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Theorem ablcmn 13100
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 13097 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
21simprbi 275 1  |-  ( G  e.  Abel  ->  G  e. CMnd
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148   Grpcgrp 12882  CMndccmn 13093   Abelcabl 13094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-abl 13096
This theorem is referenced by:  ablcom  13111  abl32  13115  ablsub4  13121  ringcmn  13221  lmodcmn  13430
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