Step |
Hyp |
Ref |
Expression |
1 |
|
limsupre3lem.1 |
⊢ Ⅎ 𝑗 𝐹 |
2 |
|
limsupre3lem.2 |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
3 |
|
limsupre3lem.3 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ* ) |
4 |
1 2 3
|
limsupre2 |
⊢ ( 𝜑 → ( ( lim sup ‘ 𝐹 ) ∈ ℝ ↔ ( ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ∧ ∃ 𝑦 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ) ) ) |
5 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ) → 𝑦 ∈ ℝ ) |
6 |
|
nfv |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑦 ∈ ℝ ) |
7 |
|
simp3l |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ) → 𝑘 ≤ 𝑗 ) |
8 |
|
simp1r |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) → 𝑦 ∈ ℝ ) |
9 |
8
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) → 𝑦 ∈ ℝ* ) |
10 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ* ) |
11 |
10
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ* ) |
12 |
11
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ* ) |
13 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) → 𝑦 < ( 𝐹 ‘ 𝑗 ) ) |
14 |
9 12 13
|
xrltled |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) → 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) |
15 |
14
|
3adant3l |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ) → 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) |
16 |
7 15
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ) → ( 𝑘 ≤ 𝑗 ∧ 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
17 |
16
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑗 ∈ 𝐴 → ( ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) → ( 𝑘 ≤ 𝑗 ∧ 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) ) |
18 |
6 17
|
reximdai |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) → ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
19 |
18
|
ralimdv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) → ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
20 |
19
|
3impia |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ) → ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
21 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ↔ 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
22 |
21
|
anbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ↔ ( 𝑘 ≤ 𝑗 ∧ 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
23 |
22
|
rexbidv |
⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ↔ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
24 |
23
|
ralbidv |
⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ↔ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
25 |
24
|
rspcev |
⊢ ( ( 𝑦 ∈ ℝ ∧ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
26 |
5 20 25
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) |
27 |
26
|
3exp |
⊢ ( 𝜑 → ( 𝑦 ∈ ℝ → ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) ) |
28 |
27
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
29 |
|
peano2rem |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 − 1 ) ∈ ℝ ) |
30 |
29
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ( 𝑥 − 1 ) ∈ ℝ ) |
31 |
|
nfv |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑥 ∈ ℝ ) |
32 |
|
simp3l |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → 𝑘 ≤ 𝑗 ) |
33 |
|
simp1r |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → 𝑥 ∈ ℝ ) |
34 |
29
|
rexrd |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 − 1 ) ∈ ℝ* ) |
35 |
33 34
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ( 𝑥 − 1 ) ∈ ℝ* ) |
36 |
33
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → 𝑥 ∈ ℝ* ) |
37 |
10
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ* ) |
38 |
37
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ* ) |
39 |
33
|
ltm1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ( 𝑥 − 1 ) < 𝑥 ) |
40 |
|
simp3r |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) |
41 |
35 36 38 39 40
|
xrltletrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) |
42 |
32 41
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ∧ ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ( 𝑘 ≤ 𝑗 ∧ ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) ) |
43 |
42
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝑗 ∈ 𝐴 → ( ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ( 𝑘 ≤ 𝑗 ∧ ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) ) ) ) |
44 |
31 43
|
reximdai |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) ) ) |
45 |
44
|
ralimdv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) ) ) |
46 |
45
|
imp |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) ) |
47 |
|
breq1 |
⊢ ( 𝑦 = ( 𝑥 − 1 ) → ( 𝑦 < ( 𝐹 ‘ 𝑗 ) ↔ ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) ) |
48 |
47
|
anbi2d |
⊢ ( 𝑦 = ( 𝑥 − 1 ) → ( ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ↔ ( 𝑘 ≤ 𝑗 ∧ ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) ) ) |
49 |
48
|
rexbidv |
⊢ ( 𝑦 = ( 𝑥 − 1 ) → ( ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ↔ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) ) ) |
50 |
49
|
ralbidv |
⊢ ( 𝑦 = ( 𝑥 − 1 ) → ( ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ↔ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) ) ) |
51 |
50
|
rspcev |
⊢ ( ( ( 𝑥 − 1 ) ∈ ℝ ∧ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ ( 𝑥 − 1 ) < ( 𝐹 ‘ 𝑗 ) ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ) |
52 |
30 46 51
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ) |
53 |
52
|
rexlimdva2 |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ) ) |
54 |
28 53
|
impbid |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ) ) |
55 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ) → 𝑦 ∈ ℝ ) |
56 |
11
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑗 ) < 𝑦 ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ* ) |
57 |
|
rexr |
⊢ ( 𝑦 ∈ ℝ → 𝑦 ∈ ℝ* ) |
58 |
57
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑗 ) < 𝑦 ) → 𝑦 ∈ ℝ* ) |
59 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑗 ) < 𝑦 ) → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) |
60 |
56 58 59
|
xrltled |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑗 ) < 𝑦 ) → ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) |
61 |
60
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑗 ) < 𝑦 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) ) |
62 |
61
|
imim2d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) → ( ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) → ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) ) ) |
63 |
62
|
ralimdva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) → ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) ) ) |
64 |
63
|
reximdv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) → ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) ) ) |
65 |
64
|
imp |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ) → ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) ) |
66 |
|
breq2 |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) ) |
67 |
66
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ↔ ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) ) ) |
68 |
67
|
ralbidv |
⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ↔ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) ) ) |
69 |
68
|
rexbidv |
⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ↔ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) ) ) |
70 |
69
|
rspcev |
⊢ ( ( 𝑦 ∈ ℝ ∧ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑦 ) ) → ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
71 |
55 65 70
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ) → ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) |
72 |
71
|
rexlimdva2 |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) → ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) |
73 |
|
peano2re |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 + 1 ) ∈ ℝ ) |
74 |
73
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) → ( 𝑥 + 1 ) ∈ ℝ ) |
75 |
37
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ* ) |
76 |
|
rexr |
⊢ ( 𝑥 ∈ ℝ → 𝑥 ∈ ℝ* ) |
77 |
76
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → 𝑥 ∈ ℝ* ) |
78 |
73
|
rexrd |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 + 1 ) ∈ ℝ* ) |
79 |
78
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ( 𝑥 + 1 ) ∈ ℝ* ) |
80 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) |
81 |
|
ltp1 |
⊢ ( 𝑥 ∈ ℝ → 𝑥 < ( 𝑥 + 1 ) ) |
82 |
81
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → 𝑥 < ( 𝑥 + 1 ) ) |
83 |
75 77 79 80 82
|
xrlelttrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) |
84 |
83
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 → ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) ) |
85 |
84
|
imim2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝐴 ) → ( ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) ) ) |
86 |
85
|
ralimdva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) ) ) |
87 |
86
|
reximdv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) ) ) |
88 |
87
|
imp |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) → ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) ) |
89 |
|
breq2 |
⊢ ( 𝑦 = ( 𝑥 + 1 ) → ( ( 𝐹 ‘ 𝑗 ) < 𝑦 ↔ ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) ) |
90 |
89
|
imbi2d |
⊢ ( 𝑦 = ( 𝑥 + 1 ) → ( ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ↔ ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) ) ) |
91 |
90
|
ralbidv |
⊢ ( 𝑦 = ( 𝑥 + 1 ) → ( ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ↔ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) ) ) |
92 |
91
|
rexbidv |
⊢ ( 𝑦 = ( 𝑥 + 1 ) → ( ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ↔ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) ) ) |
93 |
92
|
rspcev |
⊢ ( ( ( 𝑥 + 1 ) ∈ ℝ ∧ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < ( 𝑥 + 1 ) ) ) → ∃ 𝑦 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ) |
94 |
74 88 93
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) → ∃ 𝑦 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ) |
95 |
94
|
rexlimdva2 |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) → ∃ 𝑦 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ) ) |
96 |
72 95
|
impbid |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ↔ ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) |
97 |
54 96
|
anbi12d |
⊢ ( 𝜑 → ( ( ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑦 < ( 𝐹 ‘ 𝑗 ) ) ∧ ∃ 𝑦 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) < 𝑦 ) ) ↔ ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) ) |
98 |
4 97
|
bitrd |
⊢ ( 𝜑 → ( ( lim sup ‘ 𝐹 ) ∈ ℝ ↔ ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 ∧ 𝑥 ≤ ( 𝐹 ‘ 𝑗 ) ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑘 ∈ ℝ ∀ 𝑗 ∈ 𝐴 ( 𝑘 ≤ 𝑗 → ( 𝐹 ‘ 𝑗 ) ≤ 𝑥 ) ) ) ) |