Theorem rhmsscrnghm 40536

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 40387	. . . . . 6 |- ( r e. Ring -> r e. Rng )
2	1	a1i 11	. . . . 5 |- ( ph -> ( r e. Ring -> r e. Rng ) )
3	2	ssrdv 3424	. . . 4 |- ( ph -> Ring C_ Rng )
4		ssrin 3648	. . . 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 3461	. 2 |- ( ph -> R C_ S )
9		ovres 6455	. . . . . . 7 |- ( ( x e. R /\ y e. R ) -> ( x ( RingHom |` ( R X. R ) ) y ) = ( x RingHom y ) )
10	9	adantl 473	. . . . . 6 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RingHom |` ( R X. R ) ) y ) = ( x RingHom y ) )
11	10	eleq2d 2534	. . . . 5 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( h e. ( x ( RingHom |` ( R X. R ) ) y ) <-> h e. ( x RingHom y ) ) )
12		rhmisrnghm 40428	. . . . . 6 |- ( h e. ( x RingHom y ) -> h e. ( x RngHomo y ) )
13	8	sseld 3417	. . . . . . . . . 10 |- ( ph -> ( x e. R -> x e. S ) )
14	8	sseld 3417	. . . . . . . . . 10 |- ( ph -> ( y e. R -> y e. S ) )
15	13, 14	anim12d 572	. . . . . . . . 9 |- ( ph -> ( ( x e. R /\ y e. R ) -> ( x e. S /\ y e. S ) ) )
16	15	imp 436	. . . . . . . 8 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x e. S /\ y e. S ) )
17		ovres 6455	. . . . . . . 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 2534	. . . . . 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 229	. . . . 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 223	. . . 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 3424	. . 3 |- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RingHom |` ( R X. R ) ) y ) C_ ( x ( RngHomo |` ( S X. S ) ) y ) )
23	22	ralrimivva 2814	. 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 3643	. . . . . 6 |- ( Ring i^i U ) C_ Ring
25	6, 24	syl6eqss 3468	. . . . 5 |- ( ph -> R C_ Ring )
26		xpss12 4945	. . . . 5 |- ( ( R C_ Ring /\ R C_ Ring ) -> ( R X. R ) C_ ( Ring X. Ring ) )
27	25, 25, 26	syl2anc 673	. . . 4 |- ( ph -> ( R X. R ) C_ ( Ring X. Ring ) )
28		rhmfn 40426	. . . . 5 |- RingHom Fn ( Ring X. Ring )
29		fnssresb 5698	. . . . 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 240	. . 3 |- ( ph -> ( RingHom |` ( R X. R ) ) Fn ( R X. R ) )
32		inss1 3643	. . . . . 6 |- (Rng i^i U ) C_ Rng
33	7, 32	syl6eqss 3468	. . . . 5 |- ( ph -> S C_ Rng )
34		xpss12 4945	. . . . 5 |- ( ( S C_ Rng /\ S C_ Rng ) -> ( S X. S ) C_ (Rng X. Rng ) )
35	33, 33, 34	syl2anc 673	. . . 4 |- ( ph -> ( S X. S ) C_ (Rng X. Rng ) )
36		rnghmfn 40398	. . . . 5 |- RngHomo Fn (Rng X. Rng )
37		fnssresb 5698	. . . . 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 240	. . 3 |- ( ph -> ( RngHomo |` ( S X. S ) ) Fn ( S X. S ) )
40		rhmsscrnghm.u	. . . . 5 |- ( ph -> U e. V )
41		incom 3616	. . . . . 6 |- (Rng i^i U ) = ( U i^i Rng )
42		inex1g 4539	. . . . . 6 |- ( U e. V -> ( U i^i Rng ) e. _V )
43	41, 42	syl5eqel 2553	. . . . 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 2549	. . 3 |- ( ph -> S e. _V )
46	31, 39, 45	isssc 15803	. 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 936	1 |- ( ph -> ( RingHom |` ( R X. R ) ) C_cat ( RngHomo |` ( S X. S ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   _Vcvv 3031   i^i cin 3389   C_ wss 3390   class class class wbr 4395   X. cxp 4837   |` cres 4841   Fn wfn 5584  (class class class)co 6308   C__cat cssc 15790   Ringcrg 17858   RingHom crh 18018  Rngcrng 40382   RngHomo crngh 40393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-0g 15418  df-ssc 15793  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-grp 16751  df-minusg 16752  df-ghm 16959  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-rnghom 18021  df-mgmhm 40287  df-rng0 40383  df-rnghomo 40395

This theorem is referenced by:  rhmsubcrngc  40539  rhmsubc  40600  rhmsubcALTV  40619