Theorem rhmsscrnghm 33021

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

Step	Hyp	Ref	Expression
1		ringrng 32872	. . . . . 6 |- ( r e. Ring -> r e. Rng )
2	1	a1i 11	. . . . 5 |- ( ph -> ( r e. Ring -> r e. Rng ) )
3	2	ssrdv 3505	. . . 4 |- ( ph -> Ring C_ Rng )
4		ssrin 3719	. . . 4 |- ( Ring C_ Rng -> ( Ring i^i U ) C_ (Rng i^i U ) )
5	3, 4	syl 16	. . 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 3542	. 2 |- ( ph -> R C_ S )
9		ovres 6441	. . . . . . 7 |- ( ( x e. R /\ y e. R ) -> ( x ( RingHom |` ( R X. R ) ) y ) = ( x RingHom y ) )
10	9	adantl 466	. . . . . 6 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RingHom |` ( R X. R ) ) y ) = ( x RingHom y ) )
11	10	eleq2d 2527	. . . . 5 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( h e. ( x ( RingHom |` ( R X. R ) ) y ) <-> h e. ( x RingHom y ) ) )
12		rhmisrnghm 32913	. . . . . 6 |- ( h e. ( x RingHom y ) -> h e. ( x RngHomo y ) )
13	8	sseld 3498	. . . . . . . . . 10 |- ( ph -> ( x e. R -> x e. S ) )
14	8	sseld 3498	. . . . . . . . . 10 |- ( ph -> ( y e. R -> y e. S ) )
15	13, 14	anim12d 563	. . . . . . . . 9 |- ( ph -> ( ( x e. R /\ y e. R ) -> ( x e. S /\ y e. S ) ) )
16	15	imp 429	. . . . . . . 8 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x e. S /\ y e. S ) )
17		ovres 6441	. . . . . . . 8 |- ( ( x e. S /\ y e. S ) -> ( x ( RngHomo |` ( S X. S ) ) y ) = ( x RngHomo y ) )
18	16, 17	syl 16	. . . . . . 7 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RngHomo |` ( S X. S ) ) y ) = ( x RngHomo y ) )
19	18	eleq2d 2527	. . . . . 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 221	. . . . 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 215	. . . 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 3505	. . 3 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RingHom |` ( R X. R ) ) y ) C_ ( x ( RngHomo |` ( S X. S ) ) y ) )
23	22	ralrimivva 2878	. 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 3714	. . . . . 6 |- ( Ring i^i U ) C_ Ring
25	6, 24	syl6eqss 3549	. . . . 5 |- ( ph -> R C_ Ring )
26		xpss12 5117	. . . . 5 |- ( ( R C_ Ring /\ R C_ Ring ) -> ( R X. R ) C_ ( Ring X. Ring ) )
27	25, 25, 26	syl2anc 661	. . . 4 |- ( ph -> ( R X. R ) C_ ( Ring X. Ring ) )
28		rhmfn 32911	. . . . 5 |- RingHom Fn ( Ring X. Ring )
29		fnssresb 5699	. . . . 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 12	. . . 4 |- ( ph -> ( ( RingHom |` ( R X. R ) ) Fn ( R X. R ) <-> ( R X. R ) C_ ( Ring X. Ring ) ) )
31	27, 30	mpbird 232	. . 3 |- ( ph -> ( RingHom |` ( R X. R ) ) Fn ( R X. R ) )
32		inss1 3714	. . . . . 6 |- (Rng i^i U ) C_ Rng
33	7, 32	syl6eqss 3549	. . . . 5 |- ( ph -> S C_ Rng )
34		xpss12 5117	. . . . 5 |- ( ( S C_ Rng /\ S C_ Rng ) -> ( S X. S ) C_ (Rng X. Rng ) )
35	33, 33, 34	syl2anc 661	. . . 4 |- ( ph -> ( S X. S ) C_ (Rng X. Rng ) )
36		rnghmfn 32883	. . . . 5 |- RngHomo Fn (Rng X. Rng )
37		fnssresb 5699	. . . . 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 12	. . . 4 |- ( ph -> ( ( RngHomo |` ( S X. S ) ) Fn ( S X. S ) <-> ( S X. S ) C_ (Rng X. Rng ) ) )
39	35, 38	mpbird 232	. . 3 |- ( ph -> ( RngHomo |` ( S X. S ) ) Fn ( S X. S ) )
40		rhmsscrnghm.u	. . . . 5 |- ( ph -> U e. V )
41		incom 3687	. . . . . 6 |- (Rng i^i U ) = ( U i^i Rng )
42		inex1g 4599	. . . . . 6 |- ( U e. V -> ( U i^i Rng ) e. _V )
43	41, 42	syl5eqel 2549	. . . . 5 |- ( U e. V -> (Rng i^i U ) e. _V )
44	40, 43	syl 16	. . . 4 |- ( ph -> (Rng i^i U ) e. _V )
45	7, 44	eqeltrd 2545	. . 3 |- ( ph -> S e. _V )
46	31, 39, 45	isssc 15327	. 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 922	1 |- ( ph -> ( RingHom |` ( R X. R ) ) C_cat ( RngHomo |` ( S X. S ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   A.wral 2807   _Vcvv 3109   i^i cin 3470   C_ wss 3471   class class class wbr 4456   X. cxp 5006   |` cres 5010   Fn wfn 5589  (class class class)co 6296   C__cat cssc 15314   Ringcrg 17416   RingHom crh 17579  Rngcrng 32867   RngHomo crngh 32878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14738  df-slot 14739  df-base 14740  df-sets 14741  df-plusg 14816  df-0g 14950  df-ssc 15317  df-mgm 16090  df-sgrp 16129  df-mnd 16139  df-mhm 16184  df-grp 16275  df-minusg 16276  df-ghm 16483  df-cmn 17018  df-abl 17019  df-mgp 17360  df-ur 17372  df-ring 17418  df-rnghom 17582  df-mgmhm 32772  df-rng0 32868  df-rnghomo 32880

This theorem is referenced by:  rhmsubcrngc  33024  rhmsubc  33085  rhmsubcOLD  33104