Theorem ablcmnd 19857

Description: An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)

Hypothesis

Ref	Expression
ablcmnd.1	⊢ (𝜑 → 𝐺 ∈ Abel)

Assertion

Ref	Expression
ablcmnd	⊢ (𝜑 → 𝐺 ∈ CMnd)

Proof of Theorem ablcmnd

Step	Hyp	Ref	Expression
1		ablcmnd.1	. 2 ⊢ (𝜑 → 𝐺 ∈ Abel)
2		ablcmn 19856	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∈ wcel 2149  CMndccmn 19849  Abelcabl 19850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-in 3920  df-abl 19852

This theorem is referenced by:  ringcmnd  20366  elrgspnsubrunlem2  33508  primrootscoprmpow  42755  primrootscoprbij  42758  aks6d1c6isolem1  42830  aks6d1c6isolem2  42831  aks6d1c6lem5  42833