Metamath Proof Explorer


Theorem mul2negd

Description: Product of two negatives. Theorem I.12 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses mulm1d.1 ( 𝜑𝐴 ∈ ℂ )
mulnegd.2 ( 𝜑𝐵 ∈ ℂ )
Assertion mul2negd ( 𝜑 → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) )

Proof

Step Hyp Ref Expression
1 mulm1d.1 ( 𝜑𝐴 ∈ ℂ )
2 mulnegd.2 ( 𝜑𝐵 ∈ ℂ )
3 mul2neg ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) )
4 1 2 3 syl2anc ( 𝜑 → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) )