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Metamath Proof Explorer

Theorem mul32i

Description: Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999)

Ref Expression
Hypotheses mul.1 ⊢ 𝐴 ∈ ℂ
mul.2 ⊢ 𝐵 ∈ ℂ
mul.3 ⊢ 𝐶 ∈ ℂ
Assertion mul32i ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) · 𝐵 )

Proof

Step Hyp Ref Expression
1 mul.1 ⊢ 𝐴 ∈ ℂ
2 mul.2 ⊢ 𝐵 ∈ ℂ
3 mul.3 ⊢ 𝐶 ∈ ℂ
4 mul32 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) · 𝐵 ) )
5 1 2 3 4 mp3an ⊢ ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) · 𝐵 )