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 39957 |
. 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 1062 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑦 ∈ ℝ) |
6 | | nfv 1843 |
. . . . . . . . . 10
⊢
Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ ℝ) |
7 | | simp3l 1089 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑘 ≤ 𝑗) |
8 | | simp1r 1086 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ∈ ℝ) |
9 | 8 | rexrd 10089 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ∈ ℝ*) |
10 | 3 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈
ℝ*) |
11 | 10 | adantlr 751 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈
ℝ*) |
12 | 11 | 3adant3 1081 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → (𝐹‘𝑗) ∈
ℝ*) |
13 | | simp3 1063 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 < (𝐹‘𝑗)) |
14 | 9, 12, 13 | xrltled 39486 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ≤ (𝐹‘𝑗)) |
15 | 14 | 3adant3l 1322 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑦 ≤ (𝐹‘𝑗)) |
16 | 7, 15 | jca 554 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))) |
17 | 16 | 3exp 1264 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑗 ∈ 𝐴 → ((𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))) |
18 | 6, 17 | reximdai 3012 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) |
19 | 18 | ralimdv 2963 |
. . . . . . . 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 1261 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))) |
21 | | breq1 4656 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑥 ≤ (𝐹‘𝑗) ↔ 𝑦 ≤ (𝐹‘𝑗))) |
22 | 21 | anbi2d 740 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) |
23 | 22 | rexbidv 3052 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) |
24 | 23 | ralbidv 2986 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) |
25 | 24 | rspcev 3309 |
. . . . . . 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 693 |
. . . . . 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 1264 |
. . . . 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 3030 |
. . . 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 10348 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈
ℝ) |
30 | 29 | ad2antlr 763 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) ∈ ℝ) |
31 | | nfv 1843 |
. . . . . . . . . 10
⊢
Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ) |
32 | | simp3l 1089 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑘 ≤ 𝑗) |
33 | | simp1r 1086 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ∈ ℝ) |
34 | 29 | rexrd 10089 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈
ℝ*) |
35 | 33, 34 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) ∈
ℝ*) |
36 | 33 | rexrd 10089 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ∈ ℝ*) |
37 | 10 | adantlr 751 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈
ℝ*) |
38 | 37 | 3adant3 1081 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝐹‘𝑗) ∈
ℝ*) |
39 | 33 | ltm1d 10956 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) < 𝑥) |
40 | | simp3r 1090 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ≤ (𝐹‘𝑗)) |
41 | 35, 36, 38, 39, 40 | xrltletrd 11992 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) < (𝐹‘𝑗)) |
42 | 32, 41 | jca 554 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))) |
43 | 42 | 3exp 1264 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑗 ∈ 𝐴 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))))) |
44 | 31, 43 | reximdai 3012 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))) |
45 | 44 | ralimdv 2963 |
. . . . . . . 8
⊢ ((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 445 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))) |
47 | | breq1 4656 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑥 − 1) → (𝑦 < (𝐹‘𝑗) ↔ (𝑥 − 1) < (𝐹‘𝑗))) |
48 | 47 | anbi2d 740 |
. . . . . . . . . 10
⊢ (𝑦 = (𝑥 − 1) → ((𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))) |
49 | 48 | rexbidv 3052 |
. . . . . . . . 9
⊢ (𝑦 = (𝑥 − 1) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))) |
50 | 49 | ralbidv 2986 |
. . . . . . . 8
⊢ (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 3309 |
. . . . . . 7
⊢ (((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 693 |
. . . . . 6
⊢ (((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 | ex 450 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)))) |
54 | 53 | rexlimdva 3031 |
. . . 4
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)))) |
55 | 28, 54 | impbid 202 |
. . 3
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))) |
56 | | simplr 792 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) → 𝑦 ∈ ℝ) |
57 | 11 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → (𝐹‘𝑗) ∈
ℝ*) |
58 | | rexr 10085 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℝ → 𝑦 ∈
ℝ*) |
59 | 58 | ad3antlr 767 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → 𝑦 ∈ ℝ*) |
60 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → (𝐹‘𝑗) < 𝑦) |
61 | 57, 59, 60 | xrltled 39486 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → (𝐹‘𝑗) ≤ 𝑦) |
62 | 61 | ex 450 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑗) < 𝑦 → (𝐹‘𝑗) ≤ 𝑦)) |
63 | 62 | imim2d 57 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) |
64 | 63 | ralimdva 2962 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) |
65 | 64 | reximdv 3016 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) |
66 | 65 | imp 445 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)) |
67 | | breq2 4657 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑦)) |
68 | 67 | imbi2d 330 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) |
69 | 68 | ralbidv 2986 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) |
70 | 69 | rexbidv 3052 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))) |
71 | 70 | rspcev 3309 |
. . . . . . 7
⊢ ((𝑦 ∈ ℝ ∧
∃𝑘 ∈ ℝ
∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
72 | 56, 66, 71 | syl2anc 693 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
73 | 72 | ex 450 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
74 | 73 | rexlimdva 3031 |
. . . 4
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
75 | | peano2re 10209 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) |
76 | 75 | ad2antlr 763 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (𝑥 + 1) ∈ ℝ) |
77 | 37 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ∈
ℝ*) |
78 | | rexr 10085 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ*) |
79 | 78 | ad3antlr 767 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 ∈ ℝ*) |
80 | 75 | rexrd 10089 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ*) |
81 | 80 | ad3antlr 767 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝑥 + 1) ∈
ℝ*) |
82 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ≤ 𝑥) |
83 | | ltp1 10861 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1)) |
84 | 83 | ad3antlr 767 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 < (𝑥 + 1)) |
85 | 77, 79, 81, 82, 84 | xrlelttrd 11991 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) < (𝑥 + 1)) |
86 | 85 | ex 450 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑗) ≤ 𝑥 → (𝐹‘𝑗) < (𝑥 + 1))) |
87 | 86 | imim2d 57 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) |
88 | 87 | ralimdva 2962 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) |
89 | 88 | reximdv 3016 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) |
90 | 89 | imp 445 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))) |
91 | | breq2 4657 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑥 + 1) → ((𝐹‘𝑗) < 𝑦 ↔ (𝐹‘𝑗) < (𝑥 + 1))) |
92 | 91 | imbi2d 330 |
. . . . . . . . . 10
⊢ (𝑦 = (𝑥 + 1) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) |
93 | 92 | ralbidv 2986 |
. . . . . . . . 9
⊢ (𝑦 = (𝑥 + 1) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) |
94 | 93 | rexbidv 3052 |
. . . . . . . 8
⊢ (𝑦 = (𝑥 + 1) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))) |
95 | 94 | rspcev 3309 |
. . . . . . 7
⊢ (((𝑥 + 1) ∈ ℝ ∧
∃𝑘 ∈ ℝ
∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) |
96 | 76, 90, 95 | syl2anc 693 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) |
97 | 96 | ex 450 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦))) |
98 | 97 | rexlimdva 3031 |
. . . 4
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦))) |
99 | 74, 98 | impbid 202 |
. . 3
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
100 | 55, 99 | anbi12d 747 |
. 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)))) |
101 | 4, 100 | bitrd 268 |
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)))) |