Theorem muladdi

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

	Ref	Expression
Hypotheses	mulm1.1	$$⊢A\in ℂ$$
	mulneg.2	$$⊢B\in ℂ$$
	subdi.3	$$⊢C\in ℂ$$
	muladdi.4	$$⊢D\in ℂ$$
Assertion	muladdi	$$⊢\left(A+B\right)\left(C+D\right)=AC+DB+AD+CB$$

Proof

Step	Hyp	Ref	Expression
1		mulm1.1	$$⊢A\in ℂ$$
2		mulneg.2	$$⊢B\in ℂ$$
3		subdi.3	$$⊢C\in ℂ$$
4		muladdi.4	$$⊢D\in ℂ$$
5		muladd	$$⊢\left(\left(A\in ℂ\wedge B\in ℂ\right)\wedge \left(C\in ℂ\wedge D\in ℂ\right)\right)\to \left(A+B\right)\left(C+D\right)=AC+DB+AD+CB$$
6	1 2 3 4 5	mp4an	$$⊢\left(A+B\right)\left(C+D\right)=AC+DB+AD+CB$$