Step |
Hyp |
Ref |
Expression |
1 |
|
limsupreuz.1 |
⊢ Ⅎ 𝑗 𝐹 |
2 |
|
limsupreuz.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
limsupreuz.3 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
4 |
|
limsupreuz.4 |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ ) |
5 |
|
nfcv |
⊢ Ⅎ 𝑙 𝐹 |
6 |
4
|
frexr |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* ) |
7 |
5 2 3 6
|
limsupre3uzlem |
⊢ ( 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 ) ) ) |
8 |
|
breq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ≤ ( 𝐹 ‘ 𝑙 ) ↔ 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ) ) |
9 |
8
|
rexbidv |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑦 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ) ) |
10 |
9
|
ralbidv |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑦 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ) ) |
11 |
|
fveq2 |
⊢ ( 𝑖 = 𝑘 → ( ℤ≥ ‘ 𝑖 ) = ( ℤ≥ ‘ 𝑘 ) ) |
12 |
11
|
rexeqdv |
⊢ ( 𝑖 = 𝑘 → ( ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ) ) |
13 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑥 |
14 |
|
nfcv |
⊢ Ⅎ 𝑗 ≤ |
15 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑙 |
16 |
1 15
|
nffv |
⊢ Ⅎ 𝑗 ( 𝐹 ‘ 𝑙 ) |
17 |
13 14 16
|
nfbr |
⊢ Ⅎ 𝑗 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) |
18 |
|
nfv |
⊢ Ⅎ 𝑙 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) |
19 |
|
fveq2 |
⊢ ( 𝑙 = 𝑗 → ( 𝐹 ‘ 𝑙 ) = ( 𝐹 ‘ 𝑗 ) ) |
20 |
19
|
breq2d |
⊢ ( 𝑙 = 𝑗 → ( 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
21 |
17 18 20
|
cbvrexw |
⊢ ( ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
22 |
21
|
a1i |
⊢ ( 𝑖 = 𝑘 → ( ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
23 |
12 22
|
bitrd |
⊢ ( 𝑖 = 𝑘 → ( ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
24 |
23
|
cbvralvw |
⊢ ( ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
25 |
24
|
a1i |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
26 |
10 25
|
bitrd |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑦 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
27 |
26
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ ℝ ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑦 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
28 |
|
breq2 |
⊢ ( 𝑦 = 𝑥 → ( ( 𝐹 ‘ 𝑙 ) ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ) ) |
29 |
28
|
ralbidv |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑦 ↔ ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ) ) |
30 |
29
|
rexbidv |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑦 ↔ ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ) ) |
31 |
11
|
raleqdv |
⊢ ( 𝑖 = 𝑘 → ( ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ) ) |
32 |
16 14 13
|
nfbr |
⊢ Ⅎ 𝑗 ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 |
33 |
|
nfv |
⊢ Ⅎ 𝑙 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 |
34 |
19
|
breq1d |
⊢ ( 𝑙 = 𝑗 → ( ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
35 |
32 33 34
|
cbvralw |
⊢ ( ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
36 |
35
|
a1i |
⊢ ( 𝑖 = 𝑘 → ( ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
37 |
31 36
|
bitrd |
⊢ ( 𝑖 = 𝑘 → ( ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
38 |
37
|
cbvrexvw |
⊢ ( ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
39 |
38
|
a1i |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
40 |
30 39
|
bitrd |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑦 ↔ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
41 |
40
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ ℝ ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑦 ↔ ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
42 |
27 41
|
anbi12i |
⊢ ( ( ∃ 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 |
⊢ ( 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 |
7 43
|
bitrd |
⊢ ( 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 |
⊢ Ⅎ 𝑖 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 |
46 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑖 |
47 |
1 46
|
nffv |
⊢ Ⅎ 𝑗 ( 𝐹 ‘ 𝑖 ) |
48 |
47 14 13
|
nfbr |
⊢ Ⅎ 𝑗 ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 |
49 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐹 ‘ 𝑗 ) = ( 𝐹 ‘ 𝑖 ) ) |
50 |
49
|
breq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) ) |
51 |
45 48 50
|
cbvralw |
⊢ ( ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ∀ 𝑖 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) |
52 |
51
|
rexbii |
⊢ ( ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ∃ 𝑘 ∈ 𝑍 ∀ 𝑖 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) |
53 |
52
|
rexbii |
⊢ ( ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑖 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) |
54 |
53
|
a1i |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑖 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) ) |
55 |
|
nfv |
⊢ Ⅎ 𝑖 𝜑 |
56 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → 𝐹 : 𝑍 ⟶ ℝ ) |
57 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → 𝑖 ∈ 𝑍 ) |
58 |
56 57
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℝ ) |
59 |
55 2 3 58
|
uzub |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑖 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) ) |
60 |
|
eqcom |
⊢ ( 𝑗 = 𝑖 ↔ 𝑖 = 𝑗 ) |
61 |
60
|
imbi1i |
⊢ ( ( 𝑗 = 𝑖 → ( ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) ) ↔ ( 𝑖 = 𝑗 → ( ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) ) ) |
62 |
|
bicom |
⊢ ( ( ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) ↔ ( ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
63 |
62
|
imbi2i |
⊢ ( ( 𝑖 = 𝑗 → ( ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) ) ↔ ( 𝑖 = 𝑗 → ( ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) |
64 |
61 63
|
bitri |
⊢ ( ( 𝑗 = 𝑖 → ( ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ) ) ↔ ( 𝑖 = 𝑗 → ( ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) |
65 |
50 64
|
mpbi |
⊢ ( 𝑖 = 𝑗 → ( ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
66 |
48 45 65
|
cbvralw |
⊢ ( ∀ 𝑖 ∈ 𝑍 ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ↔ ∀ 𝑗 ∈ 𝑍 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
67 |
66
|
rexbii |
⊢ ( ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
68 |
67
|
a1i |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 ( 𝐹 ‘ 𝑖 ) ≤ 𝑥 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
69 |
54 59 68
|
3bitrd |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
70 |
69
|
anbi2d |
⊢ ( 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 |
⊢ ( 𝜑 → ( ( lim sup ‘ 𝐹 ) ∈ ℝ ↔ ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) |