Step |
Hyp |
Ref |
Expression |
1 |
|
neg1lt0 |
⊢ - 1 < 0 |
2 |
|
breq1 |
⊢ ( ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 → ( ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ↔ - 1 < 0 ) ) |
3 |
1 2
|
mpbiri |
⊢ ( ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 → ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) |
4 |
3
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) |
5 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ) |
6 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 0 ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 0 ) |
7 |
|
simplr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 0 ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) |
8 |
7
|
lt0ne0d |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 0 ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) ≠ 0 ) |
9 |
6 8
|
pm2.21ddne |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 0 ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ) |
10 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 1 ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 1 ) |
11 |
|
simplr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 1 ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) |
12 |
10 11
|
eqbrtrrd |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 1 ) → 1 < 0 ) |
13 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
14 |
|
nn0nlt0 |
⊢ ( 1 ∈ ℕ0 → ¬ 1 < 0 ) |
15 |
13 14
|
mp1i |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 1 ) → ¬ 1 < 0 ) |
16 |
12 15
|
pm2.21dd |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 1 ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ) |
17 |
|
remulcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) ∈ ℝ ) |
18 |
17
|
rexrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) ∈ ℝ* ) |
19 |
18
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) → ( 𝐴 · 𝐵 ) ∈ ℝ* ) |
20 |
|
sgncl |
⊢ ( ( 𝐴 · 𝐵 ) ∈ ℝ* → ( sgn ‘ ( 𝐴 · 𝐵 ) ) ∈ { - 1 , 0 , 1 } ) |
21 |
|
eltpi |
⊢ ( ( sgn ‘ ( 𝐴 · 𝐵 ) ) ∈ { - 1 , 0 , 1 } → ( ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ∨ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 0 ∨ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 1 ) ) |
22 |
19 20 21
|
3syl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) → ( ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ∨ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 0 ∨ ( sgn ‘ ( 𝐴 · 𝐵 ) ) = 1 ) ) |
23 |
5 9 16 22
|
mpjao3dan |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ) |
24 |
4 23
|
impbida |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ↔ ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ) ) |
25 |
|
sgnnbi |
⊢ ( ( 𝐴 · 𝐵 ) ∈ ℝ* → ( ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ↔ ( 𝐴 · 𝐵 ) < 0 ) ) |
26 |
18 25
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( sgn ‘ ( 𝐴 · 𝐵 ) ) = - 1 ↔ ( 𝐴 · 𝐵 ) < 0 ) ) |
27 |
|
sgnmul |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( sgn ‘ ( 𝐴 · 𝐵 ) ) = ( ( sgn ‘ 𝐴 ) · ( sgn ‘ 𝐵 ) ) ) |
28 |
27
|
breq1d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( sgn ‘ ( 𝐴 · 𝐵 ) ) < 0 ↔ ( ( sgn ‘ 𝐴 ) · ( sgn ‘ 𝐵 ) ) < 0 ) ) |
29 |
24 26 28
|
3bitr3d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 · 𝐵 ) < 0 ↔ ( ( sgn ‘ 𝐴 ) · ( sgn ‘ 𝐵 ) ) < 0 ) ) |