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	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mulnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	mulneg1d	⊢ ( 𝜑 → ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		mulm1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mulnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		mulneg1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 ) )