Step |
Hyp |
Ref |
Expression |
1 |
|
climshft.1 |
⊢ 𝐹 ∈ V |
2 |
|
zaddcl |
⊢ ( ( 𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑘 + 𝑀 ) ∈ ℤ ) |
3 |
2
|
ancoms |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( 𝑘 + 𝑀 ) ∈ ℤ ) |
4 |
|
eluzsub |
⊢ ( ( 𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ) → ( 𝑛 − 𝑀 ) ∈ ( ℤ≥ ‘ 𝑘 ) ) |
5 |
4
|
3com12 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ) → ( 𝑛 − 𝑀 ) ∈ ( ℤ≥ ‘ 𝑘 ) ) |
6 |
5
|
3expa |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ) → ( 𝑛 − 𝑀 ) ∈ ( ℤ≥ ‘ 𝑘 ) ) |
7 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑛 − 𝑀 ) → ( 𝐹 ‘ 𝑚 ) = ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) ) |
8 |
7
|
eleq1d |
⊢ ( 𝑚 = ( 𝑛 − 𝑀 ) → ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ↔ ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) ∈ ℂ ) ) |
9 |
7
|
fvoveq1d |
⊢ ( 𝑚 = ( 𝑛 − 𝑀 ) → ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) = ( abs ‘ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) − 𝐴 ) ) ) |
10 |
9
|
breq1d |
⊢ ( 𝑚 = ( 𝑛 − 𝑀 ) → ( ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ↔ ( abs ‘ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) − 𝐴 ) ) < 𝑥 ) ) |
11 |
8 10
|
anbi12d |
⊢ ( 𝑚 = ( 𝑛 − 𝑀 ) → ( ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ) ↔ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) − 𝐴 ) ) < 𝑥 ) ) ) |
12 |
11
|
rspcv |
⊢ ( ( 𝑛 − 𝑀 ) ∈ ( ℤ≥ ‘ 𝑘 ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ) → ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) − 𝐴 ) ) < 𝑥 ) ) ) |
13 |
6 12
|
syl |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ) → ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) − 𝐴 ) ) < 𝑥 ) ) ) |
14 |
|
zcn |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℂ ) |
15 |
|
eluzelcn |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) → 𝑛 ∈ ℂ ) |
16 |
1
|
shftval |
⊢ ( ( 𝑀 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) = ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) ) |
17 |
16
|
eleq1d |
⊢ ( ( 𝑀 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ↔ ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) ∈ ℂ ) ) |
18 |
16
|
fvoveq1d |
⊢ ( ( 𝑀 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) = ( abs ‘ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) − 𝐴 ) ) ) |
19 |
18
|
breq1d |
⊢ ( ( 𝑀 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ↔ ( abs ‘ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) − 𝐴 ) ) < 𝑥 ) ) |
20 |
17 19
|
anbi12d |
⊢ ( ( 𝑀 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ↔ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) − 𝐴 ) ) < 𝑥 ) ) ) |
21 |
14 15 20
|
syl2an |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ) → ( ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ↔ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) − 𝐴 ) ) < 𝑥 ) ) ) |
22 |
21
|
adantlr |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ) → ( ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ↔ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ ( 𝑛 − 𝑀 ) ) − 𝐴 ) ) < 𝑥 ) ) ) |
23 |
13 22
|
sylibrd |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ) → ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ) ) |
24 |
23
|
ralrimdva |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ) ) |
25 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑘 + 𝑀 ) → ( ℤ≥ ‘ 𝑚 ) = ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ) |
26 |
25
|
raleqdv |
⊢ ( 𝑚 = ( 𝑘 + 𝑀 ) → ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ↔ ∀ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ) ) |
27 |
26
|
rspcev |
⊢ ( ( ( 𝑘 + 𝑀 ) ∈ ℤ ∧ ∀ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑘 + 𝑀 ) ) ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ) → ∃ 𝑚 ∈ ℤ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ) |
28 |
3 24 27
|
syl6an |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ) → ∃ 𝑚 ∈ ℤ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ) ) |
29 |
28
|
rexlimdva |
⊢ ( 𝑀 ∈ ℤ → ( ∃ 𝑘 ∈ ℤ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ) → ∃ 𝑚 ∈ ℤ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ) ) |
30 |
29
|
ralimdv |
⊢ ( 𝑀 ∈ ℤ → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ ℤ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ) ) |
31 |
30
|
anim2d |
⊢ ( 𝑀 ∈ ℤ → ( ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ) ) → ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ ℤ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
32 |
1
|
a1i |
⊢ ( 𝑀 ∈ ℤ → 𝐹 ∈ V ) |
33 |
|
eqidd |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ ) → ( 𝐹 ‘ 𝑚 ) = ( 𝐹 ‘ 𝑚 ) ) |
34 |
32 33
|
clim |
⊢ ( 𝑀 ∈ ℤ → ( 𝐹 ⇝ 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
35 |
|
ovexd |
⊢ ( 𝑀 ∈ ℤ → ( 𝐹 shift 𝑀 ) ∈ V ) |
36 |
|
eqidd |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) = ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ) |
37 |
35 36
|
clim |
⊢ ( 𝑀 ∈ ℤ → ( ( 𝐹 shift 𝑀 ) ⇝ 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ ℤ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( ( 𝐹 shift 𝑀 ) ‘ 𝑛 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
38 |
31 34 37
|
3imtr4d |
⊢ ( 𝑀 ∈ ℤ → ( 𝐹 ⇝ 𝐴 → ( 𝐹 shift 𝑀 ) ⇝ 𝐴 ) ) |