Theorem isrhm 13657

Description: A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Hypotheses

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

Assertion

Ref	Expression
isrhm	|- ( F e. ( R RingHom S ) <-> ( ( R e. Ring /\ S e. Ring ) /\ ( F e. ( R GrpHom S ) /\ F e. ( M MndHom N ) ) ) )

Proof of Theorem isrhm

Dummy variables r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrhm2 13653	. . 3 |- RingHom = ( r e. Ring , s e. Ring |-> ( ( r GrpHom s ) i^i ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) ) )
2	1	elmpocl 6115	. 2 |- ( F e. ( R RingHom S ) -> ( R e. Ring /\ S e. Ring ) )
3		ringgrp 13500	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
4		ringgrp 13500	. . . . . . 7 |- ( S e. Ring -> S e. Grp )
5		ghmex 13328	. . . . . . 7 |- ( ( R e. Grp /\ S e. Grp ) -> ( R GrpHom S ) e. _V )
6	3, 4, 5	syl2an 289	. . . . . 6 |- ( ( R e. Ring /\ S e. Ring ) -> ( R GrpHom S ) e. _V )
7		inex1g 4166	. . . . . 6 |- ( ( R GrpHom S ) e. _V -> ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) e. _V )
8	6, 7	syl 14	. . . . 5 |- ( ( R e. Ring /\ S e. Ring ) -> ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) e. _V )
9		oveq12 5928	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( r GrpHom s ) = ( R GrpHom S ) )
10		fveq2 5555	. . . . . . . 8 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
11		fveq2 5555	. . . . . . . 8 |- ( s = S -> (mulGrp ` s ) = (mulGrp ` S ) )
12	10, 11	oveqan12d 5938	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) = ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
13	9, 12	ineq12d 3362	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( ( r GrpHom s ) i^i ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) ) = ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) )
14	13, 1	ovmpoga 6049	. . . . 5 |- ( ( R e. Ring /\ S e. Ring /\ ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) e. _V ) -> ( R RingHom S ) = ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) )
15	8, 14	mpd3an3 1349	. . . 4 |- ( ( R e. Ring /\ S e. Ring ) -> ( R RingHom S ) = ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) )
16	15	eleq2d 2263	. . 3 |- ( ( R e. Ring /\ S e. Ring ) -> ( F e. ( R RingHom S ) <-> F e. ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) ) )
17		elin 3343	. . . 4 |- ( F e. ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) <-> ( F e. ( R GrpHom S ) /\ F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) )
18		isrhm.m	. . . . . . . 8 |- M = (mulGrp ` R )
19		isrhm.n	. . . . . . . 8 |- N = (mulGrp ` S )
20	18, 19	oveq12i 5931	. . . . . . 7 |- ( M MndHom N ) = ( (mulGrp ` R ) MndHom (mulGrp ` S ) )
21	20	eqcomi 2197	. . . . . 6 |- ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) = ( M MndHom N )
22	21	eleq2i 2260	. . . . 5 |- ( F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) <-> F e. ( M MndHom N ) )
23	22	anbi2i 457	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) <-> ( F e. ( R GrpHom S ) /\ F e. ( M MndHom N ) ) )
24	17, 23	bitri 184	. . 3 |- ( F e. ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) <-> ( F e. ( R GrpHom S ) /\ F e. ( M MndHom N ) ) )
25	16, 24	bitrdi 196	. 2 |- ( ( R e. Ring /\ S e. Ring ) -> ( F e. ( R RingHom S ) <-> ( F e. ( R GrpHom S ) /\ F e. ( M MndHom N ) ) ) )
26	2, 25	biadanii 613	1 |- ( F e. ( R RingHom S ) <-> ( ( R e. Ring /\ S e. Ring ) /\ ( F e. ( R GrpHom S ) /\ F e. ( M MndHom N ) ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3153   ` cfv 5255  (class class class)co 5919   MndHom cmhm 13032   Grpcgrp 13075   GrpHom cghm 13313  mulGrpcmgp 13419   Ringcrg 13495   RingHom crh 13649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mhm 13034  df-grp 13078  df-ghm 13314  df-mgp 13420  df-ur 13459  df-ring 13497  df-rhm 13651

This theorem is referenced by:  rhmmhm  13658  rhmghm  13661  isrhm2d  13664  rhmf1o  13667  rhmco  13673  resrhm  13747  resrhm2b  13748  rhmpropd  13753