Theorem ablcmn 19655

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 19652	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
2	1	simprbi 498	1 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∈ wcel 2107  Grpcgrp 18819  CMndccmn 19648  Abelcabl 19649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-abl 19651

This theorem is referenced by:  ablcmnd  19656  ablcom  19667  abl32  19671  ablsub4  19678  mulgdi  19694  ghmabl  19700  ghmplusg  19714  ablcntzd  19725  prdsabld  19730  gsumsubgcl  19788  gsummulgz  19811  gsuminv  19814  gsumsub  19816  telgsumfzslem  19856  telgsums  19861  ringcmn  20099  lmodcmn  20520  clmsub4  24622  lgseisenlem4  26881