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Theorem ablcmn 15410
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 15408 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
21simprbi 451 1  |-  ( G  e.  Abel  ->  G  e. CMnd
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   Grpcgrp 14677  CMndccmn 15404   Abelcabel 15405
This theorem is referenced by:  ablcom  15421  abl32  15425  ablsub4  15429  mulgdi  15441  ghmplusg  15453  ablcntzd  15464  prdsabld  15469  gsumsubgcl  15517  gsummulgz  15530  gsuminv  15533  gsumsub  15534  rngcmn  15686  lmodcmn  15984  lgseisenlem4  21128
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 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-abl 15407
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