| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | limsupre3lem.1 | . . 3
⊢
Ⅎ𝑗𝐹 | 
| 2 |  | limsupre3lem.2 | . . 3
⊢ (𝜑 → 𝐴 ⊆ ℝ) | 
| 3 |  | limsupre3lem.3 | . . 3
⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | 
| 4 | 1, 2, 3 | limsupre2 45740 | . 2
⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔
(∃𝑦 ∈ ℝ
∀𝑘 ∈ ℝ
∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)))) | 
| 5 |  | simp2 1138 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑦 ∈ ℝ) | 
| 6 |  | nfv 1914 | . . . . . . . . . 10
⊢
Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ ℝ) | 
| 7 |  | simp3l 1202 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑘 ≤ 𝑗) | 
| 8 |  | simp1r 1199 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ∈ ℝ) | 
| 9 | 8 | rexrd 11311 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ∈ ℝ*) | 
| 10 | 3 | ffvelcdmda 7104 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈
ℝ*) | 
| 11 | 10 | adantlr 715 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈
ℝ*) | 
| 12 | 11 | 3adant3 1133 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → (𝐹‘𝑗) ∈
ℝ*) | 
| 13 |  | simp3 1139 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 < (𝐹‘𝑗)) | 
| 14 | 9, 12, 13 | xrltled 13192 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ≤ (𝐹‘𝑗)) | 
| 15 | 14 | 3adant3l 1181 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑦 ≤ (𝐹‘𝑗)) | 
| 16 | 7, 15 | jca 511 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))) | 
| 17 | 16 | 3exp 1120 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑗 ∈ 𝐴 → ((𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))) | 
| 18 | 6, 17 | reximdai 3261 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) | 
| 19 | 18 | ralimdv 3169 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) | 
| 20 | 19 | 3impia 1118 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))) | 
| 21 |  | breq1 5146 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑥 ≤ (𝐹‘𝑗) ↔ 𝑦 ≤ (𝐹‘𝑗))) | 
| 22 | 21 | anbi2d 630 | . . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) | 
| 23 | 22 | rexbidv 3179 | . . . . . . . . 9
⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) | 
| 24 | 23 | ralbidv 3178 | . . . . . . . 8
⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) | 
| 25 | 24 | rspcev 3622 | . . . . . . 7
⊢ ((𝑦 ∈ ℝ ∧
∀𝑘 ∈ ℝ
∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) | 
| 26 | 5, 20, 25 | syl2anc 584 | . . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) | 
| 27 | 26 | 3exp 1120 | . . . . 5
⊢ (𝜑 → (𝑦 ∈ ℝ → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))) | 
| 28 | 27 | rexlimdv 3153 | . . . 4
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))) | 
| 29 |  | peano2rem 11576 | . . . . . . 7
⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈
ℝ) | 
| 30 | 29 | ad2antlr 727 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) ∈ ℝ) | 
| 31 |  | nfv 1914 | . . . . . . . . 9
⊢
Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ) | 
| 32 |  | simp3l 1202 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑘 ≤ 𝑗) | 
| 33 |  | simp1r 1199 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ∈ ℝ) | 
| 34 | 29 | rexrd 11311 | . . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈
ℝ*) | 
| 35 | 33, 34 | syl 17 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) ∈
ℝ*) | 
| 36 | 33 | rexrd 11311 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ∈ ℝ*) | 
| 37 | 10 | adantlr 715 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈
ℝ*) | 
| 38 | 37 | 3adant3 1133 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝐹‘𝑗) ∈
ℝ*) | 
| 39 | 33 | ltm1d 12200 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) < 𝑥) | 
| 40 |  | simp3r 1203 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ≤ (𝐹‘𝑗)) | 
| 41 | 35, 36, 38, 39, 40 | xrltletrd 13203 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) < (𝐹‘𝑗)) | 
| 42 | 32, 41 | jca 511 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))) | 
| 43 | 42 | 3exp 1120 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑗 ∈ 𝐴 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))))) | 
| 44 | 31, 43 | reximdai 3261 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))) | 
| 45 | 44 | ralimdv 3169 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))) | 
| 46 | 45 | imp 406 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))) | 
| 47 |  | breq1 5146 | . . . . . . . . . 10
⊢ (𝑦 = (𝑥 − 1) → (𝑦 < (𝐹‘𝑗) ↔ (𝑥 − 1) < (𝐹‘𝑗))) | 
| 48 | 47 | anbi2d 630 | . . . . . . . . 9
⊢ (𝑦 = (𝑥 − 1) → ((𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))) | 
| 49 | 48 | rexbidv 3179 | . . . . . . . 8
⊢ (𝑦 = (𝑥 − 1) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))) | 
| 50 | 49 | ralbidv 3178 | . . . . . . 7
⊢ (𝑦 = (𝑥 − 1) → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))) | 
| 51 | 50 | rspcev 3622 | . . . . . 6
⊢ (((𝑥 − 1) ∈ ℝ ∧
∀𝑘 ∈ ℝ
∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) | 
| 52 | 30, 46, 51 | syl2anc 584 | . . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) | 
| 53 | 52 | rexlimdva2 3157 | . . . 4
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)))) | 
| 54 | 28, 53 | impbid 212 | . . 3
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))) | 
| 55 |  | simplr 769 | . . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) → 𝑦 ∈ ℝ) | 
| 56 | 11 | adantr 480 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → (𝐹‘𝑗) ∈
ℝ*) | 
| 57 |  | rexr 11307 | . . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℝ → 𝑦 ∈
ℝ*) | 
| 58 | 57 | ad3antlr 731 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → 𝑦 ∈ ℝ*) | 
| 59 |  | simpr 484 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → (𝐹‘𝑗) < 𝑦) | 
| 60 | 56, 58, 59 | xrltled 13192 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → (𝐹‘𝑗) ≤ 𝑦) | 
| 61 | 60 | ex 412 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑗) < 𝑦 → (𝐹‘𝑗) ≤ 𝑦)) | 
| 62 | 61 | imim2d 57 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) | 
| 63 | 62 | ralimdva 3167 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) | 
| 64 | 63 | reximdv 3170 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) | 
| 65 | 64 | imp 406 | . . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)) | 
| 66 |  | breq2 5147 | . . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑦)) | 
| 67 | 66 | imbi2d 340 | . . . . . . . . 9
⊢ (𝑥 = 𝑦 → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) | 
| 68 | 67 | ralbidv 3178 | . . . . . . . 8
⊢ (𝑥 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) | 
| 69 | 68 | rexbidv 3179 | . . . . . . 7
⊢ (𝑥 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) | 
| 70 | 69 | rspcev 3622 | . . . . . 6
⊢ ((𝑦 ∈ ℝ ∧
∃𝑘 ∈ ℝ
∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) | 
| 71 | 55, 65, 70 | syl2anc 584 | . . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) | 
| 72 | 71 | rexlimdva2 3157 | . . . 4
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) | 
| 73 |  | peano2re 11434 | . . . . . . 7
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) | 
| 74 | 73 | ad2antlr 727 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (𝑥 + 1) ∈ ℝ) | 
| 75 | 37 | adantr 480 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ∈
ℝ*) | 
| 76 |  | rexr 11307 | . . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ*) | 
| 77 | 76 | ad3antlr 731 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 ∈ ℝ*) | 
| 78 | 73 | rexrd 11311 | . . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ*) | 
| 79 | 78 | ad3antlr 731 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝑥 + 1) ∈
ℝ*) | 
| 80 |  | simpr 484 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ≤ 𝑥) | 
| 81 |  | ltp1 12107 | . . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1)) | 
| 82 | 81 | ad3antlr 731 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 < (𝑥 + 1)) | 
| 83 | 75, 77, 79, 80, 82 | xrlelttrd 13202 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) < (𝑥 + 1)) | 
| 84 | 83 | ex 412 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑗) ≤ 𝑥 → (𝐹‘𝑗) < (𝑥 + 1))) | 
| 85 | 84 | imim2d 57 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) | 
| 86 | 85 | ralimdva 3167 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) | 
| 87 | 86 | reximdv 3170 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) | 
| 88 | 87 | imp 406 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))) | 
| 89 |  | breq2 5147 | . . . . . . . . . 10
⊢ (𝑦 = (𝑥 + 1) → ((𝐹‘𝑗) < 𝑦 ↔ (𝐹‘𝑗) < (𝑥 + 1))) | 
| 90 | 89 | imbi2d 340 | . . . . . . . . 9
⊢ (𝑦 = (𝑥 + 1) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) | 
| 91 | 90 | ralbidv 3178 | . . . . . . . 8
⊢ (𝑦 = (𝑥 + 1) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) | 
| 92 | 91 | rexbidv 3179 | . . . . . . 7
⊢ (𝑦 = (𝑥 + 1) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) | 
| 93 | 92 | rspcev 3622 | . . . . . 6
⊢ (((𝑥 + 1) ∈ ℝ ∧
∃𝑘 ∈ ℝ
∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) | 
| 94 | 74, 88, 93 | syl2anc 584 | . . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) | 
| 95 | 94 | rexlimdva2 3157 | . . . 4
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦))) | 
| 96 | 72, 95 | impbid 212 | . . 3
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) | 
| 97 | 54, 96 | anbi12d 632 | . 2
⊢ (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))) | 
| 98 | 4, 97 | bitrd 279 | 1
⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔
(∃𝑥 ∈ ℝ
∀𝑘 ∈ ℝ
∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))) |