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Theorem crngmgp 18476
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
StepHypRef Expression
1 ringmgp.g . . 3 𝐺 = (mulGrp‘𝑅)
21iscrng 18475 . 2 (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
32simprbi 480 1 (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cfv 5847  CMndccmn 18114  mulGrpcmgp 18410  Ringcrg 18468  CRingccrg 18469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-cring 18471
This theorem is referenced by:  crngcom  18483  gsummgp0  18529  prdscrngd  18534  crngbinom  18542  unitabl  18589  subrgcrng  18705  sraassa  19244  mplbas2  19389  evlslem6  19432  evlslem3  19433  evlslem1  19434  evls1gsummul  19609  evl1gsummul  19643  mamuvs2  20131  matgsumcl  20185  madetsmelbas  20189  madetsmelbas2  20190  mdetleib2  20313  mdetf  20320  mdetdiaglem  20323  mdetdiag  20324  mdetdiagid  20325  mdetrlin  20327  mdetrsca  20328  mdetralt  20333  mdetuni0  20346  smadiadetlem4  20394  chpscmat  20566  chp0mat  20570  chpidmat  20571  amgmlem  24616  amgm  24617  wilthlem2  24695  wilthlem3  24696  lgseisenlem3  25002  lgseisenlem4  25003  mdetpmtr1  29668  mgpsumunsn  41425  mgpsumz  41426  mgpsumn  41427  amgmwlem  41848  amgmlemALT  41849
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