Theorem subdiri

Description: Distribution of multiplication over subtraction. Theorem I.5 of Apostol p. 18. (Contributed by NM, 8-May-1999)

	Ref	Expression
Hypotheses	mulm1.1	$$⊢A\in ℂ$$
	mulneg.2	$$⊢B\in ℂ$$
	subdi.3	$$⊢C\in ℂ$$
Assertion	subdiri	$$⊢\left(A-B\right)C=AC-BC$$

Proof

Step	Hyp	Ref	Expression
1		mulm1.1	$$⊢A\in ℂ$$
2		mulneg.2	$$⊢B\in ℂ$$
3		subdi.3	$$⊢C\in ℂ$$
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	mp3an	$$⊢\left(A-B\right)C=AC-BC$$