Theorem crngmgp 14308

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 14307	. 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
3	2	simprbi 275	1 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  ‘cfv 5377  CMndccmn 14087  mulGrpcmgp 14217  Ringcrg 14300  CRingccrg 14301

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-cring 14303

This theorem is used by:  crngcom  14318  unitabl  14424  subrgcrng  14533  lgseisenlem3  16191  lgseisenlem4  16192