Theorem ringcvalALTV 40334

Description: Value of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ringcvalALTV.c	|- C = (RingCatALTV ` U )
ringcvalALTV.u	|- ( ph -> U e. V )
ringcvalALTV.b	|- ( ph -> B = ( U i^i Ring ) )
ringcvalALTV.h	|- ( ph -> H = ( x e. B , y e. B |-> ( x RingHom y ) ) )
ringcvalALTV.o	|- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) ) )

Assertion

Ref	Expression
ringcvalALTV	|- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )

Distinct variable groups:   f, g, v, x, y, z   v, B, x, y, z   v, U, x, y, z   ph, v, x, y, z

Allowed substitution hints:   ph( f, g)   B( f, g)   C( x, y, z, v, f, g)   .x. ( x, y, z, v, f, g)   U( f, g)   H( x, y, z, v, f, g)   V( x, y, z, v, f, g)

Proof of Theorem ringcvalALTV

Dummy variables b u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringcvalALTV.c	. 2 |- C = (RingCatALTV ` U )
2		df-ringcALTV 40333	. . . 4 |- 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 ) ) ) >. } )
3	2	a1i 11	. . 3 |- ( ph -> 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 ) ) ) >. } ) )
4		vex 3016	. . . . . 6 |- u e. _V
5	4	inex1 4516	. . . . 5 |- ( u i^i Ring ) e. _V
6	5	a1i 11	. . . 4 |- ( ( ph /\ u = U ) -> ( u i^i Ring ) e. _V )
7		ineq1 3595	. . . . . 6 |- ( u = U -> ( u i^i Ring ) = ( U i^i Ring ) )
8	7	adantl 472	. . . . 5 |- ( ( ph /\ u = U ) -> ( u i^i Ring ) = ( U i^i Ring ) )
9		ringcvalALTV.b	. . . . . 6 |- ( ph -> B = ( U i^i Ring ) )
10	9	adantr 471	. . . . 5 |- ( ( ph /\ u = U ) -> B = ( U i^i Ring ) )
11	8, 10	eqtr4d 2489	. . . 4 |- ( ( ph /\ u = U ) -> ( u i^i Ring ) = B )
12		simpr 467	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> b = B )
13	12	opeq2d 4143	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
14		eqidd 2453	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x RingHom y ) = ( x RingHom y ) )
15	12, 12, 14	mpt2eq123dv 6341	. . . . . . 7 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x e. b , y e. b |-> ( x RingHom y ) ) = ( x e. B , y e. B |-> ( x RingHom y ) ) )
16		ringcvalALTV.h	. . . . . . . 8 |- ( ph -> H = ( x e. B , y e. B |-> ( x RingHom y ) ) )
17	16	ad2antrr 737	. . . . . . 7 |- ( ( ( ph /\ u = U ) /\ b = B ) -> H = ( x e. B , y e. B |-> ( x RingHom y ) ) )
18	15, 17	eqtr4d 2489	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x e. b , y e. b |-> ( x RingHom y ) ) = H )
19	18	opeq2d 4143	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x RingHom y ) ) >. = <. ( Hom ` ndx ) , H >. )
20	12	sqxpeqd 4838	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( b X. b ) = ( B X. B ) )
21		eqidd 2453	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) = ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) )
22	20, 12, 21	mpt2eq123dv 6341	. . . . . . 7 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) ) = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) ) )
23		ringcvalALTV.o	. . . . . . . 8 |- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) ) )
24	23	ad2antrr 737	. . . . . . 7 |- ( ( ( ph /\ u = U ) /\ b = B ) -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) ) )
25	22, 24	eqtr4d 2489	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) ) = .x. )
26	25	opeq2d 4143	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. (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 ) ) ) >. = <. (comp ` ndx ) , .x. >. )
27	13, 19, 26	tpeq123d 4035	. . . 4 |- ( ( ( ph /\ u = U ) /\ b = 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 ) ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
28	6, 11, 27	csbied2 3359	. . 3 |- ( ( ph /\ u = U ) -> [_ ( 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 ) ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
29		ringcvalALTV.u	. . . 4 |- ( ph -> U e. V )
30		elex 3022	. . . 4 |- ( U e. V -> U e. _V )
31	29, 30	syl 17	. . 3 |- ( ph -> U e. _V )
32		tpex 6578	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } e. _V
33	32	a1i 11	. . 3 |- ( ph -> { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } e. _V )
34	3, 28, 31, 33	fvmptd 5938	. 2 |- ( ph -> (RingCatALTV ` U ) = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
35	1, 34	syl5eq 2498	1 |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1448   e. wcel 1891   _Vcvv 3013   [_csb 3331   i^i cin 3371   {ctp 3940   <.cop 3942   |-> cmpt 4433   X. cxp 4810   o. ccom 4816   ` cfv 5561  (class class class)co 6276   |-> cmpt2 6278   1stc1st 6779   2ndc2nd 6780   ndxcnx 15129   Basecbs 15132   Hom chom 15212  compcco 15213   Ringcrg 17791   RingHom crh 17951  RingCatALTVcringcALTV 40331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pr 4612  ax-un 6571

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-iota 5525  df-fun 5563  df-fv 5569  df-oprab 6280  df-mpt2 6281  df-ringcALTV 40333

This theorem is referenced by:  ringcbasALTV  40373  ringchomfvalALTV  40374  ringccofvalALTV  40377