Theorem mulneg1d

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

	Ref	Expression
Hypotheses	mulm1d.1	$$⊢\phi \to A\in ℂ$$
	mulnegd.2	$$⊢\phi \to B\in ℂ$$
Assertion	mulneg1d	$$⊢\phi \to \left(-A\right)B=-AB$$

Proof

Step	Hyp	Ref	Expression
1		mulm1d.1	$$⊢\phi \to A\in ℂ$$
2		mulnegd.2	$$⊢\phi \to B\in ℂ$$
3		mulneg1	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-A\right)B=-AB$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(-A\right)B=-AB$$