Theorem isrhm 14035

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 14031	. . 3 |- RingHom = ( r e. Ring , s e. Ring |-> ( ( r GrpHom s ) i^i ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) ) )
2	1	elmpocl 6164	. 2 |- ( F e. ( R RingHom S ) -> ( R e. Ring /\ S e. Ring ) )
3		ringgrp 13878	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
4		ringgrp 13878	. . . . . . 7 |- ( S e. Ring -> S e. Grp )
5		ghmex 13706	. . . . . . 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 4196	. . . . . 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 5976	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( r GrpHom s ) = ( R GrpHom S ) )
10		fveq2 5599	. . . . . . . 8 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
11		fveq2 5599	. . . . . . . 8 |- ( s = S -> (mulGrp ` s ) = (mulGrp ` S ) )
12	10, 11	oveqan12d 5986	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) = ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
13	9, 12	ineq12d 3383	. . . . . 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 6098	. . . . 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 1351	. . . 4 |- ( ( R e. Ring /\ S e. Ring ) -> ( R RingHom S ) = ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) )
16	15	eleq2d 2277	. . 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 3364	. . . 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 5979	. . . . . . 7 |- ( M MndHom N ) = ( (mulGrp ` R ) MndHom (mulGrp ` S ) )
21	20	eqcomi 2211	. . . . . 6 |- ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) = ( M MndHom N )
22	21	eleq2i 2274	. . . . 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 1373   e. wcel 2178   _Vcvv 2776   i^i cin 3173   ` cfv 5290  (class class class)co 5967   MndHom cmhm 13404   Grpcgrp 13447   GrpHom cghm 13691  mulGrpcmgp 13797   Ringcrg 13873   RingHom crh 14027

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-mhm 13406  df-grp 13450  df-ghm 13692  df-mgp 13798  df-ur 13837  df-ring 13875  df-rhm 14029

This theorem is referenced by:  rhmmhm  14036  rhmghm  14039  isrhm2d  14042  rhmf1o  14045  rhmco  14051  resrhm  14125  resrhm2b  14126  rhmpropd  14131