Theorem crngmgp 14122

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

Step	Hyp	Ref	Expression
1		ringmgp.g	. . 3 |- G = (mulGrp ` R )
2	1	iscrng 14121	. 2 |- ( R e. CRing <-> ( R e. Ring /\ G e. CMnd ) )
3	2	simprbi 275	1 |- ( R e. CRing -> G e. CMnd )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   ` cfv 5343  CMndccmn 13975  mulGrpcmgp 14038   Ringcrg 14114   CRingccrg 14115

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-rab 2529  df-v 2814  df-un 3214  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-br 4103  df-iota 5303  df-fv 5351  df-cring 14117

This theorem is referenced by:  crngcom  14132  unitabl  14236  subrgcrng  14344  lgseisenlem3  15915  lgseisenlem4  15916