Proof of Theorem xsubge0
| Step | Hyp | Ref
| Expression |
| 1 | | elxr 13158 |
. 2
⊢ (𝐵 ∈ ℝ*
↔ (𝐵 ∈ ℝ
∨ 𝐵 = +∞ ∨
𝐵 =
-∞)) |
| 2 | | 0xr 11308 |
. . . . 5
⊢ 0 ∈
ℝ* |
| 3 | | rexr 11307 |
. . . . . 6
⊢ (𝐵 ∈ ℝ → 𝐵 ∈
ℝ*) |
| 4 | | xnegcl 13255 |
. . . . . . 7
⊢ (𝐵 ∈ ℝ*
→ -𝑒𝐵 ∈
ℝ*) |
| 5 | | xaddcl 13281 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ -𝑒𝐵 ∈ ℝ*) → (𝐴 +𝑒
-𝑒𝐵)
∈ ℝ*) |
| 6 | 4, 5 | sylan2 593 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (𝐴 +𝑒
-𝑒𝐵)
∈ ℝ*) |
| 7 | 3, 6 | sylan2 593 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ (𝐴
+𝑒 -𝑒𝐵) ∈
ℝ*) |
| 8 | | simpr 484 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ 𝐵 ∈
ℝ) |
| 9 | | xleadd1 13297 |
. . . . 5
⊢ ((0
∈ ℝ* ∧ (𝐴 +𝑒
-𝑒𝐵)
∈ ℝ* ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 +𝑒
-𝑒𝐵)
↔ (0 +𝑒 𝐵) ≤ ((𝐴 +𝑒
-𝑒𝐵)
+𝑒 𝐵))) |
| 10 | 2, 7, 8, 9 | mp3an2i 1468 |
. . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ (0 ≤ (𝐴
+𝑒 -𝑒𝐵) ↔ (0 +𝑒 𝐵) ≤ ((𝐴 +𝑒
-𝑒𝐵)
+𝑒 𝐵))) |
| 11 | 3 | adantl 481 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ 𝐵 ∈
ℝ*) |
| 12 | | xaddlid 13284 |
. . . . . 6
⊢ (𝐵 ∈ ℝ*
→ (0 +𝑒 𝐵) = 𝐵) |
| 13 | 11, 12 | syl 17 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ (0 +𝑒 𝐵) = 𝐵) |
| 14 | | xnpcan 13294 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ ((𝐴
+𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴) |
| 15 | 13, 14 | breq12d 5156 |
. . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ ((0 +𝑒 𝐵) ≤ ((𝐴 +𝑒
-𝑒𝐵)
+𝑒 𝐵)
↔ 𝐵 ≤ 𝐴)) |
| 16 | 10, 15 | bitrd 279 |
. . 3
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ (0 ≤ (𝐴
+𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) |
| 17 | | pnfxr 11315 |
. . . . . . 7
⊢ +∞
∈ ℝ* |
| 18 | | xrletri3 13196 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ +∞ ∈ ℝ*) → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴))) |
| 19 | 17, 18 | mpan2 691 |
. . . . . 6
⊢ (𝐴 ∈ ℝ*
→ (𝐴 = +∞ ↔
(𝐴 ≤ +∞ ∧
+∞ ≤ 𝐴))) |
| 20 | | mnflt0 13167 |
. . . . . . . . . . 11
⊢ -∞
< 0 |
| 21 | | mnfxr 11318 |
. . . . . . . . . . . 12
⊢ -∞
∈ ℝ* |
| 22 | | xrltnle 11328 |
. . . . . . . . . . . 12
⊢
((-∞ ∈ ℝ* ∧ 0 ∈
ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤
-∞)) |
| 23 | 21, 2, 22 | mp2an 692 |
. . . . . . . . . . 11
⊢ (-∞
< 0 ↔ ¬ 0 ≤ -∞) |
| 24 | 20, 23 | mpbi 230 |
. . . . . . . . . 10
⊢ ¬ 0
≤ -∞ |
| 25 | | xaddmnf1 13270 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ*
∧ 𝐴 ≠ +∞)
→ (𝐴
+𝑒 -∞) = -∞) |
| 26 | 25 | breq2d 5155 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ*
∧ 𝐴 ≠ +∞)
→ (0 ≤ (𝐴
+𝑒 -∞) ↔ 0 ≤ -∞)) |
| 27 | 24, 26 | mtbiri 327 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ*
∧ 𝐴 ≠ +∞)
→ ¬ 0 ≤ (𝐴
+𝑒 -∞)) |
| 28 | 27 | ex 412 |
. . . . . . . 8
⊢ (𝐴 ∈ ℝ*
→ (𝐴 ≠ +∞
→ ¬ 0 ≤ (𝐴
+𝑒 -∞))) |
| 29 | 28 | necon4ad 2959 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ*
→ (0 ≤ (𝐴
+𝑒 -∞) → 𝐴 = +∞)) |
| 30 | | 0le0 12367 |
. . . . . . . 8
⊢ 0 ≤
0 |
| 31 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝐴 = +∞ → (𝐴 +𝑒 -∞)
= (+∞ +𝑒 -∞)) |
| 32 | | pnfaddmnf 13272 |
. . . . . . . . 9
⊢ (+∞
+𝑒 -∞) = 0 |
| 33 | 31, 32 | eqtrdi 2793 |
. . . . . . . 8
⊢ (𝐴 = +∞ → (𝐴 +𝑒 -∞)
= 0) |
| 34 | 30, 33 | breqtrrid 5181 |
. . . . . . 7
⊢ (𝐴 = +∞ → 0 ≤ (𝐴 +𝑒
-∞)) |
| 35 | 29, 34 | impbid1 225 |
. . . . . 6
⊢ (𝐴 ∈ ℝ*
→ (0 ≤ (𝐴
+𝑒 -∞) ↔ 𝐴 = +∞)) |
| 36 | | pnfge 13172 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ*
→ 𝐴 ≤
+∞) |
| 37 | 36 | biantrurd 532 |
. . . . . 6
⊢ (𝐴 ∈ ℝ*
→ (+∞ ≤ 𝐴
↔ (𝐴 ≤ +∞
∧ +∞ ≤ 𝐴))) |
| 38 | 19, 35, 37 | 3bitr4d 311 |
. . . . 5
⊢ (𝐴 ∈ ℝ*
→ (0 ≤ (𝐴
+𝑒 -∞) ↔ +∞ ≤ 𝐴)) |
| 39 | 38 | adantr 480 |
. . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = +∞) →
(0 ≤ (𝐴
+𝑒 -∞) ↔ +∞ ≤ 𝐴)) |
| 40 | | xnegeq 13249 |
. . . . . . . 8
⊢ (𝐵 = +∞ →
-𝑒𝐵 =
-𝑒+∞) |
| 41 | | xnegpnf 13251 |
. . . . . . . 8
⊢
-𝑒+∞ = -∞ |
| 42 | 40, 41 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝐵 = +∞ →
-𝑒𝐵 =
-∞) |
| 43 | 42 | adantl 481 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = +∞) →
-𝑒𝐵 =
-∞) |
| 44 | 43 | oveq2d 7447 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = +∞) →
(𝐴 +𝑒
-𝑒𝐵) =
(𝐴 +𝑒
-∞)) |
| 45 | 44 | breq2d 5155 |
. . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = +∞) →
(0 ≤ (𝐴
+𝑒 -𝑒𝐵) ↔ 0 ≤ (𝐴 +𝑒
-∞))) |
| 46 | | breq1 5146 |
. . . . 5
⊢ (𝐵 = +∞ → (𝐵 ≤ 𝐴 ↔ +∞ ≤ 𝐴)) |
| 47 | 46 | adantl 481 |
. . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = +∞) →
(𝐵 ≤ 𝐴 ↔ +∞ ≤ 𝐴)) |
| 48 | 39, 45, 47 | 3bitr4d 311 |
. . 3
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = +∞) →
(0 ≤ (𝐴
+𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) |
| 49 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝐴 = -∞ → (𝐴 +𝑒 +∞)
= (-∞ +𝑒 +∞)) |
| 50 | | mnfaddpnf 13273 |
. . . . . . . . . 10
⊢ (-∞
+𝑒 +∞) = 0 |
| 51 | 49, 50 | eqtrdi 2793 |
. . . . . . . . 9
⊢ (𝐴 = -∞ → (𝐴 +𝑒 +∞)
= 0) |
| 52 | 51 | adantl 481 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐴 = -∞) →
(𝐴 +𝑒
+∞) = 0) |
| 53 | 30, 52 | breqtrrid 5181 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐴 = -∞) →
0 ≤ (𝐴
+𝑒 +∞)) |
| 54 | | 0lepnf 13175 |
. . . . . . . 8
⊢ 0 ≤
+∞ |
| 55 | | xaddpnf1 13268 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐴 ≠ -∞)
→ (𝐴
+𝑒 +∞) = +∞) |
| 56 | 54, 55 | breqtrrid 5181 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐴 ≠ -∞)
→ 0 ≤ (𝐴
+𝑒 +∞)) |
| 57 | 53, 56 | pm2.61dane 3029 |
. . . . . 6
⊢ (𝐴 ∈ ℝ*
→ 0 ≤ (𝐴
+𝑒 +∞)) |
| 58 | | mnfle 13177 |
. . . . . 6
⊢ (𝐴 ∈ ℝ*
→ -∞ ≤ 𝐴) |
| 59 | 57, 58 | 2thd 265 |
. . . . 5
⊢ (𝐴 ∈ ℝ*
→ (0 ≤ (𝐴
+𝑒 +∞) ↔ -∞ ≤ 𝐴)) |
| 60 | 59 | adantr 480 |
. . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = -∞) →
(0 ≤ (𝐴
+𝑒 +∞) ↔ -∞ ≤ 𝐴)) |
| 61 | | xnegeq 13249 |
. . . . . . . 8
⊢ (𝐵 = -∞ →
-𝑒𝐵 =
-𝑒-∞) |
| 62 | | xnegmnf 13252 |
. . . . . . . 8
⊢
-𝑒-∞ = +∞ |
| 63 | 61, 62 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝐵 = -∞ →
-𝑒𝐵 =
+∞) |
| 64 | 63 | adantl 481 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = -∞) →
-𝑒𝐵 =
+∞) |
| 65 | 64 | oveq2d 7447 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = -∞) →
(𝐴 +𝑒
-𝑒𝐵) =
(𝐴 +𝑒
+∞)) |
| 66 | 65 | breq2d 5155 |
. . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = -∞) →
(0 ≤ (𝐴
+𝑒 -𝑒𝐵) ↔ 0 ≤ (𝐴 +𝑒
+∞))) |
| 67 | | breq1 5146 |
. . . . 5
⊢ (𝐵 = -∞ → (𝐵 ≤ 𝐴 ↔ -∞ ≤ 𝐴)) |
| 68 | 67 | adantl 481 |
. . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = -∞) →
(𝐵 ≤ 𝐴 ↔ -∞ ≤ 𝐴)) |
| 69 | 60, 66, 68 | 3bitr4d 311 |
. . 3
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 = -∞) →
(0 ≤ (𝐴
+𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) |
| 70 | 16, 48, 69 | 3jaodan 1433 |
. 2
⊢ ((𝐴 ∈ ℝ*
∧ (𝐵 ∈ ℝ
∨ 𝐵 = +∞ ∨
𝐵 = -∞)) → (0
≤ (𝐴
+𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) |
| 71 | 1, 70 | sylan2b 594 |
1
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (0 ≤ (𝐴 +𝑒
-𝑒𝐵)
↔ 𝐵 ≤ 𝐴)) |