Proof of Theorem xmullem2
| Step | Hyp | Ref
| Expression |
| 1 | | mnfnepnf 11317 |
. . . . . . . . . . . 12
⊢ -∞
≠ +∞ |
| 2 | | eqeq1 2741 |
. . . . . . . . . . . . 13
⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ =
+∞)) |
| 3 | 2 | necon3bbid 2978 |
. . . . . . . . . . . 12
⊢ (𝐴 = -∞ → (¬ 𝐴 = +∞ ↔ -∞ ≠
+∞)) |
| 4 | 1, 3 | mpbiri 258 |
. . . . . . . . . . 11
⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞) |
| 5 | 4 | con2i 139 |
. . . . . . . . . 10
⊢ (𝐴 = +∞ → ¬ 𝐴 = -∞) |
| 6 | 5 | adantl 481 |
. . . . . . . . 9
⊢ ((0 <
𝐵 ∧ 𝐴 = +∞) → ¬ 𝐴 = -∞) |
| 7 | | 0xr 11308 |
. . . . . . . . . . . . 13
⊢ 0 ∈
ℝ* |
| 8 | | nltmnf 13171 |
. . . . . . . . . . . . 13
⊢ (0 ∈
ℝ* → ¬ 0 < -∞) |
| 9 | 7, 8 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ¬ 0
< -∞ |
| 10 | | breq2 5147 |
. . . . . . . . . . . 12
⊢ (𝐴 = -∞ → (0 < 𝐴 ↔ 0 <
-∞)) |
| 11 | 9, 10 | mtbiri 327 |
. . . . . . . . . . 11
⊢ (𝐴 = -∞ → ¬ 0 <
𝐴) |
| 12 | 11 | con2i 139 |
. . . . . . . . . 10
⊢ (0 <
𝐴 → ¬ 𝐴 = -∞) |
| 13 | 12 | adantr 480 |
. . . . . . . . 9
⊢ ((0 <
𝐴 ∧ 𝐵 = +∞) → ¬ 𝐴 = -∞) |
| 14 | 6, 13 | jaoi 858 |
. . . . . . . 8
⊢ (((0 <
𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐴 = -∞) |
| 15 | 14 | a1i 11 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐴 = -∞)) |
| 16 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → 𝐵 ∈
ℝ*) |
| 17 | | xrltnsym 13179 |
. . . . . . . . . 10
⊢ ((𝐵 ∈ ℝ*
∧ 0 ∈ ℝ*) → (𝐵 < 0 → ¬ 0 < 𝐵)) |
| 18 | 16, 7, 17 | sylancl 586 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (𝐵 < 0 → ¬ 0 < 𝐵)) |
| 19 | 18 | adantrd 491 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐵)) |
| 20 | | breq2 5147 |
. . . . . . . . . . 11
⊢ (𝐵 = -∞ → (0 < 𝐵 ↔ 0 <
-∞)) |
| 21 | 9, 20 | mtbiri 327 |
. . . . . . . . . 10
⊢ (𝐵 = -∞ → ¬ 0 <
𝐵) |
| 22 | 21 | adantl 481 |
. . . . . . . . 9
⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵) |
| 23 | 22 | a1i 11 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵)) |
| 24 | 19, 23 | jaod 860 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐵)) |
| 25 | 15, 24 | orim12d 967 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 = -∞ ∨ ¬ 0 <
𝐵))) |
| 26 | | ianor 984 |
. . . . . . 7
⊢ (¬ (0
< 𝐵 ∧ 𝐴 = -∞) ↔ (¬ 0
< 𝐵 ∨ ¬ 𝐴 = -∞)) |
| 27 | | orcom 871 |
. . . . . . 7
⊢ ((¬ 0
< 𝐵 ∨ ¬ 𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 <
𝐵)) |
| 28 | 26, 27 | bitri 275 |
. . . . . 6
⊢ (¬ (0
< 𝐵 ∧ 𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 <
𝐵)) |
| 29 | 25, 28 | imbitrrdi 252 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐵 ∧ 𝐴 = -∞))) |
| 30 | 18 | con2d 134 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (0 < 𝐵 → ¬ 𝐵 < 0)) |
| 31 | 30 | adantrd 491 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 < 0)) |
| 32 | | pnfnlt 13170 |
. . . . . . . . . . 11
⊢ (0 ∈
ℝ* → ¬ +∞ < 0) |
| 33 | 7, 32 | ax-mp 5 |
. . . . . . . . . 10
⊢ ¬
+∞ < 0 |
| 34 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((0 <
𝐴 ∧ 𝐵 = +∞) → 𝐵 = +∞) |
| 35 | 34 | breq1d 5153 |
. . . . . . . . . 10
⊢ ((0 <
𝐴 ∧ 𝐵 = +∞) → (𝐵 < 0 ↔ +∞ <
0)) |
| 36 | 33, 35 | mtbiri 327 |
. . . . . . . . 9
⊢ ((0 <
𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 < 0) |
| 37 | 36 | a1i 11 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 < 0)) |
| 38 | 31, 37 | jaod 860 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐵 < 0)) |
| 39 | 4 | a1i 11 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (𝐴 = -∞ → ¬ 𝐴 = +∞)) |
| 40 | 39 | adantld 490 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐴 = +∞)) |
| 41 | | breq1 5146 |
. . . . . . . . . . . 12
⊢ (𝐴 = +∞ → (𝐴 < 0 ↔ +∞ <
0)) |
| 42 | 33, 41 | mtbiri 327 |
. . . . . . . . . . 11
⊢ (𝐴 = +∞ → ¬ 𝐴 < 0) |
| 43 | 42 | con2i 139 |
. . . . . . . . . 10
⊢ (𝐴 < 0 → ¬ 𝐴 = +∞) |
| 44 | 43 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞) |
| 45 | 44 | a1i 11 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞)) |
| 46 | 40, 45 | jaod 860 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐴 = +∞)) |
| 47 | 38, 46 | orim12d 967 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞))) |
| 48 | | ianor 984 |
. . . . . 6
⊢ (¬
(𝐵 < 0 ∧ 𝐴 = +∞) ↔ (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞)) |
| 49 | 47, 48 | imbitrrdi 252 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞))) |
| 50 | 29, 49 | jcad 512 |
. . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))) |
| 51 | | ioran 986 |
. . . 4
⊢ (¬
((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞))) |
| 52 | 50, 51 | imbitrrdi 252 |
. . 3
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)))) |
| 53 | 21 | con2i 139 |
. . . . . . . . . 10
⊢ (0 <
𝐵 → ¬ 𝐵 = -∞) |
| 54 | 53 | adantr 480 |
. . . . . . . . 9
⊢ ((0 <
𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 = -∞) |
| 55 | 54 | a1i 11 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 = -∞)) |
| 56 | | pnfnemnf 11316 |
. . . . . . . . . . 11
⊢ +∞
≠ -∞ |
| 57 | | eqeq1 2741 |
. . . . . . . . . . . 12
⊢ (𝐵 = +∞ → (𝐵 = -∞ ↔ +∞ =
-∞)) |
| 58 | 57 | necon3bbid 2978 |
. . . . . . . . . . 11
⊢ (𝐵 = +∞ → (¬ 𝐵 = -∞ ↔ +∞ ≠
-∞)) |
| 59 | 56, 58 | mpbiri 258 |
. . . . . . . . . 10
⊢ (𝐵 = +∞ → ¬ 𝐵 = -∞) |
| 60 | 59 | adantl 481 |
. . . . . . . . 9
⊢ ((0 <
𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 = -∞) |
| 61 | 60 | a1i 11 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 = -∞)) |
| 62 | 55, 61 | jaod 860 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐵 = -∞)) |
| 63 | 11 | adantl 481 |
. . . . . . . . 9
⊢ ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴) |
| 64 | 63 | a1i 11 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴)) |
| 65 | | simpl 482 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → 𝐴 ∈
ℝ*) |
| 66 | | xrltnsym 13179 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ*
∧ 0 ∈ ℝ*) → (𝐴 < 0 → ¬ 0 < 𝐴)) |
| 67 | 65, 7, 66 | sylancl 586 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (𝐴 < 0 → ¬ 0 < 𝐴)) |
| 68 | 67 | adantrd 491 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐴)) |
| 69 | 64, 68 | jaod 860 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐴)) |
| 70 | 62, 69 | orim12d 967 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 = -∞ ∨ ¬ 0 <
𝐴))) |
| 71 | | ianor 984 |
. . . . . . 7
⊢ (¬ (0
< 𝐴 ∧ 𝐵 = -∞) ↔ (¬ 0
< 𝐴 ∨ ¬ 𝐵 = -∞)) |
| 72 | | orcom 871 |
. . . . . . 7
⊢ ((¬ 0
< 𝐴 ∨ ¬ 𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 <
𝐴)) |
| 73 | 71, 72 | bitri 275 |
. . . . . 6
⊢ (¬ (0
< 𝐴 ∧ 𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 <
𝐴)) |
| 74 | 70, 73 | imbitrrdi 252 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐴 ∧ 𝐵 = -∞))) |
| 75 | 42 | adantl 481 |
. . . . . . . . 9
⊢ ((0 <
𝐵 ∧ 𝐴 = +∞) → ¬ 𝐴 < 0) |
| 76 | 75 | a1i 11 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐴 < 0)) |
| 77 | 67 | con2d 134 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (0 < 𝐴 → ¬ 𝐴 < 0)) |
| 78 | 77 | adantrd 491 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐴 < 0)) |
| 79 | 76, 78 | jaod 860 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐴 < 0)) |
| 80 | | breq1 5146 |
. . . . . . . . . . . 12
⊢ (𝐵 = +∞ → (𝐵 < 0 ↔ +∞ <
0)) |
| 81 | 33, 80 | mtbiri 327 |
. . . . . . . . . . 11
⊢ (𝐵 = +∞ → ¬ 𝐵 < 0) |
| 82 | 81 | con2i 139 |
. . . . . . . . . 10
⊢ (𝐵 < 0 → ¬ 𝐵 = +∞) |
| 83 | 82 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐵 = +∞) |
| 84 | 59 | con2i 139 |
. . . . . . . . . 10
⊢ (𝐵 = -∞ → ¬ 𝐵 = +∞) |
| 85 | 84 | adantl 481 |
. . . . . . . . 9
⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐵 = +∞) |
| 86 | 83, 85 | jaoi 858 |
. . . . . . . 8
⊢ (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞) |
| 87 | 86 | a1i 11 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞)) |
| 88 | 79, 87 | orim12d 967 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞))) |
| 89 | | ianor 984 |
. . . . . 6
⊢ (¬
(𝐴 < 0 ∧ 𝐵 = +∞) ↔ (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞)) |
| 90 | 88, 89 | imbitrrdi 252 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐴 < 0 ∧ 𝐵 = +∞))) |
| 91 | 74, 90 | jcad 512 |
. . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) |
| 92 | | ioran 986 |
. . . 4
⊢ (¬
((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))) |
| 93 | 91, 92 | imbitrrdi 252 |
. . 3
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) |
| 94 | 52, 93 | jcad 512 |
. 2
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))) |
| 95 | | or4 927 |
. 2
⊢ ((((0
< 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) |
| 96 | | ioran 986 |
. 2
⊢ (¬
(((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) |
| 97 | 94, 95, 96 | 3imtr4g 296 |
1
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))) |