Metamath Proof Explorer
Description: Product of two negatives. 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 |
mul2negd |
⊢ ( 𝜑 → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mulm1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
mulnegd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
mul2neg |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) ) |