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Theorem crngorngo 31679
Description: A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
crngorngo  |-  ( R  e. CRingOps  ->  R  e.  RingOps )

Proof of Theorem crngorngo
StepHypRef Expression
1 iscrngo 31676 . 2  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
21simplbi 458 1  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   RingOpscrngo 25791   Com2ccm2 25826  CRingOpsccring 31674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3061  df-in 3421  df-crngo 31675
This theorem is referenced by:  crngm23  31681  crngm4  31682  crngohomfo  31685  isidlc  31694  dmnrngo  31736  prnc  31746  isfldidl  31747  isfldidl2  31748  ispridlc  31749  pridlc3  31752  isdmn3  31753
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