Theorem muladdd

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

	Ref	Expression
Hypotheses	mulm1d.1	$$⊢\phi \to A\in ℂ$$
	mulnegd.2	$$⊢\phi \to B\in ℂ$$
	subdid.3	$$⊢\phi \to C\in ℂ$$
	muladdd.4	$$⊢\phi \to D\in ℂ$$
Assertion	muladdd	$$⊢\phi \to \left(A+B\right)\left(C+D\right)=AC+DB+AD+CB$$

Proof

Step	Hyp	Ref	Expression
1		mulm1d.1	$$⊢\phi \to A\in ℂ$$
2		mulnegd.2	$$⊢\phi \to B\in ℂ$$
3		subdid.3	$$⊢\phi \to C\in ℂ$$
4		muladdd.4	$$⊢\phi \to D\in ℂ$$
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	syl22anc	$$⊢\phi \to \left(A+B\right)\left(C+D\right)=AC+DB+AD+CB$$