Theorem ablcmnd 19821

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 19820	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∈ wcel 2106  CMndccmn 19813  Abelcabl 19814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-abl 19816

This theorem is referenced by:  ringcmnd  20298  primrootscoprmpow  42081  primrootscoprbij  42084  aks6d1c6isolem1  42156  aks6d1c6isolem2  42157  aks6d1c6lem5  42159