Theorem isrhm 13714

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 13710	. . 3 |- RingHom = ( r e. Ring , s e. Ring |-> ( ( r GrpHom s ) i^i ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) ) )
2	1	elmpocl 6118	. 2 |- ( F e. ( R RingHom S ) -> ( R e. Ring /\ S e. Ring ) )
3		ringgrp 13557	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
4		ringgrp 13557	. . . . . . 7 |- ( S e. Ring -> S e. Grp )
5		ghmex 13385	. . . . . . 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 4169	. . . . . 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 5931	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( r GrpHom s ) = ( R GrpHom S ) )
10		fveq2 5558	. . . . . . . 8 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
11		fveq2 5558	. . . . . . . 8 |- ( s = S -> (mulGrp ` s ) = (mulGrp ` S ) )
12	10, 11	oveqan12d 5941	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) = ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
13	9, 12	ineq12d 3365	. . . . . 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 6052	. . . . 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 2266	. . 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 3346	. . . 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 5934	. . . . . . 7 |- ( M MndHom N ) = ( (mulGrp ` R ) MndHom (mulGrp ` S ) )
21	20	eqcomi 2200	. . . . . 6 |- ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) = ( M MndHom N )
22	21	eleq2i 2263	. . . . 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 2167   _Vcvv 2763   i^i cin 3156   ` cfv 5258  (class class class)co 5922   MndHom cmhm 13089   Grpcgrp 13132   GrpHom cghm 13370  mulGrpcmgp 13476   Ringcrg 13552   RingHom crh 13706

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mhm 13091  df-grp 13135  df-ghm 13371  df-mgp 13477  df-ur 13516  df-ring 13554  df-rhm 13708

This theorem is referenced by:  rhmmhm  13715  rhmghm  13718  isrhm2d  13721  rhmf1o  13724  rhmco  13730  resrhm  13804  resrhm2b  13805  rhmpropd  13810