Step |
Hyp |
Ref |
Expression |
1 |
|
climliminflimsupd.1 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
2 |
|
climliminflimsupd.2 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
3 |
|
climliminflimsupd.3 |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ ) |
4 |
|
climliminflimsupd.4 |
⊢ ( 𝜑 → 𝐹 ∈ dom ⇝ ) |
5 |
3
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) |
6 |
5
|
fveq2d |
⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) = ( lim inf ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) |
7 |
2
|
fvexi |
⊢ 𝑍 ∈ V |
8 |
7
|
mptex |
⊢ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ∈ V |
9 |
|
liminfcl |
⊢ ( ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ∈ V → ( lim inf ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ℝ* ) |
10 |
8 9
|
ax-mp |
⊢ ( lim inf ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ℝ* |
11 |
10
|
a1i |
⊢ ( 𝜑 → ( lim inf ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ℝ* ) |
12 |
6 11
|
eqeltrd |
⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) ∈ ℝ* ) |
13 |
|
nfv |
⊢ Ⅎ 𝑘 𝜑 |
14 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
15 |
14
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → - ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
16 |
13 1 2 15
|
limsupvaluz4 |
⊢ ( 𝜑 → ( lim sup ‘ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ) = -𝑒 ( lim inf ‘ ( 𝑘 ∈ 𝑍 ↦ - - ( 𝐹 ‘ 𝑘 ) ) ) ) |
17 |
|
climrel |
⊢ Rel ⇝ |
18 |
17
|
a1i |
⊢ ( 𝜑 → Rel ⇝ ) |
19 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐹 |
20 |
1 2 3
|
climlimsup |
⊢ ( 𝜑 → ( 𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( lim sup ‘ 𝐹 ) ) ) |
21 |
4 20
|
mpbid |
⊢ ( 𝜑 → 𝐹 ⇝ ( lim sup ‘ 𝐹 ) ) |
22 |
14
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
23 |
13 19 2 1 21 22
|
climneg |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ⇝ - ( lim sup ‘ 𝐹 ) ) |
24 |
|
releldm |
⊢ ( ( Rel ⇝ ∧ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ⇝ - ( lim sup ‘ 𝐹 ) ) → ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ∈ dom ⇝ ) |
25 |
18 23 24
|
syl2anc |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ∈ dom ⇝ ) |
26 |
15
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) : 𝑍 ⟶ ℝ ) |
27 |
1 2 26
|
climlimsup |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ∈ dom ⇝ ↔ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ⇝ ( lim sup ‘ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
28 |
25 27
|
mpbid |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ⇝ ( lim sup ‘ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ) ) |
29 |
|
climuni |
⊢ ( ( ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ⇝ ( lim sup ‘ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ) ∧ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ⇝ - ( lim sup ‘ 𝐹 ) ) → ( lim sup ‘ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ) = - ( lim sup ‘ 𝐹 ) ) |
30 |
28 23 29
|
syl2anc |
⊢ ( 𝜑 → ( lim sup ‘ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ) = - ( lim sup ‘ 𝐹 ) ) |
31 |
22
|
negnegd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → - - ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
32 |
31
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - - ( 𝐹 ‘ 𝑘 ) ) = ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) |
33 |
32 5
|
eqtr4d |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - - ( 𝐹 ‘ 𝑘 ) ) = 𝐹 ) |
34 |
33
|
fveq2d |
⊢ ( 𝜑 → ( lim inf ‘ ( 𝑘 ∈ 𝑍 ↦ - - ( 𝐹 ‘ 𝑘 ) ) ) = ( lim inf ‘ 𝐹 ) ) |
35 |
34
|
xnegeqd |
⊢ ( 𝜑 → -𝑒 ( lim inf ‘ ( 𝑘 ∈ 𝑍 ↦ - - ( 𝐹 ‘ 𝑘 ) ) ) = -𝑒 ( lim inf ‘ 𝐹 ) ) |
36 |
16 30 35
|
3eqtr3d |
⊢ ( 𝜑 → - ( lim sup ‘ 𝐹 ) = -𝑒 ( lim inf ‘ 𝐹 ) ) |
37 |
2 1 21 14
|
climrecl |
⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) ∈ ℝ ) |
38 |
37
|
renegcld |
⊢ ( 𝜑 → - ( lim sup ‘ 𝐹 ) ∈ ℝ ) |
39 |
36 38
|
eqeltrrd |
⊢ ( 𝜑 → -𝑒 ( lim inf ‘ 𝐹 ) ∈ ℝ ) |
40 |
|
xnegrecl2 |
⊢ ( ( ( lim inf ‘ 𝐹 ) ∈ ℝ* ∧ -𝑒 ( lim inf ‘ 𝐹 ) ∈ ℝ ) → ( lim inf ‘ 𝐹 ) ∈ ℝ ) |
41 |
12 39 40
|
syl2anc |
⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) ∈ ℝ ) |
42 |
41
|
recnd |
⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) ∈ ℂ ) |
43 |
37
|
recnd |
⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) ∈ ℂ ) |
44 |
41
|
rexnegd |
⊢ ( 𝜑 → -𝑒 ( lim inf ‘ 𝐹 ) = - ( lim inf ‘ 𝐹 ) ) |
45 |
36 44
|
eqtr2d |
⊢ ( 𝜑 → - ( lim inf ‘ 𝐹 ) = - ( lim sup ‘ 𝐹 ) ) |
46 |
42 43 45
|
neg11d |
⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) = ( lim sup ‘ 𝐹 ) ) |