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Theorem iscrng 17787
Description: A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypothesis
Ref Expression
ringmgp.g  |-  G  =  (mulGrp `  R )
Assertion
Ref Expression
iscrng  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  G  e. CMnd ) )

Proof of Theorem iscrng
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5865 . . . 4  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
2 ringmgp.g . . . 4  |-  G  =  (mulGrp `  R )
31, 2syl6eqr 2503 . . 3  |-  ( r  =  R  ->  (mulGrp `  r )  =  G )
43eleq1d 2513 . 2  |-  ( r  =  R  ->  (
(mulGrp `  r )  e. CMnd  <-> 
G  e. CMnd ) )
5 df-cring 17783 . 2  |-  CRing  =  {
r  e.  Ring  |  (mulGrp `  r )  e. CMnd }
64, 5elrab2 3198 1  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  G  e. CMnd ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   ` cfv 5582  CMndccmn 17430  mulGrpcmgp 17723   Ringcrg 17780   CRingccrg 17781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-iota 5546  df-fv 5590  df-cring 17783
This theorem is referenced by:  crngmgp  17788  crngring  17791  iscrng2  17796  crngpropd  17813  iscrngd  17816  prdscrngd  17841  subrgcrng  18012  psrcrng  18637
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