Theorem ablcmnd 39878

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 19071	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∈ wcel 2115  CMndccmn 19064  Abelcabl 19065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2021  ax-8 2117  ax-9 2125  ax-ext 2712

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2076  df-clab 2719  df-cleq 2732  df-clel 2813  df-v 3403  df-in 3864  df-abl 19067

This theorem is referenced by:  ringcmnd  39884