Step |
Hyp |
Ref |
Expression |
1 |
|
climisp.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
2 |
|
climisp.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
3 |
|
climisp.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℂ ) |
4 |
|
climisp.c |
⊢ ( 𝜑 → 𝐹 ⇝ 𝐴 ) |
5 |
|
climisp.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ+ ) |
6 |
|
climisp.l |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ( 𝐹 ‘ 𝑘 ) ≠ 𝐴 ) → 𝑋 ≤ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) ) |
7 |
|
nfv |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) |
8 |
|
nfra1 |
⊢ Ⅎ 𝑘 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) |
9 |
7 8
|
nfan |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) |
10 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝜑 ) |
11 |
2
|
uztrn2 |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
12 |
11
|
ad4ant24 |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
13 |
|
rspa |
⊢ ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) |
14 |
13
|
simprd |
⊢ ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) |
15 |
14
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) |
16 |
|
simpl3 |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ∧ ¬ ( 𝐹 ‘ 𝑘 ) = 𝐴 ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) |
17 |
|
neqne |
⊢ ( ¬ ( 𝐹 ‘ 𝑘 ) = 𝐴 → ( 𝐹 ‘ 𝑘 ) ≠ 𝐴 ) |
18 |
5
|
rpred |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
19 |
18
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ ( 𝐹 ‘ 𝑘 ) ≠ 𝐴 ) → 𝑋 ∈ ℝ ) |
20 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
21 |
2
|
fvexi |
⊢ 𝑍 ∈ V |
22 |
21
|
a1i |
⊢ ( 𝜑 → 𝑍 ∈ V ) |
23 |
3 22
|
fexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
24 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
25 |
23 24
|
clim |
⊢ ( 𝜑 → ( 𝐹 ⇝ 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
26 |
4 25
|
mpbid |
⊢ ( 𝜑 → ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) ) |
27 |
26
|
simpld |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐴 ∈ ℂ ) |
29 |
20 28
|
subcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ∈ ℂ ) |
30 |
29
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) ∈ ℝ ) |
31 |
30
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ ( 𝐹 ‘ 𝑘 ) ≠ 𝐴 ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) ∈ ℝ ) |
32 |
6
|
3expa |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ ( 𝐹 ‘ 𝑘 ) ≠ 𝐴 ) → 𝑋 ≤ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) ) |
33 |
19 31 32
|
lensymd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ ( 𝐹 ‘ 𝑘 ) ≠ 𝐴 ) → ¬ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) |
34 |
17 33
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ ¬ ( 𝐹 ‘ 𝑘 ) = 𝐴 ) → ¬ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) |
35 |
34
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ∧ ¬ ( 𝐹 ‘ 𝑘 ) = 𝐴 ) → ¬ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) |
36 |
16 35
|
condan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) → ( 𝐹 ‘ 𝑘 ) = 𝐴 ) |
37 |
10 12 15 36
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) = 𝐴 ) |
38 |
9 37
|
ralrimia |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) = 𝐴 ) |
39 |
|
breq2 |
⊢ ( 𝑥 = 𝑋 → ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ↔ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) |
40 |
39
|
anbi2d |
⊢ ( 𝑥 = 𝑋 → ( ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ↔ ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) ) |
41 |
40
|
rexralbidv |
⊢ ( 𝑥 = 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 ) ) ) |
42 |
26
|
simprd |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑥 ) ) |
43 |
41 42 5
|
rspcdva |
⊢ ( 𝜑 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) |
44 |
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 ) ) ) |
45 |
1 44
|
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 ) ) ) |
46 |
43 45
|
mpbird |
⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − 𝐴 ) ) < 𝑋 ) ) |
47 |
38 46
|
reximddv3 |
⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) = 𝐴 ) |