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Theorem ablcmn 16619
Description: An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
Assertion
Ref Expression
ablcmn  |-  ( G  e.  Abel  ->  G  e. CMnd
)

Proof of Theorem ablcmn
StepHypRef Expression
1 isabl 16617 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
21simprbi 464 1  |-  ( G  e.  Abel  ->  G  e. CMnd
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   Grpcgrp 15730  CMndccmn 16613   Abelcabl 16614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-abl 16616
This theorem is referenced by:  ablcom  16630  abl32  16634  ablsub4  16638  mulgdi  16650  ghmabl  16656  ghmplusg  16667  ablcntzd  16678  prdsabld  16683  gsumsubgcl  16747  gsumsubgclOLD  16748  gsummulgz  16781  gsuminv  16786  gsuminvOLD  16788  gsumsub  16789  gsumsubOLD  16790  telgsumfzslem  16832  telgsums  16837  rngcmn  17042  lmodcmn  17370  lgseisenlem4  23452
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