Theorem mul4i

Description: Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995)

	Ref	Expression
Hypotheses	mul.1	⊢ 𝐴 ∈ ℂ
	mul.2	⊢ 𝐵 ∈ ℂ
	mul.3	⊢ 𝐶 ∈ ℂ
	mul4.4	⊢ 𝐷 ∈ ℂ
Assertion	mul4i	⊢ ( ( 𝐴 · 𝐵 ) · ( 𝐶 · 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) · ( 𝐵 · 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		mul.1	⊢ 𝐴 ∈ ℂ
2		mul.2	⊢ 𝐵 ∈ ℂ
3		mul.3	⊢ 𝐶 ∈ ℂ
4		mul4.4	⊢ 𝐷 ∈ ℂ
5		mul4	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 · 𝐵 ) · ( 𝐶 · 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) · ( 𝐵 · 𝐷 ) ) )
6	1 2 3 4 5	mp4an	⊢ ( ( 𝐴 · 𝐵 ) · ( 𝐶 · 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) · ( 𝐵 · 𝐷 ) )