Step |
Hyp |
Ref |
Expression |
1 |
|
ioran |
⊢ ( ¬ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ↔ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) |
2 |
1
|
anbi2i |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ¬ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) ↔ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ) |
3 |
|
ioran |
⊢ ( ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ↔ ( ¬ ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ¬ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ) |
4 |
|
ioran |
⊢ ( ¬ ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ↔ ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ) |
5 |
|
ioran |
⊢ ( ¬ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ↔ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) |
6 |
4 5
|
anbi12i |
⊢ ( ( ¬ ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ¬ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ↔ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ) |
7 |
3 6
|
bitri |
⊢ ( ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ↔ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ) |
8 |
|
ioran |
⊢ ( ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ↔ ( ¬ ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ¬ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) |
9 |
|
ioran |
⊢ ( ¬ ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ↔ ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ) |
10 |
|
ioran |
⊢ ( ¬ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ↔ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) |
11 |
9 10
|
anbi12i |
⊢ ( ( ¬ ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ¬ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ↔ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) |
12 |
8 11
|
bitri |
⊢ ( ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ↔ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) |
13 |
7 12
|
anbi12i |
⊢ ( ( ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ↔ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) |
14 |
|
simplll |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → 𝐴 ∈ ℝ* ) |
15 |
|
elxr |
⊢ ( 𝐴 ∈ ℝ* ↔ ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞ ) ) |
16 |
14 15
|
sylib |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞ ) ) |
17 |
|
idd |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ ) ) |
18 |
|
simprlr |
⊢ ( ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) → ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) |
19 |
18
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) |
20 |
19
|
pm2.21d |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( ( 𝐵 < 0 ∧ 𝐴 = +∞ ) → 𝐴 ∈ ℝ ) ) |
21 |
20
|
expdimp |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) ∧ 𝐵 < 0 ) → ( 𝐴 = +∞ → 𝐴 ∈ ℝ ) ) |
22 |
|
simplrr |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ¬ 𝐵 = 0 ) |
23 |
22
|
pm2.21d |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( 𝐵 = 0 → ( 𝐴 = +∞ → 𝐴 ∈ ℝ ) ) ) |
24 |
23
|
imp |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) ∧ 𝐵 = 0 ) → ( 𝐴 = +∞ → 𝐴 ∈ ℝ ) ) |
25 |
|
simplll |
⊢ ( ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) → ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ) |
26 |
25
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ) |
27 |
26
|
pm2.21d |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) → 𝐴 ∈ ℝ ) ) |
28 |
27
|
expdimp |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) ∧ 0 < 𝐵 ) → ( 𝐴 = +∞ → 𝐴 ∈ ℝ ) ) |
29 |
|
simpllr |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → 𝐵 ∈ ℝ* ) |
30 |
|
0xr |
⊢ 0 ∈ ℝ* |
31 |
|
xrltso |
⊢ < Or ℝ* |
32 |
|
solin |
⊢ ( ( < Or ℝ* ∧ ( 𝐵 ∈ ℝ* ∧ 0 ∈ ℝ* ) ) → ( 𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵 ) ) |
33 |
31 32
|
mpan |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( 𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵 ) ) |
34 |
29 30 33
|
sylancl |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( 𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵 ) ) |
35 |
21 24 28 34
|
mpjao3dan |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( 𝐴 = +∞ → 𝐴 ∈ ℝ ) ) |
36 |
|
simpllr |
⊢ ( ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) → ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) |
37 |
36
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) |
38 |
37
|
pm2.21d |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → 𝐴 ∈ ℝ ) ) |
39 |
38
|
expdimp |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) ∧ 𝐵 < 0 ) → ( 𝐴 = -∞ → 𝐴 ∈ ℝ ) ) |
40 |
22
|
pm2.21d |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( 𝐵 = 0 → ( 𝐴 = -∞ → 𝐴 ∈ ℝ ) ) ) |
41 |
40
|
imp |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) ∧ 𝐵 = 0 ) → ( 𝐴 = -∞ → 𝐴 ∈ ℝ ) ) |
42 |
|
simprll |
⊢ ( ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) → ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ) |
43 |
42
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ) |
44 |
43
|
pm2.21d |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) → 𝐴 ∈ ℝ ) ) |
45 |
44
|
expdimp |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) ∧ 0 < 𝐵 ) → ( 𝐴 = -∞ → 𝐴 ∈ ℝ ) ) |
46 |
39 41 45 34
|
mpjao3dan |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( 𝐴 = -∞ → 𝐴 ∈ ℝ ) ) |
47 |
17 35 46
|
3jaod |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → ( ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞ ) → 𝐴 ∈ ℝ ) ) |
48 |
16 47
|
mpd |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( ¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0 ) ) ∧ ( ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ( ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → 𝐴 ∈ ℝ ) |
49 |
2 13 48
|
syl2anb |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ¬ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) ∧ ( ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ∧ ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) → 𝐴 ∈ ℝ ) |
50 |
49
|
anassrs |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ¬ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) ∧ ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ) ∧ ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) → 𝐴 ∈ ℝ ) |