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Theorem rngogrphom 26278
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 2387 . . 3  |-  ran  G  =  ran  G
3 rnggrphom.2 . . 3  |-  J  =  ( 1st `  S
)
4 eqid 2387 . . 3  |-  ran  J  =  ran  J
51, 2, 3, 4rngohomf 26273 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  G --> ran  J )
61, 2, 3rngohomadd 26276 . . . 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 2392 . . 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 2741 . 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 21826 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
103rngogrpo 21826 . . . 4  |-  ( S  e.  RingOps  ->  J  e.  GrpOp )
112, 4elghom 21799 . . . 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 1649    e. wcel 1717   A.wral 2649   ran crn 4819   -->wf 5390   ` cfv 5394  (class class class)co 6020   1stc1st 6286   GrpOpcgr 21622   GrpOpHom cghom 21793   RingOpscrngo 21811    RngHom crnghom 26267
This theorem is referenced by:  rngohom0  26279  rngohomsub  26280  rngokerinj  26282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-map 6956  df-ablo 21718  df-ghom 21794  df-rngo 21812  df-rngohom 26270
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