Theorem crngorngo 36857

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 36853	. 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2	1	simplbi 499	1 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2107  RingOpscrngo 36751  Com2ccm2 36846  CRingOpsccring 36850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3955  df-crngo 36851

This theorem is referenced by:  crngm23  36859  crngm4  36860  crngohomfo  36863  isidlc  36872  dmnrngo  36914  prnc  36924  isfldidl  36925  isfldidl2  36926  ispridlc  36927  pridlc3  36930  isdmn3  36931