Step |
Hyp |
Ref |
Expression |
1 |
|
stoweidlem7.1 |
⊢ 𝐹 = ( 𝑖 ∈ ℕ0 ↦ ( ( 1 / 𝐴 ) ↑ 𝑖 ) ) |
2 |
|
stoweidlem7.2 |
⊢ 𝐺 = ( 𝑖 ∈ ℕ0 ↦ ( 𝐵 ↑ 𝑖 ) ) |
3 |
|
stoweidlem7.3 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
4 |
|
stoweidlem7.4 |
⊢ ( 𝜑 → 1 < 𝐴 ) |
5 |
|
stoweidlem7.5 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
6 |
|
stoweidlem7.6 |
⊢ ( 𝜑 → 𝐵 < 1 ) |
7 |
|
stoweidlem7.7 |
⊢ ( 𝜑 → 𝐸 ∈ ℝ+ ) |
8 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
9 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
10 |
|
oveq2 |
⊢ ( 𝑖 = 𝑘 → ( 𝐵 ↑ 𝑖 ) = ( 𝐵 ↑ 𝑘 ) ) |
11 |
|
nnnn0 |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℕ0 ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℕ0 ) |
13 |
5
|
rpcnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝐵 ∈ ℂ ) |
15 |
14 12
|
expcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐵 ↑ 𝑘 ) ∈ ℂ ) |
16 |
2 10 12 15
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐵 ↑ 𝑘 ) ) |
17 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
18 |
17
|
renegcld |
⊢ ( 𝜑 → - 1 ∈ ℝ ) |
19 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
20 |
5
|
rpred |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
21 |
|
neg1lt0 |
⊢ - 1 < 0 |
22 |
21
|
a1i |
⊢ ( 𝜑 → - 1 < 0 ) |
23 |
5
|
rpgt0d |
⊢ ( 𝜑 → 0 < 𝐵 ) |
24 |
18 19 20 22 23
|
lttrd |
⊢ ( 𝜑 → - 1 < 𝐵 ) |
25 |
20 17
|
absltd |
⊢ ( 𝜑 → ( ( abs ‘ 𝐵 ) < 1 ↔ ( - 1 < 𝐵 ∧ 𝐵 < 1 ) ) ) |
26 |
24 6 25
|
mpbir2and |
⊢ ( 𝜑 → ( abs ‘ 𝐵 ) < 1 ) |
27 |
13 26
|
expcnv |
⊢ ( 𝜑 → ( 𝑖 ∈ ℕ0 ↦ ( 𝐵 ↑ 𝑖 ) ) ⇝ 0 ) |
28 |
2 27
|
eqbrtrid |
⊢ ( 𝜑 → 𝐺 ⇝ 0 ) |
29 |
8 9 7 16 28
|
climi |
⊢ ( 𝜑 → ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) |
30 |
|
r19.26 |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ↔ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) |
31 |
30
|
simprbi |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) |
32 |
31
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) |
33 |
|
oveq2 |
⊢ ( 𝑘 = 𝑖 → ( 𝐵 ↑ 𝑘 ) = ( 𝐵 ↑ 𝑖 ) ) |
34 |
33
|
oveq1d |
⊢ ( 𝑘 = 𝑖 → ( ( 𝐵 ↑ 𝑘 ) − 0 ) = ( ( 𝐵 ↑ 𝑖 ) − 0 ) ) |
35 |
34
|
fveq2d |
⊢ ( 𝑘 = 𝑖 → ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) = ( abs ‘ ( ( 𝐵 ↑ 𝑖 ) − 0 ) ) ) |
36 |
35
|
breq1d |
⊢ ( 𝑘 = 𝑖 → ( ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ↔ ( abs ‘ ( ( 𝐵 ↑ 𝑖 ) − 0 ) ) < 𝐸 ) ) |
37 |
36
|
rspccva |
⊢ ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( abs ‘ ( ( 𝐵 ↑ 𝑖 ) − 0 ) ) < 𝐸 ) |
38 |
32 37
|
sylancom |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( abs ‘ ( ( 𝐵 ↑ 𝑖 ) − 0 ) ) < 𝐸 ) |
39 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝜑 ) |
40 |
39 5
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝐵 ∈ ℝ+ ) |
41 |
40
|
rpred |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝐵 ∈ ℝ ) |
42 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑛 ∈ ℕ ) |
43 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
44 |
42 43
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑛 ∈ ℕ0 ) |
45 |
|
eluznn0 |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑖 ∈ ℕ0 ) |
46 |
44 45
|
sylancom |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑖 ∈ ℕ0 ) |
47 |
41 46
|
reexpcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐵 ↑ 𝑖 ) ∈ ℝ ) |
48 |
|
rpre |
⊢ ( 𝐸 ∈ ℝ+ → 𝐸 ∈ ℝ ) |
49 |
39 7 48
|
3syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝐸 ∈ ℝ ) |
50 |
|
recn |
⊢ ( ( 𝐵 ↑ 𝑖 ) ∈ ℝ → ( 𝐵 ↑ 𝑖 ) ∈ ℂ ) |
51 |
50
|
subid1d |
⊢ ( ( 𝐵 ↑ 𝑖 ) ∈ ℝ → ( ( 𝐵 ↑ 𝑖 ) − 0 ) = ( 𝐵 ↑ 𝑖 ) ) |
52 |
51
|
adantr |
⊢ ( ( ( 𝐵 ↑ 𝑖 ) ∈ ℝ ∧ 𝐸 ∈ ℝ ) → ( ( 𝐵 ↑ 𝑖 ) − 0 ) = ( 𝐵 ↑ 𝑖 ) ) |
53 |
52
|
fveq2d |
⊢ ( ( ( 𝐵 ↑ 𝑖 ) ∈ ℝ ∧ 𝐸 ∈ ℝ ) → ( abs ‘ ( ( 𝐵 ↑ 𝑖 ) − 0 ) ) = ( abs ‘ ( 𝐵 ↑ 𝑖 ) ) ) |
54 |
53
|
breq1d |
⊢ ( ( ( 𝐵 ↑ 𝑖 ) ∈ ℝ ∧ 𝐸 ∈ ℝ ) → ( ( abs ‘ ( ( 𝐵 ↑ 𝑖 ) − 0 ) ) < 𝐸 ↔ ( abs ‘ ( 𝐵 ↑ 𝑖 ) ) < 𝐸 ) ) |
55 |
|
abslt |
⊢ ( ( ( 𝐵 ↑ 𝑖 ) ∈ ℝ ∧ 𝐸 ∈ ℝ ) → ( ( abs ‘ ( 𝐵 ↑ 𝑖 ) ) < 𝐸 ↔ ( - 𝐸 < ( 𝐵 ↑ 𝑖 ) ∧ ( 𝐵 ↑ 𝑖 ) < 𝐸 ) ) ) |
56 |
54 55
|
bitrd |
⊢ ( ( ( 𝐵 ↑ 𝑖 ) ∈ ℝ ∧ 𝐸 ∈ ℝ ) → ( ( abs ‘ ( ( 𝐵 ↑ 𝑖 ) − 0 ) ) < 𝐸 ↔ ( - 𝐸 < ( 𝐵 ↑ 𝑖 ) ∧ ( 𝐵 ↑ 𝑖 ) < 𝐸 ) ) ) |
57 |
47 49 56
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( abs ‘ ( ( 𝐵 ↑ 𝑖 ) − 0 ) ) < 𝐸 ↔ ( - 𝐸 < ( 𝐵 ↑ 𝑖 ) ∧ ( 𝐵 ↑ 𝑖 ) < 𝐸 ) ) ) |
58 |
38 57
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( - 𝐸 < ( 𝐵 ↑ 𝑖 ) ∧ ( 𝐵 ↑ 𝑖 ) < 𝐸 ) ) |
59 |
58
|
simprd |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐵 ↑ 𝑖 ) < 𝐸 ) |
60 |
|
eluznn |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑖 ∈ ℕ ) |
61 |
42 60
|
sylancom |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑖 ∈ ℕ ) |
62 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → 𝐵 ∈ ℝ ) |
63 |
|
nnnn0 |
⊢ ( 𝑖 ∈ ℕ → 𝑖 ∈ ℕ0 ) |
64 |
63
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → 𝑖 ∈ ℕ0 ) |
65 |
62 64
|
reexpcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐵 ↑ 𝑖 ) ∈ ℝ ) |
66 |
7
|
rpred |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
67 |
66
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → 𝐸 ∈ ℝ ) |
68 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → 1 ∈ ℝ ) |
69 |
65 67 68
|
ltsub2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐵 ↑ 𝑖 ) < 𝐸 ↔ ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑖 ) ) ) ) |
70 |
39 61 69
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐵 ↑ 𝑖 ) < 𝐸 ↔ ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑖 ) ) ) ) |
71 |
59 70
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑖 ) ) ) |
72 |
71
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) → ∀ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑖 ) ) ) |
73 |
33
|
oveq2d |
⊢ ( 𝑘 = 𝑖 → ( 1 − ( 𝐵 ↑ 𝑘 ) ) = ( 1 − ( 𝐵 ↑ 𝑖 ) ) ) |
74 |
73
|
breq2d |
⊢ ( 𝑘 = 𝑖 → ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ↔ ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑖 ) ) ) ) |
75 |
74
|
cbvralvw |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ↔ ∀ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑖 ) ) ) |
76 |
72 75
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ) |
77 |
76
|
ex |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ) ) |
78 |
77
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐵 ↑ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑘 ) − 0 ) ) < 𝐸 ) → ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ) ) |
79 |
29 78
|
mpd |
⊢ ( 𝜑 → ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ) |
80 |
|
oveq2 |
⊢ ( 𝑖 = 𝑘 → ( ( 1 / 𝐴 ) ↑ 𝑖 ) = ( ( 1 / 𝐴 ) ↑ 𝑘 ) ) |
81 |
3
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
82 |
|
0lt1 |
⊢ 0 < 1 |
83 |
82
|
a1i |
⊢ ( 𝜑 → 0 < 1 ) |
84 |
19 17 3 83 4
|
lttrd |
⊢ ( 𝜑 → 0 < 𝐴 ) |
85 |
84
|
gt0ne0d |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
86 |
81 85
|
reccld |
⊢ ( 𝜑 → ( 1 / 𝐴 ) ∈ ℂ ) |
87 |
86
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 1 / 𝐴 ) ∈ ℂ ) |
88 |
87 12
|
expcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 1 / 𝐴 ) ↑ 𝑘 ) ∈ ℂ ) |
89 |
1 80 12 88
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) = ( ( 1 / 𝐴 ) ↑ 𝑘 ) ) |
90 |
3 85
|
rereccld |
⊢ ( 𝜑 → ( 1 / 𝐴 ) ∈ ℝ ) |
91 |
3 84
|
recgt0d |
⊢ ( 𝜑 → 0 < ( 1 / 𝐴 ) ) |
92 |
18 19 90 22 91
|
lttrd |
⊢ ( 𝜑 → - 1 < ( 1 / 𝐴 ) ) |
93 |
|
ltdiv23 |
⊢ ( ( 1 ∈ ℝ ∧ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 1 ∈ ℝ ∧ 0 < 1 ) ) → ( ( 1 / 𝐴 ) < 1 ↔ ( 1 / 1 ) < 𝐴 ) ) |
94 |
17 3 84 17 83 93
|
syl122anc |
⊢ ( 𝜑 → ( ( 1 / 𝐴 ) < 1 ↔ ( 1 / 1 ) < 𝐴 ) ) |
95 |
|
1cnd |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
96 |
95
|
div1d |
⊢ ( 𝜑 → ( 1 / 1 ) = 1 ) |
97 |
96
|
breq1d |
⊢ ( 𝜑 → ( ( 1 / 1 ) < 𝐴 ↔ 1 < 𝐴 ) ) |
98 |
94 97
|
bitrd |
⊢ ( 𝜑 → ( ( 1 / 𝐴 ) < 1 ↔ 1 < 𝐴 ) ) |
99 |
4 98
|
mpbird |
⊢ ( 𝜑 → ( 1 / 𝐴 ) < 1 ) |
100 |
90 17
|
absltd |
⊢ ( 𝜑 → ( ( abs ‘ ( 1 / 𝐴 ) ) < 1 ↔ ( - 1 < ( 1 / 𝐴 ) ∧ ( 1 / 𝐴 ) < 1 ) ) ) |
101 |
92 99 100
|
mpbir2and |
⊢ ( 𝜑 → ( abs ‘ ( 1 / 𝐴 ) ) < 1 ) |
102 |
86 101
|
expcnv |
⊢ ( 𝜑 → ( 𝑖 ∈ ℕ0 ↦ ( ( 1 / 𝐴 ) ↑ 𝑖 ) ) ⇝ 0 ) |
103 |
1 102
|
eqbrtrid |
⊢ ( 𝜑 → 𝐹 ⇝ 0 ) |
104 |
8 9 7 89 103
|
climi2 |
⊢ ( 𝜑 → ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( ( 1 / 𝐴 ) ↑ 𝑘 ) − 0 ) ) < 𝐸 ) |
105 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝜑 ) |
106 |
|
uznnssnn |
⊢ ( 𝑛 ∈ ℕ → ( ℤ≥ ‘ 𝑛 ) ⊆ ℕ ) |
107 |
106
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ℤ≥ ‘ 𝑛 ) ⊆ ℕ ) |
108 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
109 |
107 108
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑘 ∈ ℕ ) |
110 |
88
|
subid1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ( 1 / 𝐴 ) ↑ 𝑘 ) − 0 ) = ( ( 1 / 𝐴 ) ↑ 𝑘 ) ) |
111 |
110
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( abs ‘ ( ( ( 1 / 𝐴 ) ↑ 𝑘 ) − 0 ) ) = ( abs ‘ ( ( 1 / 𝐴 ) ↑ 𝑘 ) ) ) |
112 |
90
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 1 / 𝐴 ) ∈ ℝ ) |
113 |
112 12
|
reexpcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 1 / 𝐴 ) ↑ 𝑘 ) ∈ ℝ ) |
114 |
19 90 91
|
ltled |
⊢ ( 𝜑 → 0 ≤ ( 1 / 𝐴 ) ) |
115 |
114
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 0 ≤ ( 1 / 𝐴 ) ) |
116 |
112 12 115
|
expge0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 0 ≤ ( ( 1 / 𝐴 ) ↑ 𝑘 ) ) |
117 |
113 116
|
absidd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( abs ‘ ( ( 1 / 𝐴 ) ↑ 𝑘 ) ) = ( ( 1 / 𝐴 ) ↑ 𝑘 ) ) |
118 |
111 117
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( abs ‘ ( ( ( 1 / 𝐴 ) ↑ 𝑘 ) − 0 ) ) = ( ( 1 / 𝐴 ) ↑ 𝑘 ) ) |
119 |
118
|
breq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( abs ‘ ( ( ( 1 / 𝐴 ) ↑ 𝑘 ) − 0 ) ) < 𝐸 ↔ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) |
120 |
119
|
biimpd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( abs ‘ ( ( ( 1 / 𝐴 ) ↑ 𝑘 ) − 0 ) ) < 𝐸 → ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) |
121 |
105 109 120
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( abs ‘ ( ( ( 1 / 𝐴 ) ↑ 𝑘 ) − 0 ) ) < 𝐸 → ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) |
122 |
121
|
ralimdva |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( ( 1 / 𝐴 ) ↑ 𝑘 ) − 0 ) ) < 𝐸 → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) |
123 |
122
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( ( 1 / 𝐴 ) ↑ 𝑘 ) − 0 ) ) < 𝐸 → ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) |
124 |
104 123
|
mpd |
⊢ ( 𝜑 → ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) |
125 |
8
|
rexanuz2 |
⊢ ( ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ↔ ( ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) |
126 |
79 124 125
|
sylanbrc |
⊢ ( 𝜑 → ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) |
127 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) |
128 |
|
nnz |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℤ ) |
129 |
|
uzid |
⊢ ( 𝑛 ∈ ℤ → 𝑛 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
130 |
128 129
|
syl |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
131 |
130
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
132 |
|
oveq2 |
⊢ ( 𝑘 = 𝑛 → ( 𝐵 ↑ 𝑘 ) = ( 𝐵 ↑ 𝑛 ) ) |
133 |
132
|
oveq2d |
⊢ ( 𝑘 = 𝑛 → ( 1 − ( 𝐵 ↑ 𝑘 ) ) = ( 1 − ( 𝐵 ↑ 𝑛 ) ) ) |
134 |
133
|
breq2d |
⊢ ( 𝑘 = 𝑛 → ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ↔ ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑛 ) ) ) ) |
135 |
|
oveq2 |
⊢ ( 𝑘 = 𝑛 → ( ( 1 / 𝐴 ) ↑ 𝑘 ) = ( ( 1 / 𝐴 ) ↑ 𝑛 ) ) |
136 |
135
|
breq1d |
⊢ ( 𝑘 = 𝑛 → ( ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ↔ ( ( 1 / 𝐴 ) ↑ 𝑛 ) < 𝐸 ) ) |
137 |
134 136
|
anbi12d |
⊢ ( 𝑘 = 𝑛 → ( ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ↔ ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑛 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑛 ) < 𝐸 ) ) ) |
138 |
137
|
rspccva |
⊢ ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑛 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑛 ) < 𝐸 ) ) |
139 |
127 131 138
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) → ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑛 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑛 ) < 𝐸 ) ) |
140 |
|
1cnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 1 ∈ ℂ ) |
141 |
81 85
|
jca |
⊢ ( 𝜑 → ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ) |
142 |
141
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ) |
143 |
43
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ0 ) |
144 |
|
expdiv |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 1 / 𝐴 ) ↑ 𝑛 ) = ( ( 1 ↑ 𝑛 ) / ( 𝐴 ↑ 𝑛 ) ) ) |
145 |
140 142 143 144
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 1 / 𝐴 ) ↑ 𝑛 ) = ( ( 1 ↑ 𝑛 ) / ( 𝐴 ↑ 𝑛 ) ) ) |
146 |
128
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℤ ) |
147 |
|
1exp |
⊢ ( 𝑛 ∈ ℤ → ( 1 ↑ 𝑛 ) = 1 ) |
148 |
146 147
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1 ↑ 𝑛 ) = 1 ) |
149 |
148
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 1 ↑ 𝑛 ) / ( 𝐴 ↑ 𝑛 ) ) = ( 1 / ( 𝐴 ↑ 𝑛 ) ) ) |
150 |
145 149
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 1 / 𝐴 ) ↑ 𝑛 ) = ( 1 / ( 𝐴 ↑ 𝑛 ) ) ) |
151 |
150
|
breq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( 1 / 𝐴 ) ↑ 𝑛 ) < 𝐸 ↔ ( 1 / ( 𝐴 ↑ 𝑛 ) ) < 𝐸 ) ) |
152 |
151
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) → ( ( ( 1 / 𝐴 ) ↑ 𝑛 ) < 𝐸 ↔ ( 1 / ( 𝐴 ↑ 𝑛 ) ) < 𝐸 ) ) |
153 |
152
|
anbi2d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) → ( ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑛 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑛 ) < 𝐸 ) ↔ ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑛 ) ) ∧ ( 1 / ( 𝐴 ↑ 𝑛 ) ) < 𝐸 ) ) ) |
154 |
139 153
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) ) → ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑛 ) ) ∧ ( 1 / ( 𝐴 ↑ 𝑛 ) ) < 𝐸 ) ) |
155 |
154
|
ex |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) → ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑛 ) ) ∧ ( 1 / ( 𝐴 ↑ 𝑛 ) ) < 𝐸 ) ) ) |
156 |
155
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑘 ) ) ∧ ( ( 1 / 𝐴 ) ↑ 𝑘 ) < 𝐸 ) → ∃ 𝑛 ∈ ℕ ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑛 ) ) ∧ ( 1 / ( 𝐴 ↑ 𝑛 ) ) < 𝐸 ) ) ) |
157 |
126 156
|
mpd |
⊢ ( 𝜑 → ∃ 𝑛 ∈ ℕ ( ( 1 − 𝐸 ) < ( 1 − ( 𝐵 ↑ 𝑛 ) ) ∧ ( 1 / ( 𝐴 ↑ 𝑛 ) ) < 𝐸 ) ) |