Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  crngorngo Structured version   Unicode version

Theorem crngorngo 28823
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 28820 . 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 1756   RingOpscrngo 23881   Com2ccm2 23916  CRingOpsccring 28818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2993  df-in 3354  df-crngo 28819
This theorem is referenced by:  crngm23  28825  crngm4  28826  crngohomfo  28829  isidlc  28838  dmnrngo  28880  prnc  28890  isfldidl  28891  isfldidl2  28892  ispridlc  28893  pridlc3  28896  isdmn3  28897
  Copyright terms: Public domain W3C validator