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Metamath Proof Explorer

Theorem ablcomd

Description: An abelian group operation is commutative, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026)

Ref Expression
Hypotheses ablcomd.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
ablcomd.2 ⊢ + = ( +g ‘ 𝐺 )
ablcomd.3 ⊢ ( 𝜑 → 𝐺 ∈ Abel )
ablcomd.4 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
ablcomd.5 ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion ablcomd ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )

Proof

Step Hyp Ref Expression
1 ablcomd.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 ablcomd.2 ⊢ + = ( +g ‘ 𝐺 )
3 ablcomd.3 ⊢ ( 𝜑 → 𝐺 ∈ Abel )
4 ablcomd.4 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 ablcomd.5 ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 1 2 ablcom ⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
7 3 4 5 6 syl3anc ⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )