|- -h = ( x e. ~H , y e. ~H |-> ( x +h ( -u 1 .h y ) ) )
df-hlim 27829	|- ~~>v = { <. f , w >. | ( ( f : NN --> ~H /\ w e. ~H ) /\ A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( f ` z ) -h w ) ) < x ) }
df-hcau 27830	|- Cauchy = { f e. ( ~H ^m NN ) | A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( f ` y ) -h ( f ` z ) ) ) < x }
ax-hilex 27856	|- ~H e. _V
ax-hfvadd 27857	|- +h : ( ~H X. ~H ) --> ~H
ax-hvcom 27858	|- ( ( A e. ~H /\ B e. ~H ) -> ( A +h B ) = ( B +h A ) )
ax-hvass 27859	|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h B ) +h C ) = ( A +h ( B +h C ) ) )
ax-hv0cl 27860	|- 0h e. ~H
ax-hvaddid 27861	|- ( A e. ~H -> ( A +h 0h ) = A )
ax-hfvmul 27862	|- .h : ( CC X. ~H ) --> ~H
ax-hvmulid 27863	|- ( A e. ~H -> ( 1 .h A ) = A )
ax-hvmulass 27864	|- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A x. B ) .h C ) = ( A .h ( B .h C ) ) )
ax-hvdistr1 27865	|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( A .h ( B +h C ) ) = ( ( A .h B ) +h ( A .h C ) ) )
ax-hvdistr2 27866	|- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A + B ) .h C ) = ( ( A .h C ) +h ( B .h C ) ) )
ax-hvmul0 27867	|- ( A e. ~H -> ( 0 .h A ) = 0h )
ax-hfi 27936	|- .ih : ( ~H X. ~H ) --> CC
ax-his1 27939	|- ( ( A e. ~H /\ B e. ~H ) -> ( A .ih B ) = ( * ` ( B .ih A ) ) )
ax-his2 27940	|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h B ) .ih C ) = ( ( A .ih C ) + ( B .ih C ) ) )
ax-his3 27941	|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( A .h B ) .ih C ) = ( A x. ( B .ih C ) ) )
ax-his4 27942	|- ( ( A e. ~H /\ A =/= 0h ) -> 0 < ( A .ih A ) )
ax-hcompl 28059	|- ( F e. Cauchy -> E. x e. ~H F ~~>v x )
df-sh 28064	|- SH = { h e. ~P ~H | ( 0h e. h /\ ( +h " ( h X. h ) ) C_ h /\ ( .h " ( CC X. h ) ) C_ h ) }
df-ch 28078	|- CH = { h e. SH | ( ~~>v " ( h ^m NN ) ) C_ h }
df-oc 28109	|- _|_ = ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )
df-ch0 28110	|- 0H = { 0h }
df-shs 28167	|- +H = ( x e. SH , y e. SH |-> ( +h " ( x X. y ) ) )
df-span 28168	|- span = ( x e. ~P ~H |-> |^| { y e. SH | x C_ y } )
df-chj 28169	|- vH = ( x e. ~P ~H , y e. ~P ~H |-> ( _|_ ` ( _|_ ` ( x u. y ) ) ) )
df-chsup 28170	|- \/H = ( x e. ~P ~P ~H |-> ( _|_ ` ( _|_ ` U. x ) ) )
df-pjh 28254	|- proj h = ( h e. CH |-> ( x e. ~H |-> ( iota_ z e. h E. y e. ( _|_ ` h ) x = ( z +h y ) ) ) )
df-cm 28442	|- C_H = { <. x , y >. | ( ( x e. CH /\ y e. CH ) /\ x = ( ( x i^i y ) vH ( x i^i ( _|_ ` y ) ) ) ) }
df-hosum 28589	|- +op = ( f e. ( ~H ^m ~H ) , g e. ( ~H ^m ~H ) |-> ( x e. ~H |-> ( ( f ` x ) +h ( g ` x ) ) ) )
df-homul 28590	|- .op = ( f e. CC , g e. ( ~H ^m ~H ) |-> ( x e. ~H |-> ( f .h ( g ` x ) ) ) )
df-hodif 28591	|- -op = ( f e. ( ~H ^m ~H ) , g e. ( ~H ^m ~H ) |-> ( x e. ~H |-> ( ( f ` x ) -h ( g ` x ) ) ) )
df-hfsum 28592	|- +fn = ( f e. ( CC ^m ~H ) , g e. ( CC ^m ~H ) |-> ( x e. ~H |-> ( ( f ` x ) + ( g ` x ) ) ) )
df-hfmul 28593	|- .fn = ( f e. CC , g e. ( CC ^m ~H ) |-> ( x e. ~H |-> ( f x. ( g ` x ) ) ) )
df-h0op 28607	|- 0hop = ( proj h ` 0H )
df-iop 28608	|- Iop = ( proj h ` ~H )
df-nmop 28698	|- normop = ( t e. ( ~H ^m ~H ) |-> sup ( { x | E. z e. ~H ( ( normh ` z ) <_ 1 /\ x = ( normh ` ( t ` z ) ) ) } , RR* , < ) )
df-cnop 28699	|- ContOp = { t e. ( ~H ^m ~H ) | A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) }
df-lnop 28700	|- LinOp = { t e. ( ~H ^m ~H ) | A. x e. CC A. y e. ~H A. z e. ~H ( t ` ( ( x .h y ) +h z ) ) = ( ( x .h ( t ` y ) ) +h ( t ` z ) ) }
df-bdop 28701	|- BndLinOp = { t e. LinOp | ( normop ` t ) < +oo }
df-unop 28702	|- UniOp = { t | ( t : ~H -onto-> ~H /\ A. x e. ~H A. y e. ~H ( ( t ` x ) .ih ( t ` y ) ) = ( x .ih y ) ) }
df-hmop 28703	|- HrmOp = { t e. ( ~H ^m ~H ) | A. x e. ~H A. y e. ~H ( x .ih ( t ` y ) ) = ( ( t ` x ) .ih y ) }
df-nmfn 28704	|- normfn = ( t e. ( CC ^m ~H ) |-> sup ( { x | E. z e. ~H ( ( normh ` z ) <_ 1 /\ x = ( abs ` ( t ` z ) ) ) } , RR* , < ) )
df-nlfn 28705	|- null = ( t e. ( CC ^m ~H ) |-> ( `' t " { 0 } ) )
df-cnfn 28706	|- ContFn = { t e. ( CC ^m ~H ) | A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( t ` w ) - ( t ` x ) ) ) < y ) }
df-lnfn 28707	|- LinFn = { t e. ( CC ^m ~H ) | A. x e. CC A. y e. ~H A. z e. ~H ( t ` ( ( x .h y ) +h z ) ) = ( ( x x. ( t ` y ) ) + ( t ` z ) ) }
df-adjh 28708	|- adjh = { <. t , u >. | ( t : ~H --> ~H /\ u : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( ( t ` x ) .ih y ) = ( x .ih ( u ` y ) ) ) }
df-bra 28709	|- bra = ( x e. ~H |-> ( y e. ~H |-> ( y .ih x ) ) )
df-kb 28710	|- ketbra = ( x e. ~H , y e. ~H |-> ( z e. ~H |-> ( ( z .ih y ) .h x ) ) )
df-leop 28711	|- <_op = { <. t , u >. | ( ( u -op t ) e. HrmOp /\ A. x e. ~H 0 <_ ( ( ( u -op t ) ` x ) .ih x ) ) }
df-eigvec 28712	|- eigvec = ( t e. ( ~H ^m ~H ) |-> { x e. ( ~H \ 0H ) | E. z e. CC ( t ` x ) = ( z .h x ) } )
df-eigval 28713	|- eigval = ( t e. ( ~H ^m ~H ) |-> ( x e. ( eigvec ` t ) |-> ( ( ( t ` x ) .ih x ) / ( ( normh ` x ) ^ 2 ) ) ) )
df-spec 28714	|- Lambda = ( t e. ( ~H ^m ~H ) |-> { x e. CC | -. ( t -op ( x .op ( _I |` ~H ) ) ) : ~H -1-1-> ~H } )
df-st 29070	|- States = { f e. ( ( 0 [,] 1 ) ^m CH ) | ( ( f ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( f ` ( x vH y ) ) = ( ( f ` x ) + ( f ` y ) ) ) ) }
df-hst 29071	|- CHStates = { f e. ( ~H ^m CH ) | ( ( normh ` ( f ` ~H ) ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( ( ( f ` x ) .ih ( f ` y ) ) = 0 /\ ( f ` ( x vH y ) ) = ( ( f ` x ) +h ( f ` y ) ) ) ) ) }
df-cv 29138	|- <oH = { <. x , y >. | ( ( x e. CH /\ y e. CH ) /\ ( x C. y /\ -. E. z e. CH ( x C. z /\ z C. y ) ) ) }
df-md 29139	|- MH = { <. x , y >. | ( ( x e. CH /\ y e. CH ) /\ A. z e. CH ( z C_ y -> ( ( z vH x ) i^i y ) = ( z vH ( x i^i y ) ) ) ) }
df-dmd 29140	|- MH* = { <. x , y >. | ( ( x e. CH /\ y e. CH ) /\ A. z e. CH ( y C_ z -> ( ( z i^i x ) vH y ) = ( z i^i ( x vH y ) ) ) ) }
df-at 29197	|- HAtoms = { x e. CH | 0H <oH x }
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here
cdp2 29577	class _ A B
df-dp2 29578	|- _ A B = ( A + ( B / ; 1 0 ) )
cdp 29595	class period
df-dp 29596	|- period = ( x e. NN0 , y e. RR |-> _ x y )
cxdiv 29625	class /𝑒
df-xdiv 29626	|- /𝑒 = ( x e. RR* , y e. ( RR \ { 0 } ) |-> ( iota_ z e. RR* ( y xe z ) = x ) )
ax-xrssca 29673	|- RRfld = (Scalar ` RR*s )
ax-xrsvsca 29674	|- xe = ( .s ` RR*s )
comnd 29697	class oMnd
cogrp 29698	class oGrp
df-omnd 29699	|- oMnd = { g e. Mnd | [. ( Base ` g ) / v ]. [. ( +g ` g ) / p ]. [. ( le ` g ) / l ]. ( g e. Toset /\ A. a e. v A. b e. v A. c e. v ( a l b -> ( a p c ) l ( b p c ) ) ) }
df-ogrp 29700	|- oGrp = ( Grp i^i oMnd )
csgns 29725	class sgns
df-sgns 29726	|- sgns = ( r e. _V |-> ( x e. ( Base ` r ) |-> if ( x = ( 0g ` r ) , 0 , if ( ( 0g ` r ) ( lt ` r ) x , 1 , -u 1 ) ) ) )
cinftm 29730	class <<<
carchi 29731	class Archi
df-inftm 29732	|- <<< = ( w e. _V |-> { <. x , y >. | ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) /\ ( ( 0g ` w ) ( lt ` w ) x /\ A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y ) ) } )
df-archi 29733	|- Archi = { w | (<<< ` w ) = (/) }
cslmd 29753	class SLMod
df-slmd 29754	|- SLMod = { g e. CMnd | [. ( Base ` g ) / v ]. [. ( +g ` g ) / a ]. [. ( .s ` g ) / s ]. [. (Scalar ` g ) / f ]. [. ( Base ` f ) / k ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. ( f e. SRing /\ A. q e. k A. r e. k A. x e. v A. w e. v ( ( ( r s w ) e. v /\ ( r s ( w a x ) ) = ( ( r s w ) a ( r s x ) ) /\ ( ( q p r ) s w ) = ( ( q s w ) a ( r s w ) ) ) /\ ( ( ( q t r ) s w ) = ( q s ( r s w ) ) /\ ( ( 1r ` f ) s w ) = w /\ ( ( 0g ` f ) s w ) = ( 0g ` g ) ) ) ) }
corng 29795	class oRing
cofld 29796	class oField
df-orng 29797	|- oRing = { r e. ( Ring i^i oGrp ) | [. ( Base ` r ) / v ]. [. ( 0g ` r ) / z ]. [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. v A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) }
df-ofld 29798	|- oField = (Field i^i oRing )
cresv 29824	class ↾v
df-resv 29825	|- ↾v = ( w e. _V , x e. _V |-> if ( ( Base ` (Scalar ` w ) ) C_ x , w , ( w sSet <. (Scalar ` ndx ) , ( (Scalar ` w ) ↾s x ) >. ) ) )
csmat 29859	class subMat1
df-smat 29860	|- subMat1 = ( m e. _V |-> ( k e. NN , l e. NN |-> ( m o. ( i e. NN , j e. NN |-> <. if ( i < k , i , ( i + 1 ) ) , if ( j < l , j , ( j + 1 ) ) >. ) ) ) )
clmat 29877	class litMat
df-lmat 29878	|- litMat = ( m e. _V |-> ( i e. ( 1 ... ( # ` m ) ) , j e. ( 1 ... ( # ` ( m ` 0 ) ) ) |-> ( ( m ` ( i - 1 ) ) ` ( j - 1 ) ) ) )
ccref 29909	class CovHasRef A
df-cref 29910	|- CovHasRef A = { j e. Top | A. y e. ~P j ( U. j = U. y -> E. z e. ( ~P j i^i A ) z Ref y ) }
cldlf 29919	class Ldlf
df-ldlf 29920	|- Ldlf = CovHasRef { x | x ~<_ om }
cpcmp 29922	class Paracomp
df-pcmp 29923	|- Paracomp = { j | j e. CovHasRef ( LocFin ` j ) }
cmetid 29929	class ~Met
cpstm 29930	class pstoMet
df-metid 29931	|- ~Met = ( d e. U. ran PsMet |-> { <. x , y >. | ( ( x e. dom dom d /\ y e. dom dom d ) /\ ( x d y ) = 0 ) } )
df-pstm 29932	|- pstoMet = ( d e. U. ran PsMet |-> ( a e. ( dom dom d /. (~Met ` d ) ) , b e. ( dom dom d /. (~Met ` d ) ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) )
chcmp 30002	class HCmp
df-hcmp 30003	|- HCmp = { <. u , w >. | ( ( u e. U. ran UnifOn /\ w e. CUnifSp ) /\ ( (UnifSt ` w ) ↾t dom U. u ) = u /\ ( ( cls ` ( TopOpen ` w ) ) ` dom U. u ) = ( Base ` w ) ) }
cqqh 30016	class QQHom
df-qqh 30017	|- QQHom = ( r e. _V |-> ran ( x e. ZZ , y e. ( `' ( ZRHom ` r ) " (Unit ` r ) ) |-> <. ( x / y ) , ( ( ( ZRHom ` r ) ` x ) (/r ` r ) ( ( ZRHom ` r ) ` y ) ) >. ) )
crrh 30037	class RRHom
crrext 30038	class ℝExt
df-rrh 30039	|- RRHom = ( r e. _V |-> ( ( ( topGen ` ran (,) )CnExt ( TopOpen ` r ) ) ` (QQHom ` r ) ) )
df-rrext 30043	|- ℝExt = { r e. (NrmRing i^i DivRing ) | ( ( ( ZMod ` r ) e. NrmMod /\ (chr ` r ) = 0 ) /\ ( r e. CUnifSp /\ (UnifSt ` r ) = (metUnif ` ( ( dist ` r ) |` ( ( Base ` r ) X. ( Base ` r ) ) ) ) ) ) }
cxrh 30060	class RR*Hom
df-xrh 30061	|- RR*Hom = ( r e. _V |-> ( x e. RR* |-> if ( x e. RR , ( (RRHom ` r ) ` x ) , if ( x = +oo , ( ( lub ` r ) ` ( (RRHom ` r ) " RR ) ) , ( ( glb ` r ) ` ( (RRHom ` r ) " RR ) ) ) ) ) )
cmntop 30066	class ManTop
df-mntop 30067	|- ManTop = { <. n , j >. | ( n e. NN0 /\ ( j e. 2ndc /\ j e. Haus /\ j e. Locally [ ( TopOpen ` (𝔼hil ` n ) ) ] ~= ) ) }
cind 30072	class 𝟭
df-ind 30073	|- 𝟭 = ( o e. _V |-> ( a e. ~P o |-> ( x e. o |-> if ( x e. a , 1 , 0 ) ) ) )
cesum 30089	class Σ* k e. A B
df-esum 30090	|- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
cofc 30157	class ∘𝑓/𝑐 R
df-ofc 30158	|- ∘𝑓/𝑐 R = ( f e. _V , c e. _V |-> ( x e. dom f |-> ( ( f ` x ) R c ) ) )
csiga 30170	class sigAlgebra
df-siga 30171	|- sigAlgebra = ( o e. _V |-> { s | ( s C_ ~P o /\ ( o e. s /\ A. x e. s ( o \ x ) e. s /\ A. x e. ~P s ( x ~<_ om -> U. x e. s ) ) ) } )
csigagen 30201	class sigaGen
df-sigagen 30202	|- sigaGen = ( x e. _V |-> |^| { s e. (sigAlgebra ` U. x ) | x C_ s } )
cbrsiga 30244	class 𝔅ℝ
df-brsiga 30245	|- 𝔅ℝ = (sigaGen ` ( topGen ` ran (,) ) )
csx 30251	class ×s
df-sx 30252	|- ×s = ( s e. _V , t e. _V |-> (sigaGen ` ran ( x e. s , y e. t |-> ( x X. y ) ) ) )
cmeas 30258	class measures
df-meas 30259	|- measures = ( s e. U. ran sigAlgebra |-> { m | ( m : s --> ( 0 [,] +oo ) /\ ( m ` (/) ) = 0 /\ A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) ) } )
cdde 30295	class δ
df-dde 30296	|- δ = ( a e. ~P RR |-> if ( 0 e. a , 1 , 0 ) )
cae 30300	class a.e.
cfae 30301	class ~ a.e.
df-ae 30302	|- a.e. = { <. a , m >. | ( m ` ( U. dom m \ a ) ) = 0 }
df-fae 30308	|- ~ a.e. = ( r e. _V , m e. U. ran measures |-> { <. f , g >. | ( ( f e. ( dom r ^m U. dom m ) /\ g e. ( dom r ^m U. dom m ) ) /\ { x e. U. dom m | ( f ` x ) r ( g ` x ) }a.e. m ) } )
cmbfm 30312	class MblFnM
df-mbfm 30313	|- MblFnM = ( s e. U. ran sigAlgebra , t e. U. ran sigAlgebra |-> { f e. ( U. t ^m U. s ) | A. x e. t ( `' f " x ) e. s } )
coms 30353	class toOMeas
df-oms 30354	|- toOMeas = ( r e. _V |-> ( a e. ~P U. dom r |-> inf ( ran ( x e. { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( r ` y ) ) , ( 0 [,] +oo ) , < ) ) )
ccarsg 30363	class toCaraSiga
df-carsg 30364	|- toCaraSiga = ( m e. _V |-> { a e. ~P U. dom m | A. e e. ~P U. dom m ( ( m ` ( e i^i a ) ) +e ( m ` ( e \ a ) ) ) = ( m ` e ) } )
citgm 30389	class itgm
csitm 30390	class sitm
csitg 30391	class sitg
df-sitg 30392	|- sitg = ( w e. _V , m e. U. ran measures |-> ( f e. { g e. ( dom mMblFnM (sigaGen ` ( TopOpen ` w ) ) ) | ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( w gsumg ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( (RRHom ` (Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) ) ) ) )
df-sitm 30393	|- sitm = ( w e. _V , m e. U. ran measures |-> ( f e. dom ( wsitg m ) , g e. dom ( wsitg m ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg m ) ` ( f oF ( dist ` w ) g ) ) ) )
df-itgm 30415	|- itgm = ( w e. _V , m e. U. ran measures |-> ( ( (metUnif ` ( wsitm m ) )CnExt (UnifSt ` w ) ) ` ( wsitg m ) ) )
csseq 30445	class seqstr
df-sseq 30446	|- seqstr = ( m e. _V , f e. _V |-> ( m u. ( lastS o. seq ( # ` m ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( f ` x ) "> ) ) , ( NN0 X. { ( m ++ <" ( f ` m ) "> ) } ) ) ) ) )
cfib 30458	class Fibci
df-fib 30459	|- Fibci = ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) )
cprb 30469	class Prob
df-prob 30470	|- Prob = { p e. U. ran measures | ( p ` U. dom p ) = 1 }
ccprob 30493	class cprob
df-cndprob 30494	|- cprob = ( p e. Prob |-> ( a e. dom p , b e. dom p |-> ( ( p ` ( a i^i b ) ) / ( p ` b ) ) ) )
crrv 30502	class rRndVar
df-rrv 30503	|- rRndVar = ( p e. Prob |-> ( dom pMblFnM𝔅ℝ ) )
corvc 30517	class ∘RV/𝑐 R
df-orvc 30518	|- ∘RV/𝑐 R = ( x e. { x | Fun x } , a e. _V |-> ( `' x " { y | y R a } ) )
crepr 30686	class repr
df-repr 30687	|- repr = ( s e. NN0 |-> ( b e. ~P NN , m e. ZZ |-> { c e. ( b ^m ( 0..^ s ) ) | sum_ a e. ( 0..^ s ) ( c ` a ) = m } ) )
cvts 30713	class vts
df-vts 30714	|- vts = ( l e. ( CC ^m NN ) , n e. NN0 |-> ( x e. CC |-> sum_ a e. ( 1 ... n ) ( ( l ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) ) )
ax-hgt749 30722	|- A. n e. { z e. ZZ | -. 2 || z } ( (; 1 0 ^; 2 7 ) <_ n -> E. h e. ( ( 0 [,) +oo ) ^m NN ) E. k e. ( ( 0 [,) +oo ) ^m NN ) ( A. m e. NN ( k ` m ) <_ ( 1 period_ 0_ 7_ 9_ 9_ 5 5 ) /\ A. m e. NN ( h ` m ) <_ ( 1 period_ 4_ 1 4 ) /\ ( ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) x. ( n ^ 2 ) ) <_ S. ( 0 (,) 1 ) ( ( ( ( (Λ oF x. h )vts n ) ` x ) x. ( ( ( (Λ oF x. k )vts n ) ` x ) ^ 2 ) ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( -u n x. x ) ) ) ) _d x ) )
ax-ros335 30723	|- A. x e. RR+ (ψ ` x ) < ( ( 1 period_ 0_ 3_ 8_ 8 3 ) x. x )
ax-ros336 30724	|- A. x e. RR+ ( (ψ ` x ) - ( theta ` x ) ) < ( ( 1 period_ 4_ 2_ 6 2 ) x. ( sqr ` x ) )
cstrkg2d 30742	class TarskiG2D
df-trkg2d 30743	|- TarskiG2D = { f | [. ( Base ` f ) / p ]. [. ( dist ` f ) / d ]. [. (Itv ` f ) / i ]. ( E. x e. p E. y e. p E. z e. p -. ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) /\ A. x e. p A. y e. p A. z e. p A. u e. p A. v e. p ( ( ( ( x d u ) = ( x d v ) /\ ( y d u ) = ( y d v ) /\ ( z d u ) = ( z d v ) ) /\ u =/= v ) -> ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) ) ) }
cafs 30747	class AFS
df-afs 30748	|- AFS = ( g e. TarskiG |-> { <. e , f >. | [. ( Base ` g ) / p ]. [. ( dist ` g ) / h ]. [. (Itv ` g ) / i ]. E. a e. p E. b e. p E. c e. p E. d e. p E. x e. p E. y e. p E. z e. p E. w e. p ( e = <. <. a , b >. , <. c , d >. >. /\ f = <. <. x , y >. , <. z , w >. >. /\ ( ( b e. ( a i c ) /\ y e. ( x i z ) ) /\ ( ( a h b ) = ( x h y ) /\ ( b h c ) = ( y h z ) ) /\ ( ( a h d ) = ( x h w ) /\ ( b h d ) = ( y h w ) ) ) ) } )
w-bnj17 30752	wff ( ph /\ ps /\ ch /\ th )
df-bnj17 30753	|- ( ( ph /\ ps /\ ch /\ th ) <-> ( ( ph /\ ps /\ ch ) /\ th ) )
c-bnj14 30754	class pred ( X , A , R )
df-bnj14 30755	|- pred ( X , A , R ) = { y e. A | y R X }
w-bnj13 30756	wff R Se A
df-bnj13 30757	|- ( R Se A <-> A. x e. A pred ( x , A , R ) e. _V )
w-bnj15 30758	wff R FrSe A
df-bnj15 30759	|- ( R FrSe A <-> ( R Fr A /\ R Se A ) )
c-bnj18 30760	class trCl ( X , A , R )
df-bnj18 30761	|- trCl ( X , A , R ) = U_ f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } U_ i e. dom f ( f ` i )
w-bnj19 30762	wff TrFo ( B , A , R )
df-bnj19 30763	|- ( TrFo ( B , A , R ) <-> A. x e. B pred ( x , A , R ) C_ B )
ax-7d 31141	|- ( A. x A. y ph -> A. y A. x ph )
ax-8d 31142	|- ( x = y -> ( x = z -> y = z ) )
ax-9d1 31143	|- -. A. x -. x = x
ax-9d2 31144	|- -. A. x -. x = y
ax-10d 31145	|- ( A. x x = y -> A. y y = x )
ax-11d 31146	|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
cretr 31199	class Retr
df-retr 31200	|- Retr = ( j e. Top , k e. Top |-> { r e. ( j Cn k ) | E. s e. ( k Cn j ) ( ( r o. s ) ( j Htpy j ) ( _I |` U. j ) ) =/= (/) } )
cpconn 31201	class PConn
csconn 31202	class SConn
df-pconn 31203	|- PConn = { j e. Top | A. x e. U. j A. y e. U. j E. f e. ( II Cn j ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) }
df-sconn 31204	|- SConn = { j e. PConn | A. f e. ( II Cn j ) ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` j ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) }
ccvm 31237	class CovMap
df-cvm 31238	|- CovMap = ( c e. Top , j e. Top |-> { f e. ( c Cn j ) | A. x e. U. j E. k e. j ( x e. k /\ E. s e. ( ~P c \ { (/) } ) ( U. s = ( `' f " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( f |` u ) e. ( ( c ↾t u ) Homeo ( j ↾t k ) ) ) ) ) } )
cgoe 31315	class e.g
cgna 31316	class |g
cgol 31317	class A.g N U
csat 31318	class Sat
cfmla 31319	class Fmla
csate 31320	class SatE
cprv 31321	class |=
df-goel 31322	|- e.g = ( x e. ( om X. om ) |-> <. (/) , x >. )
df-gona 31323	|- |g = ( x e. ( _V X. _V ) |-> <. 1o , x >. )
df-goal 31324	|- A.g N U = <. 2o , <. N , U >. >.
df-sat 31325	|- Sat = ( m e. _V , e e. _V |-> ( rec ( ( f e. _V |-> ( f u. { <. x , y >. | E. u e. f ( E. v e. f ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( m ^m om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. om ( x = A.g i ( 1st ` u ) /\ y = { a e. ( m ^m om ) | A. z e. m ( { <. i , z >. } u. ( a |` ( om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) ) , { <. x , y >. | E. i e. om E. j e. om ( x = ( i e.g j ) /\ y = { a e. ( m ^m om ) | ( a ` i ) e ( a ` j ) } ) } ) |` suc om ) )
df-sate 31326	|- SatE = ( m e. _V , u e. _V |-> ( ( ( m Sat ( _E i^i ( m X. m ) ) ) ` om ) ` u ) )
df-fmla 31327	|- Fmla = ( n e. suc om |-> dom ( ( (/) Sat (/) ) ` n ) )
cgon 31328	class -.g U
cgoa 31329	class /\g
cgoi 31330	class ->g
cgoo 31331	class \/g
cgob 31332	class <->g
cgoq 31333	class =g
cgox 31334	class E.g N U
df-gonot 31335	|- -.g U = ( U |g U )
df-goan 31336	|- /\g = ( u e. _V , v e. _V |-> -.g ( u |g v ) )
df-goim 31337	|- ->g = ( u e. _V , v e. _V |-> ( u |g -.g v ) )
df-goor 31338	|- \/g = ( u e. _V , v e. _V |-> ( -.g u ->g v ) )
df-gobi 31339	|- <->g = ( u e. _V , v e. _V |-> ( ( u ->g v ) /\g ( v ->g u ) ) )
df-goeq 31340	|- =g = ( u e. om , v e. om |-> [_ suc ( u u. v ) / w ]_ A.g w ( ( w e.g u ) <->g ( w e.g v ) ) )
df-goex 31341	|- E.g N U = -.g A.g N -.g U
df-prv 31342	|- |= = { <. m , u >. | ( m SatE u ) = ( m ^m om ) }
cgze 31343	class AxExt
cgzr 31344	class AxRep
cgzp 31345	class AxPow
cgzu 31346	class AxUn
cgzg 31347	class AxReg
cgzi 31348	class AxInf
cgzf 31349	class ZF
df-gzext 31350	|- AxExt = ( A.g 2o ( ( 2o e.g (/) ) <->g ( 2o e.g 1o ) ) ->g ( (/) =g 1o ) )
df-gzrep 31351	|- AxRep = ( u e. ( Fmla ` om ) |-> ( A.g 3o E.g 1o A.g 2o ( A.g 1o u ->g ( 2o =g 1o ) ) ->g A.g 1o A.g 2o ( ( 2o e.g 1o ) <->g E.g 3o ( ( 3o e.g (/) ) /\g A.g 1o u ) ) ) )
df-gzpow 31352	|- AxPow = E.g 1o A.g 2o ( A.g 1o ( ( 1o e.g 2o ) <->g ( 1o e.g (/) ) ) ->g ( 2o e.g 1o ) )
df-gzun 31353	|- AxUn = E.g 1o A.g 2o ( E.g 1o ( ( 2o e.g 1o ) /\g ( 1o e.g (/) ) ) ->g ( 2o e.g 1o ) )
df-gzreg 31354	|- AxReg = ( E.g 1o ( 1o e.g (/) ) ->g E.g 1o ( ( 1o e.g (/) ) /\g A.g 2o ( ( 2o e.g 1o ) ->g -.g ( 2o e.g (/) ) ) ) )
df-gzinf 31355	|- AxInf = E.g 1o ( ( (/) e.g 1o ) /\g A.g 2o ( ( 2o e.g 1o ) ->g E.g (/) ( ( 2o e.g (/) ) /\g ( (/) e.g 1o ) ) ) )
df-gzf 31356	|- ZF = { m | ( ( Tr m /\ m |= AxExt /\ m |= AxPow ) /\ ( m |= AxUn /\ m |= AxReg /\ m |= AxInf ) /\ A. u e. ( Fmla ` om ) m |= ( AxRep ` u ) ) }
cmcn 31357	class mCN
cmvar 31358	class mVR
cmty 31359	class mType
cmvt 31360	class mVT
cmtc 31361	class mTC
cmax 31362	class mAx
cmrex 31363	class mREx
cmex 31364	class mEx
cmdv 31365	class mDV
cmvrs 31366	class mVars
cmrsub 31367	class mRSubst
cmsub 31368	class mSubst
cmvh 31369	class mVH
cmpst 31370	class mPreSt
cmsr 31371	class mStRed
cmsta 31372	class mStat
cmfs 31373	class mFS
cmcls 31374	class mCls
cmpps 31375	class mPPSt
cmthm 31376	class mThm
df-mcn 31377	|- mCN = Slot 1
df-mvar 31378	|- mVR = Slot 2
df-mty 31379	|- mType = Slot 3
df-mtc 31380	|- mTC = Slot 4
df-mmax 31381	|- mAx = Slot 5
df-mvt 31382	|- mVT = ( t e. _V |-> ran (mType ` t ) )
df-mrex 31383	|- mREx = ( t e. _V |-> Word ( (mCN ` t ) u. (mVR ` t ) ) )
df-mex 31384	|- mEx = ( t e. _V |-> ( (mTC ` t ) X. (mREx ` t ) ) )
df-mdv 31385	|- mDV = ( t e. _V |-> ( ( (mVR ` t ) X. (mVR ` t ) ) \ _I ) )
df-mvrs 31386	|- mVars = ( t e. _V |-> ( e e. (mEx ` t ) |-> ( ran ( 2nd ` e ) i^i (mVR ` t ) ) ) )
df-mrsub 31387	|- mRSubst = ( t e. _V |-> ( f e. ( (mREx ` t ) ^pm (mVR ` t ) ) |-> ( e e. (mREx ` t ) |-> ( (freeMnd ` ( (mCN ` t ) u. (mVR ` t ) ) ) gsumg ( ( v e. ( (mCN ` t ) u. (mVR ` t ) ) |-> if ( v e. dom f , ( f ` v ) , <" v "> ) ) o. e ) ) ) ) )
df-msub 31388	|- mSubst = ( t e. _V |-> ( f e. ( (mREx ` t ) ^pm (mVR ` t ) ) |-> ( e e. (mEx ` t ) |-> <. ( 1st ` e ) , ( ( (mRSubst ` t ) ` f ) ` ( 2nd ` e ) ) >. ) ) )
df-mvh 31389	|- mVH = ( t e. _V |-> ( v e. (mVR ` t ) |-> <. ( (mType ` t ) ` v ) , <" v "> >. ) )
df-mpst 31390	|- mPreSt = ( t e. _V |-> ( ( { d e. ~P (mDV ` t ) | `' d = d } X. ( ~P (mEx ` t ) i^i Fin ) ) X. (mEx ` t ) ) )
df-msr 31391	|- mStRed = ( t e. _V |-> ( s e. (mPreSt ` t ) |-> [_ ( 2nd ` ( 1st ` s ) ) / h ]_ [_ ( 2nd ` s ) / a ]_ <. ( ( 1st ` ( 1st ` s ) ) i^i [_ U. ( (mVars ` t ) " ( h u. { a } ) ) / z ]_ ( z X. z ) ) , h , a >. ) )
df-msta 31392	|- mStat = ( t e. _V |-> ran (mStRed ` t ) )
df-mfs 31393	|- mFS = { t | ( ( ( (mCN ` t ) i^i (mVR ` t ) ) = (/) /\ (mType ` t ) : (mVR ` t ) --> (mTC ` t ) ) /\ ( (mAx ` t ) C_ (mStat ` t ) /\ A. v e. (mVT ` t ) -. ( `' (mType ` t ) " { v } ) e. Fin ) ) }
df-mcls 31394	|- mCls = ( t e. _V |-> ( d e. ~P (mDV ` t ) , h e. ~P (mEx ` t ) |-> |^| { c | ( ( h u. ran (mVH ` t ) ) C_ c /\ A. m A. o A. p ( <. m , o , p >. e. (mAx ` t ) -> A. s e. ran (mSubst ` t ) ( ( ( s " ( o u. ran (mVH ` t ) ) ) C_ c /\ A. x A. y ( x m y -> ( ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` x ) ) ) X. ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` y ) ) ) ) C_ d ) ) -> ( s ` p ) e. c ) ) ) } ) )
df-mpps 31395	|- mPPSt = ( t e. _V |-> { <. <. d , h >. , a >. | ( <. d , h , a >. e. (mPreSt ` t ) /\ a e. ( d (mCls ` t ) h ) ) } )
df-mthm 31396	|- mThm = ( t e. _V |-> ( `' (mStRed ` t ) " ( (mStRed ` t ) " (mPPSt ` t ) ) ) )
cm0s 31482	class m0St
cmsa 31483	class mSA
cmwgfs 31484	class mWGFS
cmsy 31485	class mSyn
cmesy 31486	class mESyn
cmgfs 31487	class mGFS
cmtree 31488	class mTree
cmst 31489	class mST
cmsax 31490	class mSAX
cmufs 31491	class mUFS
df-m0s 31492	|- m0St = ( a e. _V |-> <. (/) , (/) , a >. )
df-msa 31493	|- mSA = ( t e. _V |-> { a e. (mEx ` t ) | ( (m0St ` a ) e. (mAx ` t ) /\ ( 1st ` a ) e. (mVT ` t ) /\ Fun ( `' ( 2nd ` a ) |` (mVR ` t ) ) ) } )
df-mwgfs 31494	|- mWGFS = { t e. mFS | A. d A. h A. a ( ( <. d , h , a >. e. (mAx ` t ) /\ ( 1st ` a ) e. (mVT ` t ) ) -> E. s e. ran (mSubst ` t ) a e. ( s " (mSA ` t ) ) ) }
df-msyn 31495	|- mSyn = Slot 6
df-mtree 31496	|- mTree = ( t e. _V |-> ( d e. ~P (mDV ` t ) , h e. ~P (mEx ` t ) |-> |^| { r | ( A. e e. ran (mVH ` t ) e r <. (m0St ` e ) , (/) >. /\ A. e e. h e r <. ( (mStRed ` t ) ` <. d , h , e >. ) , (/) >. /\ A. m A. o A. p ( <. m , o , p >. e. (mAx ` t ) -> A. s e. ran (mSubst ` t ) ( A. x A. y ( x m y -> ( ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` x ) ) ) X. ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` y ) ) ) ) C_ d ) -> ( { ( s ` p ) } X. X_ e e. ( o u. ( (mVH ` t ) " U. ( (mVars ` t ) " ( o u. { p } ) ) ) ) ( r " { ( s ` e ) } ) ) C_ r ) ) ) } ) )
df-mst 31497	|- mST = ( t e. _V |-> ( ( (/) (mTree ` t ) (/) ) |` ( (mEx ` t ) |` (mVT ` t ) ) ) )
df-msax 31498	|- mSAX = ( t e. _V |-> ( p e. (mSA ` t ) |-> ( (mVH ` t ) " ( (mVars ` t ) ` p ) ) ) )
df-mufs 31499	|- mUFS = { t e. mGFS | Fun (mST ` t ) }
cmuv 31500	class mUV
cmvl 31501	class mVL
cmvsb 31502	class mVSubst
cmfsh 31503	class mFresh
cmfr 31504	class mFRel
cmevl 31505	class mEval
cmdl 31506	class mMdl
cusyn 31507	class mUSyn
cgmdl 31508	class mGMdl
cmitp 31509	class mItp
cmfitp 31510	class mFromItp
df-muv 31511	|- mUV = Slot 7
df-mfsh 31512	|- mFresh = Slot 8
df-mevl 31513	|- mEval = Slot 9
df-mvl 31514	|- mVL = ( t e. _V |-> X_ v e. (mVR ` t ) ( (mUV ` t ) " { ( (mType ` t ) ` v ) } ) )
df-mvsb 31515	|- mVSubst = ( t e. _V |-> { <. <. s , m >. , x >. | ( ( s e. ran (mSubst ` t ) /\ m e. (mVL ` t ) ) /\ A. v e. (mVR ` t ) m dom (mEval ` t ) ( s ` ( (mVH ` t ) ` v ) ) /\ x = ( v e. (mVR ` t ) |-> ( m (mEval ` t ) ( s ` ( (mVH ` t ) ` v ) ) ) ) ) } )
df-mfrel 31516	|- mFRel = ( t e. _V |-> { r e. ~P ( (mUV ` t ) X. (mUV ` t ) ) | ( `' r = r /\ A. c e. (mVT ` t ) A. w e. ( ~P (mUV ` t ) i^i Fin ) E. v e. ( (mUV ` t ) " { c } ) w C_ ( r " { v } ) ) } )
df-mdl 31517	|- mMdl = { t e. mFS | [. (mUV ` t ) / u ]. [. (mEx ` t ) / x ]. [. (mVL ` t ) / v ]. [. (mEval ` t ) / n ]. [. (mFresh ` t ) / f ]. ( ( u C_ ( (mTC ` t ) X. _V ) /\ f e. (mFRel ` t ) /\ n e. ( u ^pm ( v X. (mEx ` t ) ) ) ) /\ A. m e. v ( ( A. e e. x ( n " { <. m , e >. } ) C_ ( u " { ( 1st ` e ) } ) /\ A. y e. (mVR ` t ) <. m , ( (mVH ` t ) ` y ) >. n ( m ` y ) /\ A. d A. h A. a ( <. d , h , a >. e. (mAx ` t ) -> ( ( A. y A. z ( y d z -> ( m ` y ) f ( m ` z ) ) /\ h C_ ( dom n " { m } ) ) -> m dom n a ) ) ) /\ ( A. s e. ran (mSubst ` t ) A. e e. (mEx ` t ) A. y ( <. s , m >. (mVSubst ` t ) y -> ( n " { <. m , ( s ` e ) >. } ) = ( n " { <. y , e >. } ) ) /\ A. p e. v A. e e. x ( ( m |` ( (mVars ` t ) ` e ) ) = ( p |` ( (mVars ` t ) ` e ) ) -> ( n " { <. m , e >. } ) = ( n " { <. p , e >. } ) ) /\ A. y e. u A. e e. x ( ( m " ( (mVars ` t ) ` e ) ) C_ ( f " { y } ) -> ( n " { <. m , e >. } ) C_ ( f " { y } ) ) ) ) ) }
df-musyn 31518	|- mUSyn = ( t e. _V |-> ( v e. (mUV ` t ) |-> <. ( (mSyn ` t ) ` ( 1st ` v ) ) , ( 2nd ` v ) >. ) )
df-gmdl 31519	|- mGMdl = { t e. (mGFS i^i mMdl ) | ( A. c e. (mTC ` t ) ( (mUV ` t ) " { c } ) C_ ( (mUV ` t ) " { ( (mSyn ` t ) ` c ) } ) /\ A. v e. (mUV ` c ) A. w e. (mUV ` c ) ( v (mFresh ` t ) w <-> v (mFresh ` t ) ( (mUSyn ` t ) ` w ) ) /\ A. m e. (mVL ` t ) A. e e. (mEx ` t ) ( (mEval ` t ) " { <. m , e >. } ) = ( ( (mEval ` t ) " { <. m , ( (mESyn ` t ) ` e ) >. } ) i^i ( (mUV ` t ) " { ( 1st ` e ) } ) ) ) }
df-mitp 31520	|- mItp = ( t e. _V |-> ( a e. (mSA ` t ) |-> ( g e. X_ i e. ( (mVars ` t ) ` a ) ( (mUV ` t ) " { ( (mType ` t ) ` i ) } ) |-> ( iota x E. m e. (mVL ` t ) ( g = ( m |` ( (mVars ` t ) ` a ) ) /\ x = ( m (mEval ` t ) a ) ) ) ) ) )
df-mfitp 31521	|- mFromItp = ( t e. _V |-> ( f e. X_ a e. (mSA ` t ) ( ( (mUV ` t ) " { ( ( 1st ` t ) ` a ) } ) ^m X_ i e. ( (mVars ` t ) ` a ) ( (mUV ` t ) " { ( (mType ` t ) ` i ) } ) ) |-> ( iota_ n e. ( (mUV ` t ) ^pm ( (mVL ` t ) X. (mEx ` t ) ) ) A. m e. (mVL ` t ) ( A. v e. (mVR ` t ) <. m , ( (mVH ` t ) ` v ) >. n ( m ` v ) /\ A. e A. a A. g ( e (mST ` t ) <. a , g >. -> <. m , e >. n ( f ` ( i e. ( (mVars ` t ) ` a ) |-> ( m n ( g ` ( (mVH ` t ) ` i ) ) ) ) ) ) /\ A. e e. (mEx ` t ) ( n " { <. m , e >. } ) = ( ( n " { <. m , ( (mESyn ` t ) ` e ) >. } ) i^i ( (mUV ` t ) " { ( 1st ` e ) } ) ) ) ) ) )
citr 31522	class IntgRing
ccpms 31523	class cplMetSp
chlb 31524	class HomLimB
chlim 31525	class HomLim
cpfl 31526	class polyFld
csf1 31527	class splitFld1
csf 31528	class splitFld
cpsl 31529	class polySplitLim
df-irng 31530	|- IntgRing = ( r e. _V , s e. _V |-> U_ f e. (Monic1p ` ( r ↾s s ) ) ( `' f " { ( 0g ` r ) } ) )
df-cplmet 31531	|- cplMetSp = ( w e. _V |-> [_ ( ( w ^s NN ) ↾s ( Cau ` ( dist ` w ) ) ) / r ]_ [_ ( Base ` r ) / v ]_ [_ { <. f , g >. | ( { f , g } C_ v /\ A. x e. RR+ E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( g ` j ) ( ball ` ( dist ` w ) ) x ) ) } / e ]_ ( ( r /.s e ) sSet { <. ( dist ` ndx ) , { <. <. x , y >. , z >. | E. p e. v E. q e. v ( ( x = [ p ] e /\ y = [ q ] e ) /\ ( p oF ( dist ` r ) q ) ~~> z ) } >. } ) )
df-homlimb 31532	|- HomLimB = ( f e. _V |-> [_ U_ n e. NN ( { n } X. dom ( f ` n ) ) / v ]_ [_ |^| { s | ( s Er v /\ ( x e. v |-> <. ( ( 1st ` x ) + 1 ) , ( ( f ` ( 1st ` x ) ) ` ( 2nd ` x ) ) >. ) C_ s ) } / e ]_ <. ( v /. e ) , ( n e. NN |-> ( x e. dom ( f ` n ) |-> [ <. n , x >. ] e ) ) >. )
df-homlim 31533	|- HomLim = ( r e. _V , f e. _V |-> [_ ( HomLimB ` f ) / e ]_ [_ ( 1st ` e ) / v ]_ [_ ( 2nd ` e ) / g ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , U_ n e. NN ran ( x e. dom ( g ` n ) , y e. dom ( g ` n ) |-> <. <. ( ( g ` n ) ` x ) , ( ( g ` n ) ` y ) >. , ( ( g ` n ) ` ( x ( +g ` ( r ` n ) ) y ) ) >. ) >. , <. ( .r ` ndx ) , U_ n e. NN ran ( x e. dom ( g ` n ) , y e. dom ( g ` n ) |-> <. <. ( ( g ` n ) ` x ) , ( ( g ` n ) ` y ) >. , ( ( g ` n ) ` ( x ( .r ` ( r ` n ) ) y ) ) >. ) >. } u. { <. ( TopOpen ` ndx ) , { s e. ~P v | A. n e. NN ( `' ( g ` n ) " s ) e. ( TopOpen ` ( r ` n ) ) } >. , <. ( dist ` ndx ) , U_ n e. NN ran ( x e. dom ( ( g ` n ) ` n ) , y e. dom ( ( g ` n ) ` n ) |-> <. <. ( ( g ` n ) ` x ) , ( ( g ` n ) ` y ) >. , ( x ( dist ` ( r ` n ) ) y ) >. ) >. , <. ( le ` ndx ) , U_ n e. NN ( `' ( g ` n ) o. ( ( le ` ( r ` n ) ) o. ( g ` n ) ) ) >. } ) )
df-plfl 31534	|- polyFld = ( r e. _V , p e. _V |-> [_ (Poly1 ` r ) / s ]_ [_ ( (RSpan ` s ) ` { p } ) / i ]_ [_ ( z e. ( Base ` r ) |-> [ ( z ( .s ` s ) ( 1r ` s ) ) ] ( s ~QG i ) ) / f ]_ <. [_ ( s /.s ( s ~QG i ) ) / t ]_ ( ( t toNrmGrp ( iota_ n e. (AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) >. ) , f >. )
df-sfl1 31535	|- splitFld1 = ( r e. _V , j e. _V |-> ( p e. (Poly1 ` r ) |-> ( rec ( ( s e. _V , f e. _V |-> [_ ( mPoly ` s ) / m ]_ [_ { g e. ( (Monic1p ` s ) i^i (Irred ` m ) ) | ( g ( ||r ` m ) ( p o. f ) /\ 1 < ( s deg1 g ) ) } / b ]_ if ( ( ( p o. f ) = ( 0g ` m ) \/ b = (/) ) , <. s , f >. , [_ ( glb ` b ) / h ]_ [_ ( s polyFld h ) / t ]_ <. ( 1st ` t ) , ( f o. ( 2nd ` t ) ) >. ) ) , j ) ` ( card ` ( 1 ... ( r deg1 p ) ) ) ) ) )
df-sfl 31536	|- splitFld = ( r e. _V , p e. _V |-> ( iota x E. f ( f Isom < , ( lt ` r ) ( ( 1 ... ( # ` p ) ) , p ) /\ x = ( seq 0 ( ( e e. _V , g e. _V |-> ( ( r splitFld1 e ) ` g ) ) , ( f u. { <. 0 , <. r , ( _I |` ( Base ` r ) ) >. >. } ) ) ` ( # ` p ) ) ) ) )
df-psl 31537	|- polySplitLim = ( r e. _V , p e. ( ( ~P ( Base ` r ) i^i Fin ) ^m NN ) |-> [_ ( 1st o. seq 0 ( ( g e. _V , q e. _V |-> [_ ( 1st ` g ) / e ]_ [_ ( 1st ` e ) / s ]_ [_ ( s splitFld ran ( x e. q |-> ( x o. ( 2nd ` g ) ) ) ) / f ]_ <. f , ( ( 2nd ` g ) o. ( 2nd ` f ) ) >. ) , ( p u. { <. 0 , <. <. r , (/) >. , ( _I |` ( Base ` r ) ) >. >. } ) ) ) / f ]_ ( ( 1st o. ( f shift 1 ) ) HomLim ( 2nd o. f ) ) )
czr 31538	class ZRing
cgf 31539	class GF
cgfo 31540	class GF∞
ceqp 31541	class ~Qp
crqp 31542	class /Qp
cqp 31543	class Qp
cqpOLD 31544	class QpOLD
czp 31545	class Zp
cqpa 31546	class _Qp
ccp 31547	class Cp
df-zrng 31548	|- ZRing = ( r e. _V |-> ( r IntgRing ran ( ZRHom ` r ) ) )
df-gf 31549	|- GF = ( p e. Prime , n e. NN |-> [_ (ℤ/nℤ ` p ) / r ]_ ( 1st ` ( r splitFld { [_ (Poly1 ` r ) / s ]_ [_ (var1 ` r ) / x ]_ ( ( ( p ^ n ) (.g ` (mulGrp ` s ) ) x ) ( -g ` s ) x ) } ) ) )
df-gfoo 31550	|- GF∞ = ( p e. Prime |-> [_ (ℤ/nℤ ` p ) / r ]_ ( r polySplitLim ( n e. NN |-> { [_ (Poly1 ` r ) / s ]_ [_ (var1 ` r ) / x ]_ ( ( ( p ^ n ) (.g ` (mulGrp ` s ) ) x ) ( -g ` s ) x ) } ) ) )
df-eqp 31551	|- ~Qp = ( p e. Prime |-> { <. f , g >. | ( { f , g } C_ ( ZZ ^m ZZ ) /\ A. n e. ZZ sum_ k e. ( ZZ>= ` -u n ) ( ( ( f ` -u k ) - ( g ` -u k ) ) / ( p ^ ( k + ( n + 1 ) ) ) ) e. ZZ ) } )
df-rqp 31552	|- /Qp = ( p e. Prime |-> (~Qp i^i [_ { f e. ( ZZ ^m ZZ ) | E. x e. ran ZZ>= ( `' f " ( ZZ \ { 0 } ) ) C_ x } / y ]_ ( y X. ( y i^i ( ZZ ^m ( 0 ... ( p - 1 ) ) ) ) ) ) )
df-qp 31553	|- Qp = ( p e. Prime |-> [_ { h e. ( ZZ ^m ( 0 ... ( p - 1 ) ) ) | E. x e. ran ZZ>= ( `' h " ( ZZ \ { 0 } ) ) C_ x } / b ]_ ( ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( f e. b , g e. b |-> ( (/Qp ` p ) ` ( f oF + g ) ) ) >. , <. ( .r ` ndx ) , ( f e. b , g e. b |-> ( (/Qp ` p ) ` ( n e. ZZ |-> sum_ k e. ZZ ( ( f ` k ) x. ( g ` ( n - k ) ) ) ) ) ) >. } u. { <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ b /\ sum_ k e. ZZ ( ( f ` -u k ) x. ( ( p + 1 ) ^ -u k ) ) < sum_ k e. ZZ ( ( g ` -u k ) x. ( ( p + 1 ) ^ -u k ) ) ) } >. } ) toNrmGrp ( f e. b |-> if ( f = ( ZZ X. { 0 } ) , 0 , ( p ^ -uinf ( ( `' f " ( ZZ \ { 0 } ) ) , RR , < ) ) ) ) ) )
df-qpOLD 31554	|- QpOLD = ( p e. Prime |-> [_ { h e. ( ZZ ^m ( 0 ... ( p - 1 ) ) ) | E. x e. ran ZZ>= ( `' h " ( ZZ \ { 0 } ) ) C_ x } / b ]_ ( ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( f e. b , g e. b |-> ( (/Qp ` p ) ` ( f oF + g ) ) ) >. , <. ( .r ` ndx ) , ( f e. b , g e. b |-> ( (/Qp ` p ) ` ( n e. ZZ |-> sum_ k e. ZZ ( ( f ` k ) x. ( g ` ( n - k ) ) ) ) ) ) >. } u. { <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ b /\ sum_ k e. ZZ ( ( f ` -u k ) x. ( ( p + 1 ) ^ -u k ) ) < sum_ k e. ZZ ( ( g ` -u k ) x. ( ( p + 1 ) ^ -u k ) ) ) } >. } ) toNrmGrp ( f e. b |-> if ( f = ( ZZ X. { 0 } ) , 0 , ( p ^ -u sup ( ( `' f " ( ZZ \ { 0 } ) ) , RR , `' < ) ) ) ) ) )
df-zp 31555	|- Zp = (ZRing o. Qp )
df-qpa 31556	|- _Qp = ( p e. Prime |-> [_ (Qp ` p ) / r ]_ ( r polySplitLim ( n e. NN |-> { f e. (Poly1 ` r ) | ( ( r deg1 f ) <_ n /\ A. d e. ran (coe1 ` f ) ( `' d " ( ZZ \ { 0 } ) ) C_ ( 0 ... n ) ) } ) ) )
df-cp 31557	|- Cp = ( cplMetSp o. _Qp )
ctrpred 31717	class TrPred ( R , A , X )
df-trpred 31718	|- TrPred ( R , A , X ) = U. ran ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` om )
cwsuc 31752	class wsuc ( R , A , X )
cwsucOLD 31753	class wsucOLD ( R , A , X )
cwlim 31754	class WLim ( R , A )
cwlimOLD 31755	class WLimOLD ( R , A )
df-wsuc 31756	|- wsuc ( R , A , X ) = inf ( Pred ( `' R , A , X ) , A , R )
df-wsucOLD 31757	|- wsucOLD ( R , A , X ) = sup ( Pred ( `' R , A , X ) , A , `' R )
df-wlim 31758	|- WLim ( R , A ) = { x e. A | ( x =/= inf ( A , A , R ) /\ x = sup ( Pred ( R , A , x ) , A , R ) ) }
df-wlimOLD 31759	|- WLimOLD ( R , A ) = { x e. A | ( x =/= sup ( A , A , `' R ) /\ x = sup ( Pred ( R , A , x ) , A , R ) ) }
csur 31793	class No
cslt 31794	class <s
cbday 31795	class bday
df-no 31796	|- No = { f | E. a e. On f : a --> { 1o , 2o } }
df-slt 31797	|- <s = { <. f , g >. | ( ( f e. No /\ g e. No ) /\ E. x e. On ( A. y e. x ( f ` y ) = ( g ` y ) /\ ( f ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) ) }
df-bday 31798	|- bday = ( x e. No |-> dom x )
csle 31869	class ≤s
df-sle 31870	|- ≤s = ( ( No X. No ) \ `' <s )
csslt 31896	class < <s
df-sslt 31897	|- < <s = { <. a , b >. | ( a C_ No /\ b C_ No /\ A. x e. a A. y e. b x <s y ) }
cscut 31898	class |s
df-scut 31899	|- |s = ( a e. ~P No , b e. ( < <s " { a } ) |-> ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) )
cmade 31925	class M
cold 31926	class O
cnew 31927	class N
cleft 31928	class L
cright 31929	class R
df-made 31930	|- M = recs ( ( f e. _V |-> ( |s " ( ~P U. ran f X. ~P U. ran f ) ) ) )
df-old 31931	|- O = ( x e. On |-> U. ( M " x ) )
df-new 31932	|- N = ( x e. On |-> ( ( O ` x ) \ ( M ` x ) ) )
df-left 31933	|- L = ( x e. No |-> { y e. ( O ` ( bday ` x ) ) | A. z e. No ( ( y <s z /\ z <s x ) -> ( bday ` y ) e. ( bday ` z ) ) } )
df-right 31934	|- R = ( x e. No |-> { y e. ( O ` ( bday ` x ) ) | A. z e. No ( ( x <s z /\ z <s y ) -> ( bday ` y ) e. ( bday ` z ) ) } )
ctxp 31937	class ( A (x) B )
cpprod 31938	class pprod ( R , S )
csset 31939	class SSet
ctrans 31940	class Trans
cbigcup 31941	class Bigcup
cfix 31942	class Fix A
climits 31943	class Limits
cfuns 31944	class Funs
csingle 31945	class Singleton
csingles 31946	class Singletons
cimage 31947	class Image A
ccart 31948	class Cart
cimg 31949	class Img
cdomain 31950	class Domain
crange 31951	class Range
capply 31952	class Apply
ccup 31953	class Cup
ccap 31954	class Cap
csuccf 31955	class Succ
cfunpart 31956	class Funpart F
cfullfn 31957	class FullFun F
crestrict 31958	class Restrict
cub 31959	class UB R
clb 31960	class LB R
df-txp 31961	|- ( A (x) B ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) )
df-pprod 31962	|- pprod ( A , B ) = ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) )
df-sset 31963	|- SSet = ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) )
df-trans 31964	|- Trans = ( _V \ ran ( ( _E o. _E ) \ _E ) )
df-bigcup 31965	|- Bigcup = ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. _E ) (x) _V ) ) )
df-fix 31966	|- Fix A = dom ( A i^i _I )
df-limits 31967	|- Limits = ( ( On i^i Fix Bigcup ) \ { (/) } )
df-funs 31968	|- Funs = ( ~P ( _V X. _V ) \ Fix ( _E o. ( ( 1st (x) ( ( _V \ _I ) o. 2nd ) ) o. `' _E ) ) )
df-singleton 31969	|- Singleton = ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( _I (x) _V ) ) )
df-singles 31970	|- Singletons = ran Singleton
df-image 31971	|- Image A = ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) )
df-cart 31972	|- Cart = ( ( ( _V X. _V ) X. _V ) \ ran ( ( _V (x) _E ) /_\ (pprod ( _E , _E ) (x) _V ) ) )
df-img 31973	|- Img = (Image ( ( 2nd o. 1st ) |` ( 1st |` ( _V X. _V ) ) ) o. Cart )
df-domain 31974	|- Domain = Image ( 1st |` ( _V X. _V ) )
df-range 31975	|- Range = Image ( 2nd |` ( _V X. _V ) )
df-cup 31976	|- Cup = ( ( ( _V X. _V ) X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( ( `' 1st o. _E ) u. ( `' 2nd o. _E ) ) (x) _V ) ) )
df-cap 31977	|- Cap = ( ( ( _V X. _V ) X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( ( `' 1st o. _E ) i^i ( `' 2nd o. _E ) ) (x) _V ) ) )
df-restrict 31978	|- Restrict = (Cap o. ( 1st (x) (Cart o. ( 2nd (x) (Range o. 1st ) ) ) ) )
df-succf 31979	|- Succ = (Cup o. ( _I (x) Singleton ) )
df-apply 31980	|- Apply = ( ( Bigcup o. Bigcup ) o. ( ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) o. ( (Singleton o. Img ) o. pprod ( _I , Singleton ) ) ) )
df-funpart 31981	|- Funpart F = ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
df-fullfun 31982	|- FullFun F = (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) )
df-ub 31983	|- UB R = ( ( _V X. _V ) \ ( ( _V \ R ) o. `' _E ) )
df-lb 31984	|- LB R = UB `' R
caltop 32063	class << A , B >>
caltxp 32064	class ( A XX. B )
df-altop 32065	|- << A , B >> = { { A } , { A , { B } } }
df-altxp 32066	|- ( A XX. B ) = { z | E. x e. A E. y e. B z = << x , y >> }
cofs 32089	class OuterFiveSeg
df-ofs 32090	|- OuterFiveSeg = { <. p , q >. | E. n e. NN E. a e. ( EE ` n ) E. b e. ( EE ` n ) E. c e. ( EE ` n ) E. d e. ( EE ` n ) E. x e. ( EE ` n ) E. y e. ( EE ` n ) E. z e. ( EE ` n ) E. w e. ( EE ` n ) ( p = <. <. a , b >. , <. c , d >. >. /\ q = <. <. x , y >. , <. z , w >. >. /\ ( ( b Btwn <. a , c >. /\ y Btwn <. x , z >. ) /\ ( <. a , b >.Cgr <. x , y >. /\ <. b , c >.Cgr <. y , z >. ) /\ ( <. a , d >.Cgr <. x , w >. /\ <. b , d >.Cgr <. y , w >. ) ) ) }
ctransport 32136	class TransportTo
df-transport 32137	|- TransportTo = { <. <. p , q >. , x >. | E. n e. NN ( ( p e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr p ) ) ) }
cifs 32142	class InnerFiveSeg
ccgr3 32143	class Cgr3
ccolin 32144	class Colinear
cfs 32145	class FiveSeg
df-colinear 32146	|- Colinear = `' { <. <. b , c >. , a >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ c e. ( EE ` n ) ) /\ ( a Btwn <. b , c >. \/ b Btwn <. c , a >. \/ c Btwn <. a , b >. ) ) }
df-ifs 32147	|- InnerFiveSeg = { <. p , q >. | E. n e. NN E. a e. ( EE ` n ) E. b e. ( EE ` n ) E. c e. ( EE ` n ) E. d e. ( EE ` n ) E. x e. ( EE ` n ) E. y e. ( EE ` n ) E. z e. ( EE ` n ) E. w e. ( EE ` n ) ( p = <. <. a , b >. , <. c , d >. >. /\ q = <. <. x , y >. , <. z , w >. >. /\ ( ( b Btwn <. a , c >. /\ y Btwn <. x , z >. ) /\ ( <. a , c >.Cgr <. x , z >. /\ <. b , c >.Cgr <. y , z >. ) /\ ( <. a , d >.Cgr <. x , w >. /\ <. c , d >.Cgr <. z , w >. ) ) ) }
df-cgr3 32148	|- Cgr3 = { <. p , q >. | E. n e. NN E. a e. ( EE ` n ) E. b e. ( EE ` n ) E. c e. ( EE ` n ) E. d e. ( EE ` n ) E. e e. ( EE ` n ) E. f e. ( EE ` n ) ( p = <. a , <. b , c >. >. /\ q = <. d , <. e , f >. >. /\ ( <. a , b >.Cgr <. d , e >. /\ <. a , c >.Cgr <. d , f >. /\ <. b , c >.Cgr <. e , f >. ) ) }
df-fs 32149	|- FiveSeg = { <. p , q >. | E. n e. NN E. a e. ( EE ` n ) E. b e. ( EE ` n ) E. c e. ( EE ` n ) E. d e. ( EE ` n ) E. x e. ( EE ` n ) E. y e. ( EE ` n ) E. z e. ( EE ` n ) E. w e. ( EE ` n ) ( p = <. <. a , b >. , <. c , d >. >. /\ q = <. <. x , y >. , <. z , w >. >. /\ ( a Colinear <. b , c >. /\ <. a , <. b , c >. >.Cgr3 <. x , <. y , z >. >. /\ ( <. a , d >.Cgr <. x , w >. /\ <. b , d >.Cgr <. y , w >. ) ) ) }
csegle 32213	class Seg<_
df-segle 32214	|- Seg<_ = { <. p , q >. | E. n e. NN E. a e. ( EE ` n ) E. b e. ( EE ` n ) E. c e. ( EE ` n ) E. d e. ( EE ` n ) ( p = <. a , b >. /\ q = <. c , d >. /\ E. y e. ( EE ` n ) ( y Btwn <. c , d >. /\ <. a , b >.Cgr <. c , y >. ) ) }
coutsideof 32226	class OutsideOf
df-outsideof 32227	|- OutsideOf = ( Colinear \ Btwn )
cline2 32241	class Line
cray 32242	class Ray
clines2 32243	class LinesEE
df-line2 32244	|- Line = { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) }
df-ray 32245	|- Ray = { <. <. p , a >. , r >. | E. n e. NN ( ( p e. ( EE ` n ) /\ a e. ( EE ` n ) /\ p =/= a ) /\ r = { x e. ( EE ` n ) | pOutsideOf <. a , x >. } ) }
df-lines2 32246	|- LinesEE = ran Line
cfwddif 32265	class _/_\
df-fwddif 32266	|- _/_\ = ( f e. ( CC ^pm CC ) |-> ( x e. { y e. dom f | ( y + 1 ) e. dom f } |-> ( ( f ` ( x + 1 ) ) - ( f ` x ) ) ) )
cfwddifn 32267	class _/_\^nn
df-fwddifn 32268	|- _/_\^nn = ( n e. NN0 , f e. ( CC ^pm CC ) |-> ( x e. { y e. CC | A. k e. ( 0 ... n ) ( y + k ) e. dom f } |-> sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( -u 1 ^ ( n - k ) ) x. ( f ` ( x + k ) ) ) ) ) )
chf 32279	class Hf
df-hf 32280	|- Hf = U. ( R1 " om )
cfne 32331	class Fne
df-fne 32332	|- Fne = { <. x , y >. | ( U. x = U. y /\ A. z e. x z C_ U. ( y i^i ~P z ) ) }
w3nand 32394	wff ( ph -/\ ps -/\ ch )
df-3nand 32395	|- ( ( ph -/\ ps -/\ ch ) <-> ( ph -> ( ps -> -. ch ) ) )
cgcdOLD 32458	class gcdOLD ( A , B )
df-gcdOLD 32459	|- gcdOLD ( A , B ) = sup ( { x e. NN | ( ( A / x ) e. NN /\ ( B / x ) e. NN ) } , NN , < )
cprvb 32582	wff Prv ph
ax-prv1 32583	|- ph   =>    |- Prv ph
ax-prv2 32584	|- (Prv ( ph -> ps ) -> (Prv ph -> Prv ps ) )
ax-prv3 32585	|- (Prv ph -> Prv Prv ph )
wssb 32619	wff [ t/ x]b ph
df-ssb 32620	|- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
wrnf 32930	wff F/ x e. A ph
df-bj-rnf 32931	|- ( F/ x e. A ph <-> ( E. x e. A ph -> A. x e. A ph ) )
bj-csngl 32953	class sngl A
df-bj-sngl 32954	|- sngl A = { x | E. y e. A x = { y } }
bj-ctag 32962	class tag A
df-bj-tag 32963	|- tag A = (sngl A u. { (/) } )
bj-cproj 32978	class ( A Proj B )
df-bj-proj 32979	|- ( A Proj B ) = { x | { x } e. ( B " { A } ) }
bj-c1upl 32985	class (| A|)
df-bj-1upl 32986	|- (| A|) = ( { (/) } X. tag A )
bj-cpr1 32988	class pr1 A
df-bj-pr1 32989	|- pr1 A = ( (/) Proj A )
bj-c2uple 32998	class (| A, B|)
df-bj-2upl 32999	|- (| A, B|) = ((| A|) u. ( { 1o } X. tag B ) )
bj-cpr2 33002	class pr2 A
df-bj-pr2 33003	|- pr2 A = ( 1o Proj A )
celwise 33032	class elwise
df-elwise 33033	|- elwise = ( o e. _V |-> ( x e. _V , y e. _V |-> { z | E. u e. x E. v e. y z = ( u o v ) } ) )
cmoore 33057	class Moore_
df-bj-moore 33058	|- Moore_ = { x | A. y e. ~P x ( U. x i^i |^| y ) e. x }
cmpt3 33073	class ( x e. A , y e. B , z e. C |-> D )
df-bj-mpt3 33074	|- ( x e. A , y e. B , z e. C |-> D ) = { <. s , t >. | E. x e. A E. y e. B E. z e. C ( s = <. x , y , z >. /\ t = D ) }
csethom 33075	class -Set->
ctophom 33076	class -Top->
cmagmahom 33077	class -Magma->
df-bj-sethom 33078	|- -Set-> = ( x e. _V , y e. _V |-> { f | f : x --> y } )
df-bj-tophom 33079	|- -Top-> = ( x e. TopSp , y e. TopSp |-> { f e. ( ( Base ` x ) -Set-> ( Base ` y ) ) | A. u e. ( TopOpen ` y ) ( `' f " u ) e. ( TopOpen ` x ) } )
df-bj-magmahom 33080	|- -Magma-> = ( x e. Mgm , y e. Mgm |-> { f e. ( ( Base ` x ) -Set-> ( Base ` y ) ) | A. u e. ( Base ` x ) A. v e. ( Base ` x ) ( f ` ( u ( +g ` x ) v ) ) = ( ( f ` u ) ( +g ` y ) ( f ` v ) ) } )
ccur- 33081	class curry_
df-bj-cur 33082	|- curry_ = ( x e. _V , y e. _V , z e. _V |-> ( f e. ( ( x X. y ) -Set-> z ) |-> ( a e. x |-> ( b e. y |-> ( f ` <. a , b >. ) ) ) ) )
cunc- 33083	class uncurry_
df-bj-unc 33084	|- uncurry_ = ( x e. _V , y e. _V , z e. _V |-> ( f e. ( x -Set-> ( y -Set-> z ) ) |-> ( a e. x , b e. y |-> ( ( f ` a ) ` b ) ) ) )
cdiag2 33088	class Diag
df-bj-diag 33089	|- Diag = ( x e. _V |-> ( _I i^i ( x X. x ) ) )
cinftyexpi 33093	class inftyexpi
df-bj-inftyexpi 33094	|- inftyexpi = ( x e. ( -u pi (,] pi ) |-> <. x , CC >. )
cccinfty 33098	class CCinfty
df-bj-ccinfty 33099	|- CCinfty = ran inftyexpi
cccbar 33102	class CCbar
df-bj-ccbar 33103	|- CCbar = ( CC u. CCinfty )
cpinfty 33106	class pinfty
df-bj-pinfty 33107	|- pinfty = (inftyexpi ` 0 )
cminfty 33110	class minfty
df-bj-minfty 33111	|- minfty = (inftyexpi ` pi )
crrbar 33115	class RRbar
df-bj-rrbar 33116	|- RRbar = ( RR u. {minfty , pinfty } )
cinfty 33117	class infty
df-bj-infty 33118	|- infty = ~P U. CC
ccchat 33119	class CChat
df-bj-cchat 33120	|- CChat = ( CC u. {infty } )
crrhat 33121	class RRhat
df-bj-rrhat 33122	|- RRhat = ( RR u. {infty } )
caddcc 33124	class +cc
df-bj-addc 33125	|- +cc = ( x e. ( ( ( CC X. CCbar ) u. (CCbar X. CC ) ) u. ( (CChat X. CChat ) u. (Diag ` CCinfty ) ) ) |-> if ( ( ( 1st ` x ) = infty \/ ( 2nd ` x ) = infty ) , infty , if ( ( 1st ` x ) e. CC , if ( ( 2nd ` x ) e. CC , ( ( 1st ` x ) + ( 2nd ` x ) ) , ( 2nd ` x ) ) , ( 1st ` x ) ) ) )
coppcc 33126	class -cc
df-bj-oppc 33127	|- -cc = ( x e. (CCbar u. CChat ) |-> if ( x = infty , infty , if ( x e. CC , -u x , (inftyexpi ` if ( 0 < ( 1st ` x ) , ( ( 1st ` x ) - pi ) , ( ( 1st ` x ) + pi ) ) ) ) ) )
cprcpal 33128	class prcpal
df-bj-prcpal 33129	|- prcpal = ( x e. RR |-> ( ( x mod ( 2 x. pi ) ) - if ( ( x mod ( 2 x. pi ) ) <_ pi , 0 , ( 2 x. pi ) ) ) )
carg 33130	class Arg
df-bj-arg 33131	|- Arg = ( x e. (CCbar \ { 0 } ) |-> if ( x e. CC , ( Im ` ( log ` x ) ) , ( 1st ` x ) ) )
cmulc 33132	class .cc
df-bj-mulc 33133	|- .cc = ( x e. ( (CCbar X. CCbar ) u. (CChat X. CChat ) ) |-> if ( ( ( 1st ` x ) = 0 \/ ( 2nd ` x ) = 0 ) , 0 , if ( ( ( 1st ` x ) = infty \/ ( 2nd ` x ) = infty ) , infty , if ( x e. ( CC X. CC ) , ( ( 1st ` x ) x. ( 2nd ` x ) ) , (inftyexpi ` (prcpal ` ( (Arg ` ( 1st ` x ) ) + (Arg ` ( 2nd ` x ) ) ) ) ) ) ) ) )
cinvc 33134	class invc
df-bj-invc 33135	|- invc = ( x e. (CCbar u. CChat ) |-> if ( x = 0 , infty , if ( x e. CC , ( 1 / x ) , 0 ) ) )
cfinsum 33145	class FinSum
df-bj-finsum 33146	|- FinSum = ( x e. { <. y , z >. | ( y e. CMnd /\ E. t e. Fin z : t --> ( Base ` y ) ) } |-> ( iota s E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) ) )
crrvec 33148	class RR-Vec
df-bj-rrvec 33149	|- RR-Vec = { x e. LVec | (Scalar ` x ) = RRfld }
ctau 33163	class tau
df-tau 33164	|- tau = inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < )
cfinxp 33220	class ( U ^^ ^^ N )
df-finxp 33221	|- ( U ^^ ^^ N ) = { y | ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) }
ax-luk1 33241	|- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )
ax-luk2 33242	|- ( ( -. ph -> ph ) -> ph )
ax-luk3 33243	|- ( ph -> ( -. ph -> ps ) )
ax-wl-13v 33286	|- ( -. A. x x = y -> ( y = z -> A. x y = z ) )
ax-wl-11v 33361	|- ( A. x A. y ph -> A. y A. x ph )
wcel-wl 33373	wff x e. B
wcel2-wl 33375	wff x e. B
ax-wl-8cl 33377	|- ( x = y -> ( x e. A -> y e. A ) )
df-wl-clelv2 33380	|- ( x e. A <-> A. u ( u = x -> u e. A ) )
ctotbnd 33565	class TotBnd
cbnd 33566	class Bnd
df-totbnd 33567	|- TotBnd = ( x e. _V |-> { m e. ( Met ` x ) | A. d e. RR+ E. v e. Fin ( U. v = x /\ A. b e. v E. y e. x b = ( y ( ball ` m ) d ) ) } )
df-bnd 33578	|- Bnd = ( x e. _V |-> { m e. ( Met ` x ) | A. y e. x E. r e. RR+ x = ( y ( ball ` m ) r ) } )
cismty 33597	class Ismty
df-ismty 33598	|- Ismty = ( m e. U. ran *Met , n e. U. ran *Met |-> { f | ( f : dom dom m -1-1-onto-> dom dom n /\ A. x e. dom dom m A. y e. dom dom m ( x m y ) = ( ( f ` x ) n ( f ` y ) ) ) } )
crrn 33624	class Rn
df-rrn 33625	|- Rn = ( i e. Fin |-> ( x e. ( RR ^m i ) , y e. ( RR ^m i ) |-> ( sqr ` sum_ k e. i ( ( ( x ` k ) - ( y ` k ) ) ^ 2 ) ) ) )
cass 33641	class Ass
df-ass 33642	|- Ass = { g | A. x e. dom dom g A. y e. dom dom g A. z e. dom dom g ( ( x g y ) g z ) = ( x g ( y g z ) ) }
cexid 33643	class ExId
df-exid 33644	|- ExId = { g | E. x e. dom dom g A. y e. dom dom g ( ( x g y ) = y /\ ( y g x ) = y ) }
cmagm 33647	class Magma
df-mgmOLD 33648	|- Magma = { g | E. t g : ( t X. t ) --> t }
csem 33659	class SemiGrp
df-sgrOLD 33660	|- SemiGrp = ( Magma i^i Ass )
cmndo 33665	class MndOp
df-mndo 33666	|- MndOp = ( SemiGrp i^i ExId )
cghomOLD 33682	class GrpOpHom
df-ghomOLD 33683	|- GrpOpHom = ( g e. GrpOp , h e. GrpOp |-> { f | ( f : ran g --> ran h /\ A. x e. ran g A. y e. ran g ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x g y ) ) ) } )
crngo 33693	class RingOps
df-rngo 33694	|- RingOps = { <. g , h >. | ( ( g e. AbelOp /\ h : ( ran g X. ran g ) --> ran g ) /\ ( A. x e. ran g A. y e. ran g A. z e. ran g ( ( ( x h y ) h z ) = ( x h ( y h z ) ) /\ ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) /\ ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) ) /\ E. x e. ran g A. y e. ran g ( ( x h y ) = y /\ ( y h x ) = y ) ) ) }
cdrng 33747	class DivRingOps
df-drngo 33748	|- DivRingOps = { <. g , h >. | ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) }
crnghom 33759	class RngHom
crngiso 33760	class RngIso
crisc 33761	class ~=R
df-rngohom 33762	|- RngHom = ( r e. RingOps , s e. RingOps |-> { f e. ( ran ( 1st ` s ) ^m ran ( 1st ` r ) ) | ( ( f ` (GId ` ( 2nd ` r ) ) ) = (GId ` ( 2nd ` s ) ) /\ A. x e. ran ( 1st ` r ) A. y e. ran ( 1st ` r ) ( ( f ` ( x ( 1st ` r ) y ) ) = ( ( f ` x ) ( 1st ` s ) ( f ` y ) ) /\ ( f ` ( x ( 2nd ` r ) y ) ) = ( ( f ` x ) ( 2nd ` s ) ( f ` y ) ) ) ) } )
df-rngoiso 33775	|- RngIso = ( r e. RingOps , s e. RingOps |-> { f e. ( r RngHom s ) | f : ran ( 1st ` r ) -1-1-onto-> ran ( 1st ` s ) } )
df-risc 33782	|- ~=R = { <. r , s >. | ( ( r e. RingOps /\ s e. RingOps ) /\ E. f f e. ( r RngIso s ) ) }
ccm2 33788	class Com2
df-com2 33789	|- Com2 = { <. g , h >. | A. a e. ran g A. b e. ran g ( a h b ) = ( b h a ) }
cfld 33790	class Fld
df-fld 33791	|- Fld = ( DivRingOps i^i Com2 )
ccring 33792	class CRingOps
df-crngo 33793	|- CRingOps = ( RingOps i^i Com2 )
cidl 33806	class Idl
cpridl 33807	class PrIdl
cmaxidl 33808	class MaxIdl
df-idl 33809	|- Idl = ( r e. RingOps |-> { i e. ~P ran ( 1st ` r ) | ( (GId ` ( 1st ` r ) ) e. i /\ A. x e. i ( A. y e. i ( x ( 1st ` r ) y ) e. i /\ A. z e. ran ( 1st ` r ) ( ( z ( 2nd ` r ) x ) e. i /\ ( x ( 2nd ` r ) z ) e. i ) ) ) } )
df-pridl 33810	|- PrIdl = ( r e. RingOps |-> { i e. ( Idl ` r ) | ( i =/= ran ( 1st ` r ) /\ A. a e. ( Idl ` r ) A. b e. ( Idl ` r ) ( A. x e. a A. y e. b ( x ( 2nd ` r ) y ) e. i -> ( a C_ i \/ b C_ i ) ) ) } )
df-maxidl 33811	|- MaxIdl = ( r e. RingOps |-> { i e. ( Idl ` r ) | ( i =/= ran ( 1st ` r ) /\ A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) ) } )
cprrng 33845	class PrRing
cdmn 33846	class Dmn
df-prrngo 33847	|- PrRing = { r e. RingOps | { (GId ` ( 1st ` r ) ) } e. ( PrIdl ` r ) }
df-dmn 33848	|- Dmn = ( PrRing i^i Com2 )
cigen 33858	class IdlGen
df-igen 33859	|- IdlGen = ( r e. RingOps , s e. ~P ran ( 1st ` r ) |-> |^| { j e. ( Idl ` r ) | s C_ j } )
cxrn 33982	class ( A |X. B )
df-xrn 34134	|- ( A |X. B ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) )
wprt 34156	wff Prt A
df-prt 34157	|- ( Prt A <-> A. x e. A A. y e. A ( x = y \/ ( x i^i y ) = (/) ) )
ax-c5 34168	|- ( A. x ph -> ph )
ax-c4 34169	|- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) )
ax-c7 34170	|- ( -. A. x -. A. x ph -> ph )
ax-c10 34171	|- ( A. x ( x = y -> A. x ph ) -> ph )
ax-c11 34172	|- ( A. x x = y -> ( A. x ph -> A. y ph ) )
ax-c11n 34173	|- ( A. x x = y -> A. y y = x )
ax-c15 34174	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
ax-c9 34175	|- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
ax-c14 34176	|- ( -. A. z z = x -> ( -. A. z z = y -> ( x e. y -> A. z x e. y ) ) )
ax-c16 34177	|- ( A. x x = y -> ( ph -> A. x ph ) )
ax-riotaBAD 34239	|- ( iota_ x e. A ph ) = if ( E! x e. A ph , ( iota x ( x e. A /\ ph ) ) , ( Undef ` { x | x e. A } ) )
clsa 34261	class LSAtoms
clsh 34262	class LSHyp
df-lsatoms 34263	|- LSAtoms = ( w e. _V |-> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) )
df-lshyp 34264	|- LSHyp = ( w e. _V |-> { s e. ( LSubSp ` w ) | ( s =/= ( Base ` w ) /\ E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) ) } )
clcv 34305	class <oLL
df-lcv 34306	|- <oLL = ( w e. _V |-> { <. t , u >. | ( ( t e. ( LSubSp ` w ) /\ u e. ( LSubSp ` w ) ) /\ ( t C. u /\ -. E. s e. ( LSubSp ` w ) ( t C. s /\ s C. u ) ) ) } )
clfn 34344	class LFnl
df-lfl 34345	|- LFnl = ( w e. _V |-> { f e. ( ( Base ` (Scalar ` w ) ) ^m ( Base ` w ) ) | A. r e. ( Base ` (Scalar ` w ) ) A. x e. ( Base ` w ) A. y e. ( Base ` w ) ( f ` ( ( r ( .s ` w ) x ) ( +g ` w ) y ) ) = ( ( r ( .r ` (Scalar ` w ) ) ( f ` x ) ) ( +g ` (Scalar ` w ) ) ( f ` y ) ) } )
clk 34372	class LKer
df-lkr 34373	|- LKer = ( w e. _V |-> ( f e. (LFnl ` w ) |-> ( `' f " { ( 0g ` (Scalar ` w ) ) } ) ) )
cld 34410	class LDual
df-ldual 34411	|- LDual = ( v e. _V |-> ( { <. ( Base ` ndx ) , (LFnl ` v ) >. , <. ( +g ` ndx ) , ( oF ( +g ` (Scalar ` v ) ) |` ( (LFnl ` v ) X. (LFnl ` v ) ) ) >. , <. (Scalar ` ndx ) , (oppr ` (Scalar ` v ) ) >. } u. { <. ( .s ` ndx ) , ( k e. ( Base ` (Scalar ` v ) ) , f e. (LFnl ` v ) |-> ( f oF ( .r ` (Scalar ` v ) ) ( ( Base ` v ) X. { k } ) ) ) >. } ) )
cops 34459	class OP
ccmtN 34460	class cm
col 34461	class OL
coml 34462	class OML
df-oposet 34463	|- OP = { p e. Poset | ( ( ( Base ` p ) e. dom ( lub ` p ) /\ ( Base ` p ) e. dom ( glb ` p ) ) /\ E. o ( o = ( oc ` p ) /\ A. a e. ( Base ` p ) A. b e. ( Base ` p ) ( ( ( o ` a ) e. ( Base ` p ) /\ ( o ` ( o ` a ) ) = a /\ ( a ( le ` p ) b -> ( o ` b ) ( le ` p ) ( o ` a ) ) ) /\ ( a ( join ` p ) ( o ` a ) ) = ( 1. ` p ) /\ ( a ( meet ` p ) ( o ` a ) ) = ( 0. ` p ) ) ) ) }
df-cmtN 34464	|- cm = ( p e. _V |-> { <. x , y >. | ( x e. ( Base ` p ) /\ y e. ( Base ` p ) /\ x = ( ( x ( meet ` p ) y ) ( join ` p ) ( x ( meet ` p ) ( ( oc ` p ) ` y ) ) ) ) } )
df-ol 34465	|- OL = ( Lat i^i OP )
df-oml 34466	|- OML = { l e. OL | A. a e. ( Base ` l ) A. b e. ( Base ` l ) ( a ( le ` l ) b -> b = ( a ( join ` l ) ( b ( meet ` l ) ( ( oc ` l ) ` a ) ) ) ) }
ccvr 34549	class <o
catm 34550	class Atoms
cal 34551	class AtLat
clc 34552	class CvLat
df-covers 34553	|- <o = ( p e. _V |-> { <. a , b >. | ( ( a e. ( Base ` p ) /\ b e. ( Base ` p ) ) /\ a ( lt ` p ) b /\ -. E. z e. ( Base ` p ) ( a ( lt ` p ) z /\ z ( lt ` p ) b ) ) } )
df-ats 34554	|- Atoms = ( p e. _V |-> { a e. ( Base ` p ) | ( 0. ` p ) ( <o ` p ) a } )
df-atl 34585	|- AtLat = { k e. Lat | ( ( Base ` k ) e. dom ( glb ` k ) /\ A. x e. ( Base ` k ) ( x =/= ( 0. ` k ) -> E. p e. ( Atoms ` k ) p ( le ` k ) x ) ) }
df-cvlat 34609	|- CvLat = { k e. AtLat | A. a e. ( Atoms ` k ) A. b e. ( Atoms ` k ) A. c e. ( Base ` k ) ( ( -. a ( le ` k ) c /\ a ( le ` k ) ( c ( join ` k ) b ) ) -> b ( le ` k ) ( c ( join ` k ) a ) ) }
chlt 34637	class HL
df-hlat 34638	|- HL = { l e. ( ( OML i^i CLat ) i^i CvLat ) | ( A. a e. ( Atoms ` l ) A. b e. ( Atoms ` l ) ( a =/= b -> E. c e. ( Atoms ` l ) ( c =/= a /\ c =/= b /\ c ( le ` l ) ( a ( join ` l ) b ) ) ) /\ E. a e. ( Base ` l ) E. b e. ( Base ` l ) E. c e. ( Base ` l ) ( ( ( 0. ` l ) ( lt ` l ) a /\ a ( lt ` l ) b ) /\ ( b ( lt ` l ) c /\ c ( lt ` l ) ( 1. ` l ) ) ) ) }
clln 34777	class LLines
clpl 34778	class LPlanes
clvol 34779	class LVols
clines 34780	class Lines
cpointsN 34781	class Points
cpsubsp 34782	class PSubSp
cpmap 34783	class pmap
df-llines 34784	|- LLines = ( k e. _V |-> { x e. ( Base ` k ) | E. p e. ( Atoms ` k ) p ( <o ` k ) x } )
df-lplanes 34785	|- LPlanes = ( k e. _V |-> { x e. ( Base ` k ) | E. p e. ( LLines ` k ) p ( <o ` k ) x } )
df-lvols 34786	|- LVols = ( k e. _V |-> { x e. ( Base ` k ) | E. p e. ( LPlanes ` k ) p ( <o ` k ) x } )
df-lines 34787	|- Lines = ( k e. _V |-> { s | E. q e. ( Atoms ` k ) E. r e. ( Atoms ` k ) ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) } )
df-pointsN 34788	|- Points = ( k e. _V |-> { q | E. p e. ( Atoms ` k ) q = { p } } )
df-psubsp 34789	|- PSubSp = ( k e. _V |-> { s | ( s C_ ( Atoms ` k ) /\ A. p e. s A. q e. s A. r e. ( Atoms ` k ) ( r ( le ` k ) ( p ( join ` k ) q ) -> r e. s ) ) } )
df-pmap 34790	|- pmap = ( k e. _V |-> ( a e. ( Base ` k ) |-> { p e. ( Atoms ` k ) | p ( le ` k ) a } ) )
cpadd 35081	class +P
df-padd 35082	|- +P = ( l e. _V |-> ( m e. ~P ( Atoms ` l ) , n e. ~P ( Atoms ` l ) |-> ( ( m u. n ) u. { p e. ( Atoms ` l ) | E. q e. m E. r e. n p ( le ` l ) ( q ( join ` l ) r ) } ) ) )
cpclN 35173	class PCl
df-pclN 35174	|- PCl = ( k e. _V |-> ( x e. ~P ( Atoms ` k ) |-> |^| { y e. ( PSubSp ` k ) | x C_ y } ) )
cpolN 35188	class _|_P
df-polarityN 35189	|- _|_P = ( l e. _V |-> ( m e. ~P ( Atoms ` l ) |-> ( ( Atoms ` l ) i^i |^|_ p e. m ( ( pmap ` l ) ` ( ( oc ` l ) ` p ) ) ) ) )
cpscN 35220	class PSubCl
df-psubclN 35221	|- PSubCl = ( k e. _V |-> { s | ( s C_ ( Atoms ` k ) /\ ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = s ) } )
clh 35270	class LHyp
claut 35271	class LAut
cwpointsN 35272	class WAtoms
cpautN 35273	class PAut
df-lhyp 35274	|- LHyp = ( k e. _V |-> { x e. ( Base ` k ) | x ( <o ` k ) ( 1. ` k ) } )
df-laut 35275	|- LAut = ( k e. _V |-> { f | ( f : ( Base ` k ) -1-1-onto-> ( Base ` k ) /\ A. x e. ( Base ` k ) A. y e. ( Base ` k ) ( x ( le ` k ) y <-> ( f ` x ) ( le ` k ) ( f ` y ) ) ) } )
df-watsN 35276	|- WAtoms = ( k e. _V |-> ( d e. ( Atoms ` k ) |-> ( ( Atoms ` k ) \ ( ( _|_P ` k ) ` { d } ) ) ) )
df-pautN 35277	|- PAut = ( k e. _V |-> { f | ( f : ( PSubSp ` k ) -1-1-onto-> ( PSubSp ` k ) /\ A. x e. ( PSubSp ` k ) A. y e. ( PSubSp ` k ) ( x C_ y <-> ( f ` x ) C_ ( f ` y ) ) ) } )
cldil 35386	class LDil
cltrn 35387	class LTrn
cdilN 35388	class Dil
ctrnN 35389	class Trn
df-ldil 35390	|- LDil = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { f e. ( LAut ` k ) | A. x e. ( Base ` k ) ( x ( le ` k ) w -> ( f ` x ) = x ) } ) )
df-ltrn 35391	|- LTrn = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { f e. ( ( LDil ` k ) ` w ) | A. p e. ( Atoms ` k ) A. q e. ( Atoms ` k ) ( ( -. p ( le ` k ) w /\ -. q ( le ` k ) w ) -> ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) = ( ( q ( join ` k ) ( f ` q ) ) ( meet ` k ) w ) ) } ) )
df-dilN 35392	|- Dil = ( k e. _V |-> ( d e. ( Atoms ` k ) |-> { f e. ( PAut ` k ) | A. x e. ( PSubSp ` k ) ( x C_ ( ( WAtoms ` k ) ` d ) -> ( f ` x ) = x ) } ) )
df-trnN 35393	|- Trn = ( k e. _V |-> ( d e. ( Atoms ` k ) |-> { f e. ( ( Dil ` k ) ` d ) | A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) } ) )
ctrl 35445	class trL
df-trl 35446	|- trL = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) ) )
ctgrp 36030	class TGrp
df-tgrp 36031	|- TGrp = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. } ) )
ctendo 36040	class TEndo
cedring 36041	class EDRing
cedring-rN 36042	class EDRingR
df-tendo 36043	|- TEndo = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { f | ( f : ( ( LTrn ` k ) ` w ) --> ( ( LTrn ` k ) ` w ) /\ A. x e. ( ( LTrn ` k ) ` w ) A. y e. ( ( LTrn ` k ) ` w ) ( f ` ( x o. y ) ) = ( ( f ` x ) o. ( f ` y ) ) /\ A. x e. ( ( LTrn ` k ) ` w ) ( ( ( trL ` k ) ` w ) ` ( f ` x ) ) ( le ` k ) ( ( ( trL ` k ) ` w ) ` x ) ) } ) )
df-edring-rN 36044	|- EDRingR = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( t o. s ) ) >. } ) )
df-edring 36045	|- EDRing = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( s o. t ) ) >. } ) )
cdveca 36290	class DVecA
df-dveca 36291	|- DVecA = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( LTrn ` k ) ` w ) |-> ( s ` f ) ) >. } ) ) )
cdia 36317	class DIsoA
df-disoa 36318	|- DIsoA = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. { y e. ( Base ` k ) | y ( le ` k ) w } |-> { f e. ( ( LTrn ` k ) ` w ) | ( ( ( trL ` k ) ` w ) ` f ) ( le ` k ) x } ) ) )
cdvh 36367	class DVecH
df-dvech 36368	|- DVecH = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( { <. ( Base ` ndx ) , ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) , g e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
cocaN 36408	class ocA
df-docaN 36409	|- ocA = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. ~P ( ( LTrn ` k ) ` w ) |-> ( ( ( DIsoA ` k ) ` w ) ` ( ( ( ( oc ` k ) ` ( `' ( ( DIsoA ` k ) ` w ) ` |^| { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } ) ) ( join ` k ) ( ( oc ` k ) ` w ) ) ( meet ` k ) w ) ) ) ) )
cdjaN 36420	class vA
df-djaN 36421	|- vA = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. ~P ( ( LTrn ` k ) ` w ) , y e. ~P ( ( LTrn ` k ) ` w ) |-> ( ( ( ocA ` k ) ` w ) ` ( ( ( ( ocA ` k ) ` w ) ` x ) i^i ( ( ( ocA ` k ) ` w ) ` y ) ) ) ) ) )
cdib 36427	class DIsoB
df-dib 36428	|- DIsoB = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. dom ( ( DIsoA ` k ) ` w ) |-> ( ( ( ( DIsoA ` k ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` k ) ` w ) |-> ( _I |` ( Base ` k ) ) ) } ) ) ) )
cdic 36461	class DIsoC
df-dic 36462	|- DIsoC = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( q e. { r e. ( Atoms ` k ) | -. r ( le ` k ) w } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` k ) ` w ) ( g ` ( ( oc ` k ) ` w ) ) = q ) ) /\ s e. ( ( TEndo ` k ) ` w ) ) } ) ) )
cdih 36517	class DIsoH
df-dih 36518	|- DIsoH = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. ( Base ` k ) |-> if ( x ( le ` k ) w , ( ( ( DIsoB ` k ) ` w ) ` x ) , ( iota_ u e. ( LSubSp ` ( ( DVecH ` k ) ` w ) ) A. q e. ( Atoms ` k ) ( ( -. q ( le ` k ) w /\ ( q ( join ` k ) ( x ( meet ` k ) w ) ) = x ) -> u = ( ( ( ( DIsoC ` k ) ` w ) ` q ) ( LSSum ` ( ( DVecH ` k ) ` w ) ) ( ( ( DIsoB ` k ) ` w ) ` ( x ( meet ` k ) w ) ) ) ) ) ) ) ) )
coch 36636	class ocH
df-doch 36637	|- ocH = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. ~P ( Base ` ( ( DVecH ` k ) ` w ) ) |-> ( ( ( DIsoH ` k ) ` w ) ` ( ( oc ` k ) ` ( ( glb ` k ) ` { y e. ( Base ` k ) | x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) } ) ) ) ) ) )
cdjh 36683	class joinH
df-djh 36684	|- joinH = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. ~P ( Base ` ( ( DVecH ` k ) ` w ) ) , y e. ~P ( Base ` ( ( DVecH ` k ) ` w ) ) |-> ( ( ( ocH ` k ) ` w ) ` ( ( ( ( ocH ` k ) ` w ) ` x ) i^i ( ( ( ocH ` k ) ` w ) ` y ) ) ) ) ) )
clpoN 36769	class LPol
df-lpolN 36770	|- LPol = ( w e. _V |-> { o e. ( ( LSubSp ` w ) ^m ~P ( Base ` w ) ) | ( ( o ` ( Base ` w ) ) = { ( 0g ` w ) } /\ A. x A. y ( ( x C_ ( Base ` w ) /\ y C_ ( Base ` w ) /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. (LSAtoms ` w ) ( ( o ` x ) e. (LSHyp ` w ) /\ ( o ` ( o ` x ) ) = x ) ) } )
clcd 36875	class LCDual
df-lcdual 36876	|- LCDual = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( (LDual ` ( ( DVecH ` k ) ` w ) ) ↾s { f e. (LFnl ` ( ( DVecH ` k ) ` w ) ) | ( ( ( ocH ` k ) ` w ) ` ( ( ( ocH ` k ) ` w ) ` ( (LKer ` ( ( DVecH ` k ) ` w ) ) ` f ) ) ) = ( (LKer ` ( ( DVecH ` k ) ` w ) ) ` f ) } ) ) )
cmpd 36913	class mapd
df-mapd 36914	|- mapd = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( s e. ( LSubSp ` ( ( DVecH ` k ) ` w ) ) |-> { f e. (LFnl ` ( ( DVecH ` k ) ` w ) ) | ( ( ( ( ocH ` k ) ` w ) ` ( ( ( ocH ` k ) ` w ) ` ( (LKer ` ( ( DVecH ` k ) ` w ) ) ` f ) ) ) = ( (LKer ` ( ( DVecH ` k ) ` w ) ) ` f ) /\ ( ( ( ocH ` k ) ` w ) ` ( (LKer ` ( ( DVecH ` k ) ` w ) ) ` f ) ) C_ s ) } ) ) )
chvm 37045	class HVMap
df-hvmap 37046	|- HVMap = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. ( ( Base ` ( ( DVecH ` k ) ` w ) ) \ { ( 0g ` ( ( DVecH ` k ) ` w ) ) } ) |-> ( v e. ( Base ` ( ( DVecH ` k ) ` w ) ) |-> ( iota_ j e. ( Base ` (Scalar ` ( ( DVecH ` k ) ` w ) ) ) E. t e. ( ( ( ocH ` k ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` k ) ` w ) ) ( j ( .s ` ( ( DVecH ` k ) ` w ) ) x ) ) ) ) ) ) )
chdma1 37081	class HDMap1
chdma 37082	class HDMap
df-hdmap1 37083	|- HDMap1 = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { a | [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( (LCDual ` k ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( (mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) } ) )
df-hdmap 37084	|- HDMap = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { a | [. <. ( _I |` ( Base ` k ) ) , ( _I |` ( ( LTrn ` k ) ` w ) ) >. / e ]. [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` k ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) )
chg 37175	class HGMap
df-hgmap 37176	|- HGMap = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { a | [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` k ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) ) ) } ) )
chlh 37224	class HLHil
df-hlhil 37225	|- HLHil = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) ) )
cnacs 37265	class NoeACS
df-nacs 37266	|- NoeACS = ( x e. _V |-> { c e. (ACS ` x ) | A. s e. c E. g e. ( ~P x i^i Fin ) s = ( (mrCls ` c ) ` g ) } )
cmzpcl 37284	class mzPolyCld
cmzp 37285	class mzPoly
df-mzpcl 37286	|- mzPolyCld = ( v e. _V |-> { p e. ~P ( ZZ ^m ( ZZ ^m v ) ) | ( ( A. i e. ZZ ( ( ZZ ^m v ) X. { i } ) e. p /\ A. j e. v ( x e. ( ZZ ^m v ) |-> ( x ` j ) ) e. p ) /\ A. f e. p A. g e. p ( ( f oF + g ) e. p /\ ( f oF x. g ) e. p ) ) } )
df-mzp 37287	|- mzPoly = ( v e. _V |-> |^| (mzPolyCld ` v ) )
cdioph 37318	class Dioph
df-dioph 37319	|- Dioph = ( n e. NN0 |-> ran ( k e. ( ZZ>= ` n ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... n ) ) /\ ( p ` u ) = 0 ) } ) )
csquarenn 37400	class ◻NN
cpell1qr 37401	class Pell1QR
cpell1234qr 37402	class Pell1234QR
cpell14qr 37403	class Pell14QR
cpellfund 37404	class PellFund
df-squarenn 37405	|- ◻NN = { x e. NN | ( sqr ` x ) e. QQ }
df-pell1qr 37406	|- Pell1QR = ( x e. ( NN \ ◻NN ) |-> { y e. RR | E. z e. NN0 E. w e. NN0 ( y = ( z + ( ( sqr ` x ) x. w ) ) /\ ( ( z ^ 2 ) - ( x x. ( w ^ 2 ) ) ) = 1 ) } )
df-pell14qr 37407	|- Pell14QR = ( x e. ( NN \ ◻NN ) |-> { y e. RR | E. z e. NN0 E. w e. ZZ ( y = ( z + ( ( sqr ` x ) x. w ) ) /\ ( ( z ^ 2 ) - ( x x. ( w ^ 2 ) ) ) = 1 ) } )
df-pell1234qr 37408	|- Pell1234QR = ( x e. ( NN \ ◻NN ) |-> { y e. RR | E. z e. ZZ E. w e. ZZ ( y = ( z + ( ( sqr ` x ) x. w ) ) /\ ( ( z ^ 2 ) - ( x x. ( w ^ 2 ) ) ) = 1 ) } )
df-pellfund 37409	|- PellFund = ( x e. ( NN \ ◻NN ) |-> inf ( { z e. (Pell14QR ` x ) | 1 < z } , RR , < ) )
crmx 37464	class Xrm
crmy 37465	class Yrm
df-rmx 37466	|- Xrm = ( a e. ( ZZ>= ` 2 ) , n e. ZZ |-> ( 1st ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( a + ( sqr ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) ) ) )
df-rmy 37467	|- Yrm = ( a e. ( ZZ>= ` 2 ) , n e. ZZ |-> ( 2nd ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( a + ( sqr ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) ) ) )
clfig 37637	class LFinGen
df-lfig 37638	|- LFinGen = { w e. LMod | ( Base ` w ) e. ( ( LSpan ` w ) " ( ~P ( Base ` w ) i^i Fin ) ) }
clnm 37645	class LNoeM
df-lnm 37646	|- LNoeM = { w e. LMod | A. i e. ( LSubSp ` w ) ( w ↾s i ) e. LFinGen }
clnr 37679	class LNoeR
df-lnr 37680	|- LNoeR = { a e. Ring | (ringLMod ` a ) e. LNoeM }
cldgis 37691	class ldgIdlSeq
df-ldgis 37692	|- ldgIdlSeq = ( r e. _V |-> ( i e. (LIdeal ` (Poly1 ` r ) ) |-> ( x e. NN0 |-> { j | E. k e. i ( ( ( deg1 ` r ) ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) )
cmnc 37701	class Monic
cplylt 37702	class Poly<
df-mnc 37703	|- Monic = ( s e. ~P CC |-> { p e. (Poly ` s ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } )
df-plylt 37704	|- Poly< = ( s e. ~P CC , x e. NN0 |-> { p e. (Poly ` s ) | ( p = 0p \/ (deg ` p ) < x ) } )
cdgraa 37710	class degAA
cmpaa 37711	class minPolyAA
df-dgraa 37712	|- degAA = ( x e. AA |-> inf ( { d e. NN | E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` x ) = 0 ) } , RR , < ) )
df-mpaa 37713	|- minPolyAA = ( x e. AA |-> ( iota_ p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` x ) /\ ( p ` x ) = 0 /\ ( (coeff ` p ) ` (degAA ` x ) ) = 1 ) ) )
citgo 37727	class IntgOver
cza 37728	class ℤ
df-itgo 37729	|- IntgOver = ( s e. ~P CC |-> { x e. CC | E. p e. (Poly ` s ) ( ( p ` x ) = 0 /\ ( (coeff ` p ) ` (deg ` p ) ) = 1 ) } )
df-za 37730	|- ℤ = (IntgOver ` ZZ )
cmend 37745	class MEndo
df-mend 37746	|- MEndo = ( m e. _V |-> [_ ( m LMHom m ) / b ]_ ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. , <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` m ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) >. } ) )
csdrg 37765	class SubDRing
df-sdrg 37766	|- SubDRing = ( w e. DivRing |-> { s e. (SubRing ` w ) | ( w ↾s s ) e. DivRing } )
ccytp 37780	class CytP
df-cytp 37781	|- CytP = ( n e. NN |-> ( (mulGrp ` (Poly1 ` ℂfld ) ) gsumg ( r e. ( `' ( od ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) " { n } ) |-> ( (var1 ` ℂfld ) ( -g ` (Poly1 ` ℂfld ) ) ( (algSc ` (Poly1 ` ℂfld ) ) ` r ) ) ) ) )
ctopsep 37791	class TopSep
ctoplnd 37792	class TopLnd
df-topsep 37793	|- TopSep = { j e. Top | E. x e. ~P U. j ( x ~<_ om /\ ( ( cls ` j ) ` x ) = U. j ) }
df-toplnd 37794	|- TopLnd = { x e. Top | A. y e. ~P x ( U. x = U. y -> E. z e. ~P x ( z ~<_ om /\ U. x = U. z ) ) }
crcl 37964	class r*
df-rcl 37965	|- r* = ( x e. _V |-> |^| { z | ( x C_ z /\ ( _I |` ( dom z u. ran z ) ) C_ z ) } )
whe 38066	wff R hereditary A
df-he 38067	|- ( R hereditary A <-> ( R " A ) C_ A )
ax-frege1 38084	|- ( ph -> ( ps -> ph ) )
ax-frege2 38085	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
ax-frege8 38103	|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
ax-frege28 38124	|- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )
ax-frege31 38128	|- ( -. -. ph -> ph )
ax-frege41 38139	|- ( ph -> -. -. ph )
ax-frege52a 38151	|- ( ( ph <-> ps ) -> (if- ( ph , th , ch ) -> if- ( ps , th , ch ) ) )
ax-frege54a 38156	|- ( ph <-> ph )
ax-frege58a 38169	|- ( ( ps /\ ch ) -> if- ( ph , ps , ch ) )
ax-frege52c 38182	|- ( A = B -> ( [. A / x ]. ph -> [. B / x ]. ph ) )
ax-frege54c 38186	|- A = A
ax-frege58b 38195	|- ( A. x ph -> [ y / x ] ph )
cbcc 38535	class C𝑐
df-bcc 38536	|- C𝑐 = ( c e. CC , k e. NN0 |-> ( ( c FallFac k ) / ( ! ` k ) ) )
cplusr 38661	class +r
cminusr 38662	class -r
ctimesr 38663	class .v
cptdfc 38664	class PtDf ( A , B )
crr3c 38665	class RR3
cline3 38666	class line3
df-addr 38667	|- +r = ( x e. _V , y e. _V |-> ( v e. RR |-> ( ( x ` v ) + ( y ` v ) ) ) )
df-subr 38668	|- -r = ( x e. _V , y e. _V |-> ( v e. RR |-> ( ( x ` v ) - ( y ` v ) ) ) )
df-mulv 38669	|- .v = ( x e. _V , y e. _V |-> ( v e. RR |-> ( x x. ( y ` v ) ) ) )
df-ptdf 38680	|- PtDf ( A , B ) = ( x e. RR |-> ( ( ( x .v ( B -r A ) ) +v A ) " { 1 , 2 , 3 } ) )
df-rr3 38681	|- RR3 = ( RR ^m { 1 , 2 , 3 } )
df-line3 38682	|- line3 = { x e. ~P RR3 | ( 2o ~<_ x /\ A. y e. x A. z e. x ( z =/= y -> ran PtDf ( y , z ) = x ) ) }
wvd1 38785	wff (. ph ->. ps ).
df-vd1 38786	|- ( (. ph ->. ps ). <-> ( ph -> ps ) )
wvd2 38793	wff (. ph ,. ps ->. ch ).
df-vd2 38794	|- ( (. ph ,. ps ->. ch ). <-> ( ( ph /\ ps ) -> ch ) )
wvhc2 38796	wff (. ph ,. ps ).
df-vhc2 38797	|- ( (. ph ,. ps ). <-> ( ph /\ ps ) )
wvd3 38803	wff (. ph ,. ps ,. ch ->. th ).
wvhc3 38804	wff (. ph ,. ps ,. ch ).
df-vhc3 38805	|- ( (. ph ,. ps ,. ch ). <-> ( ph /\ ps /\ ch ) )
df-vd3 38806	|- ( (. ph ,. ps ,. ch ->. th ). <-> ( ( ph /\ ps /\ ch ) -> th ) )
wvhc4 38813	wff (. ph ,. ps ,. ch ,. th ).
wvhc5 38814	wff (. ph ,. ps ,. ch ,. th ,. ta ).
wvhc6 38815	wff (. ph ,. ps ,. ch ,. th ,. ta ,. et ).
wvhc7 38816	wff (. ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ).
wvhc8 38817	wff (. ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ,. si ).
wvhc9 38818	wff (. ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ,. si ,. rh ).
wvhc10 38819	wff (. ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ,. si ,. rh ,. mu ).
wvhc11 38820	wff (. ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ,. si ,. rh ,. mu ,. la ).
wvhc12 38821	wff (. ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ,. si ,. rh ,. mu ,. la ,. ka ).
clsi 39983	class liminf
df-liminf 39984	|- liminf = ( x e. _V |-> sup ( ran ( k e. RR |-> inf ( ( ( x " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
clsxlim 40044	class ~~>*
df-xlim 40045	|- ~~>* = ( ~~> t ` (ordTop ` <_ ) )
csalg 40528	class SAlg
df-salg 40529	|- SAlg = { x | ( (/) e. x /\ A. y e. x ( U. x \ y ) e. x /\ A. y e. ~P x ( y ~<_ om -> U. y e. x ) ) }
csalon 40530	class SalOn
df-salon 40531	|- SalOn = ( x e. _V |-> { s e. SAlg | U. s = x } )
csalgen 40532	class SalGen
df-salgen 40533	|- SalGen = ( x e. _V |-> |^| { s e. SAlg | ( U. s = U. x /\ x C_ s ) } )
csumge0 40579	class Σ^
df-sumge0 40580	|- Σ^ = ( x e. _V |-> if ( +oo e. ran x , +oo , sup ( ran ( y e. ( ~P dom x i^i Fin ) |-> sum_ w e. y ( x ` w ) ) , RR* , < ) ) )
cmea 40666	class Meas
df-mea 40667	|- Meas = { x | ( ( ( x : dom x --> ( 0 [,] +oo ) /\ dom x e. SAlg ) /\ ( x ` (/) ) = 0 ) /\ A. y e. ~P dom x ( ( y ~<_ om /\ Disj w e. y w ) -> ( x ` U. y ) = (Σ^ ` ( x |` y ) ) ) ) }
come 40703	class OutMeas
df-ome 40704	|- OutMeas = { x | ( ( ( ( x : dom x --> ( 0 [,] +oo ) /\ dom x = ~P U. dom x ) /\ ( x ` (/) ) = 0 ) /\ A. y e. ~P U. dom x A. z e. ~P y ( x ` z ) <_ ( x ` y ) ) /\ A. y e. ~P dom x ( y ~<_ om -> ( x ` U. y ) <_ (Σ^ ` ( x |` y ) ) ) ) }
ccaragen 40705	class CaraGen
df-caragen 40706	|- CaraGen = ( o e. OutMeas |-> { e e. ~P U. dom o | A. a e. ~P U. dom o ( ( o ` ( a i^i e ) ) +e ( o ` ( a \ e ) ) ) = ( o ` a ) } )
covoln 40750	class voln*
df-ovoln 40751	|- voln* = ( x e. Fin |-> ( y e. ~P ( RR ^m x ) |-> if ( x = (/) , 0 , inf ( { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m x ) ^m NN ) ( y C_ U_ j e. NN X_ k e. x ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. x ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } , RR* , < ) ) ) )
cvoln 40752	class voln
df-voln 40753	|- voln = ( x e. Fin |-> ( (voln* ` x ) |` (CaraGen ` (voln* ` x ) ) ) )
csmblfn 40909	class SMblFn
df-smblfn 40910	|- SMblFn = ( s e. SAlg |-> { f e. ( RR ^pm U. s ) | A. a e. RR ( `' f " ( -oo (,) a ) ) e. ( s ↾t dom f ) } )
wdfat 41193	wff F defAt A
cafv 41194	class ( F''' A )
caov 41195	class (( A F B))
df-dfat 41196	|- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
df-afv 41197	|- ( F''' A ) = if ( F defAt A , ( iota x A F x ) , _V )
df-aov 41198	|- (( A F B)) = ( F''' <. A , B >. )
cnelbr 41288	class _ e//
df-nelbr 41289	|- _ e// = { <. x , y >. | -. x e. y }
ciccp 41349	class RePart
df-iccp 41350	|- RePart = ( m e. NN |-> { p e. ( RR* ^m ( 0 ... m ) ) | A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) } )
cpfx 41381	class prefix
df-pfx 41382	|- prefix = ( s e. _V , l e. NN0 |-> ( s substr <. 0 , l >. ) )
cfmtno 41439	class FermatNo
df-fmtno 41440	|- FermatNo = ( n e. NN0 |-> ( ( 2 ^ ( 2 ^ n ) ) + 1 ) )
ceven 41537	class Even
codd 41538	class Odd
df-even 41539	|- Even = { z e. ZZ | ( z / 2 ) e. ZZ }
df-odd 41540	|- Odd = { z e. ZZ | ( ( z + 1 ) / 2 ) e. ZZ }
cgbe 41633	class GoldbachEven
cgbow 41634	class GoldbachOddW
cgbo 41635	class GoldbachOdd
df-gbe 41636	|- GoldbachEven = { z e. Even | E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ z = ( p + q ) ) }
df-gbow 41637	|- GoldbachOddW = { z e. Odd | E. p e. Prime E. q e. Prime E. r e. Prime z = ( ( p + q ) + r ) }
df-gbo 41638	|- GoldbachOdd = { z e. Odd | E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ z = ( ( p + q ) + r ) ) }
ax-bgbltosilva 41698	|- ( ( N e. Even /\ 4 < N /\ N <_ ( 4 x. (; 1 0 ^; 1 8 ) ) ) -> N e. GoldbachEven )
ax-tgoldbachgt 41699	|- O = { z e. ZZ | -. 2 || z }   &    |- G = { z e. O | E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. O /\ q e. O /\ r e. O ) /\ z = ( ( p + q ) + r ) ) }   =>    |- E. m e. NN ( m <_ (; 1 0 ^; 2 7 ) /\ A. n e. O ( m < n -> n e. G ) )
ax-hgprmladder 41702	|- E. d e. ( ZZ>= ` 3 ) E. f e. (RePart ` d ) ( ( ( f ` 0 ) = 7 /\ ( f ` 1 ) = ; 1 3 /\ ( f ` d ) = (; 8 9 x. (; 1 0 ^; 2 9 ) ) ) /\ A. i e. ( 0..^ d ) ( ( f ` i ) e. ( Prime \ { 2 } ) /\ ( ( f ` ( i + 1 ) ) - ( f ` i ) ) < ( ( 4 x. (; 1 0 ^; 1 8 ) ) - 4 ) /\ 4 < ( ( f ` ( i + 1 ) ) - ( f ` i ) ) ) )
ax-bgbltosilvaOLD 41706	|- ( ( N e. Even /\ 4 < N /\ N <_ ( 4 x. ( 10 ^; 1 8 ) ) ) -> N e. GoldbachEven )
ax-hgprmladderOLD 41708	|- E. d e. ( ZZ>= ` 3 ) E. f e. (RePart ` d ) ( ( ( f ` 0 ) = 7 /\ ( f ` 1 ) = ; 1 3 /\ ( f ` d ) = (; 8 9 x. ( 10 ^; 2 9 ) ) ) /\ A. i e. ( 0..^ d ) ( ( f ` i ) e. ( Prime \ { 2 } ) /\ ( ( f ` ( i + 1 ) ) - ( f ` i ) ) < ( ( 4 x. ( 10 ^; 1 8 ) ) - 4 ) /\ 4 < ( ( f ` ( i + 1 ) ) - ( f ` i ) ) ) )
ax-tgoldbachgtOLD 41711	|- E. m e. NN ( m <_ ( 10 ^; 2 7 ) /\ A. n e. Odd ( m < n -> n e. GoldbachOdd ) )
cupwlks 41714	class UPWalks
df-upwlks 41715	|- UPWalks = ( g e. _V |-> { <. f , p >. | ( f e. Word dom (iEdg ` g ) /\ p : ( 0 ... ( # ` f ) ) --> (Vtx ` g ) /\ A. k e. ( 0..^ ( # ` f ) ) ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) , ( p ` ( k + 1 ) ) } ) } )
cspr 41727	class Pairs
df-spr 41728	|- Pairs = ( v e. _V |-> { p | E. a e. v E. b e. v p = { a , b } } )
cmgmhm 41777	class MgmHom
csubmgm 41778	class SubMgm
df-mgmhm 41779	|- MgmHom = ( s e. Mgm , t e. Mgm |-> { f e. ( ( Base ` t ) ^m ( Base ` s ) ) | A. x e. ( Base ` s ) A. y e. ( Base ` s ) ( f ` ( x ( +g ` s ) y ) ) = ( ( f ` x ) ( +g ` t ) ( f ` y ) ) } )
df-submgm 41780	|- SubMgm = ( s e. Mgm |-> { t e. ~P ( Base ` s ) | A. x e. t A. y e. t ( x ( +g ` s ) y ) e. t } )
ccllaw 41819	class clLaw
casslaw 41820	class assLaw
ccomlaw 41821	class comLaw
df-cllaw 41822	|- clLaw = { <. o , m >. | A. x e. m A. y e. m ( x o y ) e. m }
df-comlaw 41823	|- comLaw = { <. o , m >. | A. x e. m A. y e. m ( x o y ) = ( y o x ) }
df-asslaw 41824	|- assLaw = { <. o , m >. | A. x e. m A. y e. m A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) ) }
cintop 41832	class intOp
cclintop 41833	class clIntOp
cassintop 41834	class assIntOp
df-intop 41835	|- intOp = ( m e. _V , n e. _V |-> ( n ^m ( m X. m ) ) )
df-clintop 41836	|- clIntOp = ( m e. _V |-> ( m intOp m ) )
df-assintop 41837	|- assIntOp = ( m e. _V |-> { o e. ( clIntOp ` m ) | o assLaw m } )
cmgm2 41851	class MgmALT
ccmgm2 41852	class CMgmALT
csgrp2 41853	class SGrpALT
ccsgrp2 41854	class CSGrpALT
df-mgm2 41855	|- MgmALT = { m | ( +g ` m ) clLaw ( Base ` m ) }
df-cmgm2 41856	|- CMgmALT = { m e. MgmALT | ( +g ` m ) comLaw ( Base ` m ) }
df-sgrp2 41857	|- SGrpALT = { g e. MgmALT | ( +g ` g ) assLaw ( Base ` g ) }
df-csgrp2 41858	|- CSGrpALT = { g e. SGrpALT | ( +g ` g ) comLaw ( Base ` g ) }
crng 41874	class Rng
df-rng0 41875	|- Rng = { f e. Abel | ( (mulGrp ` f ) e. SGrp /\ [. ( Base ` f ) / b ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. A. x e. b A. y e. b A. z e. b ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) ) }
crngh 41885	class RngHomo
crngs 41886	class RngIsom
df-rnghomo 41887	|- RngHomo = ( r e. Rng , s e. Rng |-> [_ ( Base ` r ) / v ]_ [_ ( Base ` s ) / w ]_ { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } )
df-rngisom 41888	|- RngIsom = ( r e. _V , s e. _V |-> { f e. ( r RngHomo s ) | `' f e. ( s RngHomo r ) } )
crngc 41957	class RngCat
crngcALTV 41958	class RngCatALTV
df-rngc 41959	|- RngCat = ( u e. _V |-> ( (ExtStrCat ` u ) |`cat ( RngHomo |` ( ( u i^i Rng ) X. ( u i^i Rng ) ) ) ) )
df-rngcALTV 41960	|- RngCatALTV = ( u e. _V |-> [_ ( u i^i Rng ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x RngHomo y ) ) >. , <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) RngHomo z ) , f e. ( ( 1st ` v ) RngHomo ( 2nd ` v ) ) |-> ( g o. f ) ) ) >. } )
cringc 42003	class RingCat
cringcALTV 42004	class RingCatALTV
df-ringc 42005	|- RingCat = ( u e. _V |-> ( (ExtStrCat ` u ) |`cat ( RingHom |` ( ( u i^i Ring ) X. ( u i^i Ring ) ) ) ) )
df-ringcALTV 42006	|- RingCatALTV = ( u e. _V |-> [_ ( u i^i Ring ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x RingHom y ) ) >. , <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) ) >. } )
cdmatalt 42185	class DMatALT
cscmatalt 42186	class ScMatALT
df-dmatalt 42187	|- DMatALT = ( n e. Fin , r e. _V |-> [_ ( n Mat r ) / a ]_ ( a ↾s { m e. ( Base ` a ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } ) )
df-scmatalt 42188	|- ScMatALT = ( n e. Fin , r e. _V |-> [_ ( n Mat r ) / a ]_ ( a ↾s { m e. ( Base ` a ) | E. c e. ( Base ` r ) A. i e. n A. j e. n ( i m j ) = if ( i = j , c , ( 0g ` r ) ) } ) )
clinc 42193	class linC
clinco 42194	class LinCo
df-linc 42195	|- linC = ( m e. _V |-> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( x e. v |-> ( ( s ` x ) ( .s ` m ) x ) ) ) ) )
df-lco 42196	|- LinCo = ( m e. _V , v e. ~P ( Base ` m ) |-> { c e. ( Base ` m ) | E. s e. ( ( Base ` (Scalar ` m ) ) ^m v ) ( s finSupp ( 0g ` (Scalar ` m ) ) /\ c = ( s ( linC ` m ) v ) ) } )
clininds 42229	class linIndS
clindeps 42230	class linDepS
df-lininds 42231	|- linIndS = { <. s , m >. | ( s e. ~P ( Base ` m ) /\ A. f e. ( ( Base ` (Scalar ` m ) ) ^m s ) ( ( f finSupp ( 0g ` (Scalar ` m ) ) /\ ( f ( linC ` m ) s ) = ( 0g ` m ) ) -> A. x e. s ( f ` x ) = ( 0g ` (Scalar ` m ) ) ) ) }
df-lindeps 42233	|- linDepS = { <. s , m >. | -. s linIndS m }
cfdiv 42331	class /_f
df-fdiv 42332	|- /_f = ( f e. _V , g e. _V |-> ( ( f oF / g ) |` ( g supp 0 ) ) )
cbigo 42341	class _O
df-bigo 42342	|- _O = ( g e. ( RR ^pm RR ) |-> { f e. ( RR ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( f ` y ) <_ ( m x. ( g ` y ) ) } )
cblen 42363	class #b
df-blen 42364	|- #b = ( n e. _V |-> if ( n = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` n ) ) ) + 1 ) ) )
cdig 42389	class digit
df-dig 42390	|- digit = ( b e. NN |-> ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( b ^ -u k ) x. r ) ) mod b ) ) )
csetrecs 42430	class setrecs ( F )
df-setrecs 42431	|- setrecs ( F ) = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
cpg 42452	class Pg
df-pg 42453	|- Pg = setrecs ( ( x e. _V |-> ( ~P x X. ~P x ) ) )
cge-real 42461	class >_
cgt 42462	class >
df-gte 42463	|- >_ = `' <_
df-gt 42464	|- > = `' <
csinh 42471	class sinh
ccosh 42472	class cosh
ctanh 42473	class tanh
df-sinh 42474	|- sinh = ( x e. CC |-> ( ( sin ` ( _i x. x ) ) / _i ) )
df-cosh 42475	|- cosh = ( x e. CC |-> ( cos ` ( _i x. x ) ) )
df-tanh 42476	|- tanh = ( x e. ( `'cosh " ( CC \ { 0 } ) ) |-> ( ( tan ` ( _i x. x ) ) / _i ) )
csec 42482	class sec
ccsc 42483	class csc
ccot 42484	class cot
df-sec 42485	|- sec = ( x e. { y e. CC | ( cos ` y ) =/= 0 } |-> ( 1 / ( cos ` x ) ) )
df-csc 42486	|- csc = ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( 1 / ( sin ` x ) ) )
df-cot 42487	|- cot = ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( ( cos ` x ) / ( sin ` x ) ) )
clog- 42506	class log_
df-logbALT 42507	|- log_ = ( b e. ( CC \ { 0 , 1 } ) |-> ( x e. ( CC \ { 0 } ) |-> ( ( log ` x ) / ( log ` b ) ) ) )
wreflexive 42508	wff RReflexive A
df-reflexive 42509	|- ( RReflexive A <-> ( R C_ ( A X. A ) /\ A. x e. A x R x ) )
wirreflexive 42510	wff RIrreflexive A
df-irreflexive 42511	|- ( RIrreflexive A <-> ( R C_ ( A X. A ) /\ A. x e. A -. x R x ) )
walsi 42532	wff A.! x ( ph -> ps )
walsc 42533	wff A.! x e. A ph
df-alsi 42534	|- ( A.! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E. x ph ) )
df-alsc 42535	|- ( A.! x e. A ph <-> ( A. x e. A ph /\ E. x x e. A ) )