Theorem ablcmn 20052

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 20049	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
2	1	simprbi 503	1 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Grpcgrp 19194  CMndccmn 20045  Abelcabl 20046

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-abl 20048

This theorem is used by:  ablcmnd  20053  ablcom  20064  abl32  20068  ablsub4  20075  mulgdi  20091  ghmabl  20097  ghmplusg  20111  ablcntzd  20122  prdsabld  20127  gsumsubgcl  20185  gsummulgz  20208  gsuminv  20211  gsumsub  20213  telgsumfzslem  20253  telgsums  20258  ringcmn  20562  lmodcmn  21236  clmsub4  25482  lgseisenlem4  27762  primrootspoweq0  43422  aks6d1c6lem4  43489