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Metamath Proof Explorer

Theorem mul32d

Description: Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses muld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
addcomd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
addcand.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion mul32d ( 𝜑 → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) · 𝐵 ) )

Proof

Step Hyp Ref Expression
1 muld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 addcomd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 addcand.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4 mul32 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) · 𝐵 ) )
5 1 2 3 4 syl3anc ⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) · 𝐵 ) )