Proof of Theorem xltnegi
| Step | Hyp | Ref
| Expression |
| 1 | | elxr 13158 |
. . 3
⊢ (𝐴 ∈ ℝ*
↔ (𝐴 ∈ ℝ
∨ 𝐴 = +∞ ∨
𝐴 =
-∞)) |
| 2 | | elxr 13158 |
. . . . . 6
⊢ (𝐵 ∈ ℝ*
↔ (𝐵 ∈ ℝ
∨ 𝐵 = +∞ ∨
𝐵 =
-∞)) |
| 3 | | ltneg 11763 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴)) |
| 4 | | rexneg 13253 |
. . . . . . . . . 10
⊢ (𝐵 ∈ ℝ →
-𝑒𝐵 =
-𝐵) |
| 5 | | rexneg 13253 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℝ →
-𝑒𝐴 =
-𝐴) |
| 6 | 4, 5 | breqan12rd 5160 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
(-𝑒𝐵
< -𝑒𝐴
↔ -𝐵 < -𝐴)) |
| 7 | 3, 6 | bitr4d 282 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴)) |
| 8 | 7 | biimpd 229 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → -𝑒𝐵 < -𝑒𝐴)) |
| 9 | | xnegeq 13249 |
. . . . . . . . . . 11
⊢ (𝐵 = +∞ →
-𝑒𝐵 =
-𝑒+∞) |
| 10 | | xnegpnf 13251 |
. . . . . . . . . . 11
⊢
-𝑒+∞ = -∞ |
| 11 | 9, 10 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ (𝐵 = +∞ →
-𝑒𝐵 =
-∞) |
| 12 | 11 | adantl 481 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) →
-𝑒𝐵 =
-∞) |
| 13 | | renegcl 11572 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℝ → -𝐴 ∈
ℝ) |
| 14 | 5, 13 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ℝ →
-𝑒𝐴
∈ ℝ) |
| 15 | 14 | mnfltd 13166 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℝ → -∞
< -𝑒𝐴) |
| 16 | 15 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -∞
< -𝑒𝐴) |
| 17 | 12, 16 | eqbrtrd 5165 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) →
-𝑒𝐵 <
-𝑒𝐴) |
| 18 | 17 | a1d 25 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → -𝑒𝐵 < -𝑒𝐴)) |
| 19 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 = -∞) |
| 20 | 19 | breq2d 5155 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞)) |
| 21 | | rexr 11307 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ℝ → 𝐴 ∈
ℝ*) |
| 22 | | nltmnf 13171 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ℝ*
→ ¬ 𝐴 <
-∞) |
| 23 | 21, 22 | syl 17 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℝ → ¬
𝐴 <
-∞) |
| 24 | 23 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞) |
| 25 | 24 | pm2.21d 121 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < -∞ →
-𝑒𝐵 <
-𝑒𝐴)) |
| 26 | 20, 25 | sylbid 240 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → -𝑒𝐵 < -𝑒𝐴)) |
| 27 | 8, 18, 26 | 3jaodan 1433 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → -𝑒𝐵 < -𝑒𝐴)) |
| 28 | 2, 27 | sylan2b 594 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*)
→ (𝐴 < 𝐵 →
-𝑒𝐵 <
-𝑒𝐴)) |
| 29 | 28 | expimpd 453 |
. . . 4
⊢ (𝐴 ∈ ℝ → ((𝐵 ∈ ℝ*
∧ 𝐴 < 𝐵) →
-𝑒𝐵 <
-𝑒𝐴)) |
| 30 | | simpl 482 |
. . . . . . 7
⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*)
→ 𝐴 =
+∞) |
| 31 | 30 | breq1d 5153 |
. . . . . 6
⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*)
→ (𝐴 < 𝐵 ↔ +∞ < 𝐵)) |
| 32 | | pnfnlt 13170 |
. . . . . . . 8
⊢ (𝐵 ∈ ℝ*
→ ¬ +∞ < 𝐵) |
| 33 | 32 | adantl 481 |
. . . . . . 7
⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*)
→ ¬ +∞ < 𝐵) |
| 34 | 33 | pm2.21d 121 |
. . . . . 6
⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*)
→ (+∞ < 𝐵
→ -𝑒𝐵 < -𝑒𝐴)) |
| 35 | 31, 34 | sylbid 240 |
. . . . 5
⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*)
→ (𝐴 < 𝐵 →
-𝑒𝐵 <
-𝑒𝐴)) |
| 36 | 35 | expimpd 453 |
. . . 4
⊢ (𝐴 = +∞ → ((𝐵 ∈ ℝ*
∧ 𝐴 < 𝐵) →
-𝑒𝐵 <
-𝑒𝐴)) |
| 37 | | breq1 5146 |
. . . . . 6
⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵)) |
| 38 | 37 | anbi2d 630 |
. . . . 5
⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ*
∧ 𝐴 < 𝐵) ↔ (𝐵 ∈ ℝ* ∧ -∞
< 𝐵))) |
| 39 | | renegcl 11572 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ ℝ → -𝐵 ∈
ℝ) |
| 40 | 4, 39 | eqeltrd 2841 |
. . . . . . . . . 10
⊢ (𝐵 ∈ ℝ →
-𝑒𝐵
∈ ℝ) |
| 41 | 40 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐵 ∈ ℝ ∧ -∞
< 𝐵) →
-𝑒𝐵
∈ ℝ) |
| 42 | 41 | ltpnfd 13163 |
. . . . . . . 8
⊢ ((𝐵 ∈ ℝ ∧ -∞
< 𝐵) →
-𝑒𝐵 <
+∞) |
| 43 | 11 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐵 = +∞ ∧ -∞ <
𝐵) →
-𝑒𝐵 =
-∞) |
| 44 | | mnfltpnf 13168 |
. . . . . . . . 9
⊢ -∞
< +∞ |
| 45 | 43, 44 | eqbrtrdi 5182 |
. . . . . . . 8
⊢ ((𝐵 = +∞ ∧ -∞ <
𝐵) →
-𝑒𝐵 <
+∞) |
| 46 | | breq2 5147 |
. . . . . . . . . 10
⊢ (𝐵 = -∞ → (-∞
< 𝐵 ↔ -∞ <
-∞)) |
| 47 | | mnfxr 11318 |
. . . . . . . . . . . 12
⊢ -∞
∈ ℝ* |
| 48 | | nltmnf 13171 |
. . . . . . . . . . . 12
⊢ (-∞
∈ ℝ* → ¬ -∞ <
-∞) |
| 49 | 47, 48 | ax-mp 5 |
. . . . . . . . . . 11
⊢ ¬
-∞ < -∞ |
| 50 | 49 | pm2.21i 119 |
. . . . . . . . . 10
⊢ (-∞
< -∞ → -𝑒𝐵 < +∞) |
| 51 | 46, 50 | biimtrdi 253 |
. . . . . . . . 9
⊢ (𝐵 = -∞ → (-∞
< 𝐵 →
-𝑒𝐵 <
+∞)) |
| 52 | 51 | imp 406 |
. . . . . . . 8
⊢ ((𝐵 = -∞ ∧ -∞ <
𝐵) →
-𝑒𝐵 <
+∞) |
| 53 | 42, 45, 52 | 3jaoian 1432 |
. . . . . . 7
⊢ (((𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞) ∧ -∞ <
𝐵) →
-𝑒𝐵 <
+∞) |
| 54 | 2, 53 | sylanb 581 |
. . . . . 6
⊢ ((𝐵 ∈ ℝ*
∧ -∞ < 𝐵)
→ -𝑒𝐵 < +∞) |
| 55 | | xnegeq 13249 |
. . . . . . . 8
⊢ (𝐴 = -∞ →
-𝑒𝐴 =
-𝑒-∞) |
| 56 | | xnegmnf 13252 |
. . . . . . . 8
⊢
-𝑒-∞ = +∞ |
| 57 | 55, 56 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝐴 = -∞ →
-𝑒𝐴 =
+∞) |
| 58 | 57 | breq2d 5155 |
. . . . . 6
⊢ (𝐴 = -∞ →
(-𝑒𝐵
< -𝑒𝐴
↔ -𝑒𝐵 < +∞)) |
| 59 | 54, 58 | imbitrrid 246 |
. . . . 5
⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ*
∧ -∞ < 𝐵)
→ -𝑒𝐵 < -𝑒𝐴)) |
| 60 | 38, 59 | sylbid 240 |
. . . 4
⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ*
∧ 𝐴 < 𝐵) →
-𝑒𝐵 <
-𝑒𝐴)) |
| 61 | 29, 36, 60 | 3jaoi 1430 |
. . 3
⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → ((𝐵 ∈ ℝ*
∧ 𝐴 < 𝐵) →
-𝑒𝐵 <
-𝑒𝐴)) |
| 62 | 1, 61 | sylbi 217 |
. 2
⊢ (𝐴 ∈ ℝ*
→ ((𝐵 ∈
ℝ* ∧ 𝐴
< 𝐵) →
-𝑒𝐵 <
-𝑒𝐴)) |
| 63 | 62 | 3impib 1117 |
1
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
< 𝐵) →
-𝑒𝐵 <
-𝑒𝐴) |