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Theorem crngorngo 34104
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 34100 . 2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
21simplbi 487 1 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2155  RingOpscrngo 33998  Com2ccm2 34093  CRingOpsccring 34097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-v 3389  df-in 3770  df-crngo 34098
This theorem is referenced by:  crngm23  34106  crngm4  34107  crngohomfo  34110  isidlc  34119  dmnrngo  34161  prnc  34171  isfldidl  34172  isfldidl2  34173  ispridlc  34174  pridlc3  34177  isdmn3  34178
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