Theorem mulsubd

Description: Product of two differences. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	mulm1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mulnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subdid.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	muladdd.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
Assertion	mulsubd	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) · ( 𝐶 − 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐷 · 𝐵 ) ) − ( ( 𝐴 · 𝐷 ) + ( 𝐶 · 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		mulm1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mulnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subdid.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		muladdd.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
5		mulsub	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 − 𝐵 ) · ( 𝐶 − 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐷 · 𝐵 ) ) − ( ( 𝐴 · 𝐷 ) + ( 𝐶 · 𝐵 ) ) ) )
6	1 2 3 4 5	syl22anc	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) · ( 𝐶 − 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐷 · 𝐵 ) ) − ( ( 𝐴 · 𝐷 ) + ( 𝐶 · 𝐵 ) ) ) )