Theorem rhmsscrnghm 42026

Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the non-unital ring homomorphisms between non-unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)

Hypotheses

Ref	Expression
rhmsscrnghm.u	|- ( ph -> U e. V )
rhmsscrnghm.r	|- ( ph -> R = ( Ring i^i U ) )
rhmsscrnghm.s	|- ( ph -> S = (Rng i^i U ) )

Assertion

Ref	Expression
rhmsscrnghm	|- ( ph -> ( RingHom |` ( R X. R ) ) C_cat ( RngHomo |` ( S X. S ) ) )

Proof of Theorem rhmsscrnghm

Dummy variables x h y r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringrng 41879	. . . . . 6 |- ( r e. Ring -> r e. Rng )
2	1	a1i 11	. . . . 5 |- ( ph -> ( r e. Ring -> r e. Rng ) )
3	2	ssrdv 3609	. . . 4 |- ( ph -> Ring C_ Rng )
4		ssrin 3838	. . . 4 |- ( Ring C_ Rng -> ( Ring i^i U ) C_ (Rng i^i U ) )
5	3, 4	syl 17	. . 3 |- ( ph -> ( Ring i^i U ) C_ (Rng i^i U ) )
6		rhmsscrnghm.r	. . 3 |- ( ph -> R = ( Ring i^i U ) )
7		rhmsscrnghm.s	. . 3 |- ( ph -> S = (Rng i^i U ) )
8	5, 6, 7	3sstr4d 3648	. 2 |- ( ph -> R C_ S )
9		ovres 6800	. . . . . . 7 |- ( ( x e. R /\ y e. R ) -> ( x ( RingHom |` ( R X. R ) ) y ) = ( x RingHom y ) )
10	9	adantl 482	. . . . . 6 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RingHom |` ( R X. R ) ) y ) = ( x RingHom y ) )
11	10	eleq2d 2687	. . . . 5 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( h e. ( x ( RingHom |` ( R X. R ) ) y ) <-> h e. ( x RingHom y ) ) )
12		rhmisrnghm 41920	. . . . . 6 |- ( h e. ( x RingHom y ) -> h e. ( x RngHomo y ) )
13	8	sseld 3602	. . . . . . . . . 10 |- ( ph -> ( x e. R -> x e. S ) )
14	8	sseld 3602	. . . . . . . . . 10 |- ( ph -> ( y e. R -> y e. S ) )
15	13, 14	anim12d 586	. . . . . . . . 9 |- ( ph -> ( ( x e. R /\ y e. R ) -> ( x e. S /\ y e. S ) ) )
16	15	imp 445	. . . . . . . 8 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x e. S /\ y e. S ) )
17		ovres 6800	. . . . . . . 8 |- ( ( x e. S /\ y e. S ) -> ( x ( RngHomo |` ( S X. S ) ) y ) = ( x RngHomo y ) )
18	16, 17	syl 17	. . . . . . 7 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RngHomo |` ( S X. S ) ) y ) = ( x RngHomo y ) )
19	18	eleq2d 2687	. . . . . 6 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( h e. ( x ( RngHomo |` ( S X. S ) ) y ) <-> h e. ( x RngHomo y ) ) )
20	12, 19	syl5ibr 236	. . . . 5 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( h e. ( x RingHom y ) -> h e. ( x ( RngHomo |` ( S X. S ) ) y ) ) )
21	11, 20	sylbid 230	. . . 4 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( h e. ( x ( RingHom |` ( R X. R ) ) y ) -> h e. ( x ( RngHomo |` ( S X. S ) ) y ) ) )
22	21	ssrdv 3609	. . 3 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RingHom |` ( R X. R ) ) y ) C_ ( x ( RngHomo |` ( S X. S ) ) y ) )
23	22	ralrimivva 2971	. 2 |- ( ph -> A. x e. R A. y e. R ( x ( RingHom |` ( R X. R ) ) y ) C_ ( x ( RngHomo |` ( S X. S ) ) y ) )
24		inss1 3833	. . . . . 6 |- ( Ring i^i U ) C_ Ring
25	6, 24	syl6eqss 3655	. . . . 5 |- ( ph -> R C_ Ring )
26		xpss12 5225	. . . . 5 |- ( ( R C_ Ring /\ R C_ Ring ) -> ( R X. R ) C_ ( Ring X. Ring ) )
27	25, 25, 26	syl2anc 693	. . . 4 |- ( ph -> ( R X. R ) C_ ( Ring X. Ring ) )
28		rhmfn 41918	. . . . 5 |- RingHom Fn ( Ring X. Ring )
29		fnssresb 6003	. . . . 5 |- ( RingHom Fn ( Ring X. Ring ) -> ( ( RingHom |` ( R X. R ) ) Fn ( R X. R ) <-> ( R X. R ) C_ ( Ring X. Ring ) ) )
30	28, 29	mp1i 13	. . . 4 |- ( ph -> ( ( RingHom |` ( R X. R ) ) Fn ( R X. R ) <-> ( R X. R ) C_ ( Ring X. Ring ) ) )
31	27, 30	mpbird 247	. . 3 |- ( ph -> ( RingHom |` ( R X. R ) ) Fn ( R X. R ) )
32		inss1 3833	. . . . . 6 |- (Rng i^i U ) C_ Rng
33	7, 32	syl6eqss 3655	. . . . 5 |- ( ph -> S C_ Rng )
34		xpss12 5225	. . . . 5 |- ( ( S C_ Rng /\ S C_ Rng ) -> ( S X. S ) C_ (Rng X. Rng ) )
35	33, 33, 34	syl2anc 693	. . . 4 |- ( ph -> ( S X. S ) C_ (Rng X. Rng ) )
36		rnghmfn 41890	. . . . 5 |- RngHomo Fn (Rng X. Rng )
37		fnssresb 6003	. . . . 5 |- ( RngHomo Fn (Rng X. Rng ) -> ( ( RngHomo |` ( S X. S ) ) Fn ( S X. S ) <-> ( S X. S ) C_ (Rng X. Rng ) ) )
38	36, 37	mp1i 13	. . . 4 |- ( ph -> ( ( RngHomo |` ( S X. S ) ) Fn ( S X. S ) <-> ( S X. S ) C_ (Rng X. Rng ) ) )
39	35, 38	mpbird 247	. . 3 |- ( ph -> ( RngHomo |` ( S X. S ) ) Fn ( S X. S ) )
40		rhmsscrnghm.u	. . . . 5 |- ( ph -> U e. V )
41		incom 3805	. . . . . 6 |- (Rng i^i U ) = ( U i^i Rng )
42		inex1g 4801	. . . . . 6 |- ( U e. V -> ( U i^i Rng ) e. _V )
43	41, 42	syl5eqel 2705	. . . . 5 |- ( U e. V -> (Rng i^i U ) e. _V )
44	40, 43	syl 17	. . . 4 |- ( ph -> (Rng i^i U ) e. _V )
45	7, 44	eqeltrd 2701	. . 3 |- ( ph -> S e. _V )
46	31, 39, 45	isssc 16480	. 2 |- ( ph -> ( ( RingHom |` ( R X. R ) ) C_cat ( RngHomo |` ( S X. S ) ) <-> ( R C_ S /\ A. x e. R A. y e. R ( x ( RingHom |` ( R X. R ) ) y ) C_ ( x ( RngHomo |` ( S X. S ) ) y ) ) ) )
47	8, 23, 46	mpbir2and 957	1 |- ( ph -> ( RingHom |` ( R X. R ) ) C_cat ( RngHomo |` ( S X. S ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   X. cxp 5112   |` cres 5116   Fn wfn 5883  (class class class)co 6650   C__cat cssc 16467   Ringcrg 18547   RingHom crh 18712  Rngcrng 41874   RngHomo crngh 41885

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-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-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-ssc 16470  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-mgmhm 41779  df-rng0 41875  df-rnghomo 41887

This theorem is referenced by:  rhmsubcrngc  42029  rhmsubc  42090  rhmsubcALTV  42108