Theorem muladdi

Description: Product of two sums. (Contributed by NM, 17-May-1999)

	Ref	Expression
Hypotheses	mulm1.1	⊢ 𝐴 ∈ ℂ
	mulneg.2	⊢ 𝐵 ∈ ℂ
	subdi.3	⊢ 𝐶 ∈ ℂ
	muladdi.4	⊢ 𝐷 ∈ ℂ
Assertion	muladdi	⊢ ( ( 𝐴 + 𝐵 ) · ( 𝐶 + 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐷 · 𝐵 ) ) + ( ( 𝐴 · 𝐷 ) + ( 𝐶 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mulm1.1	⊢ 𝐴 ∈ ℂ
2		mulneg.2	⊢ 𝐵 ∈ ℂ
3		subdi.3	⊢ 𝐶 ∈ ℂ
4		muladdi.4	⊢ 𝐷 ∈ ℂ
5		muladd	⊢ ( ( ( 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	mp4an	⊢ ( ( 𝐴 + 𝐵 ) · ( 𝐶 + 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐷 · 𝐵 ) ) + ( ( 𝐴 · 𝐷 ) + ( 𝐶 · 𝐵 ) ) )