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Theorem iscrng 15364
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 5541 . . . 4  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
2 rngmgp.g . . . 4  |-  G  =  (mulGrp `  R )
31, 2syl6eqr 2346 . . 3  |-  ( r  =  R  ->  (mulGrp `  r )  =  G )
43eleq1d 2362 . 2  |-  ( r  =  R  ->  (
(mulGrp `  r )  e. CMnd  <-> 
G  e. CMnd ) )
5 df-cring 15357 . 2  |-  CRing  =  {
r  e.  Ring  |  (mulGrp `  r )  e. CMnd }
64, 5elrab2 2938 1  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  G  e. CMnd ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  CMndccmn 15105  mulGrpcmgp 15341   Ringcrg 15353   CRingccrg 15354
This theorem is referenced by:  crngmgp  15365  crngrng  15367  iscrng2  15372  crngpropd  15389  iscrngd  15392  prdscrngd  15412  subrgcrng  15565  psrcrng  16173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-cring 15357
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