Theorem ablcmnd 20035

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 20034	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐺 ∈ CMnd)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  CMndccmn 20027  Abelcabl 20028

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-abl 20030

This theorem is used by:  ringcmnd  20546  elrgspnsubrunlem2  33885  primrootscoprmpow  43352  primrootscoprbij  43355  aks6d1c6isolem1  43427  aks6d1c6isolem2  43428  aks6d1c6lem5  43430