Theorem crngmgp 20064

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 20063	. 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 6544  CMndccmn 19648  mulGrpcmgp 19987  Ringcrg 20056  CRingccrg 20057

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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-cring 20059

This theorem is referenced by:  crngcom  20074  crngbascntr  20078  gsummgp0  20130  prdscrngd  20135  crngbinom  20148  unitabl  20198  subrgcrng  20323  sraassaOLD  21424  mplbas2  21597  evlslem3  21643  evlslem6  21644  evlslem1  21645  evlsgsummul  21655  evls1gsummul  21844  evl1gsummul  21879  mamuvs2  21906  matgsumcl  21962  madetsmelbas  21966  madetsmelbas2  21967  mdetleib2  22090  mdetf  22097  mdetdiaglem  22100  mdetdiag  22101  mdetdiagid  22102  mdetrlin  22104  mdetrsca  22105  mdetralt  22110  mdetuni0  22123  smadiadetlem4  22171  chpscmat  22344  chp0mat  22348  chpidmat  22349  amgmlem  26494  amgm  26495  wilthlem2  26573  wilthlem3  26574  lgseisenlem3  26880  lgseisenlem4  26881  frobrhm  32382  cringm4  32565  mdetpmtr1  32803  pwsgprod  41114  evlsvvvallem  41133  selvvvval  41157  evlselv  41159  mhphf  41169  mgpsumunsn  47037  mgpsumz  47038  mgpsumn  47039  amgmwlem  47849  amgmlemALT  47850