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Theorem crngorngo 34756
 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 34752 . 2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
21simplbi 498 1 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2079  RingOpscrngo 34650  Com2ccm2 34745  CRingOpsccring 34749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-v 3434  df-in 3861  df-crngo 34750 This theorem is referenced by:  crngm23  34758  crngm4  34759  crngohomfo  34762  isidlc  34771  dmnrngo  34813  prnc  34823  isfldidl  34824  isfldidl2  34825  ispridlc  34826  pridlc3  34829  isdmn3  34830
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