Theorem mndassd

Description: A monoid operation is associative. (Contributed by Thierry Arnoux, 3-Aug-2025)

	Ref	Expression
Hypotheses	mndassd.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mndassd.2	⊢ + = ( +g ‘ 𝐺 )
	mndassd.3	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
	mndassd.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	mndassd.5	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	mndassd.6	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion	mndassd	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mndassd.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mndassd.2	⊢ + = ( +g ‘ 𝐺 )
3		mndassd.3	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
4		mndassd.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		mndassd.5	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		mndassd.6	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7	1 2	mndass	⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )
8	3 4 5 6 7	syl13anc	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )