Theorem crngorngo 38382

Description: A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Assertion

Ref	Expression
crngorngo	⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)

Proof of Theorem crngorngo

Step	Hyp	Ref	Expression
1		iscrngo 38378	. 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2	1	simplbi 498	1 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2121  RingOpscrngo 38276  Com2ccm2 38371  CRingOpsccring 38375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-in 3892  df-crngo 38376

This theorem is referenced by:  crngm23  38384  crngm4  38385  crngohomfo  38388  isidlc  38397  dmnrngo  38439  prnc  38449  isfldidl  38450  isfldidl2  38451  ispridlc  38452  pridlc3  38455  isdmn3  38456