Theorem ringcvalALTV 39628

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 39627	. . . 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 3083	. . . . . 6 |- u e. _V
5	4	inex1 4565	. . . . 5 |- ( u i^i Ring ) e. _V
6	5	a1i 11	. . . 4 |- ( ( ph /\ u = U ) -> ( u i^i Ring ) e. _V )
7		ineq1 3657	. . . . . 6 |- ( u = U -> ( u i^i Ring ) = ( U i^i Ring ) )
8	7	adantl 467	. . . . 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 466	. . . . 5 |- ( ( ph /\ u = U ) -> B = ( U i^i Ring ) )
11	8, 10	eqtr4d 2466	. . . 4 |- ( ( ph /\ u = U ) -> ( u i^i Ring ) = B )
12		simpr 462	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> b = B )
13	12	opeq2d 4194	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
14		eqidd 2423	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x RingHom y ) = ( x RingHom y ) )
15	12, 12, 14	mpt2eq123dv 6367	. . . . . . 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 730	. . . . . . 7 |- ( ( ( ph /\ u = U ) /\ b = B ) -> H = ( x e. B , y e. B |-> ( x RingHom y ) ) )
18	15, 17	eqtr4d 2466	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x e. b , y e. b |-> ( x RingHom y ) ) = H )
19	18	opeq2d 4194	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x RingHom y ) ) >. = <. ( Hom ` ndx ) , H >. )
20	12	sqxpeqd 4879	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( b X. b ) = ( B X. B ) )
21		eqidd 2423	. . . . . . . 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 6367	. . . . . . 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 730	. . . . . . 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 2466	. . . . . 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 4194	. . . . 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 4094	. . . 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 3423	. . 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 3089	. . . 4 |- ( U e. V -> U e. _V )
31	29, 30	syl 17	. . 3 |- ( ph -> U e. _V )
32		tpex 6604	. . . 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 5970	. 2 |- ( ph -> (RingCatALTV ` U ) = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
35	1, 34	syl5eq 2475	1 |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )

Syntax hints:   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1872   _Vcvv 3080   [_csb 3395   i^i cin 3435   {ctp 4002   <.cop 4004   |-> cmpt 4482   X. cxp 4851   o. ccom 4857   ` cfv 5601  (class class class)co 6305   |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   ndxcnx 15117   Basecbs 15120   Hom chom 15200  compcco 15201   Ringcrg 17779   RingHom crh 17939  RingCatALTVcringcALTV 39625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-oprab 6309  df-mpt2 6310  df-ringcALTV 39627

This theorem is referenced by:  ringcbasALTV  39667  ringchomfvalALTV  39668  ringccofvalALTV  39671