Theorem crngmgp 19706

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 ∈ CRing → 𝐺 ∈ CMnd)

Proof of Theorem crngmgp

Step	Hyp	Ref	Expression
1		ringmgp.g	. . 3 ⊢ 𝐺 = (mulGrp‘𝑅)
2	1	iscrng 19705	. 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
3	2	simprbi 496	1 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  CMndccmn 19301  mulGrpcmgp 19635  Ringcrg 19698  CRingccrg 19699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-cring 19701

This theorem is referenced by:  crngcom  19716  gsummgp0  19762  prdscrngd  19767  crngbinom  19775  unitabl  19825  subrgcrng  19943  sraassa  20984  mplbas2  21153  evlslem3  21200  evlslem6  21201  evlslem1  21202  evlsgsummul  21212  evls1gsummul  21401  evl1gsummul  21436  mamuvs2  21463  matgsumcl  21517  madetsmelbas  21521  madetsmelbas2  21522  mdetleib2  21645  mdetf  21652  mdetdiaglem  21655  mdetdiag  21656  mdetdiagid  21657  mdetrlin  21659  mdetrsca  21660  mdetralt  21665  mdetuni0  21678  smadiadetlem4  21726  chpscmat  21899  chp0mat  21903  chpidmat  21904  amgmlem  26044  amgm  26045  wilthlem2  26123  wilthlem3  26124  lgseisenlem3  26430  lgseisenlem4  26431  frobrhm  31387  cringm4  31524  mdetpmtr1  31675  pwsgprod  40194  evlsbagval  40198  mhphf  40208  mgpsumunsn  45585  mgpsumz  45586  mgpsumn  45587  amgmwlem  46392  amgmlemALT  46393