Metamath Proof Explorer

Theorem cmn4d

Description: Commutative/associative law for commutative monoids. (Contributed by Thierry Arnoux, 4-May-2025)

Ref Expression
Hypotheses cmn4d.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
cmn4d.2 ⊢ + = ( +g ‘ 𝐺 )
cmn4d.3 ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
cmn4d.4 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
cmn4d.5 ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
cmn4d.6 ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
cmn4d.7 ⊢ ( 𝜑 → 𝑊 ∈ 𝐵 )
Assertion cmn4d ( 𝜑 → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 cmn4d.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 cmn4d.2 ⊢ + = ( +g ‘ 𝐺 )
3 cmn4d.3 ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
4 cmn4d.4 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 cmn4d.5 ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 cmn4d.6 ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7 cmn4d.7 ⊢ ( 𝜑 → 𝑊 ∈ 𝐵 )
8 1 2 cmn4 ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) )
9 3 4 5 6 7 8 syl122anc ⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) )