Theorem ablcmn 14044

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 14041	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
2	1	simprbi 275	1 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∈ wcel 2205  Grpcgrp 13755  CMndccmn 14037  Abelcabl 14038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-abl 14040

This theorem is referenced by:  ablcmnd  14045  ablcom  14056  abl32  14060  ablsub4  14066  ghmabl  14081  ringcmn  14276  lmodcmn  14609  lgseisenlem4  16072