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Theorem iscrng 17523
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 5848 . . . 4  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
2 ringmgp.g . . . 4  |-  G  =  (mulGrp `  R )
31, 2syl6eqr 2461 . . 3  |-  ( r  =  R  ->  (mulGrp `  r )  =  G )
43eleq1d 2471 . 2  |-  ( r  =  R  ->  (
(mulGrp `  r )  e. CMnd  <-> 
G  e. CMnd ) )
5 df-cring 17519 . 2  |-  CRing  =  {
r  e.  Ring  |  (mulGrp `  r )  e. CMnd }
64, 5elrab2 3208 1  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  G  e. CMnd ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   ` cfv 5568  CMndccmn 17120  mulGrpcmgp 17459   Ringcrg 17516   CRingccrg 17517
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-or 368  df-an 369  df-3an 976  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-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-cring 17519
This theorem is referenced by:  crngmgp  17524  crngring  17527  iscrng2  17532  crngpropd  17549  iscrngd  17552  prdscrngd  17580  subrgcrng  17751  psrcrng  18386
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