Step | Hyp | Ref
| Expression |
1 | | nfcv 2899 |
. . 3
⊢
Ⅎ𝑙𝐹 |
2 | | limsupre3uz.2 |
. . 3
⊢ (𝜑 → 𝑀 ∈ ℤ) |
3 | | limsupre3uz.3 |
. . 3
⊢ 𝑍 =
(ℤ≥‘𝑀) |
4 | | limsupre3uz.4 |
. . 3
⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
5 | 1, 2, 3, 4 | limsupre3uzlem 42818 |
. 2
⊢ (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))) |
6 | | breq1 5033 |
. . . . . . . 8
⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙))) |
7 | 6 | rexbidv 3207 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙))) |
8 | 7 | ralbidv 3109 |
. . . . . 6
⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙))) |
9 | | fveq2 6674 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) =
(ℤ≥‘𝑘)) |
10 | 9 | rexeqdv 3317 |
. . . . . . . . 9
⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙))) |
11 | | nfcv 2899 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗𝑥 |
12 | | nfcv 2899 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗
≤ |
13 | | limsupre3uz.1 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗𝐹 |
14 | | nfcv 2899 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗𝑙 |
15 | 13, 14 | nffv 6684 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗(𝐹‘𝑙) |
16 | 11, 12, 15 | nfbr 5077 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑙) |
17 | | nfv 1921 |
. . . . . . . . . . 11
⊢
Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑗) |
18 | | fveq2 6674 |
. . . . . . . . . . . 12
⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗)) |
19 | 18 | breq2d 5042 |
. . . . . . . . . . 11
⊢ (𝑙 = 𝑗 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑗))) |
20 | 16, 17, 19 | cbvrexw 3341 |
. . . . . . . . . 10
⊢
(∃𝑙 ∈
(ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
21 | 20 | a1i 11 |
. . . . . . . . 9
⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
22 | 10, 21 | bitrd 282 |
. . . . . . . 8
⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
23 | 22 | cbvralvw 3349 |
. . . . . . 7
⊢
(∀𝑖 ∈
𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
24 | 23 | a1i 11 |
. . . . . 6
⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
25 | 8, 24 | bitrd 282 |
. . . . 5
⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
26 | 25 | cbvrexvw 3350 |
. . . 4
⊢
(∃𝑦 ∈
ℝ ∀𝑖 ∈
𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
27 | | breq2 5034 |
. . . . . . . 8
⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥)) |
28 | 27 | ralbidv 3109 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥)) |
29 | 28 | rexbidv 3207 |
. . . . . 6
⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥)) |
30 | 9 | raleqdv 3316 |
. . . . . . . . 9
⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥)) |
31 | 15, 12, 11 | nfbr 5077 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥 |
32 | | nfv 1921 |
. . . . . . . . . . 11
⊢
Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥 |
33 | 18 | breq1d 5040 |
. . . . . . . . . . 11
⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)) |
34 | 31, 32, 33 | cbvralw 3340 |
. . . . . . . . . 10
⊢
(∀𝑙 ∈
(ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
35 | 34 | a1i 11 |
. . . . . . . . 9
⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
36 | 30, 35 | bitrd 282 |
. . . . . . . 8
⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
37 | 36 | cbvrexvw 3350 |
. . . . . . 7
⊢
(∃𝑖 ∈
𝑍 ∀𝑙 ∈
(ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
38 | 37 | a1i 11 |
. . . . . 6
⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
39 | 29, 38 | bitrd 282 |
. . . . 5
⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
40 | 39 | cbvrexvw 3350 |
. . . 4
⊢
(∃𝑦 ∈
ℝ ∃𝑖 ∈
𝑍 ∀𝑙 ∈
(ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
41 | 26, 40 | anbi12i 630 |
. . 3
⊢
((∃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)) |
42 | 41 | a1i 11 |
. 2
⊢ (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))) |
43 | 5, 42 | bitrd 282 |
1
⊢ (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))) |