| Step |
Hyp |
Ref |
Expression |
| 1 |
|
limsupvaluz2.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
| 2 |
|
limsupvaluz2.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
| 3 |
|
limsupvaluz2.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ ) |
| 4 |
|
limsupvaluz2.r |
⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) ∈ ℝ ) |
| 5 |
3
|
frexr |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* ) |
| 6 |
1 2 5
|
limsupvaluz |
⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) = inf ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) , ℝ* , < ) ) |
| 7 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝐹 : 𝑍 ⟶ ℝ ) |
| 8 |
2
|
uzssd3 |
⊢ ( 𝑛 ∈ 𝑍 → ( ℤ≥ ‘ 𝑛 ) ⊆ 𝑍 ) |
| 9 |
8
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ℤ≥ ‘ 𝑛 ) ⊆ 𝑍 ) |
| 10 |
7 9
|
feqresmpt |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) = ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( 𝐹 ‘ 𝑚 ) ) ) |
| 11 |
10
|
rneqd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) = ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( 𝐹 ‘ 𝑚 ) ) ) |
| 12 |
11
|
supeq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) = sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( 𝐹 ‘ 𝑚 ) ) , ℝ* , < ) ) |
| 13 |
|
nfcv |
⊢ Ⅎ 𝑚 𝐹 |
| 14 |
4
|
renepnfd |
⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) ≠ +∞ ) |
| 15 |
13 2 3 14
|
limsupubuz |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑚 ∈ 𝑍 ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 ) |
| 16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑚 ∈ 𝑍 ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 ) |
| 17 |
|
ssralv |
⊢ ( ( ℤ≥ ‘ 𝑛 ) ⊆ 𝑍 → ( ∀ 𝑚 ∈ 𝑍 ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 ) ) |
| 18 |
8 17
|
syl |
⊢ ( 𝑛 ∈ 𝑍 → ( ∀ 𝑚 ∈ 𝑍 ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 ) ) |
| 19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ∀ 𝑚 ∈ 𝑍 ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 ) ) |
| 20 |
19
|
reximdv |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ∃ 𝑥 ∈ ℝ ∀ 𝑚 ∈ 𝑍 ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 → ∃ 𝑥 ∈ ℝ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 ) ) |
| 21 |
16 20
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 ) |
| 22 |
|
nfv |
⊢ Ⅎ 𝑚 ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) |
| 23 |
2
|
eluzelz2 |
⊢ ( 𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ ) |
| 24 |
|
uzid |
⊢ ( 𝑛 ∈ ℤ → 𝑛 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
| 25 |
|
ne0i |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑛 ) → ( ℤ≥ ‘ 𝑛 ) ≠ ∅ ) |
| 26 |
23 24 25
|
3syl |
⊢ ( 𝑛 ∈ 𝑍 → ( ℤ≥ ‘ 𝑛 ) ≠ ∅ ) |
| 27 |
26
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ℤ≥ ‘ 𝑛 ) ≠ ∅ ) |
| 28 |
7
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝐹 : 𝑍 ⟶ ℝ ) |
| 29 |
9
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑚 ∈ 𝑍 ) |
| 30 |
28 29
|
ffvelcdmd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑚 ) ∈ ℝ ) |
| 31 |
22 27 30
|
supxrre3rnmpt |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( 𝐹 ‘ 𝑚 ) ) , ℝ* , < ) ∈ ℝ ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝐹 ‘ 𝑚 ) ≤ 𝑥 ) ) |
| 32 |
21 31
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( 𝐹 ‘ 𝑚 ) ) , ℝ* , < ) ∈ ℝ ) |
| 33 |
12 32
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ∈ ℝ ) |
| 34 |
33
|
fmpttd |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) : 𝑍 ⟶ ℝ ) |
| 35 |
34
|
frnd |
⊢ ( 𝜑 → ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) ⊆ ℝ ) |
| 36 |
|
nfv |
⊢ Ⅎ 𝑛 𝜑 |
| 37 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) = ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) |
| 38 |
1 2
|
uzn0d |
⊢ ( 𝜑 → 𝑍 ≠ ∅ ) |
| 39 |
36 33 37 38
|
rnmptn0 |
⊢ ( 𝜑 → ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) ≠ ∅ ) |
| 40 |
|
nfcv |
⊢ Ⅎ 𝑗 𝐹 |
| 41 |
40 1 2 5
|
limsupre3uz |
⊢ ( 𝜑 → ( ( lim sup ‘ 𝐹 ) ∈ ℝ ↔ ( ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑖 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) |
| 42 |
4 41
|
mpbid |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑖 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
| 43 |
42
|
simpld |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
| 44 |
|
simp-4r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → 𝑥 ∈ ℝ ) |
| 45 |
44
|
rexrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → 𝑥 ∈ ℝ* ) |
| 46 |
5
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) → 𝐹 : 𝑍 ⟶ ℝ* ) |
| 47 |
2
|
uztrn2 |
⊢ ( ( 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) → 𝑗 ∈ 𝑍 ) |
| 48 |
47
|
3adant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) → 𝑗 ∈ 𝑍 ) |
| 49 |
46 48
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ* ) |
| 50 |
49
|
ad5ant134 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ* ) |
| 51 |
|
rnresss |
⊢ ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ⊆ ran 𝐹 |
| 52 |
3
|
frnd |
⊢ ( 𝜑 → ran 𝐹 ⊆ ℝ ) |
| 53 |
52
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ran 𝐹 ⊆ ℝ ) |
| 54 |
51 53
|
sstrid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ⊆ ℝ ) |
| 55 |
54
|
ssrexr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ⊆ ℝ* ) |
| 56 |
55
|
supxrcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ∈ ℝ* ) |
| 57 |
56
|
ad5ant13 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ∈ ℝ* ) |
| 58 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
| 59 |
55
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) → ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ⊆ ℝ* ) |
| 60 |
|
fvres |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) → ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ‘ 𝑗 ) = ( 𝐹 ‘ 𝑗 ) ) |
| 61 |
60
|
eqcomd |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) → ( 𝐹 ‘ 𝑗 ) = ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ‘ 𝑗 ) ) |
| 62 |
61
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) → ( 𝐹 ‘ 𝑗 ) = ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ‘ 𝑗 ) ) |
| 63 |
3
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝑍 ) |
| 64 |
2
|
uzssd3 |
⊢ ( 𝑖 ∈ 𝑍 → ( ℤ≥ ‘ 𝑖 ) ⊆ 𝑍 ) |
| 65 |
|
fnssres |
⊢ ( ( 𝐹 Fn 𝑍 ∧ ( ℤ≥ ‘ 𝑖 ) ⊆ 𝑍 ) → ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) Fn ( ℤ≥ ‘ 𝑖 ) ) |
| 66 |
63 64 65
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) Fn ( ℤ≥ ‘ 𝑖 ) ) |
| 67 |
|
fnfvelrn |
⊢ ( ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) Fn ( ℤ≥ ‘ 𝑖 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) → ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ‘ 𝑗 ) ∈ ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ) |
| 68 |
66 67
|
stoic3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) → ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ‘ 𝑗 ) ∈ ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ) |
| 69 |
62 68
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) → ( 𝐹 ‘ 𝑗 ) ∈ ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ) |
| 70 |
|
eqid |
⊢ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) = sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) |
| 71 |
59 69 70
|
supxrubd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) → ( 𝐹 ‘ 𝑗 ) ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ) |
| 72 |
71
|
ad5ant134 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑗 ) ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ) |
| 73 |
45 50 57 58 72
|
xrletrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ) |
| 74 |
73
|
rexlimdva2 |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑖 ∈ 𝑍 ) → ( ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) → 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ) ) |
| 75 |
74
|
ralimdva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑖 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) → ∀ 𝑖 ∈ 𝑍 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ) ) |
| 76 |
75
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑖 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ) ) |
| 77 |
43 76
|
mpd |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ) |
| 78 |
|
fveq2 |
⊢ ( 𝑛 = 𝑖 → ( ℤ≥ ‘ 𝑛 ) = ( ℤ≥ ‘ 𝑖 ) ) |
| 79 |
78
|
reseq2d |
⊢ ( 𝑛 = 𝑖 → ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) = ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ) |
| 80 |
79
|
rneqd |
⊢ ( 𝑛 = 𝑖 → ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) = ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) ) |
| 81 |
80
|
supeq1d |
⊢ ( 𝑛 = 𝑖 → sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) = sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ) |
| 82 |
|
eqcom |
⊢ ( 𝑛 = 𝑖 ↔ 𝑖 = 𝑛 ) |
| 83 |
|
eqcom |
⊢ ( sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) = sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ↔ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) = sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) |
| 84 |
81 82 83
|
3imtr3i |
⊢ ( 𝑖 = 𝑛 → sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) = sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) |
| 85 |
84
|
breq2d |
⊢ ( 𝑖 = 𝑛 → ( 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ↔ 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) ) |
| 86 |
85
|
cbvralvw |
⊢ ( ∀ 𝑖 ∈ 𝑍 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ↔ ∀ 𝑛 ∈ 𝑍 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) |
| 87 |
86
|
rexbii |
⊢ ( ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑖 ) ) , ℝ* , < ) ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) |
| 88 |
77 87
|
sylib |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) |
| 89 |
36 33
|
rnmptbd2 |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑥 ≤ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) 𝑥 ≤ 𝑦 ) ) |
| 90 |
88 89
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) 𝑥 ≤ 𝑦 ) |
| 91 |
|
infxrre |
⊢ ( ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) ⊆ ℝ ∧ ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) 𝑥 ≤ 𝑦 ) → inf ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) , ℝ* , < ) = inf ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) , ℝ , < ) ) |
| 92 |
35 39 90 91
|
syl3anc |
⊢ ( 𝜑 → inf ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) , ℝ* , < ) = inf ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) , ℝ , < ) ) |
| 93 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( ℤ≥ ‘ 𝑛 ) = ( ℤ≥ ‘ 𝑘 ) ) |
| 94 |
93
|
reseq2d |
⊢ ( 𝑛 = 𝑘 → ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) = ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) ) |
| 95 |
94
|
rneqd |
⊢ ( 𝑛 = 𝑘 → ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) = ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) ) |
| 96 |
95
|
supeq1d |
⊢ ( 𝑛 = 𝑘 → sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) = sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) |
| 97 |
96
|
cbvmptv |
⊢ ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) = ( 𝑘 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) |
| 98 |
97
|
rneqi |
⊢ ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) = ran ( 𝑘 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) |
| 99 |
98
|
infeq1i |
⊢ inf ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) , ℝ , < ) = inf ( ran ( 𝑘 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) , ℝ , < ) |
| 100 |
99
|
a1i |
⊢ ( 𝜑 → inf ( ran ( 𝑛 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑛 ) ) , ℝ* , < ) ) , ℝ , < ) = inf ( ran ( 𝑘 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) , ℝ , < ) ) |
| 101 |
6 92 100
|
3eqtrd |
⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) = inf ( ran ( 𝑘 ∈ 𝑍 ↦ sup ( ran ( 𝐹 ↾ ( ℤ≥ ‘ 𝑘 ) ) , ℝ* , < ) ) , ℝ , < ) ) |