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Theorem iscrng 15671
Description: A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypothesis
Ref Expression
rngmgp.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 5728 . . . 4  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
2 rngmgp.g . . . 4  |-  G  =  (mulGrp `  R )
31, 2syl6eqr 2486 . . 3  |-  ( r  =  R  ->  (mulGrp `  r )  =  G )
43eleq1d 2502 . 2  |-  ( r  =  R  ->  (
(mulGrp `  r )  e. CMnd  <-> 
G  e. CMnd ) )
5 df-cring 15664 . 2  |-  CRing  =  {
r  e.  Ring  |  (mulGrp `  r )  e. CMnd }
64, 5elrab2 3094 1  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  G  e. CMnd ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  CMndccmn 15412  mulGrpcmgp 15648   Ringcrg 15660   CRingccrg 15661
This theorem is referenced by:  crngmgp  15672  crngrng  15674  iscrng2  15679  crngpropd  15696  iscrngd  15699  prdscrngd  15719  subrgcrng  15872  psrcrng  16476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-cring 15664
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