Step |
Hyp |
Ref |
Expression |
1 |
|
limsupre3uzlem.1 |
⊢ Ⅎ 𝑗 𝐹 |
2 |
|
limsupre3uzlem.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
limsupre3uzlem.3 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
4 |
|
limsupre3uzlem.4 |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* ) |
5 |
|
uzssre |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℝ |
6 |
3 5
|
eqsstri |
⊢ 𝑍 ⊆ ℝ |
7 |
6
|
a1i |
⊢ ( 𝜑 → 𝑍 ⊆ ℝ ) |
8 |
1 7 4
|
limsupre3 |
⊢ ( 𝜑 → ( ( lim sup ‘ 𝐹 ) ∈ ℝ ↔ ( ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) ) |
9 |
|
breq1 |
⊢ ( 𝑦 = 𝑘 → ( 𝑦 ≤ 𝑗 ↔ 𝑘 ≤ 𝑗 ) ) |
10 |
9
|
anbi1d |
⊢ ( 𝑦 = 𝑘 → ( ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ↔ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
11 |
10
|
rexbidv |
⊢ ( 𝑦 = 𝑘 → ( ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ↔ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
12 |
11
|
cbvralvw |
⊢ ( ∀ 𝑦 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ↔ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
13 |
12
|
biimpi |
⊢ ( ∀ 𝑦 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
14 |
|
nfra1 |
⊢ Ⅎ 𝑘 ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
15 |
|
simpr |
⊢ ( ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑍 ) → 𝑘 ∈ 𝑍 ) |
16 |
6 15
|
sselid |
⊢ ( ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑍 ) → 𝑘 ∈ ℝ ) |
17 |
|
rspa |
⊢ ( ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑘 ∈ ℝ ) → ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
18 |
16 17
|
syldan |
⊢ ( ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑍 ) → ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
19 |
|
nfv |
⊢ Ⅎ 𝑗 𝑘 ∈ 𝑍 |
20 |
|
nfre1 |
⊢ Ⅎ 𝑗 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) |
21 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑘 ) = ( ℤ≥ ‘ 𝑘 ) |
22 |
3
|
eluzelz2 |
⊢ ( 𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ ) |
23 |
22
|
3ad2ant1 |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗 ) → 𝑘 ∈ ℤ ) |
24 |
3
|
eluzelz2 |
⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ ) |
25 |
24
|
3ad2ant2 |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗 ) → 𝑗 ∈ ℤ ) |
26 |
|
simp3 |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗 ) → 𝑘 ≤ 𝑗 ) |
27 |
21 23 25 26
|
eluzd |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ) |
28 |
27
|
3adant3r |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ) |
29 |
|
simp3r |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
30 |
|
rspe |
⊢ ( ( 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
31 |
28 29 30
|
syl2anc |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
32 |
31
|
3exp |
⊢ ( 𝑘 ∈ 𝑍 → ( 𝑗 ∈ 𝑍 → ( ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
33 |
19 20 32
|
rexlimd |
⊢ ( 𝑘 ∈ 𝑍 → ( ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
34 |
33
|
imp |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
35 |
15 18 34
|
syl2anc |
⊢ ( ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑍 ) → ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
36 |
14 35
|
ralrimia |
⊢ ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
37 |
13 36
|
syl |
⊢ ( ∀ 𝑦 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
38 |
37
|
a1i |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
39 |
|
iftrue |
⊢ ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) = ( ⌈ ‘ 𝑦 ) ) |
40 |
39
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) = ( ⌈ ‘ 𝑦 ) ) |
41 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) → 𝑀 ∈ ℤ ) |
42 |
|
ceilcl |
⊢ ( 𝑦 ∈ ℝ → ( ⌈ ‘ 𝑦 ) ∈ ℤ ) |
43 |
42
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) → ( ⌈ ‘ 𝑦 ) ∈ ℤ ) |
44 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) → 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) |
45 |
3 41 43 44
|
eluzd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) → ( ⌈ ‘ 𝑦 ) ∈ 𝑍 ) |
46 |
40 45
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ∈ 𝑍 ) |
47 |
|
iffalse |
⊢ ( ¬ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) = 𝑀 ) |
48 |
47
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) = 𝑀 ) |
49 |
2 3
|
uzidd2 |
⊢ ( 𝜑 → 𝑀 ∈ 𝑍 ) |
50 |
49
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) → 𝑀 ∈ 𝑍 ) |
51 |
48 50
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ¬ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ∈ 𝑍 ) |
52 |
51
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑀 ≤ ( ⌈ ‘ 𝑦 ) ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ∈ 𝑍 ) |
53 |
46 52
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ∈ 𝑍 ) |
54 |
53
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑦 ∈ ℝ ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ∈ 𝑍 ) |
55 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑦 ∈ ℝ ) → ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
56 |
|
fveq2 |
⊢ ( 𝑘 = if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) → ( ℤ≥ ‘ 𝑘 ) = ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) |
57 |
56
|
rexeqdv |
⊢ ( 𝑘 = if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) → ( ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ↔ ∃ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
58 |
57
|
rspcva |
⊢ ( ( if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ∈ 𝑍 ∧ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∃ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
59 |
54 55 58
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑦 ∈ ℝ ) → ∃ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
60 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
61 |
19
|
nfci |
⊢ Ⅎ 𝑗 𝑍 |
62 |
61 20
|
nfralw |
⊢ Ⅎ 𝑗 ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) |
63 |
60 62
|
nfan |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
64 |
|
nfv |
⊢ Ⅎ 𝑗 𝑦 ∈ ℝ |
65 |
63 64
|
nfan |
⊢ Ⅎ 𝑗 ( ( 𝜑 ∧ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑦 ∈ ℝ ) |
66 |
|
nfre1 |
⊢ Ⅎ 𝑗 ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
67 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑀 ∈ ℤ ) |
68 |
|
eluzelz |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) → 𝑗 ∈ ℤ ) |
69 |
68
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑗 ∈ ℤ ) |
70 |
67
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑀 ∈ ℝ ) |
71 |
6 53
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ∈ ℝ ) |
72 |
71
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ∈ ℝ ) |
73 |
69
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑗 ∈ ℝ ) |
74 |
6 49
|
sselid |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
75 |
74
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑀 ∈ ℝ ) |
76 |
42
|
zred |
⊢ ( 𝑦 ∈ ℝ → ( ⌈ ‘ 𝑦 ) ∈ ℝ ) |
77 |
76
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ⌈ ‘ 𝑦 ) ∈ ℝ ) |
78 |
|
max1 |
⊢ ( ( 𝑀 ∈ ℝ ∧ ( ⌈ ‘ 𝑦 ) ∈ ℝ ) → 𝑀 ≤ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) |
79 |
75 77 78
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑀 ≤ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) |
80 |
79
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑀 ≤ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) |
81 |
|
eluzle |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ≤ 𝑗 ) |
82 |
81
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ≤ 𝑗 ) |
83 |
70 72 73 80 82
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑀 ≤ 𝑗 ) |
84 |
3 67 69 83
|
eluzd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑗 ∈ 𝑍 ) |
85 |
84
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → 𝑗 ∈ 𝑍 ) |
86 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑦 ∈ ℝ ) |
87 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℝ ) |
88 |
|
ceilge |
⊢ ( 𝑦 ∈ ℝ → 𝑦 ≤ ( ⌈ ‘ 𝑦 ) ) |
89 |
88
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ≤ ( ⌈ ‘ 𝑦 ) ) |
90 |
|
max2 |
⊢ ( ( 𝑀 ∈ ℝ ∧ ( ⌈ ‘ 𝑦 ) ∈ ℝ ) → ( ⌈ ‘ 𝑦 ) ≤ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) |
91 |
75 77 90
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ⌈ ‘ 𝑦 ) ≤ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) |
92 |
87 77 71 89 91
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ≤ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) |
93 |
92
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑦 ≤ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) |
94 |
86 72 73 93 82
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑦 ≤ 𝑗 ) |
95 |
94
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → 𝑦 ≤ 𝑗 ) |
96 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
97 |
95 96
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
98 |
|
rspe |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
99 |
85 97 98
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
100 |
99
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) → ( 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) → ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) ) |
101 |
100
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑦 ∈ ℝ ) → ( 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) → ( 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) → ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) ) |
102 |
65 66 101
|
rexlimd |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑦 ∈ ℝ ) → ( ∃ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) → ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
103 |
59 102
|
mpd |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑦 ∈ ℝ ) → ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
104 |
103
|
ralrimiva |
⊢ ( ( 𝜑 ∧ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∀ 𝑦 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
105 |
104
|
ex |
⊢ ( 𝜑 → ( ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) → ∀ 𝑦 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
106 |
38 105
|
impbid |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ↔ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
107 |
106
|
rexbidv |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
108 |
53
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) → if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ∈ 𝑍 ) |
109 |
60 64
|
nfan |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑦 ∈ ℝ ) |
110 |
|
nfra1 |
⊢ Ⅎ 𝑗 ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
111 |
109 110
|
nfan |
⊢ Ⅎ 𝑗 ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
112 |
94
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑦 ≤ 𝑗 ) |
113 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
114 |
84
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → 𝑗 ∈ 𝑍 ) |
115 |
|
rspa |
⊢ ( ( ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ∧ 𝑗 ∈ 𝑍 ) → ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
116 |
113 114 115
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
117 |
112 116
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ) → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
118 |
117
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) → ( 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
119 |
111 118
|
ralrimi |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) → ∀ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
120 |
56
|
raleqdv |
⊢ ( 𝑘 = if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) → ( ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ∀ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
121 |
120
|
rspcev |
⊢ ( ( if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ∈ 𝑍 ∧ ∀ 𝑗 ∈ ( ℤ≥ ‘ if ( 𝑀 ≤ ( ⌈ ‘ 𝑦 ) , ( ⌈ ‘ 𝑦 ) , 𝑀 ) ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
122 |
108 119 121
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) → ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
123 |
122
|
rexlimdva2 |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
124 |
6
|
sseli |
⊢ ( 𝑘 ∈ 𝑍 → 𝑘 ∈ ℝ ) |
125 |
124
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → 𝑘 ∈ ℝ ) |
126 |
|
nfra1 |
⊢ Ⅎ 𝑗 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 |
127 |
19 126
|
nfan |
⊢ Ⅎ 𝑗 ( 𝑘 ∈ 𝑍 ∧ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
128 |
|
simp1r |
⊢ ( ( ( 𝑘 ∈ 𝑍 ∧ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗 ) → ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
129 |
27
|
3adant1r |
⊢ ( ( ( 𝑘 ∈ 𝑍 ∧ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ) |
130 |
|
rspa |
⊢ ( ( ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
131 |
128 129 130
|
syl2anc |
⊢ ( ( ( 𝑘 ∈ 𝑍 ∧ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗 ) → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
132 |
131
|
3exp |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ( 𝑗 ∈ 𝑍 → ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) |
133 |
127 132
|
ralrimi |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ∀ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
134 |
133
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ∀ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
135 |
9
|
rspceaimv |
⊢ ( ( 𝑘 ∈ ℝ ∧ ∀ 𝑗 ∈ 𝑍 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
136 |
125 134 135
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
137 |
136
|
rexlimdva2 |
⊢ ( 𝜑 → ( ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 → ∃ 𝑦 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) |
138 |
123 137
|
impbid |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ↔ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
139 |
138
|
rexbidv |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ↔ ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
140 |
107 139
|
anbi12d |
⊢ ( 𝜑 → ( ( ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ∀ 𝑗 ∈ 𝑍 ( 𝑦 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ↔ ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ∃ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ 𝑍 ∀ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) |
141 |
8 140
|
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 ) ) ) |