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Theorem crngorngo 37172
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
StepHypRef Expression
1 iscrngo 37168 . 2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
21simplbi 497 1 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  RingOpscrngo 37066  Com2ccm2 37161  CRingOpsccring 37165
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-crngo 37166
This theorem is referenced by:  crngm23  37174  crngm4  37175  crngohomfo  37178  isidlc  37187  dmnrngo  37229  prnc  37239  isfldidl  37240  isfldidl2  37241  ispridlc  37242  pridlc3  37245  isdmn3  37246
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