Theorem ringcval 42008

Description: Value of the category of unital rings (in a universe). (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.)

Hypotheses

Ref	Expression
ringcval.c	|- C = (RingCat ` U )
ringcval.u	|- ( ph -> U e. V )
ringcval.b	|- ( ph -> B = ( U i^i Ring ) )
ringcval.h	|- ( ph -> H = ( RingHom |` ( B X. B ) ) )

Assertion

Ref	Expression
ringcval	|- ( ph -> C = ( (ExtStrCat ` U ) |`cat H ) )

Proof of Theorem ringcval

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringcval.c	. 2 |- C = (RingCat ` U )
2		df-ringc 42005	. . . 4 |- RingCat = ( u e. _V |-> ( (ExtStrCat ` u ) |`cat ( RingHom |` ( ( u i^i Ring ) X. ( u i^i Ring ) ) ) ) )
3	2	a1i 11	. . 3 |- ( ph -> RingCat = ( u e. _V |-> ( (ExtStrCat ` u ) |`cat ( RingHom |` ( ( u i^i Ring ) X. ( u i^i Ring ) ) ) ) ) )
4		fveq2 6191	. . . . 5 |- ( u = U -> (ExtStrCat ` u ) = (ExtStrCat ` U ) )
5	4	adantl 482	. . . 4 |- ( ( ph /\ u = U ) -> (ExtStrCat ` u ) = (ExtStrCat ` U ) )
6		ineq1 3807	. . . . . . . 8 |- ( u = U -> ( u i^i Ring ) = ( U i^i Ring ) )
7	6	sqxpeqd 5141	. . . . . . 7 |- ( u = U -> ( ( u i^i Ring ) X. ( u i^i Ring ) ) = ( ( U i^i Ring ) X. ( U i^i Ring ) ) )
8		ringcval.b	. . . . . . . . 9 |- ( ph -> B = ( U i^i Ring ) )
9	8	sqxpeqd 5141	. . . . . . . 8 |- ( ph -> ( B X. B ) = ( ( U i^i Ring ) X. ( U i^i Ring ) ) )
10	9	eqcomd 2628	. . . . . . 7 |- ( ph -> ( ( U i^i Ring ) X. ( U i^i Ring ) ) = ( B X. B ) )
11	7, 10	sylan9eqr 2678	. . . . . 6 |- ( ( ph /\ u = U ) -> ( ( u i^i Ring ) X. ( u i^i Ring ) ) = ( B X. B ) )
12	11	reseq2d 5396	. . . . 5 |- ( ( ph /\ u = U ) -> ( RingHom |` ( ( u i^i Ring ) X. ( u i^i Ring ) ) ) = ( RingHom |` ( B X. B ) ) )
13		ringcval.h	. . . . . . 7 |- ( ph -> H = ( RingHom |` ( B X. B ) ) )
14	13	eqcomd 2628	. . . . . 6 |- ( ph -> ( RingHom |` ( B X. B ) ) = H )
15	14	adantr 481	. . . . 5 |- ( ( ph /\ u = U ) -> ( RingHom |` ( B X. B ) ) = H )
16	12, 15	eqtrd 2656	. . . 4 |- ( ( ph /\ u = U ) -> ( RingHom |` ( ( u i^i Ring ) X. ( u i^i Ring ) ) ) = H )
17	5, 16	oveq12d 6668	. . 3 |- ( ( ph /\ u = U ) -> ( (ExtStrCat ` u ) |`cat ( RingHom |` ( ( u i^i Ring ) X. ( u i^i Ring ) ) ) ) = ( (ExtStrCat ` U ) |`cat H ) )
18		ringcval.u	. . . 4 |- ( ph -> U e. V )
19		elex 3212	. . . 4 |- ( U e. V -> U e. _V )
20	18, 19	syl 17	. . 3 |- ( ph -> U e. _V )
21		ovexd 6680	. . 3 |- ( ph -> ( (ExtStrCat ` U ) |`cat H ) e. _V )
22	3, 17, 20, 21	fvmptd 6288	. 2 |- ( ph -> (RingCat ` U ) = ( (ExtStrCat ` U ) |`cat H ) )
23	1, 22	syl5eq 2668	1 |- ( ph -> C = ( (ExtStrCat ` U ) |`cat H ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   |-> cmpt 4729   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   |`_cat cresc 16468  ExtStrCatcestrc 16762   Ringcrg 18547   RingHom crh 18712  RingCatcringc 42003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-ringc 42005

This theorem is referenced by:  ringcbas  42011  ringchomfval  42012  ringccofval  42016  dfringc2  42018  ringccat  42024  ringcid  42025  funcringcsetc  42035