Theorem ablcmnd 19725

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 19724	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∈ wcel 2109  CMndccmn 19717  Abelcabl 19718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3924  df-abl 19720

This theorem is referenced by:  ringcmnd  20200  elrgspnsubrunlem2  33206  primrootscoprmpow  42094  primrootscoprbij  42097  aks6d1c6isolem1  42169  aks6d1c6isolem2  42170  aks6d1c6lem5  42172