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Theorem iscrng 16650
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 5689 . . . 4  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
2 rngmgp.g . . . 4  |-  G  =  (mulGrp `  R )
31, 2syl6eqr 2491 . . 3  |-  ( r  =  R  ->  (mulGrp `  r )  =  G )
43eleq1d 2507 . 2  |-  ( r  =  R  ->  (
(mulGrp `  r )  e. CMnd  <-> 
G  e. CMnd ) )
5 df-cring 16646 . 2  |-  CRing  =  {
r  e.  Ring  |  (mulGrp `  r )  e. CMnd }
64, 5elrab2 3117 1  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  G  e. CMnd ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5416  CMndccmn 16275  mulGrpcmgp 16589   Ringcrg 16643   CRingccrg 16644
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 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-iota 5379  df-fv 5424  df-cring 16646
This theorem is referenced by:  crngmgp  16651  crngrng  16653  iscrng2  16658  crngpropd  16675  iscrngd  16678  prdscrngd  16703  subrgcrng  16867  psrcrng  17483
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