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Metamath Proof Explorer

Theorem mul12i

Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Hypotheses	mul.1	⊢ 𝐴 ∈ ℂ
	mul.2	⊢ 𝐵 ∈ ℂ
	mul.3	⊢ 𝐶 ∈ ℂ
Assertion	mul12i	⊢ ( 𝐴 · ( 𝐵 · 𝐶 ) ) = ( 𝐵 · ( 𝐴 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		mul.1	⊢ 𝐴 ∈ ℂ
2		mul.2	⊢ 𝐵 ∈ ℂ
3		mul.3	⊢ 𝐶 ∈ ℂ
4		mul12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · ( 𝐵 · 𝐶 ) ) = ( 𝐵 · ( 𝐴 · 𝐶 ) ) )
5	1 2 3 4	mp3an	⊢ ( 𝐴 · ( 𝐵 · 𝐶 ) ) = ( 𝐵 · ( 𝐴 · 𝐶 ) )