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Theorem rngogrphom 26602
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 2283 . . 3  |-  ran  G  =  ran  G
3 rnggrphom.2 . . 3  |-  J  =  ( 1st `  S
)
4 eqid 2283 . . 3  |-  ran  J  =  ran  J
51, 2, 3, 4rngohomf 26597 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  G --> ran  J )
61, 2, 3rngohomadd 26600 . . . 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 2288 . . 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 2635 . 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 21057 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
103rngogrpo 21057 . . . 4  |-  ( S  e.  RingOps  ->  J  e.  GrpOp )
112, 4elghom 21030 . . . 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 463 . . 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 975 . 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 888 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   GrpOpcgr 20853   GrpOpHom cghom 21024   RingOpscrngo 21042    RngHom crnghom 26591
This theorem is referenced by:  rngohom0  26603  rngohomsub  26604  rngokerinj  26606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-ablo 20949  df-ghom 21025  df-rngo 21043  df-rngohom 26594
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