Theorem mulneg1i

Description: Product with negative is negative of product. Theorem I.12 of Apostol p. 18. (Contributed by NM, 10-Feb-1995) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	mulm1.1	$$⊢A\in ℂ$$
	mulneg.2	$$⊢B\in ℂ$$
Assertion	mulneg1i	$$⊢\left(-A\right)B=-AB$$

Proof

Step	Hyp	Ref	Expression
1		mulm1.1	$$⊢A\in ℂ$$
2		mulneg.2	$$⊢B\in ℂ$$
3		mulneg1	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-A\right)B=-AB$$
4	1 2 3	mp2an	$$⊢\left(-A\right)B=-AB$$