Theorem ablcmn 19701

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 19698	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
2	1	simprbi 496	1 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18847  CMndccmn 19694  Abelcabl 19695

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 3446  df-in 3918  df-abl 19697

This theorem is referenced by:  ablcmnd  19702  ablcom  19713  abl32  19717  ablsub4  19724  mulgdi  19740  ghmabl  19746  ghmplusg  19760  ablcntzd  19771  prdsabld  19776  gsumsubgcl  19834  gsummulgz  19857  gsuminv  19860  gsumsub  19862  telgsumfzslem  19902  telgsums  19907  ringcmn  20202  lmodcmn  20848  clmsub4  25039  lgseisenlem4  27322  primrootspoweq0  42087  aks6d1c6lem4  42154