Step |
Hyp |
Ref |
Expression |
1 |
|
zetacvg.1 |
⊢ ( 𝜑 → 𝑆 ∈ ℂ ) |
2 |
|
zetacvg.2 |
⊢ ( 𝜑 → 1 < ( ℜ ‘ 𝑆 ) ) |
3 |
|
zetacvg.3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) = ( 𝑘 ↑𝑐 - 𝑆 ) ) |
4 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
5 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
6 |
|
oveq1 |
⊢ ( 𝑛 = 𝑘 → ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) = ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
7 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
8 |
|
ovex |
⊢ ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ∈ V |
9 |
6 7 8
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ 𝑘 ) = ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
10 |
9
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ 𝑘 ) = ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
11 |
|
nncn |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℂ ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℂ ) |
13 |
|
nnne0 |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ≠ 0 ) |
14 |
13
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑘 ≠ 0 ) |
15 |
1
|
negcld |
⊢ ( 𝜑 → - 𝑆 ∈ ℂ ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → - 𝑆 ∈ ℂ ) |
17 |
12 14 16
|
cxpefd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 ↑𝑐 - 𝑆 ) = ( exp ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) ) |
18 |
3 17
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) = ( exp ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) ) |
19 |
18
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) = ( abs ‘ ( exp ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) ) ) |
20 |
|
nnrp |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ+ ) |
21 |
20
|
relogcld |
⊢ ( 𝑘 ∈ ℕ → ( log ‘ 𝑘 ) ∈ ℝ ) |
22 |
21
|
recnd |
⊢ ( 𝑘 ∈ ℕ → ( log ‘ 𝑘 ) ∈ ℂ ) |
23 |
|
mulcl |
⊢ ( ( - 𝑆 ∈ ℂ ∧ ( log ‘ 𝑘 ) ∈ ℂ ) → ( - 𝑆 · ( log ‘ 𝑘 ) ) ∈ ℂ ) |
24 |
15 22 23
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( - 𝑆 · ( log ‘ 𝑘 ) ) ∈ ℂ ) |
25 |
|
absef |
⊢ ( ( - 𝑆 · ( log ‘ 𝑘 ) ) ∈ ℂ → ( abs ‘ ( exp ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) ) = ( exp ‘ ( ℜ ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) ) ) |
26 |
24 25
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( abs ‘ ( exp ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) ) = ( exp ‘ ( ℜ ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) ) ) |
27 |
|
remul |
⊢ ( ( - 𝑆 ∈ ℂ ∧ ( log ‘ 𝑘 ) ∈ ℂ ) → ( ℜ ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) = ( ( ( ℜ ‘ - 𝑆 ) · ( ℜ ‘ ( log ‘ 𝑘 ) ) ) − ( ( ℑ ‘ - 𝑆 ) · ( ℑ ‘ ( log ‘ 𝑘 ) ) ) ) ) |
28 |
15 22 27
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ℜ ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) = ( ( ( ℜ ‘ - 𝑆 ) · ( ℜ ‘ ( log ‘ 𝑘 ) ) ) − ( ( ℑ ‘ - 𝑆 ) · ( ℑ ‘ ( log ‘ 𝑘 ) ) ) ) ) |
29 |
1
|
renegd |
⊢ ( 𝜑 → ( ℜ ‘ - 𝑆 ) = - ( ℜ ‘ 𝑆 ) ) |
30 |
21
|
rered |
⊢ ( 𝑘 ∈ ℕ → ( ℜ ‘ ( log ‘ 𝑘 ) ) = ( log ‘ 𝑘 ) ) |
31 |
29 30
|
oveqan12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ℜ ‘ - 𝑆 ) · ( ℜ ‘ ( log ‘ 𝑘 ) ) ) = ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) |
32 |
21
|
reim0d |
⊢ ( 𝑘 ∈ ℕ → ( ℑ ‘ ( log ‘ 𝑘 ) ) = 0 ) |
33 |
32
|
oveq2d |
⊢ ( 𝑘 ∈ ℕ → ( ( ℑ ‘ - 𝑆 ) · ( ℑ ‘ ( log ‘ 𝑘 ) ) ) = ( ( ℑ ‘ - 𝑆 ) · 0 ) ) |
34 |
|
imcl |
⊢ ( - 𝑆 ∈ ℂ → ( ℑ ‘ - 𝑆 ) ∈ ℝ ) |
35 |
34
|
recnd |
⊢ ( - 𝑆 ∈ ℂ → ( ℑ ‘ - 𝑆 ) ∈ ℂ ) |
36 |
15 35
|
syl |
⊢ ( 𝜑 → ( ℑ ‘ - 𝑆 ) ∈ ℂ ) |
37 |
36
|
mul01d |
⊢ ( 𝜑 → ( ( ℑ ‘ - 𝑆 ) · 0 ) = 0 ) |
38 |
33 37
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ℑ ‘ - 𝑆 ) · ( ℑ ‘ ( log ‘ 𝑘 ) ) ) = 0 ) |
39 |
31 38
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ( ℜ ‘ - 𝑆 ) · ( ℜ ‘ ( log ‘ 𝑘 ) ) ) − ( ( ℑ ‘ - 𝑆 ) · ( ℑ ‘ ( log ‘ 𝑘 ) ) ) ) = ( ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) − 0 ) ) |
40 |
1
|
recld |
⊢ ( 𝜑 → ( ℜ ‘ 𝑆 ) ∈ ℝ ) |
41 |
40
|
renegcld |
⊢ ( 𝜑 → - ( ℜ ‘ 𝑆 ) ∈ ℝ ) |
42 |
41
|
recnd |
⊢ ( 𝜑 → - ( ℜ ‘ 𝑆 ) ∈ ℂ ) |
43 |
|
mulcl |
⊢ ( ( - ( ℜ ‘ 𝑆 ) ∈ ℂ ∧ ( log ‘ 𝑘 ) ∈ ℂ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ∈ ℂ ) |
44 |
42 22 43
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ∈ ℂ ) |
45 |
44
|
subid1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) − 0 ) = ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) |
46 |
28 39 45
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ℜ ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) = ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) |
47 |
46
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( exp ‘ ( ℜ ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) ) = ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) ) |
48 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → - ( ℜ ‘ 𝑆 ) ∈ ℂ ) |
49 |
12 14 48
|
cxpefd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) = ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) ) |
50 |
47 49
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( exp ‘ ( ℜ ‘ ( - 𝑆 · ( log ‘ 𝑘 ) ) ) ) = ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
51 |
19 26 50
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
52 |
10 51
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ 𝑘 ) = ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
53 |
12 16
|
cxpcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 ↑𝑐 - 𝑆 ) ∈ ℂ ) |
54 |
3 53
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
55 |
|
2rp |
⊢ 2 ∈ ℝ+ |
56 |
|
1re |
⊢ 1 ∈ ℝ |
57 |
|
resubcl |
⊢ ( ( 1 ∈ ℝ ∧ ( ℜ ‘ 𝑆 ) ∈ ℝ ) → ( 1 − ( ℜ ‘ 𝑆 ) ) ∈ ℝ ) |
58 |
56 40 57
|
sylancr |
⊢ ( 𝜑 → ( 1 − ( ℜ ‘ 𝑆 ) ) ∈ ℝ ) |
59 |
|
rpcxpcl |
⊢ ( ( 2 ∈ ℝ+ ∧ ( 1 − ( ℜ ‘ 𝑆 ) ) ∈ ℝ ) → ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ∈ ℝ+ ) |
60 |
55 58 59
|
sylancr |
⊢ ( 𝜑 → ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ∈ ℝ+ ) |
61 |
60
|
rpcnd |
⊢ ( 𝜑 → ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ∈ ℂ ) |
62 |
|
recl |
⊢ ( 𝑆 ∈ ℂ → ( ℜ ‘ 𝑆 ) ∈ ℝ ) |
63 |
62
|
recnd |
⊢ ( 𝑆 ∈ ℂ → ( ℜ ‘ 𝑆 ) ∈ ℂ ) |
64 |
1 63
|
syl |
⊢ ( 𝜑 → ( ℜ ‘ 𝑆 ) ∈ ℂ ) |
65 |
64
|
addid2d |
⊢ ( 𝜑 → ( 0 + ( ℜ ‘ 𝑆 ) ) = ( ℜ ‘ 𝑆 ) ) |
66 |
2 65
|
breqtrrd |
⊢ ( 𝜑 → 1 < ( 0 + ( ℜ ‘ 𝑆 ) ) ) |
67 |
|
0re |
⊢ 0 ∈ ℝ |
68 |
|
ltsubadd |
⊢ ( ( 1 ∈ ℝ ∧ ( ℜ ‘ 𝑆 ) ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 1 − ( ℜ ‘ 𝑆 ) ) < 0 ↔ 1 < ( 0 + ( ℜ ‘ 𝑆 ) ) ) ) |
69 |
56 67 68
|
mp3an13 |
⊢ ( ( ℜ ‘ 𝑆 ) ∈ ℝ → ( ( 1 − ( ℜ ‘ 𝑆 ) ) < 0 ↔ 1 < ( 0 + ( ℜ ‘ 𝑆 ) ) ) ) |
70 |
40 69
|
syl |
⊢ ( 𝜑 → ( ( 1 − ( ℜ ‘ 𝑆 ) ) < 0 ↔ 1 < ( 0 + ( ℜ ‘ 𝑆 ) ) ) ) |
71 |
66 70
|
mpbird |
⊢ ( 𝜑 → ( 1 − ( ℜ ‘ 𝑆 ) ) < 0 ) |
72 |
|
2re |
⊢ 2 ∈ ℝ |
73 |
|
1lt2 |
⊢ 1 < 2 |
74 |
|
cxplt |
⊢ ( ( ( 2 ∈ ℝ ∧ 1 < 2 ) ∧ ( ( 1 − ( ℜ ‘ 𝑆 ) ) ∈ ℝ ∧ 0 ∈ ℝ ) ) → ( ( 1 − ( ℜ ‘ 𝑆 ) ) < 0 ↔ ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) < ( 2 ↑𝑐 0 ) ) ) |
75 |
72 73 74
|
mpanl12 |
⊢ ( ( ( 1 − ( ℜ ‘ 𝑆 ) ) ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 1 − ( ℜ ‘ 𝑆 ) ) < 0 ↔ ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) < ( 2 ↑𝑐 0 ) ) ) |
76 |
58 67 75
|
sylancl |
⊢ ( 𝜑 → ( ( 1 − ( ℜ ‘ 𝑆 ) ) < 0 ↔ ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) < ( 2 ↑𝑐 0 ) ) ) |
77 |
71 76
|
mpbid |
⊢ ( 𝜑 → ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) < ( 2 ↑𝑐 0 ) ) |
78 |
60
|
rprege0d |
⊢ ( 𝜑 → ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ∈ ℝ ∧ 0 ≤ ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) ) |
79 |
|
absid |
⊢ ( ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ∈ ℝ ∧ 0 ≤ ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) → ( abs ‘ ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) = ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) |
80 |
78 79
|
syl |
⊢ ( 𝜑 → ( abs ‘ ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) = ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) |
81 |
|
2cn |
⊢ 2 ∈ ℂ |
82 |
|
cxp0 |
⊢ ( 2 ∈ ℂ → ( 2 ↑𝑐 0 ) = 1 ) |
83 |
81 82
|
ax-mp |
⊢ ( 2 ↑𝑐 0 ) = 1 |
84 |
83
|
eqcomi |
⊢ 1 = ( 2 ↑𝑐 0 ) |
85 |
84
|
a1i |
⊢ ( 𝜑 → 1 = ( 2 ↑𝑐 0 ) ) |
86 |
77 80 85
|
3brtr4d |
⊢ ( 𝜑 → ( abs ‘ ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) < 1 ) |
87 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) = ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑚 ) ) |
88 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) |
89 |
|
ovex |
⊢ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑚 ) ∈ V |
90 |
87 88 89
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) ‘ 𝑚 ) = ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑚 ) ) |
91 |
90
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) ‘ 𝑚 ) = ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑚 ) ) |
92 |
61 86 91
|
geolim |
⊢ ( 𝜑 → seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) ) ⇝ ( 1 / ( 1 − ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) ) ) |
93 |
|
seqex |
⊢ seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) ) ∈ V |
94 |
|
ovex |
⊢ ( 1 / ( 1 − ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) ) ∈ V |
95 |
93 94
|
breldm |
⊢ ( seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) ) ⇝ ( 1 / ( 1 − ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) ) → seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) ) ∈ dom ⇝ ) |
96 |
92 95
|
syl |
⊢ ( 𝜑 → seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) ) ∈ dom ⇝ ) |
97 |
|
rpcxpcl |
⊢ ( ( 𝑘 ∈ ℝ+ ∧ - ( ℜ ‘ 𝑆 ) ∈ ℝ ) → ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ∈ ℝ+ ) |
98 |
20 41 97
|
syl2anr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ∈ ℝ+ ) |
99 |
98
|
rpred |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ∈ ℝ ) |
100 |
10 99
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ 𝑘 ) ∈ ℝ ) |
101 |
98
|
rpge0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 0 ≤ ( 𝑘 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
102 |
101 10
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 0 ≤ ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ 𝑘 ) ) |
103 |
|
nnre |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ ) |
104 |
103
|
lep1d |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ≤ ( 𝑘 + 1 ) ) |
105 |
20
|
reeflogd |
⊢ ( 𝑘 ∈ ℕ → ( exp ‘ ( log ‘ 𝑘 ) ) = 𝑘 ) |
106 |
|
peano2nn |
⊢ ( 𝑘 ∈ ℕ → ( 𝑘 + 1 ) ∈ ℕ ) |
107 |
106
|
nnrpd |
⊢ ( 𝑘 ∈ ℕ → ( 𝑘 + 1 ) ∈ ℝ+ ) |
108 |
107
|
reeflogd |
⊢ ( 𝑘 ∈ ℕ → ( exp ‘ ( log ‘ ( 𝑘 + 1 ) ) ) = ( 𝑘 + 1 ) ) |
109 |
104 105 108
|
3brtr4d |
⊢ ( 𝑘 ∈ ℕ → ( exp ‘ ( log ‘ 𝑘 ) ) ≤ ( exp ‘ ( log ‘ ( 𝑘 + 1 ) ) ) ) |
110 |
107
|
relogcld |
⊢ ( 𝑘 ∈ ℕ → ( log ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) |
111 |
|
efle |
⊢ ( ( ( log ‘ 𝑘 ) ∈ ℝ ∧ ( log ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) → ( ( log ‘ 𝑘 ) ≤ ( log ‘ ( 𝑘 + 1 ) ) ↔ ( exp ‘ ( log ‘ 𝑘 ) ) ≤ ( exp ‘ ( log ‘ ( 𝑘 + 1 ) ) ) ) ) |
112 |
21 110 111
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( log ‘ 𝑘 ) ≤ ( log ‘ ( 𝑘 + 1 ) ) ↔ ( exp ‘ ( log ‘ 𝑘 ) ) ≤ ( exp ‘ ( log ‘ ( 𝑘 + 1 ) ) ) ) ) |
113 |
109 112
|
mpbird |
⊢ ( 𝑘 ∈ ℕ → ( log ‘ 𝑘 ) ≤ ( log ‘ ( 𝑘 + 1 ) ) ) |
114 |
113
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( log ‘ 𝑘 ) ≤ ( log ‘ ( 𝑘 + 1 ) ) ) |
115 |
21
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( log ‘ 𝑘 ) ∈ ℝ ) |
116 |
106
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 + 1 ) ∈ ℕ ) |
117 |
116
|
nnrpd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 + 1 ) ∈ ℝ+ ) |
118 |
117
|
relogcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( log ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) |
119 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ℜ ‘ 𝑆 ) ∈ ℝ ) |
120 |
67
|
a1i |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
121 |
56
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
122 |
|
0lt1 |
⊢ 0 < 1 |
123 |
122
|
a1i |
⊢ ( 𝜑 → 0 < 1 ) |
124 |
120 121 40 123 2
|
lttrd |
⊢ ( 𝜑 → 0 < ( ℜ ‘ 𝑆 ) ) |
125 |
124
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 0 < ( ℜ ‘ 𝑆 ) ) |
126 |
|
lemul2 |
⊢ ( ( ( log ‘ 𝑘 ) ∈ ℝ ∧ ( log ‘ ( 𝑘 + 1 ) ) ∈ ℝ ∧ ( ( ℜ ‘ 𝑆 ) ∈ ℝ ∧ 0 < ( ℜ ‘ 𝑆 ) ) ) → ( ( log ‘ 𝑘 ) ≤ ( log ‘ ( 𝑘 + 1 ) ) ↔ ( ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ≤ ( ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ) ) |
127 |
115 118 119 125 126
|
syl112anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( log ‘ 𝑘 ) ≤ ( log ‘ ( 𝑘 + 1 ) ) ↔ ( ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ≤ ( ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ) ) |
128 |
114 127
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ≤ ( ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ) |
129 |
|
remulcl |
⊢ ( ( ( ℜ ‘ 𝑆 ) ∈ ℝ ∧ ( log ‘ 𝑘 ) ∈ ℝ ) → ( ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ∈ ℝ ) |
130 |
40 21 129
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ∈ ℝ ) |
131 |
|
remulcl |
⊢ ( ( ( ℜ ‘ 𝑆 ) ∈ ℝ ∧ ( log ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) → ( ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
132 |
40 110 131
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
133 |
130 132
|
lenegd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ≤ ( ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ↔ - ( ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ≤ - ( ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) ) |
134 |
128 133
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → - ( ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ≤ - ( ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) |
135 |
110
|
recnd |
⊢ ( 𝑘 ∈ ℕ → ( log ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
136 |
|
mulneg1 |
⊢ ( ( ( ℜ ‘ 𝑆 ) ∈ ℂ ∧ ( log ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) = - ( ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ) |
137 |
64 135 136
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) = - ( ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ) |
138 |
|
mulneg1 |
⊢ ( ( ( ℜ ‘ 𝑆 ) ∈ ℂ ∧ ( log ‘ 𝑘 ) ∈ ℂ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) = - ( ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) |
139 |
64 22 138
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) = - ( ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) |
140 |
134 137 139
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ≤ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) |
141 |
|
remulcl |
⊢ ( ( - ( ℜ ‘ 𝑆 ) ∈ ℝ ∧ ( log ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
142 |
41 110 141
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
143 |
|
remulcl |
⊢ ( ( - ( ℜ ‘ 𝑆 ) ∈ ℝ ∧ ( log ‘ 𝑘 ) ∈ ℝ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ∈ ℝ ) |
144 |
41 21 143
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ∈ ℝ ) |
145 |
|
efle |
⊢ ( ( ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ∧ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ∈ ℝ ) → ( ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ≤ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ↔ ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ) ≤ ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) ) ) |
146 |
142 144 145
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ≤ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ↔ ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ) ≤ ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) ) ) |
147 |
140 146
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ) ≤ ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) ) |
148 |
|
oveq1 |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) = ( ( 𝑘 + 1 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
149 |
|
ovex |
⊢ ( ( 𝑘 + 1 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ∈ V |
150 |
148 7 149
|
fvmpt |
⊢ ( ( 𝑘 + 1 ) ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝑘 + 1 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
151 |
116 150
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝑘 + 1 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
152 |
116
|
nncnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 + 1 ) ∈ ℂ ) |
153 |
116
|
nnne0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 + 1 ) ≠ 0 ) |
154 |
152 153 48
|
cxpefd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑘 + 1 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) = ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ) ) |
155 |
151 154
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ ( 𝑘 + 1 ) ) = ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ ( 𝑘 + 1 ) ) ) ) ) |
156 |
10 49
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ 𝑘 ) = ( exp ‘ ( - ( ℜ ‘ 𝑆 ) · ( log ‘ 𝑘 ) ) ) ) |
157 |
147 155 156
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ ( 𝑘 + 1 ) ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ 𝑘 ) ) |
158 |
58
|
recnd |
⊢ ( 𝜑 → ( 1 − ( ℜ ‘ 𝑆 ) ) ∈ ℂ ) |
159 |
158
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 1 − ( ℜ ‘ 𝑆 ) ) ∈ ℂ ) |
160 |
|
nn0re |
⊢ ( 𝑚 ∈ ℕ0 → 𝑚 ∈ ℝ ) |
161 |
160
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → 𝑚 ∈ ℝ ) |
162 |
161
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → 𝑚 ∈ ℂ ) |
163 |
159 162
|
mulcomd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 1 − ( ℜ ‘ 𝑆 ) ) · 𝑚 ) = ( 𝑚 · ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) |
164 |
163
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑𝑐 ( ( 1 − ( ℜ ‘ 𝑆 ) ) · 𝑚 ) ) = ( 2 ↑𝑐 ( 𝑚 · ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) ) |
165 |
55
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → 2 ∈ ℝ+ ) |
166 |
165 161 159
|
cxpmuld |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑𝑐 ( 𝑚 · ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) = ( ( 2 ↑𝑐 𝑚 ) ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ) |
167 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → 𝑚 ∈ ℕ0 ) |
168 |
|
cxpexp |
⊢ ( ( 2 ∈ ℂ ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑𝑐 𝑚 ) = ( 2 ↑ 𝑚 ) ) |
169 |
81 167 168
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑𝑐 𝑚 ) = ( 2 ↑ 𝑚 ) ) |
170 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
171 |
64
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ℜ ‘ 𝑆 ) ∈ ℂ ) |
172 |
|
negsub |
⊢ ( ( 1 ∈ ℂ ∧ ( ℜ ‘ 𝑆 ) ∈ ℂ ) → ( 1 + - ( ℜ ‘ 𝑆 ) ) = ( 1 − ( ℜ ‘ 𝑆 ) ) ) |
173 |
170 171 172
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 1 + - ( ℜ ‘ 𝑆 ) ) = ( 1 − ( ℜ ‘ 𝑆 ) ) ) |
174 |
173
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 1 − ( ℜ ‘ 𝑆 ) ) = ( 1 + - ( ℜ ‘ 𝑆 ) ) ) |
175 |
169 174
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 2 ↑𝑐 𝑚 ) ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) = ( ( 2 ↑ 𝑚 ) ↑𝑐 ( 1 + - ( ℜ ‘ 𝑆 ) ) ) ) |
176 |
164 166 175
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑𝑐 ( ( 1 − ( ℜ ‘ 𝑆 ) ) · 𝑚 ) ) = ( ( 2 ↑ 𝑚 ) ↑𝑐 ( 1 + - ( ℜ ‘ 𝑆 ) ) ) ) |
177 |
58
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 1 − ( ℜ ‘ 𝑆 ) ) ∈ ℝ ) |
178 |
165 177 162
|
cxpmuld |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑𝑐 ( ( 1 − ( ℜ ‘ 𝑆 ) ) · 𝑚 ) ) = ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑𝑐 𝑚 ) ) |
179 |
|
2nn |
⊢ 2 ∈ ℕ |
180 |
|
nnexpcl |
⊢ ( ( 2 ∈ ℕ ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑ 𝑚 ) ∈ ℕ ) |
181 |
179 180
|
mpan |
⊢ ( 𝑚 ∈ ℕ0 → ( 2 ↑ 𝑚 ) ∈ ℕ ) |
182 |
181
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑ 𝑚 ) ∈ ℕ ) |
183 |
182
|
nncnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑ 𝑚 ) ∈ ℂ ) |
184 |
182
|
nnne0d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑ 𝑚 ) ≠ 0 ) |
185 |
|
1cnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → 1 ∈ ℂ ) |
186 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → - ( ℜ ‘ 𝑆 ) ∈ ℂ ) |
187 |
183 184 185 186
|
cxpaddd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 2 ↑ 𝑚 ) ↑𝑐 ( 1 + - ( ℜ ‘ 𝑆 ) ) ) = ( ( ( 2 ↑ 𝑚 ) ↑𝑐 1 ) · ( ( 2 ↑ 𝑚 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ) |
188 |
176 178 187
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑𝑐 𝑚 ) = ( ( ( 2 ↑ 𝑚 ) ↑𝑐 1 ) · ( ( 2 ↑ 𝑚 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ) |
189 |
|
cxpexp |
⊢ ( ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ∈ ℂ ∧ 𝑚 ∈ ℕ0 ) → ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑𝑐 𝑚 ) = ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑚 ) ) |
190 |
61 189
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑𝑐 𝑚 ) = ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑚 ) ) |
191 |
183
|
cxp1d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 2 ↑ 𝑚 ) ↑𝑐 1 ) = ( 2 ↑ 𝑚 ) ) |
192 |
191
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( ( 2 ↑ 𝑚 ) ↑𝑐 1 ) · ( ( 2 ↑ 𝑚 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) = ( ( 2 ↑ 𝑚 ) · ( ( 2 ↑ 𝑚 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ) |
193 |
188 190 192
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑚 ) = ( ( 2 ↑ 𝑚 ) · ( ( 2 ↑ 𝑚 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ) |
194 |
179 167 180
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( 2 ↑ 𝑚 ) ∈ ℕ ) |
195 |
|
oveq1 |
⊢ ( 𝑛 = ( 2 ↑ 𝑚 ) → ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) = ( ( 2 ↑ 𝑚 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
196 |
|
ovex |
⊢ ( ( 2 ↑ 𝑚 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ∈ V |
197 |
195 7 196
|
fvmpt |
⊢ ( ( 2 ↑ 𝑚 ) ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ ( 2 ↑ 𝑚 ) ) = ( ( 2 ↑ 𝑚 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
198 |
194 197
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ ( 2 ↑ 𝑚 ) ) = ( ( 2 ↑ 𝑚 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) |
199 |
198
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 2 ↑ 𝑚 ) · ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ ( 2 ↑ 𝑚 ) ) ) = ( ( 2 ↑ 𝑚 ) · ( ( 2 ↑ 𝑚 ) ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ) |
200 |
193 91 199
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ0 ) → ( ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) ‘ 𝑚 ) = ( ( 2 ↑ 𝑚 ) · ( ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ‘ ( 2 ↑ 𝑚 ) ) ) ) |
201 |
100 102 157 200
|
climcnds |
⊢ ( 𝜑 → ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ) ∈ dom ⇝ ↔ seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑𝑐 ( 1 − ( ℜ ‘ 𝑆 ) ) ) ↑ 𝑛 ) ) ) ∈ dom ⇝ ) ) |
202 |
96 201
|
mpbird |
⊢ ( 𝜑 → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑛 ↑𝑐 - ( ℜ ‘ 𝑆 ) ) ) ) ∈ dom ⇝ ) |
203 |
4 5 52 54 202
|
abscvgcvg |
⊢ ( 𝜑 → seq 1 ( + , 𝐹 ) ∈ dom ⇝ ) |