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Theorem ablcmn 18120
 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
StepHypRef Expression
1 isabl 18118 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
21simprbi 480 1 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1987  Grpcgrp 17343  CMndccmn 18114  Abelcabl 18115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-in 3562  df-abl 18117 This theorem is referenced by:  ablcom  18131  abl32  18135  ablsub4  18139  mulgdi  18153  ghmabl  18159  ghmplusg  18170  ablcntzd  18181  prdsabld  18186  gsumsubgcl  18241  gsummulgz  18264  gsuminv  18267  gsumsub  18269  telgsumfzslem  18306  telgsums  18311  ringcmn  18502  lmodcmn  18832  clmsub4  22814  lgseisenlem4  25003
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