Step | Hyp | Ref
| Expression |
1 | | lmxrge0.j |
. . . . . . 7
⊢ 𝐽 =
(TopOpen‘(ℝ*𝑠 ↾s
(0[,]+∞))) |
2 | | eqid 2738 |
. . . . . . . 8
⊢
(ℝ*𝑠 ↾s
(0[,]+∞)) = (ℝ*𝑠 ↾s
(0[,]+∞)) |
3 | | xrstopn 21959 |
. . . . . . . 8
⊢
(ordTop‘ ≤ ) =
(TopOpen‘ℝ*𝑠) |
4 | 2, 3 | resstopn 21937 |
. . . . . . 7
⊢
((ordTop‘ ≤ ) ↾t (0[,]+∞)) =
(TopOpen‘(ℝ*𝑠 ↾s
(0[,]+∞))) |
5 | 1, 4 | eqtr4i 2764 |
. . . . . 6
⊢ 𝐽 = ((ordTop‘ ≤ )
↾t (0[,]+∞)) |
6 | | letopon 21956 |
. . . . . . 7
⊢
(ordTop‘ ≤ ) ∈
(TopOn‘ℝ*) |
7 | | iccssxr 12904 |
. . . . . . 7
⊢
(0[,]+∞) ⊆ ℝ* |
8 | | resttopon 21912 |
. . . . . . 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 2829 |
. . . . 5
⊢ 𝐽 ∈
(TopOn‘(0[,]+∞)) |
11 | 10 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐽 ∈
(TopOn‘(0[,]+∞))) |
12 | | nnuz 12363 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
13 | | 1zzd 12094 |
. . . 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 22011 |
. . 3
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)+∞ ↔ (+∞
∈ (0[,]+∞) ∧ ∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)))) |
17 | | 0xr 10766 |
. . . . 5
⊢ 0 ∈
ℝ* |
18 | | pnfxr 10773 |
. . . . 5
⊢ +∞
∈ ℝ* |
19 | | 0lepnf 12610 |
. . . . 5
⊢ 0 ≤
+∞ |
20 | | ubicc2 12939 |
. . . . 5
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0
≤ +∞) → +∞ ∈ (0[,]+∞)) |
21 | 17, 18, 19, 20 | mp3an 1462 |
. . . 4
⊢ +∞
∈ (0[,]+∞) |
22 | 21 | biantrur 534 |
. . 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 292 |
. 2
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)+∞ ↔ ∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎))) |
24 | | rexr 10765 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ*) |
25 | 18 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ → +∞
∈ ℝ*) |
26 | | ltpnf 12598 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) |
27 | | ubioc1 12874 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ*
∧ +∞ ∈ ℝ* ∧ 𝑥 < +∞) → +∞ ∈ (𝑥(,]+∞)) |
28 | 24, 25, 26, 27 | syl3anc 1372 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ → +∞
∈ (𝑥(,]+∞)) |
29 | | 0ltpnf 12600 |
. . . . . . . . . 10
⊢ 0 <
+∞ |
30 | | ubioc1 12874 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0
< +∞) → +∞ ∈ (0(,]+∞)) |
31 | 17, 18, 29, 30 | mp3an 1462 |
. . . . . . . . 9
⊢ +∞
∈ (0(,]+∞) |
32 | 28, 31 | jctir 524 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ → (+∞
∈ (𝑥(,]+∞) ∧
+∞ ∈ (0(,]+∞))) |
33 | | elin 3859 |
. . . . . . . 8
⊢ (+∞
∈ ((𝑥(,]+∞)
∩ (0(,]+∞)) ↔ (+∞ ∈ (𝑥(,]+∞) ∧ +∞ ∈
(0(,]+∞))) |
34 | 32, 33 | sylibr 237 |
. . . . . . 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 21957 |
. . . . . . . . . . 11
⊢
(ordTop‘ ≤ ) ∈ Top |
37 | | ovex 7203 |
. . . . . . . . . . 11
⊢
(0[,]+∞) ∈ V |
38 | | iocpnfordt 21966 |
. . . . . . . . . . . 12
⊢ (𝑥(,]+∞) ∈
(ordTop‘ ≤ ) |
39 | | iocpnfordt 21966 |
. . . . . . . . . . . 12
⊢
(0(,]+∞) ∈ (ordTop‘ ≤ ) |
40 | | inopn 21650 |
. . . . . . . . . . . 12
⊢
(((ordTop‘ ≤ ) ∈ Top ∧ (𝑥(,]+∞) ∈ (ordTop‘ ≤ )
∧ (0(,]+∞) ∈ (ordTop‘ ≤ )) → ((𝑥(,]+∞) ∩ (0(,]+∞)) ∈
(ordTop‘ ≤ )) |
41 | 36, 38, 39, 40 | mp3an 1462 |
. . . . . . . . . . 11
⊢ ((𝑥(,]+∞) ∩
(0(,]+∞)) ∈ (ordTop‘ ≤ ) |
42 | | elrestr 16805 |
. . . . . . . . . . 11
⊢
(((ordTop‘ ≤ ) ∈ Top ∧ (0[,]+∞) ∈ V ∧
((𝑥(,]+∞) ∩
(0(,]+∞)) ∈ (ordTop‘ ≤ )) → (((𝑥(,]+∞) ∩ (0(,]+∞)) ∩
(0[,]+∞)) ∈ ((ordTop‘ ≤ ) ↾t
(0[,]+∞))) |
43 | 36, 37, 41, 42 | mp3an 1462 |
. . . . . . . . . 10
⊢ (((𝑥(,]+∞) ∩
(0(,]+∞)) ∩ (0[,]+∞)) ∈ ((ordTop‘ ≤ )
↾t (0[,]+∞)) |
44 | | inss2 4120 |
. . . . . . . . . . . . 13
⊢ ((𝑥(,]+∞) ∩
(0(,]+∞)) ⊆ (0(,]+∞) |
45 | | iocssicc 12911 |
. . . . . . . . . . . . 13
⊢
(0(,]+∞) ⊆ (0[,]+∞) |
46 | 44, 45 | sstri 3886 |
. . . . . . . . . . . 12
⊢ ((𝑥(,]+∞) ∩
(0(,]+∞)) ⊆ (0[,]+∞) |
47 | | sseqin2 4106 |
. . . . . . . . . . . 12
⊢ (((𝑥(,]+∞) ∩
(0(,]+∞)) ⊆ (0[,]+∞) ↔ ((0[,]+∞) ∩ ((𝑥(,]+∞) ∩
(0(,]+∞))) = ((𝑥(,]+∞) ∩
(0(,]+∞))) |
48 | 46, 47 | mpbi 233 |
. . . . . . . . . . 11
⊢
((0[,]+∞) ∩ ((𝑥(,]+∞) ∩ (0(,]+∞))) = ((𝑥(,]+∞) ∩
(0(,]+∞)) |
49 | | incom 4091 |
. . . . . . . . . . 11
⊢
((0[,]+∞) ∩ ((𝑥(,]+∞) ∩ (0(,]+∞))) =
(((𝑥(,]+∞) ∩
(0(,]+∞)) ∩ (0[,]+∞)) |
50 | 48, 49 | eqtr3i 2763 |
. . . . . . . . . 10
⊢ ((𝑥(,]+∞) ∩
(0(,]+∞)) = (((𝑥(,]+∞) ∩ (0(,]+∞)) ∩
(0[,]+∞)) |
51 | 43, 50, 5 | 3eltr4i 2846 |
. . . . . . . . 9
⊢ ((𝑥(,]+∞) ∩
(0(,]+∞)) ∈ 𝐽 |
52 | 51 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑥(,]+∞) ∩ (0(,]+∞)) ∈
𝐽) |
53 | | eleq2 2821 |
. . . . . . . . . . 11
⊢ (𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞)) →
(+∞ ∈ 𝑎 ↔
+∞ ∈ ((𝑥(,]+∞) ∩
(0(,]+∞)))) |
54 | 53 | adantl 485 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) →
(+∞ ∈ 𝑎 ↔
+∞ ∈ ((𝑥(,]+∞) ∩
(0(,]+∞)))) |
55 | 54 | biimprd 251 |
. . . . . . . . 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 10769 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) ∧ 𝐴 ∈ 𝑎) → 𝑥 ∈ ℝ*) |
58 | | simpr 488 |
. . . . . . . . . . . . . . . 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 2835 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) ∧ 𝐴 ∈ 𝑎) → 𝐴 ∈ ((𝑥(,]+∞) ∩
(0(,]+∞))) |
61 | | elin 3859 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ((𝑥(,]+∞) ∩ (0(,]+∞)) ↔
(𝐴 ∈ (𝑥(,]+∞) ∧ 𝐴 ∈
(0(,]+∞))) |
62 | 61 | simplbi 501 |
. . . . . . . . . . . . . . 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 12863 |
. . . . . . . . . . . . . . . . 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 480 |
. . . . . . . . . . . . . . 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 587 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) ∧ 𝐴 ∈ 𝑎) → 𝑥 < 𝐴) |
69 | 68 | ex 416 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘𝑙)) → (𝐴 ∈ 𝑎 → 𝑥 < 𝐴)) |
70 | 69 | ralimdva 3091 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) ∧
𝑙 ∈ ℕ) →
(∀𝑘 ∈
(ℤ≥‘𝑙)𝐴 ∈ 𝑎 → ∀𝑘 ∈ (ℤ≥‘𝑙)𝑥 < 𝐴)) |
71 | 70 | reximdva 3184 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑎 = ((𝑥(,]+∞) ∩ (0(,]+∞))) →
(∃𝑙 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑙)𝐴 ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝑥 < 𝐴)) |
72 | | fveq2 6674 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑙 → (ℤ≥‘𝑗) =
(ℤ≥‘𝑙)) |
73 | 72 | raleqdv 3316 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑙 → (∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴 ↔ ∀𝑘 ∈ (ℤ≥‘𝑙)𝑥 < 𝐴)) |
74 | 73 | cbvrexvw 3350 |
. . . . . . . . . 10
⊢
(∃𝑗 ∈
ℕ ∀𝑘 ∈
(ℤ≥‘𝑗)𝑥 < 𝐴 ↔ ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝑥 < 𝐴) |
75 | 71, 74 | syl6ibr 255 |
. . . . . . . . 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 3516 |
. . . . . . 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 410 |
. . . . . 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 416 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑎 ∈ 𝐽 (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴)) |
81 | 80 | ralrimdva 3101 |
. . 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 488 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐽) ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) ∧ +∞ ∈ 𝑎) → +∞ ∈ 𝑎) |
85 | 1 | pnfneige0 31473 |
. . . . . . 7
⊢ ((𝑎 ∈ 𝐽 ∧ +∞ ∈ 𝑎) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑎) |
86 | 83, 84, 85 | syl2anc 587 |
. . . . . 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 3168 |
. . . . . . . 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 12377 |
. . . . . . . . . . . . . . . . 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 488 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) → 𝑘 ∈ (ℤ≥‘𝑙)) |
93 | 91, 92 | sseldd 3878 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) → 𝑘 ∈ ℕ) |
94 | 89, 93 | jca 515 |
. . . . . . . . . . . . . 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 10769 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < 𝐴) → 𝑥 ∈ ℝ*) |
100 | 14 | ffvelrnda 6861 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞)) |
101 | 15, 100 | eqeltrrd 2834 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) |
102 | 7, 101 | sseldi 3875 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈
ℝ*) |
103 | 102 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < 𝐴) → 𝐴 ∈
ℝ*) |
104 | | simpr 488 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < 𝐴) → 𝑥 < 𝐴) |
105 | | pnfge 12608 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℝ*
→ 𝐴 ≤
+∞) |
106 | 103, 105 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < 𝐴) → 𝐴 ≤ +∞) |
107 | 65 | biimpar 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ*
∧ (𝐴 ∈
ℝ* ∧ 𝑥
< 𝐴 ∧ 𝐴 ≤ +∞)) → 𝐴 ∈ (𝑥(,]+∞)) |
108 | 99, 103, 104, 106, 107 | syl13anc 1373 |
. . . . . . . . . . . . . . . . 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 3878 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑥 < 𝐴) → 𝐴 ∈ 𝑎) |
111 | 110 | ex 416 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) → (𝑥 < 𝐴 → 𝐴 ∈ 𝑎)) |
112 | 94, 95, 96, 111 | syl21anc 837 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑙)) → (𝑥 < 𝐴 → 𝐴 ∈ 𝑎)) |
113 | 112 | ralimdva 3091 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) ∧ 𝑙 ∈ ℕ) → (∀𝑘 ∈
(ℤ≥‘𝑙)𝑥 < 𝐴 → ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) |
114 | 113 | reximdva 3184 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) → (∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝑥 < 𝐴 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) |
115 | 74, 114 | syl5bi 245 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑎) → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) |
116 | 115 | expimpd 457 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (((𝑥(,]+∞) ⊆ 𝑎 ∧ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎)) |
117 | 116 | rexlimdva 3194 |
. . . . . . . 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 410 |
. . . . . 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 836 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐽) ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴) ∧ +∞ ∈ 𝑎) → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎) |
121 | 120 | exp31 423 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴 → (+∞ ∈ 𝑎 → ∃𝑙 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑙)𝐴 ∈ 𝑎))) |
122 | 121 | ralrimdva 3101 |
. . 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 215 |
. 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 282 |
1
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑗)𝑥 < 𝐴)) |