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Metamath Proof Explorer

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 ) ) )