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Theorem crngmgp 15349
Description: A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypothesis
Ref Expression
rngmgp.g  |-  G  =  (mulGrp `  R )
Assertion
Ref Expression
crngmgp  |-  ( R  e.  CRing  ->  G  e. CMnd )

Proof of Theorem crngmgp
StepHypRef Expression
1 rngmgp.g . . 3  |-  G  =  (mulGrp `  R )
21iscrng 15348 . 2  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  G  e. CMnd ) )
32simprbi 450 1  |-  ( R  e.  CRing  ->  G  e. CMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  CMndccmn 15089  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338
This theorem is referenced by:  crngcom  15355  prdscrngd  15396  unitabl  15450  subrgcrng  15549  sraassa  16065  mplcoe2  16211  mplbas2  16212  ply1coe  16368  evlslem6  19397  evlslem3  19398  evlslem1  19399  amgmlem  20284  amgm  20285  wilthlem2  20307  wilthlem3  20308  lgseisenlem3  20590  lgseisenlem4  20591  mamuvs2  27464
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-cring 15341
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