Theorem crngorngo 36085

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 36081	. 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2	1	simplbi 497	1 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2108  RingOpscrngo 35979  Com2ccm2 36074  CRingOpsccring 36078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-crngo 36079

This theorem is referenced by:  crngm23  36087  crngm4  36088  crngohomfo  36091  isidlc  36100  dmnrngo  36142  prnc  36152  isfldidl  36153  isfldidl2  36154  ispridlc  36155  pridlc3  36158  isdmn3  36159