Theorem crngmgp 14291

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 14290	. 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 1402   e. wcel 2209   ` cfv 5375  CMndccmn 14070  mulGrpcmgp 14200   Ringcrg 14283   CRingccrg 14284

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383  df-cring 14286

This theorem is referenced by:  crngcom  14301  unitabl  14407  subrgcrng  14516  lgseisenlem3  16174  lgseisenlem4  16175