Theorem crngmgp 19791

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 19790	. 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
3	2	simprbi 497	1 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  CMndccmn 19386  mulGrpcmgp 19720  Ringcrg 19783  CRingccrg 19784

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-cring 19786

This theorem is referenced by:  crngcom  19801  gsummgp0  19847  prdscrngd  19852  crngbinom  19860  unitabl  19910  subrgcrng  20028  sraassa  21074  mplbas2  21243  evlslem3  21290  evlslem6  21291  evlslem1  21292  evlsgsummul  21302  evls1gsummul  21491  evl1gsummul  21526  mamuvs2  21553  matgsumcl  21609  madetsmelbas  21613  madetsmelbas2  21614  mdetleib2  21737  mdetf  21744  mdetdiaglem  21747  mdetdiag  21748  mdetdiagid  21749  mdetrlin  21751  mdetrsca  21752  mdetralt  21757  mdetuni0  21770  smadiadetlem4  21818  chpscmat  21991  chp0mat  21995  chpidmat  21996  amgmlem  26139  amgm  26140  wilthlem2  26218  wilthlem3  26219  lgseisenlem3  26525  lgseisenlem4  26526  frobrhm  31485  cringm4  31622  mdetpmtr1  31773  pwsgprod  40269  evlsbagval  40275  mhphf  40285  mgpsumunsn  45697  mgpsumz  45698  mgpsumn  45699  amgmwlem  46506  amgmlemALT  46507