Theorem rngogrphom 33770

Description: A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)

Hypotheses

Ref	Expression
rnggrphom.1	|- G = ( 1st ` R )
rnggrphom.2	|- J = ( 1st ` S )

Assertion

Ref	Expression
rngogrphom	|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> F e. ( G GrpOpHom J ) )

Proof of Theorem rngogrphom

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

Step	Hyp	Ref	Expression
1		rnggrphom.1	. . 3 |- G = ( 1st ` R )
2		eqid 2622	. . 3 |- ran G = ran G
3		rnggrphom.2	. . 3 |- J = ( 1st ` S )
4		eqid 2622	. . 3 |- ran J = ran J
5	1, 2, 3, 4	rngohomf 33765	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> F : ran G --> ran J )
6	1, 2, 3	rngohomadd 33768	. . . 4 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran G /\ y e. ran G ) ) -> ( F ` ( x G y ) ) = ( ( F ` x ) J ( F ` y ) ) )
7	6	eqcomd 2628	. . 3 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran G /\ y e. ran G ) ) -> ( ( F ` x ) J ( F ` y ) ) = ( F ` ( x G y ) ) )
8	7	ralrimivva 2971	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> A. x e. ran G A. y e. ran G ( ( F ` x ) J ( F ` y ) ) = ( F ` ( x G y ) ) )
9	1	rngogrpo 33709	. . . 4 |- ( R e. RingOps -> G e. GrpOp )
10	3	rngogrpo 33709	. . . 4 |- ( S e. RingOps -> J e. GrpOp )
11	2, 4	elghomOLD 33686	. . . 4 |- ( ( G e. GrpOp /\ J e. GrpOp ) -> ( F e. ( G GrpOpHom J ) <-> ( F : ran G --> ran J /\ A. x e. ran G A. y e. ran G ( ( F ` x ) J ( F ` y ) ) = ( F ` ( x G y ) ) ) ) )
12	9, 10, 11	syl2an 494	. . 3 |- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( G GrpOpHom J ) <-> ( F : ran G --> ran J /\ A. x e. ran G A. y e. ran G ( ( F ` x ) J ( F ` y ) ) = ( F ` ( x G y ) ) ) ) )
13	12	3adant3 1081	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( F e. ( G GrpOpHom J ) <-> ( F : ran G --> ran J /\ A. x e. ran G A. y e. ran G ( ( F ` x ) J ( F ` y ) ) = ( F ` ( x G y ) ) ) ) )
14	5, 8, 13	mpbir2and 957	1 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> F e. ( G GrpOpHom J ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   GrpOpcgr 27343   GrpOpHom cghomOLD 33682   RingOpscrngo 33693   RngHom crnghom 33759

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ablo 27399  df-ghomOLD 33683  df-rngo 33694  df-rngohom 33762

This theorem is referenced by:  rngohom0  33771  rngohomsub  33772  rngokerinj  33774