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Theorem crngmgp 13118
Description: A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypothesis
Ref Expression
ringmgp.g  |-  G  =  (mulGrp `  R )
Assertion
Ref Expression
crngmgp  |-  ( R  e.  CRing  ->  G  e. CMnd )

Proof of Theorem crngmgp
StepHypRef Expression
1 ringmgp.g . . 3  |-  G  =  (mulGrp `  R )
21iscrng 13117 . 2  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  G  e. CMnd ) )
32simprbi 275 1  |-  ( R  e.  CRing  ->  G  e. CMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   ` cfv 5215  CMndccmn 13019  mulGrpcmgp 13061   Ringcrg 13110   CRingccrg 13111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-cring 13113
This theorem is referenced by:  crngcom  13128  unitabl  13217  subrgcrng  13284
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