Step |
Hyp |
Ref |
Expression |
1 |
|
limsupre3uz.1 |
⊢ Ⅎ 𝑗 𝐹 |
2 |
|
limsupre3uz.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
limsupre3uz.3 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
4 |
|
limsupre3uz.4 |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* ) |
5 |
|
nfcv |
⊢ Ⅎ 𝑙 𝐹 |
6 |
5 2 3 4
|
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 ) ) ) |
7 |
|
breq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ≤ ( 𝐹 ‘ 𝑙 ) ↔ 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ) ) |
8 |
7
|
rexbidv |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑦 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ) ) |
9 |
8
|
ralbidv |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑦 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ) ) |
10 |
|
fveq2 |
⊢ ( 𝑖 = 𝑘 → ( ℤ≥ ‘ 𝑖 ) = ( ℤ≥ ‘ 𝑘 ) ) |
11 |
10
|
rexeqdv |
⊢ ( 𝑖 = 𝑘 → ( ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ) ) |
12 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑥 |
13 |
|
nfcv |
⊢ Ⅎ 𝑗 ≤ |
14 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑙 |
15 |
1 14
|
nffv |
⊢ Ⅎ 𝑗 ( 𝐹 ‘ 𝑙 ) |
16 |
12 13 15
|
nfbr |
⊢ Ⅎ 𝑗 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) |
17 |
|
nfv |
⊢ Ⅎ 𝑙 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) |
18 |
|
fveq2 |
⊢ ( 𝑙 = 𝑗 → ( 𝐹 ‘ 𝑙 ) = ( 𝐹 ‘ 𝑗 ) ) |
19 |
18
|
breq2d |
⊢ ( 𝑙 = 𝑗 → ( 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
20 |
16 17 19
|
cbvrexw |
⊢ ( ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
21 |
20
|
a1i |
⊢ ( 𝑖 = 𝑘 → ( ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
22 |
11 21
|
bitrd |
⊢ ( 𝑖 = 𝑘 → ( ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
23 |
22
|
cbvralvw |
⊢ ( ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
24 |
23
|
a1i |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
25 |
9 24
|
bitrd |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑦 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
26 |
25
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ ℝ ∀ 𝑖 ∈ 𝑍 ∃ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑦 ≤ ( 𝐹 ‘ 𝑙 ) ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
27 |
|
breq2 |
⊢ ( 𝑦 = 𝑥 → ( ( 𝐹 ‘ 𝑙 ) ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ) ) |
28 |
27
|
ralbidv |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑦 ↔ ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ) ) |
29 |
28
|
rexbidv |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑦 ↔ ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ) ) |
30 |
10
|
raleqdv |
⊢ ( 𝑖 = 𝑘 → ( ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ) ) |
31 |
15 13 12
|
nfbr |
⊢ Ⅎ 𝑗 ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 |
32 |
|
nfv |
⊢ Ⅎ 𝑙 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 |
33 |
18
|
breq1d |
⊢ ( 𝑙 = 𝑗 → ( ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
34 |
31 32 33
|
cbvralw |
⊢ ( ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
35 |
34
|
a1i |
⊢ ( 𝑖 = 𝑘 → ( ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
36 |
30 35
|
bitrd |
⊢ ( 𝑖 = 𝑘 → ( ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
37 |
36
|
cbvrexvw |
⊢ ( ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
38 |
37
|
a1i |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑥 ↔ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
39 |
29 38
|
bitrd |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑦 ↔ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
40 |
39
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ ℝ ∃ 𝑖 ∈ 𝑍 ∀ 𝑙 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑙 ) ≤ 𝑦 ↔ ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
41 |
26 40
|
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 ) ) |
42 |
41
|
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 ) ) ) |
43 |
6 42
|
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 ) ) ) |