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Theorem crngorngo 36462
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 36458 . 2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
21simplbi 499 1 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  RingOpscrngo 36356  Com2ccm2 36451  CRingOpsccring 36455
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 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3448  df-in 3918  df-crngo 36456
This theorem is referenced by:  crngm23  36464  crngm4  36465  crngohomfo  36468  isidlc  36477  dmnrngo  36519  prnc  36529  isfldidl  36530  isfldidl2  36531  ispridlc  36532  pridlc3  36535  isdmn3  36536
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