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Metamath Proof Explorer

Theorem mul2lt0rlt0

Description: If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018)

Ref Expression
Hypotheses mul2lt0.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
mul2lt0.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
mul2lt0.3 ⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) < 0 )
Assertion mul2lt0rlt0 ( ( 𝜑 ∧ 𝐵 < 0 ) → 0 < 𝐴 )

Proof

Step Hyp Ref Expression
1 mul2lt0.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 mul2lt0.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 mul2lt0.3 ⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) < 0 )
4 1 2 remulcld ⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℝ )
5 4 adantr ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → ( 𝐴 · 𝐵 ) ∈ ℝ )
6 0red ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → 0 ∈ ℝ )
7 negelrp ⊢ ( 𝐵 ∈ ℝ → ( - 𝐵 ∈ ℝ+ ↔ 𝐵 < 0 ) )
8 2 7 syl ⊢ ( 𝜑 → ( - 𝐵 ∈ ℝ+ ↔ 𝐵 < 0 ) )
9 8 biimpar ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → - 𝐵 ∈ ℝ+ )
10 3 adantr ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → ( 𝐴 · 𝐵 ) < 0 )
11 5 6 9 10 ltdiv1dd ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → ( ( 𝐴 · 𝐵 ) / - 𝐵 ) < ( 0 / - 𝐵 ) )
12 1 recnd ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
13 12 adantr ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → 𝐴 ∈ ℂ )
14 2 recnd ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
15 14 adantr ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → 𝐵 ∈ ℂ )
16 13 15 mulcld ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → ( 𝐴 · 𝐵 ) ∈ ℂ )
17 simpr ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → 𝐵 < 0 )
18 17 lt0ne0d ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → 𝐵 ≠ 0 )
19 16 15 18 divneg2d ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → - ( ( 𝐴 · 𝐵 ) / 𝐵 ) = ( ( 𝐴 · 𝐵 ) / - 𝐵 ) )
20 13 15 18 divcan4d ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → ( ( 𝐴 · 𝐵 ) / 𝐵 ) = 𝐴 )
21 20 negeqd ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → - ( ( 𝐴 · 𝐵 ) / 𝐵 ) = - 𝐴 )
22 19 21 eqtr3d ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → ( ( 𝐴 · 𝐵 ) / - 𝐵 ) = - 𝐴 )
23 15 negcld ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → - 𝐵 ∈ ℂ )
24 15 18 negne0d ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → - 𝐵 ≠ 0 )
25 23 24 div0d ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → ( 0 / - 𝐵 ) = 0 )
26 11 22 25 3brtr3d ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → - 𝐴 < 0 )
27 1 adantr ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → 𝐴 ∈ ℝ )
28 27 lt0neg2d ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → ( 0 < 𝐴 ↔ - 𝐴 < 0 ) )
29 26 28 mpbird ⊢ ( ( 𝜑 ∧ 𝐵 < 0 ) → 0 < 𝐴 )