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Theorem crngorngo 29987
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 29984 . 2  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
21simplbi 460 1  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1762   RingOpscrngo 25039   Com2ccm2 25074  CRingOpsccring 29982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-in 3476  df-crngo 29983
This theorem is referenced by:  crngm23  29989  crngm4  29990  crngohomfo  29993  isidlc  30002  dmnrngo  30044  prnc  30054  isfldidl  30055  isfldidl2  30056  ispridlc  30057  pridlc3  30060  isdmn3  30061
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