Step |
Hyp |
Ref |
Expression |
1 |
|
liminfpnfuz.1 |
⊢ Ⅎ 𝑗 𝐹 |
2 |
|
liminfpnfuz.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
liminfpnfuz.3 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
4 |
|
liminfpnfuz.4 |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* ) |
5 |
|
nfv |
⊢ Ⅎ 𝑙 𝜑 |
6 |
|
nfcv |
⊢ Ⅎ 𝑙 𝐹 |
7 |
5 6 2 3 4
|
liminfvaluz3 |
⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) = -𝑒 ( lim sup ‘ ( 𝑙 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑙 ) ) ) ) |
8 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑙 |
9 |
1 8
|
nffv |
⊢ Ⅎ 𝑗 ( 𝐹 ‘ 𝑙 ) |
10 |
9
|
nfxneg |
⊢ Ⅎ 𝑗 -𝑒 ( 𝐹 ‘ 𝑙 ) |
11 |
|
nfcv |
⊢ Ⅎ 𝑙 -𝑒 ( 𝐹 ‘ 𝑗 ) |
12 |
|
fveq2 |
⊢ ( 𝑙 = 𝑗 → ( 𝐹 ‘ 𝑙 ) = ( 𝐹 ‘ 𝑗 ) ) |
13 |
12
|
xnegeqd |
⊢ ( 𝑙 = 𝑗 → -𝑒 ( 𝐹 ‘ 𝑙 ) = -𝑒 ( 𝐹 ‘ 𝑗 ) ) |
14 |
10 11 13
|
cbvmpt |
⊢ ( 𝑙 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑙 ) ) = ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) |
15 |
14
|
fveq2i |
⊢ ( lim sup ‘ ( 𝑙 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑙 ) ) ) = ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) |
16 |
15
|
xnegeqi |
⊢ -𝑒 ( lim sup ‘ ( 𝑙 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑙 ) ) ) = -𝑒 ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) |
17 |
7 16
|
eqtrdi |
⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) = -𝑒 ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) ) |
18 |
17
|
eqeq1d |
⊢ ( 𝜑 → ( ( lim inf ‘ 𝐹 ) = +∞ ↔ -𝑒 ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = +∞ ) ) |
19 |
|
xnegmnf |
⊢ -𝑒 -∞ = +∞ |
20 |
19
|
eqcomi |
⊢ +∞ = -𝑒 -∞ |
21 |
20
|
a1i |
⊢ ( 𝜑 → +∞ = -𝑒 -∞ ) |
22 |
21
|
eqeq2d |
⊢ ( 𝜑 → ( -𝑒 ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = +∞ ↔ -𝑒 ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = -𝑒 -∞ ) ) |
23 |
3
|
fvexi |
⊢ 𝑍 ∈ V |
24 |
23
|
mptex |
⊢ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ∈ V |
25 |
24
|
a1i |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ∈ V ) |
26 |
25
|
limsupcld |
⊢ ( 𝜑 → ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) ∈ ℝ* ) |
27 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
28 |
|
xneg11 |
⊢ ( ( ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) ∈ ℝ* ∧ -∞ ∈ ℝ* ) → ( -𝑒 ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = -𝑒 -∞ ↔ ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = -∞ ) ) |
29 |
26 27 28
|
sylancl |
⊢ ( 𝜑 → ( -𝑒 ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = -𝑒 -∞ ↔ ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = -∞ ) ) |
30 |
22 29
|
bitrd |
⊢ ( 𝜑 → ( -𝑒 ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = +∞ ↔ ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = -∞ ) ) |
31 |
3
|
uztrn2 |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ) → 𝑗 ∈ 𝑍 ) |
32 |
|
xnegex |
⊢ -𝑒 ( 𝐹 ‘ 𝑗 ) ∈ V |
33 |
|
fvmpt4 |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ -𝑒 ( 𝐹 ‘ 𝑗 ) ∈ V ) → ( ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ‘ 𝑗 ) = -𝑒 ( 𝐹 ‘ 𝑗 ) ) |
34 |
31 32 33
|
sylancl |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ) → ( ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ‘ 𝑗 ) = -𝑒 ( 𝐹 ‘ 𝑗 ) ) |
35 |
34
|
breq1d |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ) → ( ( ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ‘ 𝑗 ) ≤ 𝑥 ↔ -𝑒 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
36 |
35
|
ralbidva |
⊢ ( 𝑘 ∈ 𝑍 → ( ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ‘ 𝑗 ) ≤ 𝑥 ↔ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) -𝑒 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
37 |
36
|
rexbiia |
⊢ ( ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ‘ 𝑗 ) ≤ 𝑥 ↔ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) -𝑒 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
38 |
37
|
ralbii |
⊢ ( ∀ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ‘ 𝑗 ) ≤ 𝑥 ↔ ∀ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) -𝑒 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
39 |
38
|
a1i |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ‘ 𝑗 ) ≤ 𝑥 ↔ ∀ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) -𝑒 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
40 |
|
nfmpt1 |
⊢ Ⅎ 𝑗 ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) |
41 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑙 ) ∈ ℝ* ) |
42 |
41
|
xnegcld |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑍 ) → -𝑒 ( 𝐹 ‘ 𝑙 ) ∈ ℝ* ) |
43 |
14
|
eqcomi |
⊢ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) = ( 𝑙 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑙 ) ) |
44 |
42 43
|
fmptd |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) : 𝑍 ⟶ ℝ* ) |
45 |
40 2 3 44
|
limsupmnfuz |
⊢ ( 𝜑 → ( ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = -∞ ↔ ∀ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ‘ 𝑗 ) ≤ 𝑥 ) ) |
46 |
1 3 4
|
xlimpnfxnegmnf |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ↔ ∀ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) -𝑒 ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
47 |
39 45 46
|
3bitr4d |
⊢ ( 𝜑 → ( ( lim sup ‘ ( 𝑗 ∈ 𝑍 ↦ -𝑒 ( 𝐹 ‘ 𝑗 ) ) ) = -∞ ↔ ∀ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
48 |
18 30 47
|
3bitrd |
⊢ ( 𝜑 → ( ( lim inf ‘ 𝐹 ) = +∞ ↔ ∀ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |