| Step | Hyp | Ref
| Expression |
| 1 | | lmxrge0.j |
. . . . . . 7
⊢ 𝐽 =
(TopOpen‘(ℝ*𝑠 ↾s
(0[,]+∞))) |
| 2 | | eqid 2737 |
. . . . . . . 8
⊢
(ℝ*𝑠 ↾s
(0[,]+∞)) = (ℝ*𝑠 ↾s
(0[,]+∞)) |
| 3 | | xrstopn 23216 |
. . . . . . . 8
⊢
(ordTop‘ ≤ ) =
(TopOpen‘ℝ*𝑠) |
| 4 | 2, 3 | resstopn 23194 |
. . . . . . 7
⊢
((ordTop‘ ≤ ) ↾t (0[,]+∞)) =
(TopOpen‘(ℝ*𝑠 ↾s
(0[,]+∞))) |
| 5 | 1, 4 | eqtr4i 2768 |
. . . . . 6
⊢ 𝐽 = ((ordTop‘ ≤ )
↾t (0[,]+∞)) |
| 6 | | letopon 23213 |
. . . . . . 7
⊢
(ordTop‘ ≤ ) ∈
(TopOn‘ℝ*) |
| 7 | | iccssxr 13470 |
. . . . . . 7
⊢
(0[,]+∞) ⊆ ℝ* |
| 8 | | resttopon 23169 |
. . . . . . 7
⊢
(((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧
(0[,]+∞) ⊆ ℝ*) → ((ordTop‘ ≤ )
↾t (0[,]+∞)) ∈
(TopOn‘(0[,]+∞))) |
| 9 | 6, 7, 8 | mp2an 692 |
. . . . . 6
⊢
((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈
(TopOn‘(0[,]+∞)) |
| 10 | 5, 9 | eqeltri 2837 |
. . . . 5
⊢ 𝐽 ∈
(TopOn‘(0[,]+∞)) |
| 11 | 10 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐽 ∈
(TopOn‘(0[,]+∞))) |
| 12 | | nnuz 12921 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
| 13 | | 1zzd 12648 |
. . . 4
⊢ (𝜑 → 1 ∈
ℤ) |
| 14 | | lmxrge0.6 |
. . . 4
⊢ (𝜑 → 𝐹:ℕ⟶(0[,]+∞)) |
| 15 | | lmxrge0.7 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 𝐴) |
| 16 | 11, 12, 13, 14, 15 | lmbrf 23268 |
. . 3
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)+∞ ↔ (+∞
∈ (0[,]+∞) ∧ ∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)))) |
| 17 | | 0xr 11308 |
. . . . 5
⊢ 0 ∈
ℝ* |
| 18 | | pnfxr 11315 |
. . . . 5
⊢ +∞
∈ ℝ* |
| 19 | | 0lepnf 13175 |
. . . . 5
⊢ 0 ≤
+∞ |
| 20 | | ubicc2 13505 |
. . . . 5
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0
≤ +∞) → +∞ ∈ (0[,]+∞)) |
| 21 | 17, 18, 19, 20 | mp3an 1463 |
. . . 4
⊢ +∞
∈ (0[,]+∞) |
| 22 | 21 | biantrur 530 |
. . 3
⊢
(∀𝑎 ∈
𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑙)𝐴 ∈ 𝑎) ↔ (+∞ ∈ (0[,]+∞)
∧ ∀𝑎 ∈
𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑙)𝐴 ∈ 𝑎))) |
| 23 | 16, 22 | bitr4di 289 |
. 2
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)+∞ ↔ ∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎))) |
| 24 | | rexr 11307 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ*) |
| 25 | 18 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ → +∞
∈ ℝ*) |
| 26 | | ltpnf 13162 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) |
| 27 | | ubioc1 13440 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ*
∧ +∞ ∈ ℝ* ∧ 𝑥 < +∞) → +∞ ∈ (𝑥(,]+∞)) |
| 28 | 24, 25, 26, 27 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ → +∞
∈ (𝑥(,]+∞)) |
| 29 | | 0ltpnf 13164 |
. . . . . . . . . 10
⊢ 0 <
+∞ |
| 30 | | ubioc1 13440 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0
< +∞) → +∞ ∈ (0(,]+∞)) |
| 31 | 17, 18, 29, 30 | mp3an 1463 |
. . . . . . . . 9
⊢ +∞
∈ (0(,]+∞) |
| 32 | 28, 31 | jctir 520 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ → (+∞
∈ (𝑥(,]+∞) ∧
+∞ ∈ (0(,]+∞))) |
| 33 | | elin 3967 |
. . . . . . . 8
⊢ (+∞
∈ ((𝑥(,]+∞)
∩ (0(,]+∞)) ↔ (+∞ ∈ (𝑥(,]+∞) ∧ +∞ ∈
(0(,]+∞))) |
| 34 | 32, 33 | sylibr 234 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ → +∞
∈ ((𝑥(,]+∞)
∩ (0(,]+∞))) |
| 35 | 34 | ad2antlr 727 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) → +∞ ∈ ((𝑥(,]+∞) ∩
(0(,]+∞))) |
| 36 | | letop 23214 |
. . . . . . . . . . 11
⊢
(ordTop‘ ≤ ) ∈ Top |
| 37 | | ovex 7464 |
. . . . . . . . . . 11
⊢
(0[,]+∞) ∈ V |
| 38 | | iocpnfordt 23223 |
. . . . . . . . . . . 12
⊢ (𝑥(,]+∞) ∈
(ordTop‘ ≤ ) |
| 39 | | iocpnfordt 23223 |
. . . . . . . . . . . 12
⊢
(0(,]+∞) ∈ (ordTop‘ ≤ ) |
| 40 | | inopn 22905 |
. . . . . . . . . . . 12
⊢
(((ordTop‘ ≤ ) ∈ Top ∧ (𝑥(,]+∞) ∈ (ordTop‘ ≤ )
∧ (0(,]+∞) ∈ (ordTop‘ ≤ )) → ((𝑥(,]+∞) ∩ (0(,]+∞)) ∈
(ordTop‘ ≤ )) |
| 41 | 36, 38, 39, 40 | mp3an 1463 |
. . . . . . . . . . 11
⊢ ((𝑥(,]+∞) ∩
(0(,]+∞)) ∈ (ordTop‘ ≤ ) |
| 42 | | elrestr 17473 |
. . . . . . . . . . 11
⊢
(((ordTop‘ ≤ ) ∈ Top ∧ (0[,]+∞) ∈ V ∧
((𝑥(,]+∞) ∩
(0(,]+∞)) ∈ (ordTop‘ ≤ )) → (((𝑥(,]+∞) ∩ (0(,]+∞)) ∩
(0[,]+∞)) ∈ ((ordTop‘ ≤ ) ↾t
(0[,]+∞))) |
| 43 | 36, 37, 41, 42 | mp3an 1463 |
. . . . . . . . . 10
⊢ (((𝑥(,]+∞) ∩
(0(,]+∞)) ∩ (0[,]+∞)) ∈ ((ordTop‘ ≤ )
↾t (0[,]+∞)) |
| 44 | | inss2 4238 |
. . . . . . . . . . . . 13
⊢ ((𝑥(,]+∞) ∩
(0(,]+∞)) ⊆ (0(,]+∞) |
| 45 | | iocssicc 13477 |
. . . . . . . . . . . . 13
⊢
(0(,]+∞) ⊆ (0[,]+∞) |
| 46 | 44, 45 | sstri 3993 |
. . . . . . . . . . . 12
⊢ ((𝑥(,]+∞) ∩
(0(,]+∞)) ⊆ (0[,]+∞) |
| 47 | | sseqin2 4223 |
. . . . . . . . . . . 12
⊢ (((𝑥(,]+∞) ∩
(0(,]+∞)) ⊆ (0[,]+∞) ↔ ((0[,]+∞) ∩ ((𝑥(,]+∞) ∩
(0(,]+∞))) = ((𝑥(,]+∞) ∩
(0(,]+∞))) |
| 48 | 46, 47 | mpbi 230 |
. . . . . . . . . . 11
⊢
((0[,]+∞) ∩ ((𝑥(,]+∞) ∩ (0(,]+∞))) = ((𝑥(,]+∞) ∩
(0(,]+∞)) |
| 49 | | incom 4209 |
. . . . . . . . . . 11
⊢
((0[,]+∞) ∩ ((𝑥(,]+∞) ∩ (0(,]+∞))) =
(((𝑥(,]+∞) ∩
(0(,]+∞)) ∩ (0[,]+∞)) |
| 50 | 48, 49 | eqtr3i 2767 |
. . . . . . . . . 10
⊢ ((𝑥(,]+∞) ∩
(0(,]+∞)) = (((𝑥(,]+∞) ∩ (0(,]+∞)) ∩
(0[,]+∞)) |
| 51 | 43, 50, 5 | 3eltr4i 2854 |
. . . . . . . . 9
⊢ ((𝑥(,]+∞) ∩
(0(,]+∞)) ∈ 𝐽 |
| 52 | 51 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑥(,]+∞) ∩ (0(,]+∞)) ∈
𝐽) |
| 53 | | eleq2 2830 |
. . . . . . . . . . 11
⊢ (𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞)) →
(+∞ ∈ 𝑎 ↔
+∞ ∈ ((𝑥(,]+∞) ∩
(0(,]+∞)))) |
| 54 | 53 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) →
(+∞ ∈ 𝑎 ↔
+∞ ∈ ((𝑥(,]+∞) ∩
(0(,]+∞)))) |
| 55 | 54 | biimprd 248 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) →
(+∞ ∈ ((𝑥(,]+∞) ∩ (0(,]+∞)) →
+∞ ∈ 𝑎)) |
| 56 | | simp-5r 786 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) ∧ 𝐴 ∈ 𝑎) → 𝑥 ∈ ℝ) |
| 57 | 56 | rexrd 11311 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) ∧ 𝐴 ∈ 𝑎) → 𝑥 ∈ ℝ*) |
| 58 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) ∧ 𝐴 ∈ 𝑎) → 𝐴 ∈ 𝑎) |
| 59 | | simp-4r 784 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) ∧ 𝐴 ∈ 𝑎) → 𝑎 = ((𝑥(,]+∞) ∩
(0(,]+∞))) |
| 60 | 58, 59 | eleqtrd 2843 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) ∧ 𝐴 ∈ 𝑎) → 𝐴 ∈ ((𝑥(,]+∞) ∩
(0(,]+∞))) |
| 61 | | elin 3967 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ((𝑥(,]+∞) ∩ (0(,]+∞)) ↔
(𝐴 ∈ (𝑥(,]+∞) ∧ 𝐴 ∈
(0(,]+∞))) |
| 62 | 61 | simplbi 497 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ ((𝑥(,]+∞) ∩ (0(,]+∞)) →
𝐴 ∈ (𝑥(,]+∞)) |
| 63 | 60, 62 | syl 17 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) ∧ 𝐴 ∈ 𝑎) → 𝐴 ∈ (𝑥(,]+∞)) |
| 64 | | elioc1 13429 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ*
∧ +∞ ∈ ℝ*) → (𝐴 ∈ (𝑥(,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 𝑥 < 𝐴 ∧ 𝐴 ≤ +∞))) |
| 65 | 18, 64 | mpan2 691 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℝ*
→ (𝐴 ∈ (𝑥(,]+∞) ↔ (𝐴 ∈ ℝ*
∧ 𝑥 < 𝐴 ∧ 𝐴 ≤ +∞))) |
| 66 | 65 | biimpa 476 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ*
∧ 𝐴 ∈ (𝑥(,]+∞)) → (𝐴 ∈ ℝ*
∧ 𝑥 < 𝐴 ∧ 𝐴 ≤ +∞)) |
| 67 | 66 | simp2d 1144 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ*
∧ 𝐴 ∈ (𝑥(,]+∞)) → 𝑥 < 𝐴) |
| 68 | 57, 63, 67 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) ∧ 𝐴 ∈ 𝑎) → 𝑥 < 𝐴) |
| 69 | 68 | ex 412 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) → (𝐴 ∈ 𝑎 → 𝑥 < 𝐴)) |
| 70 | 69 | ralimdva 3167 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) →
(∀𝑘 ∈
(ℤ≥‘𝑙)𝐴 ∈ 𝑎 → ∀𝑘 ∈ (ℤ≥‘𝑙)𝑥 < 𝐴)) |
| 71 | 70 | reximdva 3168 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) →
(∃𝑙 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑙)𝐴 ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝑥 < 𝐴)) |
| 72 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑙 → (ℤ≥‘𝑗) =
(ℤ≥‘𝑙)) |
| 73 | 72 | raleqdv 3326 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑙 → (∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴 ↔ ∀𝑘 ∈ (ℤ≥‘𝑙)𝑥 < 𝐴)) |
| 74 | 73 | cbvrexvw 3238 |
. . . . . . . . . 10
⊢
(∃𝑗 ∈
ℕ ∀𝑘 ∈
(ℤ≥‘𝑗)𝑥 < 𝐴 ↔ ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝑥 < 𝐴) |
| 75 | 71, 74 | imbitrrdi 252 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) →
(∃𝑙 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑙)𝐴 ∈ 𝑎 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴)) |
| 76 | 55, 75 | imim12d 81 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) →
((+∞ ∈ 𝑎 →
∃𝑙 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑙)𝐴 ∈ 𝑎) → (+∞ ∈ ((𝑥(,]+∞) ∩
(0(,]+∞)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴))) |
| 77 | 52, 76 | rspcimdv 3612 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎) → (+∞ ∈ ((𝑥(,]+∞) ∩
(0(,]+∞)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴))) |
| 78 | 77 | imp 406 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) → (+∞ ∈ ((𝑥(,]+∞) ∩
(0(,]+∞)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴)) |
| 79 | 35, 78 | mpd 15 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) |
| 80 | 79 | ex 412 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴)) |
| 81 | 80 | ralrimdva 3154 |
. . 3
⊢ (𝜑 → (∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴)) |
| 82 | | simplll 775 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐽) ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) ∧ +∞ ∈ 𝑎) → 𝜑) |
| 83 | | simpllr 776 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐽) ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) ∧ +∞ ∈ 𝑎) → 𝑎 ∈ 𝐽) |
| 84 | | simpr 484 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐽) ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) ∧ +∞ ∈ 𝑎) → +∞ ∈ 𝑎) |
| 85 | 1 | pnfneige0 33950 |
. . . . . . 7
⊢ ((𝑎 ∈ 𝐽 ∧ +∞ ∈ 𝑎) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑎) |
| 86 | 83, 84, 85 | syl2anc 584 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐽) ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) ∧ +∞ ∈ 𝑎) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑎) |
| 87 | | simplr 769 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐽) ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) ∧ +∞ ∈ 𝑎) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) |
| 88 | | r19.29r 3116 |
. . . . . . . 8
⊢
((∃𝑥 ∈
ℝ (𝑥(,]+∞)
⊆ 𝑎 ∧
∀𝑥 ∈ ℝ
∃𝑗 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑗)𝑥 < 𝐴) → ∃𝑥 ∈ ℝ ((𝑥(,]+∞) ⊆ 𝑎 ∧ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴)) |
| 89 | | simp-4l 783 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) → 𝜑) |
| 90 | | uznnssnn 12937 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑙 ∈ ℕ →
(ℤ≥‘𝑙) ⊆ ℕ) |
| 91 | 90 | ad2antlr 727 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) →
(ℤ≥‘𝑙) ⊆ ℕ) |
| 92 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) → 𝑘 ∈ (ℤ≥‘𝑙)) |
| 93 | 91, 92 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) → 𝑘 ∈ ℕ) |
| 94 | 89, 93 | jca 511 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) → (𝜑 ∧ 𝑘 ∈ ℕ)) |
| 95 | | simp-4r 784 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) → 𝑥 ∈ ℝ) |
| 96 | | simpllr 776 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) → (𝑥(,]+∞) ⊆ 𝑎) |
| 97 | | simplr 769 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑥 < 𝐴) → (𝑥(,]+∞) ⊆ 𝑎) |
| 98 | | simplr 769 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < 𝐴) → 𝑥 ∈ ℝ) |
| 99 | 98 | rexrd 11311 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < 𝐴) → 𝑥 ∈ ℝ*) |
| 100 | 14 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞)) |
| 101 | 15, 100 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) |
| 102 | 7, 101 | sselid 3981 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈
ℝ*) |
| 103 | 102 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < 𝐴) → 𝐴 ∈
ℝ*) |
| 104 | | simpr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < 𝐴) → 𝑥 < 𝐴) |
| 105 | | pnfge 13172 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℝ*
→ 𝐴 ≤
+∞) |
| 106 | 103, 105 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < 𝐴) → 𝐴 ≤ +∞) |
| 107 | 65 | biimpar 477 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ*
∧ (𝐴 ∈
ℝ* ∧ 𝑥
< 𝐴 ∧ 𝐴 ≤ +∞)) → 𝐴 ∈ (𝑥(,]+∞)) |
| 108 | 99, 103, 104, 106, 107 | syl13anc 1374 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < 𝐴) → 𝐴 ∈ (𝑥(,]+∞)) |
| 109 | 108 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑥 < 𝐴) → 𝐴 ∈ (𝑥(,]+∞)) |
| 110 | 97, 109 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑥 < 𝐴) → 𝐴 ∈ 𝑎) |
| 111 | 110 | ex 412 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) → (𝑥 < 𝐴 → 𝐴 ∈ 𝑎)) |
| 112 | 94, 95, 96, 111 | syl21anc 838 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) → (𝑥 < 𝐴 → 𝐴 ∈ 𝑎)) |
| 113 | 112 | ralimdva 3167 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) → (∀𝑘 ∈
(ℤ≥‘𝑙)𝑥 < 𝐴 → ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) |
| 114 | 113 | reximdva 3168 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) → (∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝑥 < 𝐴 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) |
| 115 | 74, 114 | biimtrid 242 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) |
| 116 | 115 | expimpd 453 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (((𝑥(,]+∞) ⊆ 𝑎 ∧ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) |
| 117 | 116 | rexlimdva 3155 |
. . . . . . . 8
⊢ (𝜑 → (∃𝑥 ∈ ℝ ((𝑥(,]+∞) ⊆ 𝑎 ∧ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) |
| 118 | 88, 117 | syl5 34 |
. . . . . . 7
⊢ (𝜑 → ((∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑎 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑗)𝑥 < 𝐴) → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) |
| 119 | 118 | imp 406 |
. . . . . 6
⊢ ((𝜑 ∧ (∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑎 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴)) → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎) |
| 120 | 82, 86, 87, 119 | syl12anc 837 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐽) ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) ∧ +∞ ∈ 𝑎) → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎) |
| 121 | 120 | exp31 419 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴 → (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎))) |
| 122 | 121 | ralrimdva 3154 |
. . 3
⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑗)𝑥 < 𝐴 → ∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎))) |
| 123 | 81, 122 | impbid 212 |
. 2
⊢ (𝜑 → (∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎) ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴)) |
| 124 | 23, 123 | bitrd 279 |
1
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑗)𝑥 < 𝐴)) |