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Theorem rngogrphom 26568
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.
StepHypRef Expression
1 rnggrphom.1 . . 3  |-  G  =  ( 1st `  R
)
2 eqid 2435 . . 3  |-  ran  G  =  ran  G
3 rnggrphom.2 . . 3  |-  J  =  ( 1st `  S
)
4 eqid 2435 . . 3  |-  ran  J  =  ran  J
51, 2, 3, 4rngohomf 26563 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  G --> ran  J )
61, 2, 3rngohomadd 26566 . . . 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 ) ) )
76eqcomd 2440 . . 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 ) ) )
87ralrimivva 2790 . 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 ) ) )
91rngogrpo 21970 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
103rngogrpo 21970 . . . 4  |-  ( S  e.  RingOps  ->  J  e.  GrpOp )
112, 4elghom 21943 . . . 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 ) ) ) ) )
129, 10, 11syl2an 464 . . 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 ) ) ) ) )
13123adant3 977 . 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 ) ) ) ) )
145, 8, 13mpbir2and 889 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F  e.  ( G GrpOpHom  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   GrpOpcgr 21766   GrpOpHom cghom 21937   RingOpscrngo 21955    RngHom crnghom 26557
This theorem is referenced by:  rngohom0  26569  rngohomsub  26570  rngokerinj  26572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-ablo 21862  df-ghom 21938  df-rngo 21956  df-rngohom 26560
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