Theorem crngorngo 38140

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 38136	. 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2	1	simplbi 497	1 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2113  RingOpscrngo 38034  Com2ccm2 38129  CRingOpsccring 38133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-in 3906  df-crngo 38134

This theorem is referenced by:  crngm23  38142  crngm4  38143  crngohomfo  38146  isidlc  38155  dmnrngo  38197  prnc  38207  isfldidl  38208  isfldidl2  38209  ispridlc  38210  pridlc3  38213  isdmn3  38214