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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 mulGrp
Assertion
Ref Expression
iscrng CMnd

Proof of Theorem iscrng
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . . . 4 mulGrp mulGrp
2 rngmgp.g . . . 4 mulGrp
31, 2syl6eqr 2486 . . 3 mulGrp
43eleq1d 2502 . 2 mulGrp CMnd CMnd
5 df-cring 15664 . 2 mulGrp CMnd
64, 5elrab2 3094 1 CMnd
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  cfv 5454  CMndccmn 15412  mulGrpcmgp 15648  crg 15660  ccrg 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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