Theorem mul4d

Description: Rearrangement of 4 factors. (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 )
	mul4d.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
Assertion	mul4d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) · ( 𝐶 · 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) · ( 𝐵 · 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		muld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		addcomd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		addcand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		mul4d.4	⊢ ( 𝜑 → 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	syl22anc	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) · ( 𝐶 · 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) · ( 𝐵 · 𝐷 ) ) )