Step |
Hyp |
Ref |
Expression |
1 |
|
id |
⊢ ( 𝐴 ∈ ℝ* → 𝐴 ∈ ℝ* ) |
2 |
|
eqeq1 |
⊢ ( ( sgn ‘ 𝐴 ) = 0 → ( ( sgn ‘ 𝐴 ) = - 1 ↔ 0 = - 1 ) ) |
3 |
2
|
imbi1d |
⊢ ( ( sgn ‘ 𝐴 ) = 0 → ( ( ( sgn ‘ 𝐴 ) = - 1 → 𝐴 < 0 ) ↔ ( 0 = - 1 → 𝐴 < 0 ) ) ) |
4 |
|
eqeq1 |
⊢ ( ( sgn ‘ 𝐴 ) = 1 → ( ( sgn ‘ 𝐴 ) = - 1 ↔ 1 = - 1 ) ) |
5 |
4
|
imbi1d |
⊢ ( ( sgn ‘ 𝐴 ) = 1 → ( ( ( sgn ‘ 𝐴 ) = - 1 → 𝐴 < 0 ) ↔ ( 1 = - 1 → 𝐴 < 0 ) ) ) |
6 |
|
eqeq1 |
⊢ ( ( sgn ‘ 𝐴 ) = - 1 → ( ( sgn ‘ 𝐴 ) = - 1 ↔ - 1 = - 1 ) ) |
7 |
6
|
imbi1d |
⊢ ( ( sgn ‘ 𝐴 ) = - 1 → ( ( ( sgn ‘ 𝐴 ) = - 1 → 𝐴 < 0 ) ↔ ( - 1 = - 1 → 𝐴 < 0 ) ) ) |
8 |
|
neg1ne0 |
⊢ - 1 ≠ 0 |
9 |
8
|
nesymi |
⊢ ¬ 0 = - 1 |
10 |
9
|
pm2.21i |
⊢ ( 0 = - 1 → 𝐴 < 0 ) |
11 |
10
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 = 0 ) → ( 0 = - 1 → 𝐴 < 0 ) ) |
12 |
|
neg1rr |
⊢ - 1 ∈ ℝ |
13 |
|
neg1lt0 |
⊢ - 1 < 0 |
14 |
|
0lt1 |
⊢ 0 < 1 |
15 |
|
0re |
⊢ 0 ∈ ℝ |
16 |
|
1re |
⊢ 1 ∈ ℝ |
17 |
12 15 16
|
lttri |
⊢ ( ( - 1 < 0 ∧ 0 < 1 ) → - 1 < 1 ) |
18 |
13 14 17
|
mp2an |
⊢ - 1 < 1 |
19 |
12 18
|
gtneii |
⊢ 1 ≠ - 1 |
20 |
19
|
neii |
⊢ ¬ 1 = - 1 |
21 |
20
|
pm2.21i |
⊢ ( 1 = - 1 → 𝐴 < 0 ) |
22 |
21
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 < 𝐴 ) → ( 1 = - 1 → 𝐴 < 0 ) ) |
23 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 < 0 ∧ - 1 = - 1 ) → 𝐴 < 0 ) |
24 |
23
|
3expia |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 < 0 ) → ( - 1 = - 1 → 𝐴 < 0 ) ) |
25 |
1 3 5 7 11 22 24
|
sgn3da |
⊢ ( 𝐴 ∈ ℝ* → ( ( sgn ‘ 𝐴 ) = - 1 → 𝐴 < 0 ) ) |
26 |
25
|
imp |
⊢ ( ( 𝐴 ∈ ℝ* ∧ ( sgn ‘ 𝐴 ) = - 1 ) → 𝐴 < 0 ) |
27 |
|
sgnn |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 < 0 ) → ( sgn ‘ 𝐴 ) = - 1 ) |
28 |
26 27
|
impbida |
⊢ ( 𝐴 ∈ ℝ* → ( ( sgn ‘ 𝐴 ) = - 1 ↔ 𝐴 < 0 ) ) |