Theorem isrnghmmul 41893

Description: A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020.)

Hypotheses

Ref	Expression
isrnghmmul.m	|- M = (mulGrp ` R )
isrnghmmul.n	|- N = (mulGrp ` S )

Assertion

Ref	Expression
isrnghmmul	|- ( F e. ( R RngHomo S ) <-> ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ F e. ( M MgmHom N ) ) ) )

Proof of Theorem isrnghmmul

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
2		eqid 2622	. . 3 |- ( .r ` R ) = ( .r ` R )
3		eqid 2622	. . 3 |- ( .r ` S ) = ( .r ` S )
4	1, 2, 3	isrnghm 41892	. 2 |- ( F e. ( R RngHomo S ) <-> ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) ) ) )
5		isrnghmmul.m	. . . . . . . . . . 11 |- M = (mulGrp ` R )
6	5	rngmgp 41878	. . . . . . . . . 10 |- ( R e. Rng -> M e. SGrp )
7		sgrpmgm 17289	. . . . . . . . . 10 |- ( M e. SGrp -> M e. Mgm )
8	6, 7	syl 17	. . . . . . . . 9 |- ( R e. Rng -> M e. Mgm )
9		isrnghmmul.n	. . . . . . . . . . 11 |- N = (mulGrp ` S )
10	9	rngmgp 41878	. . . . . . . . . 10 |- ( S e. Rng -> N e. SGrp )
11		sgrpmgm 17289	. . . . . . . . . 10 |- ( N e. SGrp -> N e. Mgm )
12	10, 11	syl 17	. . . . . . . . 9 |- ( S e. Rng -> N e. Mgm )
13	8, 12	anim12i 590	. . . . . . . 8 |- ( ( R e. Rng /\ S e. Rng ) -> ( M e. Mgm /\ N e. Mgm ) )
14		eqid 2622	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
15	1, 14	ghmf 17664	. . . . . . . 8 |- ( F e. ( R GrpHom S ) -> F : ( Base ` R ) --> ( Base ` S ) )
16	13, 15	anim12i 590	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. Rng ) /\ F e. ( R GrpHom S ) ) -> ( ( M e. Mgm /\ N e. Mgm ) /\ F : ( Base ` R ) --> ( Base ` S ) ) )
17	16	biantrurd 529	. . . . . 6 |- ( ( ( R e. Rng /\ S e. Rng ) /\ F e. ( R GrpHom S ) ) -> ( A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) <-> ( ( ( M e. Mgm /\ N e. Mgm ) /\ F : ( Base ` R ) --> ( Base ` S ) ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) ) ) )
18		anass 681	. . . . . 6 |- ( ( ( ( M e. Mgm /\ N e. Mgm ) /\ F : ( Base ` R ) --> ( Base ` S ) ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) ) <-> ( ( M e. Mgm /\ N e. Mgm ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) ) ) )
19	17, 18	syl6bb 276	. . . . 5 |- ( ( ( R e. Rng /\ S e. Rng ) /\ F e. ( R GrpHom S ) ) -> ( A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) <-> ( ( M e. Mgm /\ N e. Mgm ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) ) ) ) )
20	5, 1	mgpbas 18495	. . . . . 6 |- ( Base ` R ) = ( Base ` M )
21	9, 14	mgpbas 18495	. . . . . 6 |- ( Base ` S ) = ( Base ` N )
22	5, 2	mgpplusg 18493	. . . . . 6 |- ( .r ` R ) = ( +g ` M )
23	9, 3	mgpplusg 18493	. . . . . 6 |- ( .r ` S ) = ( +g ` N )
24	20, 21, 22, 23	ismgmhm 41783	. . . . 5 |- ( F e. ( M MgmHom N ) <-> ( ( M e. Mgm /\ N e. Mgm ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) ) ) )
25	19, 24	syl6bbr 278	. . . 4 |- ( ( ( R e. Rng /\ S e. Rng ) /\ F e. ( R GrpHom S ) ) -> ( A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) <-> F e. ( M MgmHom N ) ) )
26	25	pm5.32da 673	. . 3 |- ( ( R e. Rng /\ S e. Rng ) -> ( ( F e. ( R GrpHom S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) ) <-> ( F e. ( R GrpHom S ) /\ F e. ( M MgmHom N ) ) ) )
27	26	pm5.32i 669	. 2 |- ( ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) ) ) <-> ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ F e. ( M MgmHom N ) ) ) )
28	4, 27	bitri 264	1 |- ( F e. ( R RngHomo S ) <-> ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ F e. ( M MgmHom N ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942  Mgmcmgm 17240  SGrpcsgrp 17283   GrpHom cghm 17657  mulGrpcmgp 18489   MgmHom cmgmhm 41777  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-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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  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-sgrp 17284  df-ghm 17658  df-abl 18196  df-mgp 18490  df-mgmhm 41779  df-rng0 41875  df-rnghomo 41887

This theorem is referenced by:  rnghmmgmhm  41894  rnghmval2  41895  rnghmf1o  41903  rnghmco  41907  idrnghm  41908  c0rnghm  41913  rhmisrnghm  41920