Theorem adddid

Description: Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	addcld.1	$$⊢\phi \to A\in ℂ$$
	addcld.2	$$⊢\phi \to B\in ℂ$$
	addassd.3	$$⊢\phi \to C\in ℂ$$
Assertion	adddid	$$⊢\phi \to A\left(B+C\right)=AB+AC$$

Proof

Step	Hyp	Ref	Expression
1		addcld.1	$$⊢\phi \to A\in ℂ$$
2		addcld.2	$$⊢\phi \to B\in ℂ$$
3		addassd.3	$$⊢\phi \to C\in ℂ$$
4		adddi	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to A\left(B+C\right)=AB+AC$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A\left(B+C\right)=AB+AC$$