Theorem ablcmn 19666

Description: An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)

Assertion

Ref	Expression
ablcmn	⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)

Proof of Theorem ablcmn

Step	Hyp	Ref	Expression
1		isabl 19663	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
2	1	simprbi 496	1 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18812  CMndccmn 19659  Abelcabl 19660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-in 3910  df-abl 19662

This theorem is referenced by:  ablcmnd  19667  ablcom  19678  abl32  19682  ablsub4  19689  mulgdi  19705  ghmabl  19711  ghmplusg  19725  ablcntzd  19736  prdsabld  19741  gsumsubgcl  19799  gsummulgz  19822  gsuminv  19825  gsumsub  19827  telgsumfzslem  19867  telgsums  19872  ringcmn  20167  lmodcmn  20813  clmsub4  25004  lgseisenlem4  27287  primrootspoweq0  42079  aks6d1c6lem4  42146