Step |
Hyp |
Ref |
Expression |
1 |
|
climuzlem.1 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
2 |
|
climuzlem.2 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
3 |
|
climuzlem.3 |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℂ ) |
4 |
|
climcl |
⊢ ( 𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ ) |
5 |
4
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 𝐴 ) → 𝐴 ∈ ℂ ) |
6 |
|
id |
⊢ ( 𝐹 ⇝ 𝐴 → 𝐹 ⇝ 𝐴 ) |
7 |
|
climrel |
⊢ Rel ⇝ |
8 |
7
|
brrelex1i |
⊢ ( 𝐹 ⇝ 𝐴 → 𝐹 ∈ V ) |
9 |
|
eqidd |
⊢ ( ( 𝐹 ⇝ 𝐴 ∧ 𝑘 ∈ ℤ ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
10 |
8 9
|
clim |
⊢ ( 𝐹 ⇝ 𝐴 → ( 𝐹 ⇝ 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
11 |
6 10
|
mpbid |
⊢ ( 𝐹 ⇝ 𝐴 → ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |
12 |
11
|
simprd |
⊢ ( 𝐹 ⇝ 𝐴 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
13 |
12
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 𝐴 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
15 |
2
|
rexuz3 |
⊢ ( 𝑀 ∈ ℤ → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |
16 |
1 15
|
syl |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |
18 |
14 17
|
mpbird |
⊢ ( ( 𝜑 ∧ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
19 |
|
simpr |
⊢ ( ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) |
20 |
19
|
ralimi |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) |
21 |
20
|
reximi |
⊢ ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) |
22 |
21
|
a1i |
⊢ ( ( 𝜑 ∧ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
23 |
18 22
|
mpd |
⊢ ( ( 𝜑 ∧ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) |
24 |
23
|
ex |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
25 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
26 |
25
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 𝐴 ) → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
28 |
13 27
|
mpd |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 𝐴 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) |
29 |
5 28
|
jca |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 𝐴 ) → ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
30 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → 𝐴 ∈ ℂ ) |
31 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
32 |
|
nfre1 |
⊢ Ⅎ 𝑗 ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) |
33 |
2
|
uzssz2 |
⊢ 𝑍 ⊆ ℤ |
34 |
33
|
sseli |
⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ ) |
35 |
34
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → 𝑗 ∈ ℤ ) |
36 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝜑 ) |
37 |
2
|
uztrn2 |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
38 |
37
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
39 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
40 |
36 38 39
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
41 |
40
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
42 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) |
43 |
41 42
|
jca |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
44 |
43
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 → ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |
45 |
44
|
ralimdva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |
46 |
45
|
3impia |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
47 |
|
rspe |
⊢ ( ( 𝑗 ∈ ℤ ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
48 |
35 46 47
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
49 |
48
|
3exp |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑍 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
50 |
31 32 49
|
rexlimd |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |
51 |
50
|
ralimdv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |
52 |
51
|
imp |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
53 |
52
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
54 |
30 53
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |
55 |
2
|
fvexi |
⊢ 𝑍 ∈ V |
56 |
55
|
a1i |
⊢ ( 𝜑 → 𝑍 ∈ V ) |
57 |
3 56
|
fexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
58 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
59 |
57 58
|
clim |
⊢ ( 𝜑 → ( 𝐹 ⇝ 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
60 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → ( 𝐹 ⇝ 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
61 |
54 60
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) → 𝐹 ⇝ 𝐴 ) |
62 |
29 61
|
impbida |
⊢ ( 𝜑 → ( 𝐹 ⇝ 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |