Theorem ablcmn 19724

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 19721	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
2	1	simprbi 496	1 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18872  CMndccmn 19717  Abelcabl 19718

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 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3924  df-abl 19720

This theorem is referenced by:  ablcmnd  19725  ablcom  19736  abl32  19740  ablsub4  19747  mulgdi  19763  ghmabl  19769  ghmplusg  19783  ablcntzd  19794  prdsabld  19799  gsumsubgcl  19857  gsummulgz  19880  gsuminv  19883  gsumsub  19885  telgsumfzslem  19925  telgsums  19930  ringcmn  20198  lmodcmn  20823  clmsub4  25013  lgseisenlem4  27296  primrootspoweq0  42101  aks6d1c6lem4  42168