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Theorem crngmgp 13484
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 13483 . 2 (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
32simprbi 275 1 (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2164  cfv 5246  CMndccmn 13343  mulGrpcmgp 13400  Ringcrg 13476  CRingccrg 13477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5207  df-fv 5254  df-cring 13479
This theorem is referenced by:  crngcom  13494  unitabl  13597  subrgcrng  13705  lgseisenlem3  15136
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