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Theorem crngmgp 15664
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 15663 . 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 1652    e. wcel 1725   ` cfv 5446  CMndccmn 15404  mulGrpcmgp 15640   Ringcrg 15652   CRingccrg 15653
This theorem is referenced by:  crngcom  15670  prdscrngd  15711  unitabl  15765  subrgcrng  15864  sraassa  16376  mplcoe2  16522  mplbas2  16523  ply1coe  16676  evlslem6  19926  evlslem3  19927  evlslem1  19928  amgmlem  20820  amgm  20821  wilthlem2  20844  wilthlem3  20845  lgseisenlem3  21127  lgseisenlem4  21128  mamuvs2  27432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-cring 15656
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