Metamath Proof Explorer


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 ∈ ℂ
mulneg.2 𝐵 ∈ ℂ
Assertion mulneg1i ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 )

Proof

Step Hyp Ref Expression
1 mulm1.1 𝐴 ∈ ℂ
2 mulneg.2 𝐵 ∈ ℂ
3 mulneg1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 ) )
4 1 2 3 mp2an ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 )