Theorem crngmgp 20058

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 20057	. 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
3	2	simprbi 498	1 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  ‘cfv 6541  CMndccmn 19643  mulGrpcmgp 19982  Ringcrg 20050  CRingccrg 20051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6493  df-fv 6549  df-cring 20053

This theorem is referenced by:  crngcom  20068  crngbascntr  20072  gsummgp0  20124  prdscrngd  20129  crngbinom  20141  unitabl  20191  subrgcrng  20360  sraassaOLD  21416  mplbas2  21589  evlslem3  21635  evlslem6  21636  evlslem1  21637  evlsgsummul  21647  evls1gsummul  21836  evl1gsummul  21871  mamuvs2  21898  matgsumcl  21954  madetsmelbas  21958  madetsmelbas2  21959  mdetleib2  22082  mdetf  22089  mdetdiaglem  22092  mdetdiag  22093  mdetdiagid  22094  mdetrlin  22096  mdetrsca  22097  mdetralt  22102  mdetuni0  22115  smadiadetlem4  22163  chpscmat  22336  chp0mat  22340  chpidmat  22341  amgmlem  26484  amgm  26485  wilthlem2  26563  wilthlem3  26564  lgseisenlem3  26870  lgseisenlem4  26871  frobrhm  32371  cringm4  32554  mdetpmtr1  32792  pwsgprod  41112  evlsvvvallem  41131  selvvvval  41155  evlselv  41157  mhphf  41167  mgpsumunsn  46991  mgpsumz  46992  mgpsumn  46993  amgmwlem  47803  amgmlemALT  47804