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Theorem crngmgp 15365
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 15364 . 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 1632    e. wcel 1696   ` cfv 5271  CMndccmn 15105  mulGrpcmgp 15341   Ringcrg 15353   CRingccrg 15354
This theorem is referenced by:  crngcom  15371  prdscrngd  15412  unitabl  15466  subrgcrng  15565  sraassa  16081  mplcoe2  16227  mplbas2  16228  ply1coe  16384  evlslem6  19413  evlslem3  19414  evlslem1  19415  amgmlem  20300  amgm  20301  wilthlem2  20323  wilthlem3  20324  lgseisenlem3  20606  lgseisenlem4  20607  mamuvs2  27567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-cring 15357
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