# Metamath Proof Explorer

## Theorem mullt0

Description: The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009)

Ref Expression
Assertion mullt0 ( ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 0 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 < 0 ) ) → 0 < ( 𝐴 · 𝐵 ) )

### Proof

Step Hyp Ref Expression
1 renegcl ( 𝐴 ∈ ℝ → - 𝐴 ∈ ℝ )
2 1 adantr ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 0 ) → - 𝐴 ∈ ℝ )
3 lt0neg1 ( 𝐴 ∈ ℝ → ( 𝐴 < 0 ↔ 0 < - 𝐴 ) )
4 3 biimpa ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 0 ) → 0 < - 𝐴 )
5 2 4 jca ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 0 ) → ( - 𝐴 ∈ ℝ ∧ 0 < - 𝐴 ) )
6 renegcl ( 𝐵 ∈ ℝ → - 𝐵 ∈ ℝ )
7 6 adantr ( ( 𝐵 ∈ ℝ ∧ 𝐵 < 0 ) → - 𝐵 ∈ ℝ )
8 lt0neg1 ( 𝐵 ∈ ℝ → ( 𝐵 < 0 ↔ 0 < - 𝐵 ) )
9 8 biimpa ( ( 𝐵 ∈ ℝ ∧ 𝐵 < 0 ) → 0 < - 𝐵 )
10 7 9 jca ( ( 𝐵 ∈ ℝ ∧ 𝐵 < 0 ) → ( - 𝐵 ∈ ℝ ∧ 0 < - 𝐵 ) )
11 mulgt0 ( ( ( - 𝐴 ∈ ℝ ∧ 0 < - 𝐴 ) ∧ ( - 𝐵 ∈ ℝ ∧ 0 < - 𝐵 ) ) → 0 < ( - 𝐴 · - 𝐵 ) )
12 5 10 11 syl2an ( ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 0 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 < 0 ) ) → 0 < ( - 𝐴 · - 𝐵 ) )
13 recn ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
14 recn ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
15 mul2neg ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) )
16 13 14 15 syl2an ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) )
17 16 ad2ant2r ( ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 0 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 < 0 ) ) → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) )
18 12 17 breqtrd ( ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 0 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 < 0 ) ) → 0 < ( 𝐴 · 𝐵 ) )