Step | Hyp | Ref
| Expression |
1 | | nfcv 2904 |
. . . 4
⊢
Ⅎ𝑙𝐹 |
2 | | limsupreuz.2 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℤ) |
3 | | limsupreuz.3 |
. . . 4
⊢ 𝑍 =
(ℤ≥‘𝑀) |
4 | | limsupreuz.4 |
. . . . 5
⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
5 | 4 | frexr 42597 |
. . . 4
⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
6 | 1, 2, 3, 5 | limsupre3uzlem 42951 |
. . 3
⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔
(∃𝑦 ∈ ℝ
∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦))) |
7 | | breq1 5056 |
. . . . . . . . 9
⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙))) |
8 | 7 | rexbidv 3216 |
. . . . . . . 8
⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙))) |
9 | 8 | ralbidv 3118 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙))) |
10 | | fveq2 6717 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) =
(ℤ≥‘𝑘)) |
11 | 10 | rexeqdv 3326 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙))) |
12 | | nfcv 2904 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗𝑥 |
13 | | nfcv 2904 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗
≤ |
14 | | limsupreuz.1 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗𝐹 |
15 | | nfcv 2904 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗𝑙 |
16 | 14, 15 | nffv 6727 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗(𝐹‘𝑙) |
17 | 12, 13, 16 | nfbr 5100 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑙) |
18 | | nfv 1922 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑗) |
19 | | fveq2 6717 |
. . . . . . . . . . . . 13
⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗)) |
20 | 19 | breq2d 5065 |
. . . . . . . . . . . 12
⊢ (𝑙 = 𝑗 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑗))) |
21 | 17, 18, 20 | cbvrexw 3350 |
. . . . . . . . . . 11
⊢
(∃𝑙 ∈
(ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
22 | 21 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
23 | 11, 22 | bitrd 282 |
. . . . . . . . 9
⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
24 | 23 | cbvralvw 3358 |
. . . . . . . 8
⊢
(∀𝑖 ∈
𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
25 | 24 | a1i 11 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
26 | 9, 25 | bitrd 282 |
. . . . . 6
⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
27 | 26 | cbvrexvw 3359 |
. . . . 5
⊢
(∃𝑦 ∈
ℝ ∀𝑖 ∈
𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
28 | | breq2 5057 |
. . . . . . . . 9
⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥)) |
29 | 28 | ralbidv 3118 |
. . . . . . . 8
⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥)) |
30 | 29 | rexbidv 3216 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥)) |
31 | 10 | raleqdv 3325 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥)) |
32 | 16, 13, 12 | nfbr 5100 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥 |
33 | | nfv 1922 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥 |
34 | 19 | breq1d 5063 |
. . . . . . . . . . . 12
⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)) |
35 | 32, 33, 34 | cbvralw 3349 |
. . . . . . . . . . 11
⊢
(∀𝑙 ∈
(ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
36 | 35 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
37 | 31, 36 | bitrd 282 |
. . . . . . . . 9
⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
38 | 37 | cbvrexvw 3359 |
. . . . . . . 8
⊢
(∃𝑖 ∈
𝑍 ∀𝑙 ∈
(ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
39 | 38 | a1i 11 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
40 | 30, 39 | bitrd 282 |
. . . . . 6
⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
41 | 40 | cbvrexvw 3359 |
. . . . 5
⊢
(∃𝑦 ∈
ℝ ∃𝑖 ∈
𝑍 ∀𝑙 ∈
(ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
42 | 27, 41 | anbi12i 630 |
. . . 4
⊢
((∃𝑦 ∈
ℝ ∀𝑖 ∈
𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
43 | 42 | a1i 11 |
. . 3
⊢ (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))) |
44 | 6, 43 | bitrd 282 |
. 2
⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔
(∃𝑥 ∈ ℝ
∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))) |
45 | | nfv 1922 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝐹‘𝑗) ≤ 𝑥 |
46 | | nfcv 2904 |
. . . . . . . . . 10
⊢
Ⅎ𝑗𝑖 |
47 | 14, 46 | nffv 6727 |
. . . . . . . . 9
⊢
Ⅎ𝑗(𝐹‘𝑖) |
48 | 47, 13, 12 | nfbr 5100 |
. . . . . . . 8
⊢
Ⅎ𝑗(𝐹‘𝑖) ≤ 𝑥 |
49 | | fveq2 6717 |
. . . . . . . . 9
⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖)) |
50 | 49 | breq1d 5063 |
. . . . . . . 8
⊢ (𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) |
51 | 45, 48, 50 | cbvralw 3349 |
. . . . . . 7
⊢
(∀𝑗 ∈
(ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥) |
52 | 51 | rexbii 3170 |
. . . . . 6
⊢
(∃𝑘 ∈
𝑍 ∀𝑗 ∈
(ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥) |
53 | 52 | rexbii 3170 |
. . . . 5
⊢
(∃𝑥 ∈
ℝ ∃𝑘 ∈
𝑍 ∀𝑗 ∈
(ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥) |
54 | 53 | a1i 11 |
. . . 4
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥)) |
55 | | nfv 1922 |
. . . . 5
⊢
Ⅎ𝑖𝜑 |
56 | 4 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹:𝑍⟶ℝ) |
57 | | simpr 488 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍) |
58 | 56, 57 | ffvelrnd 6905 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖) ∈ ℝ) |
59 | 55, 2, 3, 58 | uzub 42644 |
. . . 4
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥)) |
60 | | eqcom 2744 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑖 ↔ 𝑖 = 𝑗) |
61 | 60 | imbi1i 353 |
. . . . . . . . 9
⊢ ((𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥))) |
62 | | bicom 225 |
. . . . . . . . . 10
⊢ (((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥) ↔ ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)) |
63 | 62 | imbi2i 339 |
. . . . . . . . 9
⊢ ((𝑖 = 𝑗 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))) |
64 | 61, 63 | bitri 278 |
. . . . . . . 8
⊢ ((𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))) |
65 | 50, 64 | mpbi 233 |
. . . . . . 7
⊢ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)) |
66 | 48, 45, 65 | cbvralw 3349 |
. . . . . 6
⊢
(∀𝑖 ∈
𝑍 (𝐹‘𝑖) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥) |
67 | 66 | rexbii 3170 |
. . . . 5
⊢
(∃𝑥 ∈
ℝ ∀𝑖 ∈
𝑍 (𝐹‘𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥) |
68 | 67 | a1i 11 |
. . . 4
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)) |
69 | 54, 59, 68 | 3bitrd 308 |
. . 3
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)) |
70 | 69 | anbi2d 632 |
. 2
⊢ (𝜑 → ((∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥))) |
71 | 44, 70 | bitrd 282 |
1
⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔
(∃𝑥 ∈ ℝ
∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥))) |