Step |
Hyp |
Ref |
Expression |
1 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
2 |
|
0zd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 0 ∈ ℤ ) |
3 |
|
eqeq1 |
⊢ ( 𝑘 = 𝑛 → ( 𝑘 = 0 ↔ 𝑛 = 0 ) ) |
4 |
|
oveq2 |
⊢ ( 𝑘 = 𝑛 → ( 1 / 𝑘 ) = ( 1 / 𝑛 ) ) |
5 |
3 4
|
ifbieq2d |
⊢ ( 𝑘 = 𝑛 → if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) ) |
6 |
|
oveq2 |
⊢ ( 𝑘 = 𝑛 → ( 𝐴 ↑ 𝑘 ) = ( 𝐴 ↑ 𝑛 ) ) |
7 |
5 6
|
oveq12d |
⊢ ( 𝑘 = 𝑛 → ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ) |
8 |
|
eqid |
⊢ ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) = ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) |
9 |
|
ovex |
⊢ ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ∈ V |
10 |
7 8 9
|
fvmpt |
⊢ ( 𝑛 ∈ ℕ0 → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ) |
11 |
10
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ) |
12 |
|
0cnd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → 0 ∈ ℂ ) |
13 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → 𝑛 ∈ ℕ0 ) |
14 |
|
elnn0 |
⊢ ( 𝑛 ∈ ℕ0 ↔ ( 𝑛 ∈ ℕ ∨ 𝑛 = 0 ) ) |
15 |
13 14
|
sylib |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 ∈ ℕ ∨ 𝑛 = 0 ) ) |
16 |
15
|
ord |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( ¬ 𝑛 ∈ ℕ → 𝑛 = 0 ) ) |
17 |
16
|
con1d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( ¬ 𝑛 = 0 → 𝑛 ∈ ℕ ) ) |
18 |
17
|
imp |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → 𝑛 ∈ ℕ ) |
19 |
18
|
nnrecred |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 1 / 𝑛 ) ∈ ℝ ) |
20 |
19
|
recnd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 1 / 𝑛 ) ∈ ℂ ) |
21 |
12 20
|
ifclda |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) ∈ ℂ ) |
22 |
|
expcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑛 ) ∈ ℂ ) |
23 |
22
|
adantlr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑛 ) ∈ ℂ ) |
24 |
21 23
|
mulcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ∈ ℂ ) |
25 |
|
logtayllem |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ∈ dom ⇝ ) |
26 |
1 2 11 24 25
|
isumclim2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ⇝ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ) |
27 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 𝐴 ∈ ℂ ) |
28 |
|
0cn |
⊢ 0 ∈ ℂ |
29 |
|
eqid |
⊢ ( abs ∘ − ) = ( abs ∘ − ) |
30 |
29
|
cnmetdval |
⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ) → ( 𝐴 ( abs ∘ − ) 0 ) = ( abs ‘ ( 𝐴 − 0 ) ) ) |
31 |
27 28 30
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ( abs ∘ − ) 0 ) = ( abs ‘ ( 𝐴 − 0 ) ) ) |
32 |
|
subid1 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 − 0 ) = 𝐴 ) |
33 |
32
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 − 0 ) = 𝐴 ) |
34 |
33
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ ( 𝐴 − 0 ) ) = ( abs ‘ 𝐴 ) ) |
35 |
31 34
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ( abs ∘ − ) 0 ) = ( abs ‘ 𝐴 ) ) |
36 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ 𝐴 ) < 1 ) |
37 |
35 36
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ( abs ∘ − ) 0 ) < 1 ) |
38 |
|
cnxmet |
⊢ ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) |
39 |
|
1xr |
⊢ 1 ∈ ℝ* |
40 |
|
elbl3 |
⊢ ( ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 1 ∈ ℝ* ) ∧ ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ) ) → ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝐴 ( abs ∘ − ) 0 ) < 1 ) ) |
41 |
38 39 40
|
mpanl12 |
⊢ ( ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝐴 ( abs ∘ − ) 0 ) < 1 ) ) |
42 |
28 27 41
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝐴 ( abs ∘ − ) 0 ) < 1 ) ) |
43 |
37 42
|
mpbird |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) |
44 |
|
tru |
⊢ ⊤ |
45 |
|
eqid |
⊢ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) = ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) |
46 |
|
0cnd |
⊢ ( ⊤ → 0 ∈ ℂ ) |
47 |
39
|
a1i |
⊢ ( ⊤ → 1 ∈ ℝ* ) |
48 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
49 |
|
blssm |
⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ* ) → ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ℂ ) |
50 |
38 28 39 49
|
mp3an |
⊢ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ℂ |
51 |
50
|
sseli |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 𝑦 ∈ ℂ ) |
52 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 1 − 𝑦 ) ∈ ℂ ) |
53 |
48 51 52
|
sylancr |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 − 𝑦 ) ∈ ℂ ) |
54 |
51
|
abscld |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) ∈ ℝ ) |
55 |
29
|
cnmetdval |
⊢ ( ( 𝑦 ∈ ℂ ∧ 0 ∈ ℂ ) → ( 𝑦 ( abs ∘ − ) 0 ) = ( abs ‘ ( 𝑦 − 0 ) ) ) |
56 |
51 28 55
|
sylancl |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑦 ( abs ∘ − ) 0 ) = ( abs ‘ ( 𝑦 − 0 ) ) ) |
57 |
51
|
subid1d |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑦 − 0 ) = 𝑦 ) |
58 |
57
|
fveq2d |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ ( 𝑦 − 0 ) ) = ( abs ‘ 𝑦 ) ) |
59 |
56 58
|
eqtrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑦 ( abs ∘ − ) 0 ) = ( abs ‘ 𝑦 ) ) |
60 |
|
elbl3 |
⊢ ( ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 1 ∈ ℝ* ) ∧ ( 0 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝑦 ( abs ∘ − ) 0 ) < 1 ) ) |
61 |
38 39 60
|
mpanl12 |
⊢ ( ( 0 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝑦 ( abs ∘ − ) 0 ) < 1 ) ) |
62 |
28 51 61
|
sylancr |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝑦 ( abs ∘ − ) 0 ) < 1 ) ) |
63 |
62
|
ibi |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑦 ( abs ∘ − ) 0 ) < 1 ) |
64 |
59 63
|
eqbrtrrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) < 1 ) |
65 |
54 64
|
gtned |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 1 ≠ ( abs ‘ 𝑦 ) ) |
66 |
|
abs1 |
⊢ ( abs ‘ 1 ) = 1 |
67 |
|
fveq2 |
⊢ ( 1 = 𝑦 → ( abs ‘ 1 ) = ( abs ‘ 𝑦 ) ) |
68 |
66 67
|
eqtr3id |
⊢ ( 1 = 𝑦 → 1 = ( abs ‘ 𝑦 ) ) |
69 |
68
|
necon3i |
⊢ ( 1 ≠ ( abs ‘ 𝑦 ) → 1 ≠ 𝑦 ) |
70 |
65 69
|
syl |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 1 ≠ 𝑦 ) |
71 |
|
subeq0 |
⊢ ( ( 1 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( 1 − 𝑦 ) = 0 ↔ 1 = 𝑦 ) ) |
72 |
71
|
necon3bid |
⊢ ( ( 1 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( 1 − 𝑦 ) ≠ 0 ↔ 1 ≠ 𝑦 ) ) |
73 |
48 51 72
|
sylancr |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 1 − 𝑦 ) ≠ 0 ↔ 1 ≠ 𝑦 ) ) |
74 |
70 73
|
mpbird |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 − 𝑦 ) ≠ 0 ) |
75 |
53 74
|
logcld |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( log ‘ ( 1 − 𝑦 ) ) ∈ ℂ ) |
76 |
75
|
negcld |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → - ( log ‘ ( 1 − 𝑦 ) ) ∈ ℂ ) |
77 |
76
|
adantl |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → - ( log ‘ ( 1 − 𝑦 ) ) ∈ ℂ ) |
78 |
77
|
fmpttd |
⊢ ( ⊤ → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) : ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⟶ ℂ ) |
79 |
51
|
absge0d |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 0 ≤ ( abs ‘ 𝑦 ) ) |
80 |
54
|
rexrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) ∈ ℝ* ) |
81 |
|
peano2re |
⊢ ( ( abs ‘ 𝑦 ) ∈ ℝ → ( ( abs ‘ 𝑦 ) + 1 ) ∈ ℝ ) |
82 |
54 81
|
syl |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( abs ‘ 𝑦 ) + 1 ) ∈ ℝ ) |
83 |
82
|
rehalfcld |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ∈ ℝ ) |
84 |
83
|
rexrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ∈ ℝ* ) |
85 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
86 |
|
eqeq1 |
⊢ ( 𝑚 = 𝑗 → ( 𝑚 = 0 ↔ 𝑗 = 0 ) ) |
87 |
|
oveq2 |
⊢ ( 𝑚 = 𝑗 → ( 1 / 𝑚 ) = ( 1 / 𝑗 ) ) |
88 |
86 87
|
ifbieq2d |
⊢ ( 𝑚 = 𝑗 → if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) = if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) ) |
89 |
|
eqid |
⊢ ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) = ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) |
90 |
|
c0ex |
⊢ 0 ∈ V |
91 |
|
ovex |
⊢ ( 1 / 𝑗 ) ∈ V |
92 |
90 91
|
ifex |
⊢ if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) ∈ V |
93 |
88 89 92
|
fvmpt |
⊢ ( 𝑗 ∈ ℕ0 → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑗 ) = if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) ) |
94 |
93
|
eqcomd |
⊢ ( 𝑗 ∈ ℕ0 → if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) = ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑗 ) ) |
95 |
94
|
oveq1d |
⊢ ( 𝑗 ∈ ℕ0 → ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) = ( ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑗 ) · ( 𝑥 ↑ 𝑗 ) ) ) |
96 |
95
|
mpteq2ia |
⊢ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑗 ) · ( 𝑥 ↑ 𝑗 ) ) ) |
97 |
96
|
mpteq2i |
⊢ ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) = ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑗 ) · ( 𝑥 ↑ 𝑗 ) ) ) ) |
98 |
|
0cnd |
⊢ ( ( ( ⊤ ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 = 0 ) → 0 ∈ ℂ ) |
99 |
|
nn0cn |
⊢ ( 𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ ) |
100 |
99
|
adantl |
⊢ ( ( ⊤ ∧ 𝑚 ∈ ℕ0 ) → 𝑚 ∈ ℂ ) |
101 |
|
neqne |
⊢ ( ¬ 𝑚 = 0 → 𝑚 ≠ 0 ) |
102 |
|
reccl |
⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑚 ≠ 0 ) → ( 1 / 𝑚 ) ∈ ℂ ) |
103 |
100 101 102
|
syl2an |
⊢ ( ( ( ⊤ ∧ 𝑚 ∈ ℕ0 ) ∧ ¬ 𝑚 = 0 ) → ( 1 / 𝑚 ) ∈ ℂ ) |
104 |
98 103
|
ifclda |
⊢ ( ( ⊤ ∧ 𝑚 ∈ ℕ0 ) → if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ∈ ℂ ) |
105 |
104
|
fmpttd |
⊢ ( ⊤ → ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) : ℕ0 ⟶ ℂ ) |
106 |
|
recn |
⊢ ( 𝑟 ∈ ℝ → 𝑟 ∈ ℂ ) |
107 |
|
oveq1 |
⊢ ( 𝑥 = 𝑟 → ( 𝑥 ↑ 𝑗 ) = ( 𝑟 ↑ 𝑗 ) ) |
108 |
107
|
oveq2d |
⊢ ( 𝑥 = 𝑟 → ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) = ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) |
109 |
108
|
mpteq2dv |
⊢ ( 𝑥 = 𝑟 → ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) |
110 |
|
eqid |
⊢ ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) = ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) |
111 |
|
nn0ex |
⊢ ℕ0 ∈ V |
112 |
111
|
mptex |
⊢ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ∈ V |
113 |
109 110 112
|
fvmpt |
⊢ ( 𝑟 ∈ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) |
114 |
106 113
|
syl |
⊢ ( 𝑟 ∈ ℝ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) |
115 |
114
|
eqcomd |
⊢ ( 𝑟 ∈ ℝ → ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) = ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) ) |
116 |
115
|
seqeq3d |
⊢ ( 𝑟 ∈ ℝ → seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) = seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) ) ) |
117 |
116
|
eleq1d |
⊢ ( 𝑟 ∈ ℝ → ( seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ ↔ seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) ) ∈ dom ⇝ ) ) |
118 |
117
|
rabbiia |
⊢ { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } = { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) ) ∈ dom ⇝ } |
119 |
118
|
supeq1i |
⊢ sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) = sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) ) ∈ dom ⇝ } , ℝ* , < ) |
120 |
97 105 119
|
radcnvcl |
⊢ ( ⊤ → sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ( 0 [,] +∞ ) ) |
121 |
85 120
|
sselid |
⊢ ( ⊤ → sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ* ) |
122 |
44 121
|
mp1i |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ* ) |
123 |
|
1re |
⊢ 1 ∈ ℝ |
124 |
|
avglt1 |
⊢ ( ( ( abs ‘ 𝑦 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( abs ‘ 𝑦 ) < 1 ↔ ( abs ‘ 𝑦 ) < ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) ) |
125 |
54 123 124
|
sylancl |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( abs ‘ 𝑦 ) < 1 ↔ ( abs ‘ 𝑦 ) < ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) ) |
126 |
64 125
|
mpbid |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) < ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) |
127 |
|
0red |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 0 ∈ ℝ ) |
128 |
127 54 83 79 126
|
lelttrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 0 < ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) |
129 |
127 83 128
|
ltled |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 0 ≤ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) |
130 |
83 129
|
absidd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) = ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) |
131 |
44 105
|
mp1i |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) : ℕ0 ⟶ ℂ ) |
132 |
83
|
recnd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ∈ ℂ ) |
133 |
|
oveq1 |
⊢ ( 𝑥 = ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) → ( 𝑥 ↑ 𝑗 ) = ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) |
134 |
133
|
oveq2d |
⊢ ( 𝑥 = ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) → ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) = ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) |
135 |
134
|
mpteq2dv |
⊢ ( 𝑥 = ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) → ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ) |
136 |
111
|
mptex |
⊢ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ∈ V |
137 |
135 110 136
|
fvmpt |
⊢ ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ∈ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ) |
138 |
132 137
|
syl |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ) |
139 |
138
|
seqeq3d |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) ) = seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ) ) |
140 |
|
avglt2 |
⊢ ( ( ( abs ‘ 𝑦 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( abs ‘ 𝑦 ) < 1 ↔ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) < 1 ) ) |
141 |
54 123 140
|
sylancl |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( abs ‘ 𝑦 ) < 1 ↔ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) < 1 ) ) |
142 |
64 141
|
mpbid |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) < 1 ) |
143 |
130 142
|
eqbrtrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) < 1 ) |
144 |
|
logtayllem |
⊢ ( ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ∈ ℂ ∧ ( abs ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) < 1 ) → seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ) ∈ dom ⇝ ) |
145 |
132 143 144
|
syl2anc |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ) ∈ dom ⇝ ) |
146 |
139 145
|
eqeltrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) ) ∈ dom ⇝ ) |
147 |
97 131 119 132 146
|
radcnvle |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) ≤ sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) |
148 |
130 147
|
eqbrtrrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ≤ sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) |
149 |
80 84 122 126 148
|
xrltletrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) < sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) |
150 |
|
0re |
⊢ 0 ∈ ℝ |
151 |
|
elico2 |
⊢ ( ( 0 ∈ ℝ ∧ sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ* ) → ( ( abs ‘ 𝑦 ) ∈ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ↔ ( ( abs ‘ 𝑦 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝑦 ) ∧ ( abs ‘ 𝑦 ) < sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) |
152 |
150 122 151
|
sylancr |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( abs ‘ 𝑦 ) ∈ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ↔ ( ( abs ‘ 𝑦 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝑦 ) ∧ ( abs ‘ 𝑦 ) < sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) |
153 |
54 79 149 152
|
mpbir3and |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) ∈ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) |
154 |
|
absf |
⊢ abs : ℂ ⟶ ℝ |
155 |
|
ffn |
⊢ ( abs : ℂ ⟶ ℝ → abs Fn ℂ ) |
156 |
|
elpreima |
⊢ ( abs Fn ℂ → ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↔ ( 𝑦 ∈ ℂ ∧ ( abs ‘ 𝑦 ) ∈ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) ) |
157 |
154 155 156
|
mp2b |
⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↔ ( 𝑦 ∈ ℂ ∧ ( abs ‘ 𝑦 ) ∈ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) |
158 |
51 153 157
|
sylanbrc |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) |
159 |
|
cnvimass |
⊢ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ⊆ dom abs |
160 |
154
|
fdmi |
⊢ dom abs = ℂ |
161 |
159 160
|
sseqtri |
⊢ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ⊆ ℂ |
162 |
161
|
sseli |
⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) → 𝑦 ∈ ℂ ) |
163 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ↑ 𝑗 ) = ( 𝑦 ↑ 𝑗 ) ) |
164 |
163
|
oveq2d |
⊢ ( 𝑥 = 𝑦 → ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) = ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) |
165 |
164
|
mpteq2dv |
⊢ ( 𝑥 = 𝑦 → ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ) |
166 |
111
|
mptex |
⊢ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ∈ V |
167 |
165 110 166
|
fvmpt |
⊢ ( 𝑦 ∈ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ) |
168 |
167
|
adantr |
⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ) |
169 |
168
|
fveq1d |
⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) = ( ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ‘ 𝑛 ) ) |
170 |
|
eqeq1 |
⊢ ( 𝑗 = 𝑛 → ( 𝑗 = 0 ↔ 𝑛 = 0 ) ) |
171 |
|
oveq2 |
⊢ ( 𝑗 = 𝑛 → ( 1 / 𝑗 ) = ( 1 / 𝑛 ) ) |
172 |
170 171
|
ifbieq2d |
⊢ ( 𝑗 = 𝑛 → if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) ) |
173 |
|
oveq2 |
⊢ ( 𝑗 = 𝑛 → ( 𝑦 ↑ 𝑗 ) = ( 𝑦 ↑ 𝑛 ) ) |
174 |
172 173
|
oveq12d |
⊢ ( 𝑗 = 𝑛 → ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) |
175 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) |
176 |
|
ovex |
⊢ ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ∈ V |
177 |
174 175 176
|
fvmpt |
⊢ ( 𝑛 ∈ ℕ0 → ( ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ‘ 𝑛 ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) |
178 |
177
|
adantl |
⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ‘ 𝑛 ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) |
179 |
169 178
|
eqtr2d |
⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) ) |
180 |
179
|
sumeq2dv |
⊢ ( 𝑦 ∈ ℂ → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = Σ 𝑛 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) ) |
181 |
162 180
|
syl |
⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = Σ 𝑛 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) ) |
182 |
181
|
mpteq2ia |
⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) = ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) ) |
183 |
|
eqid |
⊢ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) = ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) |
184 |
|
eqid |
⊢ if ( sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ , ( ( ( abs ‘ 𝑧 ) + sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) / 2 ) , ( ( abs ‘ 𝑧 ) + 1 ) ) = if ( sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ , ( ( ( abs ‘ 𝑧 ) + sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) / 2 ) , ( ( abs ‘ 𝑧 ) + 1 ) ) |
185 |
97 182 105 119 183 184
|
psercn |
⊢ ( ⊤ → ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ∈ ( ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) –cn→ ℂ ) ) |
186 |
|
cncff |
⊢ ( ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ∈ ( ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) –cn→ ℂ ) → ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) : ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ⟶ ℂ ) |
187 |
185 186
|
syl |
⊢ ( ⊤ → ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) : ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ⟶ ℂ ) |
188 |
187
|
fvmptelrn |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ∈ ℂ ) |
189 |
158 188
|
sylan2 |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ∈ ℂ ) |
190 |
189
|
fmpttd |
⊢ ( ⊤ → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) : ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⟶ ℂ ) |
191 |
|
cnelprrecn |
⊢ ℂ ∈ { ℝ , ℂ } |
192 |
191
|
a1i |
⊢ ( ⊤ → ℂ ∈ { ℝ , ℂ } ) |
193 |
75
|
adantl |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → ( log ‘ ( 1 − 𝑦 ) ) ∈ ℂ ) |
194 |
|
ovexd |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) ∈ V ) |
195 |
29
|
cnmetdval |
⊢ ( ( 1 ∈ ℂ ∧ ( 1 − 𝑦 ) ∈ ℂ ) → ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) = ( abs ‘ ( 1 − ( 1 − 𝑦 ) ) ) ) |
196 |
48 53 195
|
sylancr |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) = ( abs ‘ ( 1 − ( 1 − 𝑦 ) ) ) ) |
197 |
|
nncan |
⊢ ( ( 1 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 1 − ( 1 − 𝑦 ) ) = 𝑦 ) |
198 |
48 51 197
|
sylancr |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 − ( 1 − 𝑦 ) ) = 𝑦 ) |
199 |
198
|
fveq2d |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ ( 1 − ( 1 − 𝑦 ) ) ) = ( abs ‘ 𝑦 ) ) |
200 |
196 199
|
eqtrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) = ( abs ‘ 𝑦 ) ) |
201 |
200 64
|
eqbrtrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) < 1 ) |
202 |
|
elbl |
⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ* ) → ( ( 1 − 𝑦 ) ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( ( 1 − 𝑦 ) ∈ ℂ ∧ ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) < 1 ) ) ) |
203 |
38 48 39 202
|
mp3an |
⊢ ( ( 1 − 𝑦 ) ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( ( 1 − 𝑦 ) ∈ ℂ ∧ ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) < 1 ) ) |
204 |
53 201 203
|
sylanbrc |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 − 𝑦 ) ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) |
205 |
204
|
adantl |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → ( 1 − 𝑦 ) ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) |
206 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
207 |
206
|
a1i |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → - 1 ∈ ℂ ) |
208 |
|
eqid |
⊢ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) = ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) |
209 |
208
|
dvlog2lem |
⊢ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ( ℂ ∖ ( -∞ (,] 0 ) ) |
210 |
209
|
sseli |
⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → 𝑥 ∈ ( ℂ ∖ ( -∞ (,] 0 ) ) ) |
211 |
210
|
eldifad |
⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → 𝑥 ∈ ℂ ) |
212 |
|
eqid |
⊢ ( ℂ ∖ ( -∞ (,] 0 ) ) = ( ℂ ∖ ( -∞ (,] 0 ) ) |
213 |
212
|
logdmn0 |
⊢ ( 𝑥 ∈ ( ℂ ∖ ( -∞ (,] 0 ) ) → 𝑥 ≠ 0 ) |
214 |
210 213
|
syl |
⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → 𝑥 ≠ 0 ) |
215 |
211 214
|
logcld |
⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( log ‘ 𝑥 ) ∈ ℂ ) |
216 |
215
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → ( log ‘ 𝑥 ) ∈ ℂ ) |
217 |
|
ovexd |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → ( 1 / 𝑥 ) ∈ V ) |
218 |
|
simpr |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ℂ ) → 𝑦 ∈ ℂ ) |
219 |
48 218 52
|
sylancr |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ℂ ) → ( 1 − 𝑦 ) ∈ ℂ ) |
220 |
206
|
a1i |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ℂ ) → - 1 ∈ ℂ ) |
221 |
|
1cnd |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ℂ ) → 1 ∈ ℂ ) |
222 |
|
0cnd |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ℂ ) → 0 ∈ ℂ ) |
223 |
|
1cnd |
⊢ ( ⊤ → 1 ∈ ℂ ) |
224 |
192 223
|
dvmptc |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ℂ ↦ 1 ) ) = ( 𝑦 ∈ ℂ ↦ 0 ) ) |
225 |
192
|
dvmptid |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ℂ ↦ 𝑦 ) ) = ( 𝑦 ∈ ℂ ↦ 1 ) ) |
226 |
192 221 222 224 218 221 225
|
dvmptsub |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( 1 − 𝑦 ) ) ) = ( 𝑦 ∈ ℂ ↦ ( 0 − 1 ) ) ) |
227 |
|
df-neg |
⊢ - 1 = ( 0 − 1 ) |
228 |
227
|
mpteq2i |
⊢ ( 𝑦 ∈ ℂ ↦ - 1 ) = ( 𝑦 ∈ ℂ ↦ ( 0 − 1 ) ) |
229 |
226 228
|
eqtr4di |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( 1 − 𝑦 ) ) ) = ( 𝑦 ∈ ℂ ↦ - 1 ) ) |
230 |
50
|
a1i |
⊢ ( ⊤ → ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ℂ ) |
231 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
232 |
231
|
cnfldtopon |
⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) |
233 |
232
|
toponrestid |
⊢ ( TopOpen ‘ ℂfld ) = ( ( TopOpen ‘ ℂfld ) ↾t ℂ ) |
234 |
231
|
cnfldtopn |
⊢ ( TopOpen ‘ ℂfld ) = ( MetOpen ‘ ( abs ∘ − ) ) |
235 |
234
|
blopn |
⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ* ) → ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∈ ( TopOpen ‘ ℂfld ) ) |
236 |
38 28 39 235
|
mp3an |
⊢ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∈ ( TopOpen ‘ ℂfld ) |
237 |
236
|
a1i |
⊢ ( ⊤ → ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∈ ( TopOpen ‘ ℂfld ) ) |
238 |
192 219 220 229 230 233 231 237
|
dvmptres |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 − 𝑦 ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - 1 ) ) |
239 |
|
logf1o |
⊢ log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log |
240 |
|
f1of |
⊢ ( log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log → log : ( ℂ ∖ { 0 } ) ⟶ ran log ) |
241 |
239 240
|
ax-mp |
⊢ log : ( ℂ ∖ { 0 } ) ⟶ ran log |
242 |
212
|
logdmss |
⊢ ( ℂ ∖ ( -∞ (,] 0 ) ) ⊆ ( ℂ ∖ { 0 } ) |
243 |
209 242
|
sstri |
⊢ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ( ℂ ∖ { 0 } ) |
244 |
|
fssres |
⊢ ( ( log : ( ℂ ∖ { 0 } ) ⟶ ran log ∧ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ( ℂ ∖ { 0 } ) ) → ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) : ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⟶ ran log ) |
245 |
241 243 244
|
mp2an |
⊢ ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) : ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⟶ ran log |
246 |
245
|
a1i |
⊢ ( ⊤ → ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) : ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⟶ ran log ) |
247 |
246
|
feqmptd |
⊢ ( ⊤ → ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) = ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) ‘ 𝑥 ) ) ) |
248 |
|
fvres |
⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) ‘ 𝑥 ) = ( log ‘ 𝑥 ) ) |
249 |
248
|
mpteq2ia |
⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) ‘ 𝑥 ) ) = ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( log ‘ 𝑥 ) ) |
250 |
247 249
|
eqtrdi |
⊢ ( ⊤ → ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) = ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( log ‘ 𝑥 ) ) ) |
251 |
250
|
oveq2d |
⊢ ( ⊤ → ( ℂ D ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) ) = ( ℂ D ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( log ‘ 𝑥 ) ) ) ) |
252 |
208
|
dvlog2 |
⊢ ( ℂ D ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) ) = ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / 𝑥 ) ) |
253 |
251 252
|
eqtr3di |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( log ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / 𝑥 ) ) ) |
254 |
|
fveq2 |
⊢ ( 𝑥 = ( 1 − 𝑦 ) → ( log ‘ 𝑥 ) = ( log ‘ ( 1 − 𝑦 ) ) ) |
255 |
|
oveq2 |
⊢ ( 𝑥 = ( 1 − 𝑦 ) → ( 1 / 𝑥 ) = ( 1 / ( 1 − 𝑦 ) ) ) |
256 |
192 192 205 207 216 217 238 253 254 255
|
dvmptco |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( log ‘ ( 1 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) ) ) |
257 |
192 193 194 256
|
dvmptneg |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) ) ) |
258 |
53 74
|
reccld |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 / ( 1 − 𝑦 ) ) ∈ ℂ ) |
259 |
|
mulcom |
⊢ ( ( ( 1 / ( 1 − 𝑦 ) ) ∈ ℂ ∧ - 1 ∈ ℂ ) → ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) = ( - 1 · ( 1 / ( 1 − 𝑦 ) ) ) ) |
260 |
258 206 259
|
sylancl |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) = ( - 1 · ( 1 / ( 1 − 𝑦 ) ) ) ) |
261 |
258
|
mulm1d |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( - 1 · ( 1 / ( 1 − 𝑦 ) ) ) = - ( 1 / ( 1 − 𝑦 ) ) ) |
262 |
260 261
|
eqtrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) = - ( 1 / ( 1 − 𝑦 ) ) ) |
263 |
262
|
negeqd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → - ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) = - - ( 1 / ( 1 − 𝑦 ) ) ) |
264 |
258
|
negnegd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → - - ( 1 / ( 1 − 𝑦 ) ) = ( 1 / ( 1 − 𝑦 ) ) ) |
265 |
263 264
|
eqtrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → - ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) = ( 1 / ( 1 − 𝑦 ) ) ) |
266 |
265
|
mpteq2ia |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) |
267 |
257 266
|
eqtrdi |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) ) |
268 |
267
|
dmeqd |
⊢ ( ⊤ → dom ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ) = dom ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) ) |
269 |
|
dmmptg |
⊢ ( ∀ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ( 1 / ( 1 − 𝑦 ) ) ∈ V → dom ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) = ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) |
270 |
|
ovexd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 / ( 1 − 𝑦 ) ) ∈ V ) |
271 |
269 270
|
mprg |
⊢ dom ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) = ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) |
272 |
268 271
|
eqtrdi |
⊢ ( ⊤ → dom ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ) = ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) |
273 |
|
sumex |
⊢ Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ∈ V |
274 |
273
|
a1i |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) → Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ∈ V ) |
275 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) = ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑘 ) ) |
276 |
275
|
cbvsumv |
⊢ Σ 𝑛 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) = Σ 𝑘 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑘 ) |
277 |
181 276
|
eqtrdi |
⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = Σ 𝑘 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑘 ) ) |
278 |
277
|
mpteq2ia |
⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) = ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑘 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑘 ) ) |
279 |
|
eqid |
⊢ ( 0 ( ball ‘ ( abs ∘ − ) ) ( ( ( abs ‘ 𝑧 ) + if ( sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ , ( ( ( abs ‘ 𝑧 ) + sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) / 2 ) , ( ( abs ‘ 𝑧 ) + 1 ) ) ) / 2 ) ) = ( 0 ( ball ‘ ( abs ∘ − ) ) ( ( ( abs ‘ 𝑧 ) + if ( sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ , ( ( ( abs ‘ 𝑧 ) + sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) / 2 ) , ( ( abs ‘ 𝑧 ) + 1 ) ) ) / 2 ) ) |
280 |
97 278 105 119 183 184 279
|
pserdv2 |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ) = ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ) ) |
281 |
158
|
ssriv |
⊢ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) |
282 |
281
|
a1i |
⊢ ( ⊤ → ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) |
283 |
192 188 274 280 282 233 231 237
|
dvmptres |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ) ) |
284 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
285 |
284
|
adantl |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ0 ) |
286 |
|
eqeq1 |
⊢ ( 𝑚 = 𝑛 → ( 𝑚 = 0 ↔ 𝑛 = 0 ) ) |
287 |
|
oveq2 |
⊢ ( 𝑚 = 𝑛 → ( 1 / 𝑚 ) = ( 1 / 𝑛 ) ) |
288 |
286 287
|
ifbieq2d |
⊢ ( 𝑚 = 𝑛 → if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) ) |
289 |
|
ovex |
⊢ ( 1 / 𝑛 ) ∈ V |
290 |
90 289
|
ifex |
⊢ if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) ∈ V |
291 |
288 89 290
|
fvmpt |
⊢ ( 𝑛 ∈ ℕ0 → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) ) |
292 |
285 291
|
syl |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) ) |
293 |
|
nnne0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ≠ 0 ) |
294 |
293
|
adantl |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ≠ 0 ) |
295 |
294
|
neneqd |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ¬ 𝑛 = 0 ) |
296 |
295
|
iffalsed |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) = ( 1 / 𝑛 ) ) |
297 |
292 296
|
eqtrd |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) = ( 1 / 𝑛 ) ) |
298 |
297
|
oveq2d |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) = ( 𝑛 · ( 1 / 𝑛 ) ) ) |
299 |
|
nncn |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℂ ) |
300 |
299
|
adantl |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℂ ) |
301 |
300 294
|
recidd |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( 𝑛 · ( 1 / 𝑛 ) ) = 1 ) |
302 |
298 301
|
eqtrd |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) = 1 ) |
303 |
302
|
oveq1d |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) = ( 1 · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ) |
304 |
|
nnm1nn0 |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 − 1 ) ∈ ℕ0 ) |
305 |
|
expcl |
⊢ ( ( 𝑦 ∈ ℂ ∧ ( 𝑛 − 1 ) ∈ ℕ0 ) → ( 𝑦 ↑ ( 𝑛 − 1 ) ) ∈ ℂ ) |
306 |
51 304 305
|
syl2an |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( 𝑦 ↑ ( 𝑛 − 1 ) ) ∈ ℂ ) |
307 |
306
|
mulid2d |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( 1 · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) = ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) |
308 |
303 307
|
eqtrd |
⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) = ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) |
309 |
308
|
sumeq2dv |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) = Σ 𝑛 ∈ ℕ ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) |
310 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
311 |
|
1e0p1 |
⊢ 1 = ( 0 + 1 ) |
312 |
311
|
fveq2i |
⊢ ( ℤ≥ ‘ 1 ) = ( ℤ≥ ‘ ( 0 + 1 ) ) |
313 |
310 312
|
eqtri |
⊢ ℕ = ( ℤ≥ ‘ ( 0 + 1 ) ) |
314 |
|
oveq1 |
⊢ ( 𝑛 = ( 1 + 𝑚 ) → ( 𝑛 − 1 ) = ( ( 1 + 𝑚 ) − 1 ) ) |
315 |
314
|
oveq2d |
⊢ ( 𝑛 = ( 1 + 𝑚 ) → ( 𝑦 ↑ ( 𝑛 − 1 ) ) = ( 𝑦 ↑ ( ( 1 + 𝑚 ) − 1 ) ) ) |
316 |
|
1zzd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 1 ∈ ℤ ) |
317 |
|
0zd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 0 ∈ ℤ ) |
318 |
1 313 315 316 317 306
|
isumshft |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → Σ 𝑛 ∈ ℕ ( 𝑦 ↑ ( 𝑛 − 1 ) ) = Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ ( ( 1 + 𝑚 ) − 1 ) ) ) |
319 |
|
pncan2 |
⊢ ( ( 1 ∈ ℂ ∧ 𝑚 ∈ ℂ ) → ( ( 1 + 𝑚 ) − 1 ) = 𝑚 ) |
320 |
48 99 319
|
sylancr |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 1 + 𝑚 ) − 1 ) = 𝑚 ) |
321 |
320
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑦 ↑ ( ( 1 + 𝑚 ) − 1 ) ) = ( 𝑦 ↑ 𝑚 ) ) |
322 |
321
|
sumeq2i |
⊢ Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ ( ( 1 + 𝑚 ) − 1 ) ) = Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ 𝑚 ) |
323 |
318 322
|
eqtrdi |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → Σ 𝑛 ∈ ℕ ( 𝑦 ↑ ( 𝑛 − 1 ) ) = Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ 𝑚 ) ) |
324 |
|
geoisum |
⊢ ( ( 𝑦 ∈ ℂ ∧ ( abs ‘ 𝑦 ) < 1 ) → Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ 𝑚 ) = ( 1 / ( 1 − 𝑦 ) ) ) |
325 |
51 64 324
|
syl2anc |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ 𝑚 ) = ( 1 / ( 1 − 𝑦 ) ) ) |
326 |
309 323 325
|
3eqtrd |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) = ( 1 / ( 1 − 𝑦 ) ) ) |
327 |
326
|
mpteq2ia |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) |
328 |
283 327
|
eqtrdi |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) ) |
329 |
267 328
|
eqtr4d |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ) = ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ) ) |
330 |
|
1rp |
⊢ 1 ∈ ℝ+ |
331 |
|
blcntr |
⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ+ ) → 0 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) |
332 |
38 28 330 331
|
mp3an |
⊢ 0 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) |
333 |
332
|
a1i |
⊢ ( ⊤ → 0 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) |
334 |
|
oveq2 |
⊢ ( 𝑦 = 0 → ( 1 − 𝑦 ) = ( 1 − 0 ) ) |
335 |
|
1m0e1 |
⊢ ( 1 − 0 ) = 1 |
336 |
334 335
|
eqtrdi |
⊢ ( 𝑦 = 0 → ( 1 − 𝑦 ) = 1 ) |
337 |
336
|
fveq2d |
⊢ ( 𝑦 = 0 → ( log ‘ ( 1 − 𝑦 ) ) = ( log ‘ 1 ) ) |
338 |
|
log1 |
⊢ ( log ‘ 1 ) = 0 |
339 |
337 338
|
eqtrdi |
⊢ ( 𝑦 = 0 → ( log ‘ ( 1 − 𝑦 ) ) = 0 ) |
340 |
339
|
negeqd |
⊢ ( 𝑦 = 0 → - ( log ‘ ( 1 − 𝑦 ) ) = - 0 ) |
341 |
|
neg0 |
⊢ - 0 = 0 |
342 |
340 341
|
eqtrdi |
⊢ ( 𝑦 = 0 → - ( log ‘ ( 1 − 𝑦 ) ) = 0 ) |
343 |
|
eqid |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) |
344 |
342 343 90
|
fvmpt |
⊢ ( 0 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 0 ) = 0 ) |
345 |
332 344
|
mp1i |
⊢ ( ⊤ → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 0 ) = 0 ) |
346 |
|
oveq1 |
⊢ ( 0 = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) → ( 0 · ( 𝑦 ↑ 𝑛 ) ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) |
347 |
346
|
eqeq1d |
⊢ ( 0 = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) → ( ( 0 · ( 𝑦 ↑ 𝑛 ) ) = 0 ↔ ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = 0 ) ) |
348 |
|
oveq1 |
⊢ ( ( 1 / 𝑛 ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) → ( ( 1 / 𝑛 ) · ( 𝑦 ↑ 𝑛 ) ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) |
349 |
348
|
eqeq1d |
⊢ ( ( 1 / 𝑛 ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) → ( ( ( 1 / 𝑛 ) · ( 𝑦 ↑ 𝑛 ) ) = 0 ↔ ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = 0 ) ) |
350 |
|
simpll |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → 𝑦 = 0 ) |
351 |
350 28
|
eqeltrdi |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → 𝑦 ∈ ℂ ) |
352 |
|
simplr |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → 𝑛 ∈ ℕ0 ) |
353 |
351 352
|
expcld |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → ( 𝑦 ↑ 𝑛 ) ∈ ℂ ) |
354 |
353
|
mul02d |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → ( 0 · ( 𝑦 ↑ 𝑛 ) ) = 0 ) |
355 |
|
simpll |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → 𝑦 = 0 ) |
356 |
355
|
oveq1d |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 𝑦 ↑ 𝑛 ) = ( 0 ↑ 𝑛 ) ) |
357 |
|
simpr |
⊢ ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) → 𝑛 ∈ ℕ0 ) |
358 |
357 14
|
sylib |
⊢ ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 ∈ ℕ ∨ 𝑛 = 0 ) ) |
359 |
358
|
ord |
⊢ ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) → ( ¬ 𝑛 ∈ ℕ → 𝑛 = 0 ) ) |
360 |
359
|
con1d |
⊢ ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) → ( ¬ 𝑛 = 0 → 𝑛 ∈ ℕ ) ) |
361 |
360
|
imp |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → 𝑛 ∈ ℕ ) |
362 |
361
|
0expd |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 0 ↑ 𝑛 ) = 0 ) |
363 |
356 362
|
eqtrd |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 𝑦 ↑ 𝑛 ) = 0 ) |
364 |
363
|
oveq2d |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( ( 1 / 𝑛 ) · ( 𝑦 ↑ 𝑛 ) ) = ( ( 1 / 𝑛 ) · 0 ) ) |
365 |
361
|
nnrecred |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 1 / 𝑛 ) ∈ ℝ ) |
366 |
365
|
recnd |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 1 / 𝑛 ) ∈ ℂ ) |
367 |
366
|
mul01d |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( ( 1 / 𝑛 ) · 0 ) = 0 ) |
368 |
364 367
|
eqtrd |
⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( ( 1 / 𝑛 ) · ( 𝑦 ↑ 𝑛 ) ) = 0 ) |
369 |
347 349 354 368
|
ifbothda |
⊢ ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = 0 ) |
370 |
369
|
sumeq2dv |
⊢ ( 𝑦 = 0 → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = Σ 𝑛 ∈ ℕ0 0 ) |
371 |
1
|
eqimssi |
⊢ ℕ0 ⊆ ( ℤ≥ ‘ 0 ) |
372 |
371
|
orci |
⊢ ( ℕ0 ⊆ ( ℤ≥ ‘ 0 ) ∨ ℕ0 ∈ Fin ) |
373 |
|
sumz |
⊢ ( ( ℕ0 ⊆ ( ℤ≥ ‘ 0 ) ∨ ℕ0 ∈ Fin ) → Σ 𝑛 ∈ ℕ0 0 = 0 ) |
374 |
372 373
|
ax-mp |
⊢ Σ 𝑛 ∈ ℕ0 0 = 0 |
375 |
370 374
|
eqtrdi |
⊢ ( 𝑦 = 0 → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = 0 ) |
376 |
|
eqid |
⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) |
377 |
375 376 90
|
fvmpt |
⊢ ( 0 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 0 ) = 0 ) |
378 |
332 377
|
mp1i |
⊢ ( ⊤ → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 0 ) = 0 ) |
379 |
345 378
|
eqtr4d |
⊢ ( ⊤ → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 0 ) = ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 0 ) ) |
380 |
45 46 47 78 190 272 329 333 379
|
dv11cn |
⊢ ( ⊤ → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ) |
381 |
380
|
fveq1d |
⊢ ( ⊤ → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 𝐴 ) = ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 𝐴 ) ) |
382 |
44 381
|
mp1i |
⊢ ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 𝐴 ) = ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 𝐴 ) ) |
383 |
|
oveq2 |
⊢ ( 𝑦 = 𝐴 → ( 1 − 𝑦 ) = ( 1 − 𝐴 ) ) |
384 |
383
|
fveq2d |
⊢ ( 𝑦 = 𝐴 → ( log ‘ ( 1 − 𝑦 ) ) = ( log ‘ ( 1 − 𝐴 ) ) ) |
385 |
384
|
negeqd |
⊢ ( 𝑦 = 𝐴 → - ( log ‘ ( 1 − 𝑦 ) ) = - ( log ‘ ( 1 − 𝐴 ) ) ) |
386 |
|
negex |
⊢ - ( log ‘ ( 1 − 𝐴 ) ) ∈ V |
387 |
385 343 386
|
fvmpt |
⊢ ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 𝐴 ) = - ( log ‘ ( 1 − 𝐴 ) ) ) |
388 |
|
oveq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 ↑ 𝑛 ) = ( 𝐴 ↑ 𝑛 ) ) |
389 |
388
|
oveq2d |
⊢ ( 𝑦 = 𝐴 → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ) |
390 |
389
|
sumeq2sdv |
⊢ ( 𝑦 = 𝐴 → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ) |
391 |
|
sumex |
⊢ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ∈ V |
392 |
390 376 391
|
fvmpt |
⊢ ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 𝐴 ) = Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ) |
393 |
382 387 392
|
3eqtr3d |
⊢ ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → - ( log ‘ ( 1 − 𝐴 ) ) = Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ) |
394 |
43 393
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → - ( log ‘ ( 1 − 𝐴 ) ) = Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ) |
395 |
26 394
|
breqtrrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ⇝ - ( log ‘ ( 1 − 𝐴 ) ) ) |
396 |
|
seqex |
⊢ seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ∈ V |
397 |
396
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ∈ V ) |
398 |
|
seqex |
⊢ seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ∈ V |
399 |
398
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ∈ V ) |
400 |
|
1zzd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 1 ∈ ℤ ) |
401 |
|
elnnuz |
⊢ ( 𝑛 ∈ ℕ ↔ 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) |
402 |
|
fvres |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 1 ) → ( ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) ‘ 𝑛 ) = ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ‘ 𝑛 ) ) |
403 |
401 402
|
sylbi |
⊢ ( 𝑛 ∈ ℕ → ( ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) ‘ 𝑛 ) = ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ‘ 𝑛 ) ) |
404 |
403
|
eqcomd |
⊢ ( 𝑛 ∈ ℕ → ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ‘ 𝑛 ) = ( ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) ‘ 𝑛 ) ) |
405 |
|
addid2 |
⊢ ( 𝑛 ∈ ℂ → ( 0 + 𝑛 ) = 𝑛 ) |
406 |
405
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℂ ) → ( 0 + 𝑛 ) = 𝑛 ) |
407 |
|
0cnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 0 ∈ ℂ ) |
408 |
|
1eluzge0 |
⊢ 1 ∈ ( ℤ≥ ‘ 0 ) |
409 |
408
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 1 ∈ ( ℤ≥ ‘ 0 ) ) |
410 |
|
0cnd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑘 = 0 ) → 0 ∈ ℂ ) |
411 |
|
nn0cn |
⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ ) |
412 |
411
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℂ ) |
413 |
|
neqne |
⊢ ( ¬ 𝑘 = 0 → 𝑘 ≠ 0 ) |
414 |
|
reccl |
⊢ ( ( 𝑘 ∈ ℂ ∧ 𝑘 ≠ 0 ) → ( 1 / 𝑘 ) ∈ ℂ ) |
415 |
412 413 414
|
syl2an |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) ∧ ¬ 𝑘 = 0 ) → ( 1 / 𝑘 ) ∈ ℂ ) |
416 |
410 415
|
ifclda |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) → if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) ∈ ℂ ) |
417 |
|
expcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑘 ) ∈ ℂ ) |
418 |
417
|
adantlr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑘 ) ∈ ℂ ) |
419 |
416 418
|
mulcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) → ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ∈ ℂ ) |
420 |
419
|
fmpttd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) : ℕ0 ⟶ ℂ ) |
421 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
422 |
|
ffvelrn |
⊢ ( ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) : ℕ0 ⟶ ℂ ∧ 1 ∈ ℕ0 ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 1 ) ∈ ℂ ) |
423 |
420 421 422
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 1 ) ∈ ℂ ) |
424 |
|
elfz1eq |
⊢ ( 𝑛 ∈ ( 0 ... 0 ) → 𝑛 = 0 ) |
425 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
426 |
425
|
oveq2i |
⊢ ( 0 ... ( 1 − 1 ) ) = ( 0 ... 0 ) |
427 |
424 426
|
eleq2s |
⊢ ( 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) → 𝑛 = 0 ) |
428 |
427
|
fveq2d |
⊢ ( 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 0 ) ) |
429 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
430 |
|
iftrue |
⊢ ( 𝑘 = 0 → if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) = 0 ) |
431 |
|
oveq2 |
⊢ ( 𝑘 = 0 → ( 𝐴 ↑ 𝑘 ) = ( 𝐴 ↑ 0 ) ) |
432 |
430 431
|
oveq12d |
⊢ ( 𝑘 = 0 → ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) = ( 0 · ( 𝐴 ↑ 0 ) ) ) |
433 |
|
ovex |
⊢ ( 0 · ( 𝐴 ↑ 0 ) ) ∈ V |
434 |
432 8 433
|
fvmpt |
⊢ ( 0 ∈ ℕ0 → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 0 ) = ( 0 · ( 𝐴 ↑ 0 ) ) ) |
435 |
429 434
|
ax-mp |
⊢ ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 0 ) = ( 0 · ( 𝐴 ↑ 0 ) ) |
436 |
|
expcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℕ0 ) → ( 𝐴 ↑ 0 ) ∈ ℂ ) |
437 |
27 429 436
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ↑ 0 ) ∈ ℂ ) |
438 |
437
|
mul02d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 0 · ( 𝐴 ↑ 0 ) ) = 0 ) |
439 |
435 438
|
syl5eq |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 0 ) = 0 ) |
440 |
428 439
|
sylan9eqr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = 0 ) |
441 |
406 407 409 423 440
|
seqid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ) |
442 |
293
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ≠ 0 ) |
443 |
442
|
neneqd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ¬ 𝑛 = 0 ) |
444 |
443
|
iffalsed |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) = ( 1 / 𝑛 ) ) |
445 |
444
|
oveq1d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) = ( ( 1 / 𝑛 ) · ( 𝐴 ↑ 𝑛 ) ) ) |
446 |
284 23
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ↑ 𝑛 ) ∈ ℂ ) |
447 |
299
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℂ ) |
448 |
446 447 442
|
divrec2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) = ( ( 1 / 𝑛 ) · ( 𝐴 ↑ 𝑛 ) ) ) |
449 |
445 448
|
eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) = ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) |
450 |
284 11
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ) |
451 |
|
id |
⊢ ( 𝑘 = 𝑛 → 𝑘 = 𝑛 ) |
452 |
6 451
|
oveq12d |
⊢ ( 𝑘 = 𝑛 → ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) = ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) |
453 |
|
eqid |
⊢ ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) = ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) |
454 |
|
ovex |
⊢ ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ∈ V |
455 |
452 453 454
|
fvmpt |
⊢ ( 𝑛 ∈ ℕ → ( ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ‘ 𝑛 ) = ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) |
456 |
455
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ‘ 𝑛 ) = ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) |
457 |
449 450 456
|
3eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ‘ 𝑛 ) ) |
458 |
401 457
|
sylan2br |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ‘ 𝑛 ) ) |
459 |
400 458
|
seqfeq |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ) |
460 |
441 459
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ) |
461 |
460
|
fveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) ‘ 𝑛 ) = ( seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ‘ 𝑛 ) ) |
462 |
404 461
|
sylan9eqr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ‘ 𝑛 ) = ( seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ‘ 𝑛 ) ) |
463 |
310 397 399 400 462
|
climeq |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ⇝ - ( log ‘ ( 1 − 𝐴 ) ) ↔ seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ⇝ - ( log ‘ ( 1 − 𝐴 ) ) ) ) |
464 |
395 463
|
mpbid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ⇝ - ( log ‘ ( 1 − 𝐴 ) ) ) |