Theorem crngmgp 19524

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 19523	. 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
3	2	simprbi 500	1 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112  ‘cfv 6358  CMndccmn 19124  mulGrpcmgp 19458  Ringcrg 19516  CRingccrg 19517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-cring 19519

This theorem is referenced by:  crngcom  19534  gsummgp0  19580  prdscrngd  19585  crngbinom  19593  unitabl  19640  subrgcrng  19758  sraassa  20783  mplbas2  20953  evlslem3  20994  evlslem6  20995  evlslem1  20996  evlsgsummul  21006  evls1gsummul  21195  evl1gsummul  21230  mamuvs2  21257  matgsumcl  21311  madetsmelbas  21315  madetsmelbas2  21316  mdetleib2  21439  mdetf  21446  mdetdiaglem  21449  mdetdiag  21450  mdetdiagid  21451  mdetrlin  21453  mdetrsca  21454  mdetralt  21459  mdetuni0  21472  smadiadetlem4  21520  chpscmat  21693  chp0mat  21697  chpidmat  21698  amgmlem  25826  amgm  25827  wilthlem2  25905  wilthlem3  25906  lgseisenlem3  26212  lgseisenlem4  26213  frobrhm  31158  cringm4  31290  mdetpmtr1  31441  pwsgprod  39922  evlsbagval  39926  mhphf  39936  mgpsumunsn  45313  mgpsumz  45314  mgpsumn  45315  amgmwlem  46120  amgmlemALT  46121