| Step |
Hyp |
Ref |
Expression |
| 1 |
|
atantayl.1 |
⊢ 𝐹 = ( 𝑛 ∈ ℕ ↦ ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) |
| 2 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
| 3 |
|
1zzd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 1 ∈ ℤ ) |
| 4 |
|
ax-icn |
⊢ i ∈ ℂ |
| 5 |
|
halfcl |
⊢ ( i ∈ ℂ → ( i / 2 ) ∈ ℂ ) |
| 6 |
4 5
|
mp1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( i / 2 ) ∈ ℂ ) |
| 7 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 𝐴 ∈ ℂ ) |
| 8 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( i · 𝐴 ) ∈ ℂ ) |
| 9 |
4 7 8
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( i · 𝐴 ) ∈ ℂ ) |
| 10 |
9
|
negcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → - ( i · 𝐴 ) ∈ ℂ ) |
| 11 |
9
|
absnegd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ - ( i · 𝐴 ) ) = ( abs ‘ ( i · 𝐴 ) ) ) |
| 12 |
|
absmul |
⊢ ( ( i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( abs ‘ ( i · 𝐴 ) ) = ( ( abs ‘ i ) · ( abs ‘ 𝐴 ) ) ) |
| 13 |
4 7 12
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ ( i · 𝐴 ) ) = ( ( abs ‘ i ) · ( abs ‘ 𝐴 ) ) ) |
| 14 |
|
absi |
⊢ ( abs ‘ i ) = 1 |
| 15 |
14
|
oveq1i |
⊢ ( ( abs ‘ i ) · ( abs ‘ 𝐴 ) ) = ( 1 · ( abs ‘ 𝐴 ) ) |
| 16 |
|
abscl |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℝ ) |
| 17 |
16
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ 𝐴 ) ∈ ℝ ) |
| 18 |
17
|
recnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ 𝐴 ) ∈ ℂ ) |
| 19 |
18
|
mullidd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 1 · ( abs ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) ) |
| 20 |
15 19
|
eqtrid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( ( abs ‘ i ) · ( abs ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) ) |
| 21 |
11 13 20
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ - ( i · 𝐴 ) ) = ( abs ‘ 𝐴 ) ) |
| 22 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ 𝐴 ) < 1 ) |
| 23 |
21 22
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ - ( i · 𝐴 ) ) < 1 ) |
| 24 |
|
logtayl |
⊢ ( ( - ( i · 𝐴 ) ∈ ℂ ∧ ( abs ‘ - ( i · 𝐴 ) ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ⇝ - ( log ‘ ( 1 − - ( i · 𝐴 ) ) ) ) |
| 25 |
10 23 24
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ⇝ - ( log ‘ ( 1 − - ( i · 𝐴 ) ) ) ) |
| 26 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 27 |
|
subneg |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( 1 − - ( i · 𝐴 ) ) = ( 1 + ( i · 𝐴 ) ) ) |
| 28 |
26 9 27
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 1 − - ( i · 𝐴 ) ) = ( 1 + ( i · 𝐴 ) ) ) |
| 29 |
28
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( log ‘ ( 1 − - ( i · 𝐴 ) ) ) = ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) |
| 30 |
29
|
negeqd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → - ( log ‘ ( 1 − - ( i · 𝐴 ) ) ) = - ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) |
| 31 |
25 30
|
breqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ⇝ - ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) |
| 32 |
|
seqex |
⊢ seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ) ∈ V |
| 33 |
32
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ) ∈ V ) |
| 34 |
11 23
|
eqbrtrrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ ( i · 𝐴 ) ) < 1 ) |
| 35 |
|
logtayl |
⊢ ( ( ( i · 𝐴 ) ∈ ℂ ∧ ( abs ‘ ( i · 𝐴 ) ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ⇝ - ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) |
| 36 |
9 34 35
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ⇝ - ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) |
| 37 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( - ( i · 𝐴 ) ↑ 𝑛 ) = ( - ( i · 𝐴 ) ↑ 𝑚 ) ) |
| 38 |
|
id |
⊢ ( 𝑛 = 𝑚 → 𝑛 = 𝑚 ) |
| 39 |
37 38
|
oveq12d |
⊢ ( 𝑛 = 𝑚 → ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) = ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) |
| 40 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) |
| 41 |
|
ovex |
⊢ ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ∈ V |
| 42 |
39 40 41
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) = ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) |
| 43 |
42
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) = ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) |
| 44 |
|
nnnn0 |
⊢ ( 𝑚 ∈ ℕ → 𝑚 ∈ ℕ0 ) |
| 45 |
|
expcl |
⊢ ( ( - ( i · 𝐴 ) ∈ ℂ ∧ 𝑚 ∈ ℕ0 ) → ( - ( i · 𝐴 ) ↑ 𝑚 ) ∈ ℂ ) |
| 46 |
10 44 45
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( - ( i · 𝐴 ) ↑ 𝑚 ) ∈ ℂ ) |
| 47 |
|
nncn |
⊢ ( 𝑚 ∈ ℕ → 𝑚 ∈ ℂ ) |
| 48 |
47
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℂ ) |
| 49 |
|
nnne0 |
⊢ ( 𝑚 ∈ ℕ → 𝑚 ≠ 0 ) |
| 50 |
49
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ≠ 0 ) |
| 51 |
46 48 50
|
divcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ∈ ℂ ) |
| 52 |
43 51
|
eqeltrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) ∈ ℂ ) |
| 53 |
2 3 52
|
serf |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) : ℕ ⟶ ℂ ) |
| 54 |
53
|
ffvelcdmda |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ ) → ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑘 ) ∈ ℂ ) |
| 55 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( ( i · 𝐴 ) ↑ 𝑛 ) = ( ( i · 𝐴 ) ↑ 𝑚 ) ) |
| 56 |
55 38
|
oveq12d |
⊢ ( 𝑛 = 𝑚 → ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) = ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) |
| 57 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) |
| 58 |
|
ovex |
⊢ ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ∈ V |
| 59 |
56 57 58
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) = ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) |
| 60 |
59
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) = ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) |
| 61 |
|
expcl |
⊢ ( ( ( i · 𝐴 ) ∈ ℂ ∧ 𝑚 ∈ ℕ0 ) → ( ( i · 𝐴 ) ↑ 𝑚 ) ∈ ℂ ) |
| 62 |
9 44 61
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( i · 𝐴 ) ↑ 𝑚 ) ∈ ℂ ) |
| 63 |
62 48 50
|
divcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ∈ ℂ ) |
| 64 |
60 63
|
eqeltrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) ∈ ℂ ) |
| 65 |
2 3 64
|
serf |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) : ℕ ⟶ ℂ ) |
| 66 |
65
|
ffvelcdmda |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ ) → ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑘 ) ∈ ℂ ) |
| 67 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℕ ) |
| 68 |
67 2
|
eleqtrdi |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) |
| 69 |
|
simpl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ) |
| 70 |
|
elfznn |
⊢ ( 𝑚 ∈ ( 1 ... 𝑘 ) → 𝑚 ∈ ℕ ) |
| 71 |
69 70 52
|
syl2an |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ ) ∧ 𝑚 ∈ ( 1 ... 𝑘 ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) ∈ ℂ ) |
| 72 |
69 70 64
|
syl2an |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ ) ∧ 𝑚 ∈ ( 1 ... 𝑘 ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) ∈ ℂ ) |
| 73 |
39 56
|
oveq12d |
⊢ ( 𝑛 = 𝑚 → ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) = ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ) |
| 74 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) |
| 75 |
|
ovex |
⊢ ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ∈ V |
| 76 |
73 74 75
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑚 ) = ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ) |
| 77 |
76
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑚 ) = ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ) |
| 78 |
43 60
|
oveq12d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) − ( ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) ) = ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ) |
| 79 |
77 78
|
eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑚 ) = ( ( ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) − ( ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) ) ) |
| 80 |
69 70 79
|
syl2an |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ ) ∧ 𝑚 ∈ ( 1 ... 𝑘 ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑚 ) = ( ( ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) − ( ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ‘ 𝑚 ) ) ) |
| 81 |
68 71 72 80
|
sersub |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ ) → ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ) ‘ 𝑘 ) = ( ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑘 ) − ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑘 ) ) ) |
| 82 |
2 3 31 33 36 54 66 81
|
climsub |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ) ⇝ ( - ( log ‘ ( 1 + ( i · 𝐴 ) ) ) − - ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) |
| 83 |
|
addcl |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( 1 + ( i · 𝐴 ) ) ∈ ℂ ) |
| 84 |
26 9 83
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 1 + ( i · 𝐴 ) ) ∈ ℂ ) |
| 85 |
|
bndatandm |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 𝐴 ∈ dom arctan ) |
| 86 |
|
atandm2 |
⊢ ( 𝐴 ∈ dom arctan ↔ ( 𝐴 ∈ ℂ ∧ ( 1 − ( i · 𝐴 ) ) ≠ 0 ∧ ( 1 + ( i · 𝐴 ) ) ≠ 0 ) ) |
| 87 |
85 86
|
sylib |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ∈ ℂ ∧ ( 1 − ( i · 𝐴 ) ) ≠ 0 ∧ ( 1 + ( i · 𝐴 ) ) ≠ 0 ) ) |
| 88 |
87
|
simp3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 1 + ( i · 𝐴 ) ) ≠ 0 ) |
| 89 |
84 88
|
logcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ∈ ℂ ) |
| 90 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( 1 − ( i · 𝐴 ) ) ∈ ℂ ) |
| 91 |
26 9 90
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 1 − ( i · 𝐴 ) ) ∈ ℂ ) |
| 92 |
87
|
simp2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 1 − ( i · 𝐴 ) ) ≠ 0 ) |
| 93 |
91 92
|
logcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ∈ ℂ ) |
| 94 |
89 93
|
neg2subd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( - ( log ‘ ( 1 + ( i · 𝐴 ) ) ) − - ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) = ( ( log ‘ ( 1 − ( i · 𝐴 ) ) ) − ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ) |
| 95 |
82 94
|
breqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ) ⇝ ( ( log ‘ ( 1 − ( i · 𝐴 ) ) ) − ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ) |
| 96 |
51 63
|
subcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ∈ ℂ ) |
| 97 |
77 96
|
eqeltrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑚 ) ∈ ℂ ) |
| 98 |
4
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → i ∈ ℂ ) |
| 99 |
|
negicn |
⊢ - i ∈ ℂ |
| 100 |
44
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℕ0 ) |
| 101 |
|
expcl |
⊢ ( ( - i ∈ ℂ ∧ 𝑚 ∈ ℕ0 ) → ( - i ↑ 𝑚 ) ∈ ℂ ) |
| 102 |
99 100 101
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( - i ↑ 𝑚 ) ∈ ℂ ) |
| 103 |
|
expcl |
⊢ ( ( i ∈ ℂ ∧ 𝑚 ∈ ℕ0 ) → ( i ↑ 𝑚 ) ∈ ℂ ) |
| 104 |
4 100 103
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( i ↑ 𝑚 ) ∈ ℂ ) |
| 105 |
102 104
|
subcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ∈ ℂ ) |
| 106 |
|
2cnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → 2 ∈ ℂ ) |
| 107 |
|
2ne0 |
⊢ 2 ≠ 0 |
| 108 |
107
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → 2 ≠ 0 ) |
| 109 |
98 105 106 108
|
div23d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( i · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) / 2 ) = ( ( i / 2 ) · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) ) |
| 110 |
109
|
oveq1d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( i · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) = ( ( ( i / 2 ) · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) ) |
| 111 |
6
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( i / 2 ) ∈ ℂ ) |
| 112 |
|
expcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑚 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑚 ) ∈ ℂ ) |
| 113 |
7 44 112
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( 𝐴 ↑ 𝑚 ) ∈ ℂ ) |
| 114 |
113 48 50
|
divcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ∈ ℂ ) |
| 115 |
111 105 114
|
mulassd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( i / 2 ) · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) = ( ( i / 2 ) · ( ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) ) ) |
| 116 |
102 104 113
|
subdird |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) · ( 𝐴 ↑ 𝑚 ) ) = ( ( ( - i ↑ 𝑚 ) · ( 𝐴 ↑ 𝑚 ) ) − ( ( i ↑ 𝑚 ) · ( 𝐴 ↑ 𝑚 ) ) ) ) |
| 117 |
7
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → 𝐴 ∈ ℂ ) |
| 118 |
|
mulneg1 |
⊢ ( ( i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( - i · 𝐴 ) = - ( i · 𝐴 ) ) |
| 119 |
4 117 118
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( - i · 𝐴 ) = - ( i · 𝐴 ) ) |
| 120 |
119
|
oveq1d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( - i · 𝐴 ) ↑ 𝑚 ) = ( - ( i · 𝐴 ) ↑ 𝑚 ) ) |
| 121 |
99
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → - i ∈ ℂ ) |
| 122 |
121 117 100
|
mulexpd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( - i · 𝐴 ) ↑ 𝑚 ) = ( ( - i ↑ 𝑚 ) · ( 𝐴 ↑ 𝑚 ) ) ) |
| 123 |
120 122
|
eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( - ( i · 𝐴 ) ↑ 𝑚 ) = ( ( - i ↑ 𝑚 ) · ( 𝐴 ↑ 𝑚 ) ) ) |
| 124 |
98 117 100
|
mulexpd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( i · 𝐴 ) ↑ 𝑚 ) = ( ( i ↑ 𝑚 ) · ( 𝐴 ↑ 𝑚 ) ) ) |
| 125 |
123 124
|
oveq12d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( - ( i · 𝐴 ) ↑ 𝑚 ) − ( ( i · 𝐴 ) ↑ 𝑚 ) ) = ( ( ( - i ↑ 𝑚 ) · ( 𝐴 ↑ 𝑚 ) ) − ( ( i ↑ 𝑚 ) · ( 𝐴 ↑ 𝑚 ) ) ) ) |
| 126 |
116 125
|
eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) · ( 𝐴 ↑ 𝑚 ) ) = ( ( - ( i · 𝐴 ) ↑ 𝑚 ) − ( ( i · 𝐴 ) ↑ 𝑚 ) ) ) |
| 127 |
126
|
oveq1d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) · ( 𝐴 ↑ 𝑚 ) ) / 𝑚 ) = ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) − ( ( i · 𝐴 ) ↑ 𝑚 ) ) / 𝑚 ) ) |
| 128 |
105 113 48 50
|
divassd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) · ( 𝐴 ↑ 𝑚 ) ) / 𝑚 ) = ( ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) ) |
| 129 |
46 62 48 50
|
divsubdird |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) − ( ( i · 𝐴 ) ↑ 𝑚 ) ) / 𝑚 ) = ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ) |
| 130 |
127 128 129
|
3eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) = ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ) |
| 131 |
130
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( i / 2 ) · ( ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) ) = ( ( i / 2 ) · ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ) ) |
| 132 |
110 115 131
|
3eqtrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( i · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) = ( ( i / 2 ) · ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ) ) |
| 133 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( - i ↑ 𝑛 ) = ( - i ↑ 𝑚 ) ) |
| 134 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( i ↑ 𝑛 ) = ( i ↑ 𝑚 ) ) |
| 135 |
133 134
|
oveq12d |
⊢ ( 𝑛 = 𝑚 → ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) = ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) |
| 136 |
135
|
oveq2d |
⊢ ( 𝑛 = 𝑚 → ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) = ( i · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) ) |
| 137 |
136
|
oveq1d |
⊢ ( 𝑛 = 𝑚 → ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) = ( ( i · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) / 2 ) ) |
| 138 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝐴 ↑ 𝑛 ) = ( 𝐴 ↑ 𝑚 ) ) |
| 139 |
138 38
|
oveq12d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) = ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) |
| 140 |
137 139
|
oveq12d |
⊢ ( 𝑛 = 𝑚 → ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) = ( ( ( i · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) ) |
| 141 |
|
ovex |
⊢ ( ( ( i · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) ∈ V |
| 142 |
140 1 141
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( 𝐹 ‘ 𝑚 ) = ( ( ( i · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) ) |
| 143 |
142
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( 𝐹 ‘ 𝑚 ) = ( ( ( i · ( ( - i ↑ 𝑚 ) − ( i ↑ 𝑚 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑚 ) / 𝑚 ) ) ) |
| 144 |
77
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( ( i / 2 ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑚 ) ) = ( ( i / 2 ) · ( ( ( - ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) − ( ( ( i · 𝐴 ) ↑ 𝑚 ) / 𝑚 ) ) ) ) |
| 145 |
132 143 144
|
3eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑚 ∈ ℕ ) → ( 𝐹 ‘ 𝑚 ) = ( ( i / 2 ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( - ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) − ( ( ( i · 𝐴 ) ↑ 𝑛 ) / 𝑛 ) ) ) ‘ 𝑚 ) ) ) |
| 146 |
2 3 6 95 97 145
|
isermulc2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , 𝐹 ) ⇝ ( ( i / 2 ) · ( ( log ‘ ( 1 − ( i · 𝐴 ) ) ) − ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ) ) |
| 147 |
|
atanval |
⊢ ( 𝐴 ∈ dom arctan → ( arctan ‘ 𝐴 ) = ( ( i / 2 ) · ( ( log ‘ ( 1 − ( i · 𝐴 ) ) ) − ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ) ) |
| 148 |
85 147
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( arctan ‘ 𝐴 ) = ( ( i / 2 ) · ( ( log ‘ ( 1 − ( i · 𝐴 ) ) ) − ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ) ) |
| 149 |
146 148
|
breqtrrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , 𝐹 ) ⇝ ( arctan ‘ 𝐴 ) ) |