| 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 45401 | . . . 4
⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | 
| 6 | 1, 2, 3, 5 | limsupre3uzlem 45755 | . . 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 5145 | . . . . . . . . 9
⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙))) | 
| 8 | 7 | rexbidv 3178 | . . . . . . . 8
⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙))) | 
| 9 | 8 | ralbidv 3177 | . . . . . . 7
⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙))) | 
| 10 |  | fveq2 6905 | . . . . . . . . . . 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 6915 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑗(𝐹‘𝑙) | 
| 17 | 12, 13, 16 | nfbr 5189 | . . . . . . . . . . . 12
⊢
Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑙) | 
| 18 |  | nfv 1913 | . . . . . . . . . . . 12
⊢
Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑗) | 
| 19 |  | fveq2 6905 | . . . . . . . . . . . . 13
⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗)) | 
| 20 | 19 | breq2d 5154 | . . . . . . . . . . . 12
⊢ (𝑙 = 𝑗 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑗))) | 
| 21 | 17, 18, 20 | cbvrexw 3306 | . . . . . . . . . . 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 279 | . . . . . . . . 9
⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) | 
| 24 | 23 | cbvralvw 3236 | . . . . . . . 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 279 | . . . . . 6
⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) | 
| 27 | 26 | cbvrexvw 3237 | . . . . 5
⊢
(∃𝑦 ∈
ℝ ∀𝑖 ∈
𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) | 
| 28 |  | breq2 5146 | . . . . . . . . 9
⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥)) | 
| 29 | 28 | ralbidv 3177 | . . . . . . . 8
⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥)) | 
| 30 | 29 | rexbidv 3178 | . . . . . . 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 5189 | . . . . . . . . . . . 12
⊢
Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥 | 
| 33 |  | nfv 1913 | . . . . . . . . . . . 12
⊢
Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥 | 
| 34 | 19 | breq1d 5152 | . . . . . . . . . . . 12
⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)) | 
| 35 | 32, 33, 34 | cbvralw 3305 | . . . . . . . . . . 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 279 | . . . . . . . . 9
⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) | 
| 38 | 37 | cbvrexvw 3237 | . . . . . . . 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 279 | . . . . . 6
⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) | 
| 41 | 40 | cbvrexvw 3237 | . . . . 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 628 | . . . 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 279 | . 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 1913 | . . . . . . . 8
⊢
Ⅎ𝑖(𝐹‘𝑗) ≤ 𝑥 | 
| 46 |  | nfcv 2904 | . . . . . . . . . 10
⊢
Ⅎ𝑗𝑖 | 
| 47 | 14, 46 | nffv 6915 | . . . . . . . . 9
⊢
Ⅎ𝑗(𝐹‘𝑖) | 
| 48 | 47, 13, 12 | nfbr 5189 | . . . . . . . 8
⊢
Ⅎ𝑗(𝐹‘𝑖) ≤ 𝑥 | 
| 49 |  | fveq2 6905 | . . . . . . . . 9
⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖)) | 
| 50 | 49 | breq1d 5152 | . . . . . . . 8
⊢ (𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) | 
| 51 | 45, 48, 50 | cbvralw 3305 | . . . . . . 7
⊢
(∀𝑗 ∈
(ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥) | 
| 52 | 51 | rexbii 3093 | . . . . . 6
⊢
(∃𝑘 ∈
𝑍 ∀𝑗 ∈
(ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥) | 
| 53 | 52 | rexbii 3093 | . . . . 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 1913 | . . . . 5
⊢
Ⅎ𝑖𝜑 | 
| 56 | 4 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹:𝑍⟶ℝ) | 
| 57 |  | simpr 484 | . . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍) | 
| 58 | 56, 57 | ffvelcdmd 7104 | . . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖) ∈ ℝ) | 
| 59 | 55, 2, 3, 58 | uzub 45447 | . . . 4
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥)) | 
| 60 |  | eqcom 2743 | . . . . . . . . . 10
⊢ (𝑗 = 𝑖 ↔ 𝑖 = 𝑗) | 
| 61 | 60 | imbi1i 349 | . . . . . . . . 9
⊢ ((𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥))) | 
| 62 |  | bicom 222 | . . . . . . . . . 10
⊢ (((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥) ↔ ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)) | 
| 63 | 62 | imbi2i 336 | . . . . . . . . 9
⊢ ((𝑖 = 𝑗 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))) | 
| 64 | 61, 63 | bitri 275 | . . . . . . . 8
⊢ ((𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))) | 
| 65 | 50, 64 | mpbi 230 | . . . . . . 7
⊢ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)) | 
| 66 | 48, 45, 65 | cbvralw 3305 | . . . . . 6
⊢
(∀𝑖 ∈
𝑍 (𝐹‘𝑖) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥) | 
| 67 | 66 | rexbii 3093 | . . . . 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 305 | . . 3
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)) | 
| 70 | 69 | anbi2d 630 | . 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 279 | 1
⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔
(∃𝑥 ∈ ℝ
∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥))) |