Theorem crngmgp 13193

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 13192	. 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 1353   e. wcel 2148   ` cfv 5218  CMndccmn 13094  mulGrpcmgp 13136   Ringcrg 13185   CRingccrg 13186

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 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-cring 13188

This theorem is referenced by:  crngcom  13203  unitabl  13292  subrgcrng  13352