Theorem subdird

Description: Distribution of multiplication over subtraction. Theorem I.5 of Apostol p. 18. (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	$$⊢\phi \to C\in ℂ$$
Assertion	subdird	$$⊢\phi \to \left(A-B\right)C=AC-BC$$

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		subdir	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to \left(A-B\right)C=AC-BC$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A-B\right)C=AC-BC$$