Theorem crngmgp 14252

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 14251	. 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 2205   ` cfv 5358  CMndccmn 14042  mulGrpcmgp 14164   Ringcrg 14244   CRingccrg 14245

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-iota 5318  df-fv 5366  df-cring 14247

This theorem is referenced by:  crngcom  14262  unitabl  14367  subrgcrng  14476  lgseisenlem3  16076  lgseisenlem4  16077