Theorem ablcmn 14008

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

Step	Hyp	Ref	Expression
1		isabl 14005	. 2 |- ( G e. Abel <-> ( G e. Grp /\ G e. CMnd ) )
2	1	simprbi 275	1 |- ( G e. Abel -> G e. CMnd )

Syntax hints:   -> wi 4   e. wcel 2203   Grpcgrp 13713  CMndccmn 14001   Abelcabl 14002

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-abl 14004

This theorem is referenced by:  ablcmnd  14009  ablcom  14020  abl32  14024  ablsub4  14030  ghmabl  14045  ringcmn  14177  lmodcmn  14483  lgseisenlem4  15946