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 )