Theorem ablcmn 18907

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 18904	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
2	1	simprbi 499	1 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∈ wcel 2110  Grpcgrp 18097  CMndccmn 18900  Abelcabl 18901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-in 3943  df-abl 18903

This theorem is referenced by:  ablcom  18918  abl32  18922  ablsub4  18927  mulgdi  18941  ghmabl  18947  ghmplusg  18960  ablcntzd  18971  prdsabld  18976  gsumsubgcl  19034  gsummulgz  19057  gsuminv  19060  gsumsub  19062  telgsumfzslem  19102  telgsums  19107  ringcmn  19325  lmodcmn  19676  clmsub4  23704  lgseisenlem4  25948