| Step |
Hyp |
Ref |
Expression |
| 1 |
|
atantayl2.1 |
⊢ 𝐹 = ( 𝑛 ∈ ℕ ↦ if ( 2 ∥ 𝑛 , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) ) |
| 2 |
|
ax-icn |
⊢ i ∈ ℂ |
| 3 |
2
|
negcli |
⊢ - i ∈ ℂ |
| 4 |
3
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → - i ∈ ℂ ) |
| 5 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
| 6 |
5
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → 𝑛 ∈ ℕ0 ) |
| 7 |
4 6
|
expcld |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( - i ↑ 𝑛 ) ∈ ℂ ) |
| 8 |
|
sqneg |
⊢ ( i ∈ ℂ → ( - i ↑ 2 ) = ( i ↑ 2 ) ) |
| 9 |
2 8
|
ax-mp |
⊢ ( - i ↑ 2 ) = ( i ↑ 2 ) |
| 10 |
9
|
oveq1i |
⊢ ( ( - i ↑ 2 ) ↑ ( 𝑛 / 2 ) ) = ( ( i ↑ 2 ) ↑ ( 𝑛 / 2 ) ) |
| 11 |
|
ine0 |
⊢ i ≠ 0 |
| 12 |
2 11
|
negne0i |
⊢ - i ≠ 0 |
| 13 |
12
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → - i ≠ 0 ) |
| 14 |
|
2z |
⊢ 2 ∈ ℤ |
| 15 |
14
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → 2 ∈ ℤ ) |
| 16 |
|
2ne0 |
⊢ 2 ≠ 0 |
| 17 |
|
nnz |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℤ ) |
| 18 |
17
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℤ ) |
| 19 |
|
dvdsval2 |
⊢ ( ( 2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑛 ∈ ℤ ) → ( 2 ∥ 𝑛 ↔ ( 𝑛 / 2 ) ∈ ℤ ) ) |
| 20 |
14 16 18 19
|
mp3an12i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( 2 ∥ 𝑛 ↔ ( 𝑛 / 2 ) ∈ ℤ ) ) |
| 21 |
20
|
biimpa |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( 𝑛 / 2 ) ∈ ℤ ) |
| 22 |
|
expmulz |
⊢ ( ( ( - i ∈ ℂ ∧ - i ≠ 0 ) ∧ ( 2 ∈ ℤ ∧ ( 𝑛 / 2 ) ∈ ℤ ) ) → ( - i ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( ( - i ↑ 2 ) ↑ ( 𝑛 / 2 ) ) ) |
| 23 |
4 13 15 21 22
|
syl22anc |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( - i ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( ( - i ↑ 2 ) ↑ ( 𝑛 / 2 ) ) ) |
| 24 |
2
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → i ∈ ℂ ) |
| 25 |
11
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → i ≠ 0 ) |
| 26 |
|
expmulz |
⊢ ( ( ( i ∈ ℂ ∧ i ≠ 0 ) ∧ ( 2 ∈ ℤ ∧ ( 𝑛 / 2 ) ∈ ℤ ) ) → ( i ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( ( i ↑ 2 ) ↑ ( 𝑛 / 2 ) ) ) |
| 27 |
24 25 15 21 26
|
syl22anc |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( i ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( ( i ↑ 2 ) ↑ ( 𝑛 / 2 ) ) ) |
| 28 |
10 23 27
|
3eqtr4a |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( - i ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( i ↑ ( 2 · ( 𝑛 / 2 ) ) ) ) |
| 29 |
|
nncn |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℂ ) |
| 30 |
29
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → 𝑛 ∈ ℂ ) |
| 31 |
|
2cnd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → 2 ∈ ℂ ) |
| 32 |
16
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → 2 ≠ 0 ) |
| 33 |
30 31 32
|
divcan2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( 2 · ( 𝑛 / 2 ) ) = 𝑛 ) |
| 34 |
33
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( - i ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( - i ↑ 𝑛 ) ) |
| 35 |
33
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( i ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( i ↑ 𝑛 ) ) |
| 36 |
28 34 35
|
3eqtr3d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( - i ↑ 𝑛 ) = ( i ↑ 𝑛 ) ) |
| 37 |
7 36
|
subeq0bd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) = 0 ) |
| 38 |
37
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) = ( i · 0 ) ) |
| 39 |
|
it0e0 |
⊢ ( i · 0 ) = 0 |
| 40 |
38 39
|
eqtrdi |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) = 0 ) |
| 41 |
40
|
oveq1d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) = ( 0 / 2 ) ) |
| 42 |
|
2cn |
⊢ 2 ∈ ℂ |
| 43 |
42 16
|
div0i |
⊢ ( 0 / 2 ) = 0 |
| 44 |
41 43
|
eqtrdi |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) = 0 ) |
| 45 |
44
|
oveq1d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) = ( 0 · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) |
| 46 |
|
simplll |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → 𝐴 ∈ ℂ ) |
| 47 |
46 6
|
expcld |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( 𝐴 ↑ 𝑛 ) ∈ ℂ ) |
| 48 |
|
nnne0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ≠ 0 ) |
| 49 |
48
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → 𝑛 ≠ 0 ) |
| 50 |
47 30 49
|
divcld |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ∈ ℂ ) |
| 51 |
50
|
mul02d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → ( 0 · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) = 0 ) |
| 52 |
45 51
|
eqtr2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ 2 ∥ 𝑛 ) → 0 = ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) |
| 53 |
|
2cnd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → 2 ∈ ℂ ) |
| 54 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 55 |
54
|
negcli |
⊢ - 1 ∈ ℂ |
| 56 |
55
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → - 1 ∈ ℂ ) |
| 57 |
|
neg1ne0 |
⊢ - 1 ≠ 0 |
| 58 |
57
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → - 1 ≠ 0 ) |
| 59 |
29
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → 𝑛 ∈ ℂ ) |
| 60 |
|
peano2cn |
⊢ ( 𝑛 ∈ ℂ → ( 𝑛 + 1 ) ∈ ℂ ) |
| 61 |
59 60
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( 𝑛 + 1 ) ∈ ℂ ) |
| 62 |
16
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → 2 ≠ 0 ) |
| 63 |
61 53 53 62
|
divsubdird |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( ( 𝑛 + 1 ) − 2 ) / 2 ) = ( ( ( 𝑛 + 1 ) / 2 ) − ( 2 / 2 ) ) ) |
| 64 |
|
2div2e1 |
⊢ ( 2 / 2 ) = 1 |
| 65 |
64
|
oveq2i |
⊢ ( ( ( 𝑛 + 1 ) / 2 ) − ( 2 / 2 ) ) = ( ( ( 𝑛 + 1 ) / 2 ) − 1 ) |
| 66 |
63 65
|
eqtrdi |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( ( 𝑛 + 1 ) − 2 ) / 2 ) = ( ( ( 𝑛 + 1 ) / 2 ) − 1 ) ) |
| 67 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
| 68 |
67
|
oveq2i |
⊢ ( ( 𝑛 + 1 ) − 2 ) = ( ( 𝑛 + 1 ) − ( 1 + 1 ) ) |
| 69 |
54
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → 1 ∈ ℂ ) |
| 70 |
59 69 69
|
pnpcan2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( 𝑛 + 1 ) − ( 1 + 1 ) ) = ( 𝑛 − 1 ) ) |
| 71 |
68 70
|
syl5eq |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( 𝑛 + 1 ) − 2 ) = ( 𝑛 − 1 ) ) |
| 72 |
71
|
oveq1d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( ( 𝑛 + 1 ) − 2 ) / 2 ) = ( ( 𝑛 − 1 ) / 2 ) ) |
| 73 |
66 72
|
eqtr3d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( ( 𝑛 + 1 ) / 2 ) − 1 ) = ( ( 𝑛 − 1 ) / 2 ) ) |
| 74 |
20
|
notbid |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ¬ 2 ∥ 𝑛 ↔ ¬ ( 𝑛 / 2 ) ∈ ℤ ) ) |
| 75 |
|
zeo |
⊢ ( 𝑛 ∈ ℤ → ( ( 𝑛 / 2 ) ∈ ℤ ∨ ( ( 𝑛 + 1 ) / 2 ) ∈ ℤ ) ) |
| 76 |
18 75
|
syl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑛 / 2 ) ∈ ℤ ∨ ( ( 𝑛 + 1 ) / 2 ) ∈ ℤ ) ) |
| 77 |
76
|
ord |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ¬ ( 𝑛 / 2 ) ∈ ℤ → ( ( 𝑛 + 1 ) / 2 ) ∈ ℤ ) ) |
| 78 |
74 77
|
sylbid |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ¬ 2 ∥ 𝑛 → ( ( 𝑛 + 1 ) / 2 ) ∈ ℤ ) ) |
| 79 |
78
|
imp |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( 𝑛 + 1 ) / 2 ) ∈ ℤ ) |
| 80 |
|
peano2zm |
⊢ ( ( ( 𝑛 + 1 ) / 2 ) ∈ ℤ → ( ( ( 𝑛 + 1 ) / 2 ) − 1 ) ∈ ℤ ) |
| 81 |
79 80
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( ( 𝑛 + 1 ) / 2 ) − 1 ) ∈ ℤ ) |
| 82 |
73 81
|
eqeltrrd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( 𝑛 − 1 ) / 2 ) ∈ ℤ ) |
| 83 |
56 58 82
|
expclzd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) ∈ ℂ ) |
| 84 |
83
|
2timesd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( 2 · ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) ) = ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) + ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) ) ) |
| 85 |
|
subcl |
⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 𝑛 − 1 ) ∈ ℂ ) |
| 86 |
59 54 85
|
sylancl |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( 𝑛 − 1 ) ∈ ℂ ) |
| 87 |
86 53 62
|
divcan2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( 2 · ( ( 𝑛 − 1 ) / 2 ) ) = ( 𝑛 − 1 ) ) |
| 88 |
87
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - i ↑ ( 2 · ( ( 𝑛 − 1 ) / 2 ) ) ) = ( - i ↑ ( 𝑛 − 1 ) ) ) |
| 89 |
3
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → - i ∈ ℂ ) |
| 90 |
12
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → - i ≠ 0 ) |
| 91 |
17
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → 𝑛 ∈ ℤ ) |
| 92 |
89 90 91
|
expm1d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - i ↑ ( 𝑛 − 1 ) ) = ( ( - i ↑ 𝑛 ) / - i ) ) |
| 93 |
88 92
|
eqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - i ↑ ( 2 · ( ( 𝑛 − 1 ) / 2 ) ) ) = ( ( - i ↑ 𝑛 ) / - i ) ) |
| 94 |
14
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → 2 ∈ ℤ ) |
| 95 |
|
expmulz |
⊢ ( ( ( - i ∈ ℂ ∧ - i ≠ 0 ) ∧ ( 2 ∈ ℤ ∧ ( ( 𝑛 − 1 ) / 2 ) ∈ ℤ ) ) → ( - i ↑ ( 2 · ( ( 𝑛 − 1 ) / 2 ) ) ) = ( ( - i ↑ 2 ) ↑ ( ( 𝑛 − 1 ) / 2 ) ) ) |
| 96 |
89 90 94 82 95
|
syl22anc |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - i ↑ ( 2 · ( ( 𝑛 − 1 ) / 2 ) ) ) = ( ( - i ↑ 2 ) ↑ ( ( 𝑛 − 1 ) / 2 ) ) ) |
| 97 |
5
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → 𝑛 ∈ ℕ0 ) |
| 98 |
|
expcl |
⊢ ( ( - i ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( - i ↑ 𝑛 ) ∈ ℂ ) |
| 99 |
3 97 98
|
sylancr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - i ↑ 𝑛 ) ∈ ℂ ) |
| 100 |
99 89 90
|
divrec2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( - i ↑ 𝑛 ) / - i ) = ( ( 1 / - i ) · ( - i ↑ 𝑛 ) ) ) |
| 101 |
93 96 100
|
3eqtr3d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( - i ↑ 2 ) ↑ ( ( 𝑛 − 1 ) / 2 ) ) = ( ( 1 / - i ) · ( - i ↑ 𝑛 ) ) ) |
| 102 |
|
i2 |
⊢ ( i ↑ 2 ) = - 1 |
| 103 |
9 102
|
eqtri |
⊢ ( - i ↑ 2 ) = - 1 |
| 104 |
103
|
oveq1i |
⊢ ( ( - i ↑ 2 ) ↑ ( ( 𝑛 − 1 ) / 2 ) ) = ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) |
| 105 |
|
irec |
⊢ ( 1 / i ) = - i |
| 106 |
105
|
negeqi |
⊢ - ( 1 / i ) = - - i |
| 107 |
|
divneg2 |
⊢ ( ( 1 ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0 ) → - ( 1 / i ) = ( 1 / - i ) ) |
| 108 |
54 2 11 107
|
mp3an |
⊢ - ( 1 / i ) = ( 1 / - i ) |
| 109 |
2
|
negnegi |
⊢ - - i = i |
| 110 |
106 108 109
|
3eqtr3i |
⊢ ( 1 / - i ) = i |
| 111 |
110
|
oveq1i |
⊢ ( ( 1 / - i ) · ( - i ↑ 𝑛 ) ) = ( i · ( - i ↑ 𝑛 ) ) |
| 112 |
101 104 111
|
3eqtr3g |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) = ( i · ( - i ↑ 𝑛 ) ) ) |
| 113 |
87
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( i ↑ ( 2 · ( ( 𝑛 − 1 ) / 2 ) ) ) = ( i ↑ ( 𝑛 − 1 ) ) ) |
| 114 |
2
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → i ∈ ℂ ) |
| 115 |
11
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → i ≠ 0 ) |
| 116 |
114 115 91
|
expm1d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( i ↑ ( 𝑛 − 1 ) ) = ( ( i ↑ 𝑛 ) / i ) ) |
| 117 |
113 116
|
eqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( i ↑ ( 2 · ( ( 𝑛 − 1 ) / 2 ) ) ) = ( ( i ↑ 𝑛 ) / i ) ) |
| 118 |
|
expmulz |
⊢ ( ( ( i ∈ ℂ ∧ i ≠ 0 ) ∧ ( 2 ∈ ℤ ∧ ( ( 𝑛 − 1 ) / 2 ) ∈ ℤ ) ) → ( i ↑ ( 2 · ( ( 𝑛 − 1 ) / 2 ) ) ) = ( ( i ↑ 2 ) ↑ ( ( 𝑛 − 1 ) / 2 ) ) ) |
| 119 |
114 115 94 82 118
|
syl22anc |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( i ↑ ( 2 · ( ( 𝑛 − 1 ) / 2 ) ) ) = ( ( i ↑ 2 ) ↑ ( ( 𝑛 − 1 ) / 2 ) ) ) |
| 120 |
|
expcl |
⊢ ( ( i ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( i ↑ 𝑛 ) ∈ ℂ ) |
| 121 |
2 97 120
|
sylancr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( i ↑ 𝑛 ) ∈ ℂ ) |
| 122 |
121 114 115
|
divrec2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( i ↑ 𝑛 ) / i ) = ( ( 1 / i ) · ( i ↑ 𝑛 ) ) ) |
| 123 |
117 119 122
|
3eqtr3d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( i ↑ 2 ) ↑ ( ( 𝑛 − 1 ) / 2 ) ) = ( ( 1 / i ) · ( i ↑ 𝑛 ) ) ) |
| 124 |
102
|
oveq1i |
⊢ ( ( i ↑ 2 ) ↑ ( ( 𝑛 − 1 ) / 2 ) ) = ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) |
| 125 |
105
|
oveq1i |
⊢ ( ( 1 / i ) · ( i ↑ 𝑛 ) ) = ( - i · ( i ↑ 𝑛 ) ) |
| 126 |
123 124 125
|
3eqtr3g |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) = ( - i · ( i ↑ 𝑛 ) ) ) |
| 127 |
|
mulneg1 |
⊢ ( ( i ∈ ℂ ∧ ( i ↑ 𝑛 ) ∈ ℂ ) → ( - i · ( i ↑ 𝑛 ) ) = - ( i · ( i ↑ 𝑛 ) ) ) |
| 128 |
2 121 127
|
sylancr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - i · ( i ↑ 𝑛 ) ) = - ( i · ( i ↑ 𝑛 ) ) ) |
| 129 |
126 128
|
eqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) = - ( i · ( i ↑ 𝑛 ) ) ) |
| 130 |
112 129
|
oveq12d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) + ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) ) = ( ( i · ( - i ↑ 𝑛 ) ) + - ( i · ( i ↑ 𝑛 ) ) ) ) |
| 131 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ ( - i ↑ 𝑛 ) ∈ ℂ ) → ( i · ( - i ↑ 𝑛 ) ) ∈ ℂ ) |
| 132 |
2 99 131
|
sylancr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( i · ( - i ↑ 𝑛 ) ) ∈ ℂ ) |
| 133 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ ( i ↑ 𝑛 ) ∈ ℂ ) → ( i · ( i ↑ 𝑛 ) ) ∈ ℂ ) |
| 134 |
2 121 133
|
sylancr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( i · ( i ↑ 𝑛 ) ) ∈ ℂ ) |
| 135 |
132 134
|
negsubd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( i · ( - i ↑ 𝑛 ) ) + - ( i · ( i ↑ 𝑛 ) ) ) = ( ( i · ( - i ↑ 𝑛 ) ) − ( i · ( i ↑ 𝑛 ) ) ) ) |
| 136 |
114 99 121
|
subdid |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) = ( ( i · ( - i ↑ 𝑛 ) ) − ( i · ( i ↑ 𝑛 ) ) ) ) |
| 137 |
135 136
|
eqtr4d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( i · ( - i ↑ 𝑛 ) ) + - ( i · ( i ↑ 𝑛 ) ) ) = ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) ) |
| 138 |
84 130 137
|
3eqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( 2 · ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) ) = ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) ) |
| 139 |
53 83 62 138
|
mvllmuld |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) = ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) ) |
| 140 |
139
|
oveq1d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ 2 ∥ 𝑛 ) → ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) = ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) |
| 141 |
52 140
|
ifeqda |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → if ( 2 ∥ 𝑛 , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) = ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) |
| 142 |
141
|
mpteq2dva |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝑛 ∈ ℕ ↦ if ( 2 ∥ 𝑛 , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) ) = ( 𝑛 ∈ ℕ ↦ ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) ) |
| 143 |
1 142
|
syl5eq |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 𝐹 = ( 𝑛 ∈ ℕ ↦ ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) ) |
| 144 |
143
|
seqeq3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , 𝐹 ) = seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) ) ) |
| 145 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) |
| 146 |
145
|
atantayl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( ( ( i · ( ( - i ↑ 𝑛 ) − ( i ↑ 𝑛 ) ) ) / 2 ) · ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ) ) ) ⇝ ( arctan ‘ 𝐴 ) ) |
| 147 |
144 146
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , 𝐹 ) ⇝ ( arctan ‘ 𝐴 ) ) |