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Theorem ablcmn 16295
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 16293 . 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 1756   Grpcgrp 15422  CMndccmn 16289   Abelcabel 16290
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 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2986  df-in 3347  df-abl 16292
This theorem is referenced by:  ablcom  16306  abl32  16310  ablsub4  16314  mulgdi  16326  ghmplusg  16340  ablcntzd  16351  prdsabld  16356  gsumsubgcl  16418  gsumsubgclOLD  16419  gsummulgz  16451  gsuminv  16456  gsuminvOLD  16458  gsumsub  16459  gsumsubOLD  16460  rngcmn  16687  lmodcmn  17005  lgseisenlem4  22703  telescfzgsumlem  30821  telescgsum  30823
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