Step |
Hyp |
Ref |
Expression |
1 |
|
limsupvaluz.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
2 |
|
limsupvaluz.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
3 |
|
limsupvaluz.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* ) |
4 |
|
eqid |
⊢ ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) = ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
5 |
2
|
fvexi |
⊢ 𝑍 ∈ V |
6 |
5
|
a1i |
⊢ ( 𝜑 → 𝑍 ∈ V ) |
7 |
3 6
|
fexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
8 |
7
|
elexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
9 |
|
uzssre |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℝ |
10 |
2 9
|
eqsstri |
⊢ 𝑍 ⊆ ℝ |
11 |
10
|
a1i |
⊢ ( 𝜑 → 𝑍 ⊆ ℝ ) |
12 |
2
|
uzsup |
⊢ ( 𝑀 ∈ ℤ → sup ( 𝑍 , ℝ* , < ) = +∞ ) |
13 |
1 12
|
syl |
⊢ ( 𝜑 → sup ( 𝑍 , ℝ* , < ) = +∞ ) |
14 |
4 8 11 13
|
limsupval2 |
⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) = inf ( ( ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) “ 𝑍 ) , ℝ* , < ) ) |
15 |
11
|
mptima2 |
⊢ ( 𝜑 → ( ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) “ 𝑍 ) = ran ( 𝑖 ∈ 𝑍 ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ) |
16 |
|
oveq1 |
⊢ ( 𝑖 = 𝑛 → ( 𝑖 [,) +∞ ) = ( 𝑛 [,) +∞ ) ) |
17 |
16
|
imaeq2d |
⊢ ( 𝑖 = 𝑛 → ( 𝐹 “ ( 𝑖 [,) +∞ ) ) = ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ) |
18 |
17
|
ineq1d |
⊢ ( 𝑖 = 𝑛 → ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) = ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) ) |
19 |
18
|
supeq1d |
⊢ ( 𝑖 = 𝑛 → sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) = sup ( ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
20 |
19
|
cbvmptv |
⊢ ( 𝑖 ∈ 𝑍 ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) = ( 𝑛 ∈ 𝑍 ↦ sup ( ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
21 |
20
|
a1i |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝑍 ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) = ( 𝑛 ∈ 𝑍 ↦ sup ( ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ) |
22 |
|
fimass |
⊢ ( 𝐹 : 𝑍 ⟶ ℝ* → ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ⊆ ℝ* ) |
23 |
3 22
|
syl |
⊢ ( 𝜑 → ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ⊆ ℝ* ) |
24 |
|
df-ss |
⊢ ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ⊆ ℝ* ↔ ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) = ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ) |
25 |
24
|
biimpi |
⊢ ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ⊆ ℝ* → ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) = ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ) |
26 |
23 25
|
syl |
⊢ ( 𝜑 → ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) = ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) = ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ) |
28 |
|
df-ima |
⊢ ( 𝐹 “ ( 𝑛 [,) +∞ ) ) = ran ( 𝐹 ↾ ( 𝑛 [,) +∞ ) ) |
29 |
28
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 “ ( 𝑛 [,) +∞ ) ) = ran ( 𝐹 ↾ ( 𝑛 [,) +∞ ) ) ) |
30 |
3
|
freld |
⊢ ( 𝜑 → Rel 𝐹 ) |
31 |
|
resindm |
⊢ ( Rel 𝐹 → ( 𝐹 ↾ ( ( 𝑛 [,) +∞ ) ∩ dom 𝐹 ) ) = ( 𝐹 ↾ ( 𝑛 [,) +∞ ) ) ) |
32 |
30 31
|
syl |
⊢ ( 𝜑 → ( 𝐹 ↾ ( ( 𝑛 [,) +∞ ) ∩ dom 𝐹 ) ) = ( 𝐹 ↾ ( 𝑛 [,) +∞ ) ) ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ↾ ( ( 𝑛 [,) +∞ ) ∩ dom 𝐹 ) ) = ( 𝐹 ↾ ( 𝑛 [,) +∞ ) ) ) |
34 |
|
incom |
⊢ ( ( 𝑛 [,) +∞ ) ∩ 𝑍 ) = ( 𝑍 ∩ ( 𝑛 [,) +∞ ) ) |
35 |
2
|
ineq1i |
⊢ ( 𝑍 ∩ ( 𝑛 [,) +∞ ) ) = ( ( ℤ≥ ‘ 𝑀 ) ∩ ( 𝑛 [,) +∞ ) ) |
36 |
34 35
|
eqtri |
⊢ ( ( 𝑛 [,) +∞ ) ∩ 𝑍 ) = ( ( ℤ≥ ‘ 𝑀 ) ∩ ( 𝑛 [,) +∞ ) ) |
37 |
36
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 [,) +∞ ) ∩ 𝑍 ) = ( ( ℤ≥ ‘ 𝑀 ) ∩ ( 𝑛 [,) +∞ ) ) ) |
38 |
3
|
fdmd |
⊢ ( 𝜑 → dom 𝐹 = 𝑍 ) |
39 |
38
|
ineq2d |
⊢ ( 𝜑 → ( ( 𝑛 [,) +∞ ) ∩ dom 𝐹 ) = ( ( 𝑛 [,) +∞ ) ∩ 𝑍 ) ) |
40 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 [,) +∞ ) ∩ dom 𝐹 ) = ( ( 𝑛 [,) +∞ ) ∩ 𝑍 ) ) |
41 |
2
|
eleq2i |
⊢ ( 𝑛 ∈ 𝑍 ↔ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
42 |
41
|
biimpi |
⊢ ( 𝑛 ∈ 𝑍 → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
43 |
42
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
44 |
43
|
uzinico2 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ℤ≥ ‘ 𝑛 ) = ( ( ℤ≥ ‘ 𝑀 ) ∩ ( 𝑛 [,) +∞ ) ) ) |
45 |
37 40 44
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 [,) +∞ ) ∩ dom 𝐹 ) = ( ℤ≥ ‘ 𝑛 ) ) |
46 |
45
|
reseq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ↾ ( ( 𝑛 [,) +∞ ) ∩ dom 𝐹 ) ) = ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) ) |
47 |
33 46
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ↾ ( 𝑛 [,) +∞ ) ) = ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) ) |
48 |
47
|
rneqd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ran ( 𝐹 ↾ ( 𝑛 [,) +∞ ) ) = ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) ) |
49 |
27 29 48
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) = ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) ) |
50 |
49
|
supeq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → sup ( ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) = sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) |
51 |
50
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ sup ( ( ( 𝐹 “ ( 𝑛 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) = ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) ) |
52 |
21 51
|
eqtrd |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝑍 ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) = ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) ) |
53 |
52
|
rneqd |
⊢ ( 𝜑 → ran ( 𝑖 ∈ 𝑍 ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) = ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) ) |
54 |
15 53
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) “ 𝑍 ) = ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) ) |
55 |
54
|
infeq1d |
⊢ ( 𝜑 → inf ( ( ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) “ 𝑍 ) , ℝ* , < ) = inf ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) , ℝ* , < ) ) |
56 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( ℤ≥ ‘ 𝑛 ) = ( ℤ≥ ‘ 𝑘 ) ) |
57 |
56
|
reseq2d |
⊢ ( 𝑛 = 𝑘 → ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) = ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) ) |
58 |
57
|
rneqd |
⊢ ( 𝑛 = 𝑘 → ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) = ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) ) |
59 |
58
|
supeq1d |
⊢ ( 𝑛 = 𝑘 → sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) = sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) |
60 |
59
|
cbvmptv |
⊢ ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) = ( 𝑘 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) |
61 |
60
|
rneqi |
⊢ ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) = ran ( 𝑘 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) |
62 |
61
|
infeq1i |
⊢ inf ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) , ℝ* , < ) = inf ( ran ( 𝑘 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) , ℝ* , < ) |
63 |
62
|
a1i |
⊢ ( 𝜑 → inf ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) , ℝ* , < ) = inf ( ran ( 𝑘 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) , ℝ* , < ) ) |
64 |
14 55 63
|
3eqtrd |
⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) = inf ( ran ( 𝑘 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) , ℝ* , < ) ) |