Theorem crngorngo 38007

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 38003	. 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2	1	simplbi 497	1 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2108  RingOpscrngo 37901  Com2ccm2 37996  CRingOpsccring 38000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-crngo 38001

This theorem is referenced by:  crngm23  38009  crngm4  38010  crngohomfo  38013  isidlc  38022  dmnrngo  38064  prnc  38074  isfldidl  38075  isfldidl2  38076  ispridlc  38077  pridlc3  38080  isdmn3  38081