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Theorem crngmgp 15600
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 15599 . 2  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  G  e. CMnd ) )
32simprbi 451 1  |-  ( R  e.  CRing  ->  G  e. CMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5395  CMndccmn 15340  mulGrpcmgp 15576   Ringcrg 15588   CRingccrg 15589
This theorem is referenced by:  crngcom  15606  prdscrngd  15647  unitabl  15701  subrgcrng  15800  sraassa  16312  mplcoe2  16458  mplbas2  16459  ply1coe  16612  evlslem6  19802  evlslem3  19803  evlslem1  19804  amgmlem  20696  amgm  20697  wilthlem2  20720  wilthlem3  20721  lgseisenlem3  21003  lgseisenlem4  21004  mamuvs2  27134
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
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-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-cring 15592
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