Step | Hyp | Ref
| Expression |
1 | | limsupre3uzlem.1 |
. . 3
⊢
Ⅎ𝑗𝐹 |
2 | | limsupre3uzlem.3 |
. . . . 5
⊢ 𝑍 =
(ℤ≥‘𝑀) |
3 | | uzssre 12604 |
. . . . 5
⊢
(ℤ≥‘𝑀) ⊆ ℝ |
4 | 2, 3 | eqsstri 3955 |
. . . 4
⊢ 𝑍 ⊆
ℝ |
5 | 4 | a1i 11 |
. . 3
⊢ (𝜑 → 𝑍 ⊆ ℝ) |
6 | | limsupre3uzlem.4 |
. . 3
⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
7 | 1, 5, 6 | limsupre3 43274 |
. 2
⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔
(∃𝑥 ∈ ℝ
∀𝑦 ∈ ℝ
∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))) |
8 | | breq1 5077 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑘 → (𝑦 ≤ 𝑗 ↔ 𝑘 ≤ 𝑗)) |
9 | 8 | anbi1d 630 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑘 → ((𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))) |
10 | 9 | rexbidv 3226 |
. . . . . . . . 9
⊢ (𝑦 = 𝑘 → (∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))) |
11 | 10 | cbvralvw 3383 |
. . . . . . . 8
⊢
(∀𝑦 ∈
ℝ ∃𝑗 ∈
𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) |
12 | 11 | biimpi 215 |
. . . . . . 7
⊢
(∀𝑦 ∈
ℝ ∃𝑗 ∈
𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) |
13 | | nfra1 3144 |
. . . . . . . 8
⊢
Ⅎ𝑘∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) |
14 | | simpr 485 |
. . . . . . . . 9
⊢
((∀𝑘 ∈
ℝ ∃𝑗 ∈
𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) |
15 | 4, 14 | sselid 3919 |
. . . . . . . . . 10
⊢
((∀𝑘 ∈
ℝ ∃𝑗 ∈
𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℝ) |
16 | | rspa 3132 |
. . . . . . . . . 10
⊢
((∀𝑘 ∈
ℝ ∃𝑗 ∈
𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ∈ ℝ) → ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) |
17 | 15, 16 | syldan 591 |
. . . . . . . . 9
⊢
((∀𝑘 ∈
ℝ ∃𝑗 ∈
𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ∈ 𝑍) → ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) |
18 | | nfv 1917 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗 𝑘 ∈ 𝑍 |
19 | | nfre1 3239 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗∃𝑗 ∈
(ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) |
20 | | eqid 2738 |
. . . . . . . . . . . . . . 15
⊢
(ℤ≥‘𝑘) = (ℤ≥‘𝑘) |
21 | 2 | eluzelz2 42943 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
22 | 21 | 3ad2ant1 1132 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → 𝑘 ∈ ℤ) |
23 | 2 | eluzelz2 42943 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ) |
24 | 23 | 3ad2ant2 1133 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ ℤ) |
25 | | simp3 1137 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → 𝑘 ≤ 𝑗) |
26 | 20, 22, 24, 25 | eluzd 42949 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ (ℤ≥‘𝑘)) |
27 | 26 | 3adant3r 1180 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑗 ∈ (ℤ≥‘𝑘)) |
28 | | simp3r 1201 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ≤ (𝐹‘𝑗)) |
29 | | rspe 3237 |
. . . . . . . . . . . . 13
⊢ ((𝑗 ∈
(ℤ≥‘𝑘) ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
30 | 27, 28, 29 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
31 | 30 | 3exp 1118 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ 𝑍 → (𝑗 ∈ 𝑍 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))) |
32 | 18, 19, 31 | rexlimd 3250 |
. . . . . . . . . 10
⊢ (𝑘 ∈ 𝑍 → (∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
33 | 32 | imp 407 |
. . . . . . . . 9
⊢ ((𝑘 ∈ 𝑍 ∧ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
34 | 14, 17, 33 | syl2anc 584 |
. . . . . . . 8
⊢
((∀𝑘 ∈
ℝ ∃𝑗 ∈
𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ∈ 𝑍) → ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
35 | 13, 34 | ralrimia 3430 |
. . . . . . 7
⊢
(∀𝑘 ∈
ℝ ∃𝑗 ∈
𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
36 | 12, 35 | syl 17 |
. . . . . 6
⊢
(∀𝑦 ∈
ℝ ∃𝑗 ∈
𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
37 | 36 | a1i 11 |
. . . . 5
⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
38 | | iftrue 4465 |
. . . . . . . . . . . . 13
⊢ (𝑀 ≤ (⌈‘𝑦) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = (⌈‘𝑦)) |
39 | 38 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = (⌈‘𝑦)) |
40 | | limsupre3uzlem.2 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ ℤ) |
41 | 40 | ad2antrr 723 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ∈ ℤ) |
42 | | ceilcl 13562 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℝ →
(⌈‘𝑦) ∈
ℤ) |
43 | 42 | ad2antlr 724 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → (⌈‘𝑦) ∈ ℤ) |
44 | | simpr 485 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ≤ (⌈‘𝑦)) |
45 | 2, 41, 43, 44 | eluzd 42949 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → (⌈‘𝑦) ∈ 𝑍) |
46 | 39, 45 | eqeltrd 2839 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍) |
47 | | iffalse 4468 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑀 ≤ (⌈‘𝑦) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = 𝑀) |
48 | 47 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = 𝑀) |
49 | 40, 2 | uzidd2 42956 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ 𝑍) |
50 | 49 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ∈ 𝑍) |
51 | 48, 50 | eqeltrd 2839 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍) |
52 | 51 | adantlr 712 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍) |
53 | 46, 52 | pm2.61dan 810 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍) |
54 | 53 | adantlr 712 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍) |
55 | | simplr 766 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
56 | | fveq2 6774 |
. . . . . . . . . . 11
⊢ (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (ℤ≥‘𝑘) =
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) |
57 | 56 | rexeqdv 3349 |
. . . . . . . . . 10
⊢ (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ↔ ∃𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹‘𝑗))) |
58 | 57 | rspcva 3559 |
. . . . . . . . 9
⊢
((if(𝑀 ≤
(⌈‘𝑦),
(⌈‘𝑦), 𝑀) ∈ 𝑍 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹‘𝑗)) |
59 | 54, 55, 58 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → ∃𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹‘𝑗)) |
60 | | nfv 1917 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗𝜑 |
61 | 18 | nfci 2890 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗𝑍 |
62 | 61, 19 | nfralw 3151 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) |
63 | 60, 62 | nfan 1902 |
. . . . . . . . . 10
⊢
Ⅎ𝑗(𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) |
64 | | nfv 1917 |
. . . . . . . . . 10
⊢
Ⅎ𝑗 𝑦 ∈ ℝ |
65 | 63, 64 | nfan 1902 |
. . . . . . . . 9
⊢
Ⅎ𝑗((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) |
66 | | nfre1 3239 |
. . . . . . . . 9
⊢
Ⅎ𝑗∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) |
67 | 40 | ad2antrr 723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ∈ ℤ) |
68 | | eluzelz 12592 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → 𝑗 ∈ ℤ) |
69 | 68 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ ℤ) |
70 | 67 | zred 12426 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ∈ ℝ) |
71 | 4, 53 | sselid 3919 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ ℝ) |
72 | 71 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ ℝ) |
73 | 69 | zred 12426 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ ℝ) |
74 | 4, 49 | sselid 3919 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑀 ∈ ℝ) |
75 | 74 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑀 ∈ ℝ) |
76 | 42 | zred 12426 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ℝ →
(⌈‘𝑦) ∈
ℝ) |
77 | 76 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (⌈‘𝑦) ∈
ℝ) |
78 | | max1 12919 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ ℝ ∧
(⌈‘𝑦) ∈
ℝ) → 𝑀 ≤
if(𝑀 ≤
(⌈‘𝑦),
(⌈‘𝑦), 𝑀)) |
79 | 75, 77, 78 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) |
80 | 79 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) |
81 | | eluzle 12595 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ≤ 𝑗) |
82 | 81 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ≤ 𝑗) |
83 | 70, 72, 73, 80, 82 | letrd 11132 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ≤ 𝑗) |
84 | 2, 67, 69, 83 | eluzd 42949 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ 𝑍) |
85 | 84 | 3adant3 1131 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑗 ∈ 𝑍) |
86 | | simplr 766 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ∈ ℝ) |
87 | | simpr 485 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) |
88 | | ceilge 13565 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ℝ → 𝑦 ≤ (⌈‘𝑦)) |
89 | 88 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ (⌈‘𝑦)) |
90 | | max2 12921 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑀 ∈ ℝ ∧
(⌈‘𝑦) ∈
ℝ) → (⌈‘𝑦) ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) |
91 | 75, 77, 90 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (⌈‘𝑦) ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) |
92 | 87, 77, 71, 89, 91 | letrd 11132 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) |
93 | 92 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) |
94 | 86, 72, 73, 93, 82 | letrd 11132 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ≤ 𝑗) |
95 | 94 | 3adant3 1131 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑦 ≤ 𝑗) |
96 | | simp3 1137 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ (𝐹‘𝑗)) |
97 | 95, 96 | jca 512 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) |
98 | | rspe 3237 |
. . . . . . . . . . . 12
⊢ ((𝑗 ∈ 𝑍 ∧ (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) |
99 | 85, 97, 98 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) |
100 | 99 | 3exp 1118 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝑥 ≤ (𝐹‘𝑗) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))) |
101 | 100 | adantlr 712 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → (𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝑥 ≤ (𝐹‘𝑗) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))) |
102 | 65, 66, 101 | rexlimd 3250 |
. . . . . . . 8
⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → (∃𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹‘𝑗) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))) |
103 | 59, 102 | mpd 15 |
. . . . . . 7
⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) |
104 | 103 | ralrimiva 3103 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) → ∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) |
105 | 104 | ex 413 |
. . . . 5
⊢ (𝜑 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) → ∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))) |
106 | 37, 105 | impbid 211 |
. . . 4
⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
107 | 106 | rexbidv 3226 |
. . 3
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
108 | 53 | adantr 481 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍) |
109 | 60, 64 | nfan 1902 |
. . . . . . . . 9
⊢
Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ ℝ) |
110 | | nfra1 3144 |
. . . . . . . . 9
⊢
Ⅎ𝑗∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) |
111 | 109, 110 | nfan 1902 |
. . . . . . . 8
⊢
Ⅎ𝑗((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
112 | 94 | adantlr 712 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ≤ 𝑗) |
113 | | simplr 766 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
114 | 84 | adantlr 712 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ 𝑍) |
115 | | rspa 3132 |
. . . . . . . . . . 11
⊢
((∀𝑗 ∈
𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝑍) → (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
116 | 113, 114,
115 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
117 | 112, 116 | mpd 15 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → (𝐹‘𝑗) ≤ 𝑥) |
118 | 117 | ex 413 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝐹‘𝑗) ≤ 𝑥)) |
119 | 111, 118 | ralrimi 3141 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∀𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹‘𝑗) ≤ 𝑥) |
120 | 56 | raleqdv 3348 |
. . . . . . . 8
⊢ (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹‘𝑗) ≤ 𝑥)) |
121 | 120 | rspcev 3561 |
. . . . . . 7
⊢
((if(𝑀 ≤
(⌈‘𝑦),
(⌈‘𝑦), 𝑀) ∈ 𝑍 ∧ ∀𝑗 ∈
(ℤ≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹‘𝑗) ≤ 𝑥) → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
122 | 108, 119,
121 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
123 | 122 | rexlimdva2 3216 |
. . . . 5
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
124 | 4 | sseli 3917 |
. . . . . . . 8
⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℝ) |
125 | 124 | ad2antlr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → 𝑘 ∈ ℝ) |
126 | | nfra1 3144 |
. . . . . . . . . 10
⊢
Ⅎ𝑗∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 |
127 | 18, 126 | nfan 1902 |
. . . . . . . . 9
⊢
Ⅎ𝑗(𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
128 | | simp1r 1197 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
129 | 26 | 3adant1r 1176 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ (ℤ≥‘𝑘)) |
130 | | rspa 3132 |
. . . . . . . . . . 11
⊢
((∀𝑗 ∈
(ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ≤ 𝑥) |
131 | 128, 129,
130 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥) |
132 | 131 | 3exp 1118 |
. . . . . . . . 9
⊢ ((𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → (𝑗 ∈ 𝑍 → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
133 | 127, 132 | ralrimi 3141 |
. . . . . . . 8
⊢ ((𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
134 | 133 | adantll 711 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
135 | 8 | rspceaimv 3565 |
. . . . . . 7
⊢ ((𝑘 ∈ ℝ ∧
∀𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
136 | 125, 134,
135 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
137 | 136 | rexlimdva2 3216 |
. . . . 5
⊢ (𝜑 → (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
138 | 123, 137 | impbid 211 |
. . . 4
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
139 | 138 | rexbidv 3226 |
. . 3
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
140 | 107, 139 | anbi12d 631 |
. 2
⊢ (𝜑 → ((∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))) |
141 | 7, 140 | bitrd 278 |
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))) |