Step |
Hyp |
Ref |
Expression |
1 |
|
mulgt0con2d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
mulgt0con2d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
mulgt0con2d.1 |
⊢ ( 𝜑 → 0 < 𝐴 ) |
4 |
|
mulgt0con2d.2 |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) < 0 ) |
5 |
1 2
|
remulcld |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℝ ) |
6 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 𝐴 ∈ ℝ ) |
7 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 𝐵 ∈ ℝ ) |
8 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 0 < 𝐴 ) |
9 |
|
simpr |
⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 0 < 𝐵 ) |
10 |
6 7 8 9
|
mulgt0d |
⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 · 𝐵 ) ) |
11 |
10
|
ex |
⊢ ( 𝜑 → ( 0 < 𝐵 → 0 < ( 𝐴 · 𝐵 ) ) ) |
12 |
|
remul01 |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 0 ) = 0 ) |
13 |
1 12
|
syl |
⊢ ( 𝜑 → ( 𝐴 · 0 ) = 0 ) |
14 |
|
oveq2 |
⊢ ( 𝐵 = 0 → ( 𝐴 · 𝐵 ) = ( 𝐴 · 0 ) ) |
15 |
14
|
eqeq1d |
⊢ ( 𝐵 = 0 → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 · 0 ) = 0 ) ) |
16 |
13 15
|
syl5ibrcom |
⊢ ( 𝜑 → ( 𝐵 = 0 → ( 𝐴 · 𝐵 ) = 0 ) ) |
17 |
2 5 11 16
|
mulgt0con1dlem |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) < 0 → 𝐵 < 0 ) ) |
18 |
4 17
|
mpd |
⊢ ( 𝜑 → 𝐵 < 0 ) |