Theorem isrhm 14438

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 14434	. . 3 |- RingHom = ( r e. Ring , s e. Ring |-> ( ( r GrpHom s ) i^i ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) ) )
2	1	elmpocl 6274	. 2 |- ( F e. ( R RingHom S ) -> ( R e. Ring /\ S e. Ring ) )
3		ringgrp 14279	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
4		ringgrp 14279	. . . . . . 7 |- ( S e. Ring -> S e. Grp )
5		ghmex 14035	. . . . . . 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 4264	. . . . . 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 6084	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( r GrpHom s ) = ( R GrpHom S ) )
10		fveq2 5690	. . . . . . . 8 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
11		fveq2 5690	. . . . . . . 8 |- ( s = S -> (mulGrp ` s ) = (mulGrp ` S ) )
12	10, 11	oveqan12d 6094	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) = ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
13	9, 12	ineq12d 3433	. . . . . 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 6208	. . . . 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 1379	. . . 4 |- ( ( R e. Ring /\ S e. Ring ) -> ( R RingHom S ) = ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) )
16	15	eleq2d 2308	. . 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 3412	. . . 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 6087	. . . . . . 7 |- ( M MndHom N ) = ( (mulGrp ` R ) MndHom (mulGrp ` S ) )
21	20	eqcomi 2242	. . . . . 6 |- ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) = ( M MndHom N )
22	21	eleq2i 2305	. . . . 5 |- ( F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) <-> F e. ( M MndHom N ) )
23	22	anbi2i 461	. . . 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 621	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 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   ` cfv 5372  (class class class)co 6075   MndHom cmhm 13741   Grpcgrp 13782   GrpHom cghm 14020  mulGrpcmgp 14194   Ringcrg 14274   RingHom crh 14430

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743  df-grp 13785  df-ghm 14021  df-mgp 14195  df-ur 14238  df-ring 14276  df-rhm 14432

This theorem is referenced by:  rhmmhm  14439  rhmghm  14442  isrhm2d  14445  rhmf1o  14448  rhmco  14454  resrhm  14529  resrhm2b  14530  rhmpropd  14535