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Theorem crngorngo 32969
 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 32965 . 2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
21simplbi 475 1 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  RingOpscrngo 32863  Com2ccm2 32958  CRingOpsccring 32962 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-crngo 32963 This theorem is referenced by:  crngm23  32971  crngm4  32972  crngohomfo  32975  isidlc  32984  dmnrngo  33026  prnc  33036  isfldidl  33037  isfldidl2  33038  ispridlc  33039  pridlc3  33042  isdmn3  33043
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