Theorem rngcresringcat 42030

Description: The restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings. (Contributed by AV, 16-Mar-2020.)

Hypotheses

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

Assertion

Ref	Expression
rngcresringcat	|- ( ph -> ( C |`cat H ) = (RingCat ` U ) )

Proof of Theorem rngcresringcat

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmsubcrngc.c	. . . 4 |- C = (RngCat ` U )
2		rhmsubcrngc.u	. . . 4 |- ( ph -> U e. V )
3		eqidd 2623	. . . 4 |- ( ph -> ( U i^i Rng ) = ( U i^i Rng ) )
4		eqidd 2623	. . . 4 |- ( ph -> ( RngHomo |` ( ( U i^i Rng ) X. ( U i^i Rng ) ) ) = ( RngHomo |` ( ( U i^i Rng ) X. ( U i^i Rng ) ) ) )
5		eqidd 2623	. . . 4 |- ( ph -> (comp ` (ExtStrCat ` U ) ) = (comp ` (ExtStrCat ` U ) ) )
6	1, 2, 3, 4, 5	dfrngc2 41972	. . 3 |- ( ph -> C = { <. ( Base ` ndx ) , ( U i^i Rng ) >. , <. ( Hom ` ndx ) , ( RngHomo |` ( ( U i^i Rng ) X. ( U i^i Rng ) ) ) >. , <. (comp ` ndx ) , (comp ` (ExtStrCat ` U ) ) >. } )
7		inex1g 4801	. . . 4 |- ( U e. V -> ( U i^i Rng ) e. _V )
8	2, 7	syl 17	. . 3 |- ( ph -> ( U i^i Rng ) e. _V )
9		rnghmfn 41890	. . . . 5 |- RngHomo Fn (Rng X. Rng )
10		fnfun 5988	. . . . 5 |- ( RngHomo Fn (Rng X. Rng ) -> Fun RngHomo )
11	9, 10	mp1i 13	. . . 4 |- ( ph -> Fun RngHomo )
12		sqxpexg 6963	. . . . 5 |- ( ( U i^i Rng ) e. _V -> ( ( U i^i Rng ) X. ( U i^i Rng ) ) e. _V )
13	8, 12	syl 17	. . . 4 |- ( ph -> ( ( U i^i Rng ) X. ( U i^i Rng ) ) e. _V )
14		resfunexg 6479	. . . 4 |- ( ( Fun RngHomo /\ ( ( U i^i Rng ) X. ( U i^i Rng ) ) e. _V ) -> ( RngHomo |` ( ( U i^i Rng ) X. ( U i^i Rng ) ) ) e. _V )
15	11, 13, 14	syl2anc 693	. . 3 |- ( ph -> ( RngHomo |` ( ( U i^i Rng ) X. ( U i^i Rng ) ) ) e. _V )
16		fvexd 6203	. . 3 |- ( ph -> (comp ` (ExtStrCat ` U ) ) e. _V )
17		rhmsubcrngc.b	. . . . 5 |- ( ph -> B = ( Ring i^i U ) )
18		incom 3805	. . . . 5 |- ( Ring i^i U ) = ( U i^i Ring )
19	17, 18	syl6eq 2672	. . . 4 |- ( ph -> B = ( U i^i Ring ) )
20		inex1g 4801	. . . . 5 |- ( U e. V -> ( U i^i Ring ) e. _V )
21	2, 20	syl 17	. . . 4 |- ( ph -> ( U i^i Ring ) e. _V )
22	19, 21	eqeltrd 2701	. . 3 |- ( ph -> B e. _V )
23		rhmsubcrngc.h	. . . 4 |- ( ph -> H = ( RingHom |` ( B X. B ) ) )
24		rhmfn 41918	. . . . . 6 |- RingHom Fn ( Ring X. Ring )
25		fnfun 5988	. . . . . 6 |- ( RingHom Fn ( Ring X. Ring ) -> Fun RingHom )
26	24, 25	mp1i 13	. . . . 5 |- ( ph -> Fun RingHom )
27		sqxpexg 6963	. . . . . 6 |- ( B e. _V -> ( B X. B ) e. _V )
28	22, 27	syl 17	. . . . 5 |- ( ph -> ( B X. B ) e. _V )
29		resfunexg 6479	. . . . 5 |- ( ( Fun RingHom /\ ( B X. B ) e. _V ) -> ( RingHom |` ( B X. B ) ) e. _V )
30	26, 28, 29	syl2anc 693	. . . 4 |- ( ph -> ( RingHom |` ( B X. B ) ) e. _V )
31	23, 30	eqeltrd 2701	. . 3 |- ( ph -> H e. _V )
32		ringrng 41879	. . . . . . 7 |- ( r e. Ring -> r e. Rng )
33	32	a1i 11	. . . . . 6 |- ( ph -> ( r e. Ring -> r e. Rng ) )
34	33	ssrdv 3609	. . . . 5 |- ( ph -> Ring C_ Rng )
35		ssrin 3838	. . . . 5 |- ( Ring C_ Rng -> ( Ring i^i U ) C_ (Rng i^i U ) )
36	34, 35	syl 17	. . . 4 |- ( ph -> ( Ring i^i U ) C_ (Rng i^i U ) )
37		incom 3805	. . . . 5 |- ( U i^i Rng ) = (Rng i^i U )
38	37	a1i 11	. . . 4 |- ( ph -> ( U i^i Rng ) = (Rng i^i U ) )
39	36, 17, 38	3sstr4d 3648	. . 3 |- ( ph -> B C_ ( U i^i Rng ) )
40	6, 8, 15, 16, 22, 31, 39	estrres 16779	. 2 |- ( ph -> ( ( C ↾s B ) sSet <. ( Hom ` ndx ) , H >. ) = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , (comp ` (ExtStrCat ` U ) ) >. } )
41		eqid 2622	. . 3 |- ( C |`cat H ) = ( C |`cat H )
42	1	a1i 11	. . . 4 |- ( ph -> C = (RngCat ` U ) )
43		fvexd 6203	. . . 4 |- ( ph -> (RngCat ` U ) e. _V )
44	42, 43	eqeltrd 2701	. . 3 |- ( ph -> C e. _V )
45	19, 23	rhmresfn 42009	. . 3 |- ( ph -> H Fn ( B X. B ) )
46	41, 44, 22, 45	rescval2 16488	. 2 |- ( ph -> ( C |`cat H ) = ( ( C ↾s B ) sSet <. ( Hom ` ndx ) , H >. ) )
47		eqid 2622	. . 3 |- (RingCat ` U ) = (RingCat ` U )
48	47, 2, 19, 23, 5	dfringc2 42018	. 2 |- ( ph -> (RingCat ` U ) = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , (comp ` (ExtStrCat ` U ) ) >. } )
49	40, 46, 48	3eqtr4d 2666	1 |- ( ph -> ( C |`cat H ) = (RingCat ` U ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {ctp 4181   <.cop 4183   X. cxp 5112   |` cres 5116   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   Basecbs 15857   ↾_s cress 15858   Hom chom 15952  compcco 15953   |`_cat cresc 16468  ExtStrCatcestrc 16762   Ringcrg 18547   RingHom crh 18712  Rngcrng 41874   RngHomo crngh 41885  RngCatcrngc 41957  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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-hom 15966  df-cco 15967  df-0g 16102  df-resc 16471  df-estrc 16763  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-ghm 17658  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-rnghom 18715  df-rng0 41875  df-rnghomo 41887  df-rngc 41959  df-ringc 42005

This theorem is referenced by: (None)