| Step |
Hyp |
Ref |
Expression |
| 1 |
|
atancl |
⊢ ( 𝐴 ∈ dom arctan → ( arctan ‘ 𝐴 ) ∈ ℂ ) |
| 2 |
|
cosval |
⊢ ( ( arctan ‘ 𝐴 ) ∈ ℂ → ( cos ‘ ( arctan ‘ 𝐴 ) ) = ( ( ( exp ‘ ( i · ( arctan ‘ 𝐴 ) ) ) + ( exp ‘ ( - i · ( arctan ‘ 𝐴 ) ) ) ) / 2 ) ) |
| 3 |
1 2
|
syl |
⊢ ( 𝐴 ∈ dom arctan → ( cos ‘ ( arctan ‘ 𝐴 ) ) = ( ( ( exp ‘ ( i · ( arctan ‘ 𝐴 ) ) ) + ( exp ‘ ( - i · ( arctan ‘ 𝐴 ) ) ) ) / 2 ) ) |
| 4 |
|
efiatan2 |
⊢ ( 𝐴 ∈ dom arctan → ( exp ‘ ( i · ( arctan ‘ 𝐴 ) ) ) = ( ( 1 + ( i · 𝐴 ) ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) ) |
| 5 |
|
ax-icn |
⊢ i ∈ ℂ |
| 6 |
|
mulneg12 |
⊢ ( ( i ∈ ℂ ∧ ( arctan ‘ 𝐴 ) ∈ ℂ ) → ( - i · ( arctan ‘ 𝐴 ) ) = ( i · - ( arctan ‘ 𝐴 ) ) ) |
| 7 |
5 1 6
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( - i · ( arctan ‘ 𝐴 ) ) = ( i · - ( arctan ‘ 𝐴 ) ) ) |
| 8 |
|
atanneg |
⊢ ( 𝐴 ∈ dom arctan → ( arctan ‘ - 𝐴 ) = - ( arctan ‘ 𝐴 ) ) |
| 9 |
8
|
oveq2d |
⊢ ( 𝐴 ∈ dom arctan → ( i · ( arctan ‘ - 𝐴 ) ) = ( i · - ( arctan ‘ 𝐴 ) ) ) |
| 10 |
7 9
|
eqtr4d |
⊢ ( 𝐴 ∈ dom arctan → ( - i · ( arctan ‘ 𝐴 ) ) = ( i · ( arctan ‘ - 𝐴 ) ) ) |
| 11 |
10
|
fveq2d |
⊢ ( 𝐴 ∈ dom arctan → ( exp ‘ ( - i · ( arctan ‘ 𝐴 ) ) ) = ( exp ‘ ( i · ( arctan ‘ - 𝐴 ) ) ) ) |
| 12 |
|
atandmneg |
⊢ ( 𝐴 ∈ dom arctan → - 𝐴 ∈ dom arctan ) |
| 13 |
|
efiatan2 |
⊢ ( - 𝐴 ∈ dom arctan → ( exp ‘ ( i · ( arctan ‘ - 𝐴 ) ) ) = ( ( 1 + ( i · - 𝐴 ) ) / ( √ ‘ ( 1 + ( - 𝐴 ↑ 2 ) ) ) ) ) |
| 14 |
12 13
|
syl |
⊢ ( 𝐴 ∈ dom arctan → ( exp ‘ ( i · ( arctan ‘ - 𝐴 ) ) ) = ( ( 1 + ( i · - 𝐴 ) ) / ( √ ‘ ( 1 + ( - 𝐴 ↑ 2 ) ) ) ) ) |
| 15 |
|
atandm4 |
⊢ ( 𝐴 ∈ dom arctan ↔ ( 𝐴 ∈ ℂ ∧ ( 1 + ( 𝐴 ↑ 2 ) ) ≠ 0 ) ) |
| 16 |
15
|
simplbi |
⊢ ( 𝐴 ∈ dom arctan → 𝐴 ∈ ℂ ) |
| 17 |
|
mulneg2 |
⊢ ( ( i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( i · - 𝐴 ) = - ( i · 𝐴 ) ) |
| 18 |
5 16 17
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( i · - 𝐴 ) = - ( i · 𝐴 ) ) |
| 19 |
18
|
oveq2d |
⊢ ( 𝐴 ∈ dom arctan → ( 1 + ( i · - 𝐴 ) ) = ( 1 + - ( i · 𝐴 ) ) ) |
| 20 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 21 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( i · 𝐴 ) ∈ ℂ ) |
| 22 |
5 16 21
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( i · 𝐴 ) ∈ ℂ ) |
| 23 |
|
negsub |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( 1 + - ( i · 𝐴 ) ) = ( 1 − ( i · 𝐴 ) ) ) |
| 24 |
20 22 23
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( 1 + - ( i · 𝐴 ) ) = ( 1 − ( i · 𝐴 ) ) ) |
| 25 |
19 24
|
eqtrd |
⊢ ( 𝐴 ∈ dom arctan → ( 1 + ( i · - 𝐴 ) ) = ( 1 − ( i · 𝐴 ) ) ) |
| 26 |
|
sqneg |
⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 ↑ 2 ) = ( 𝐴 ↑ 2 ) ) |
| 27 |
16 26
|
syl |
⊢ ( 𝐴 ∈ dom arctan → ( - 𝐴 ↑ 2 ) = ( 𝐴 ↑ 2 ) ) |
| 28 |
27
|
oveq2d |
⊢ ( 𝐴 ∈ dom arctan → ( 1 + ( - 𝐴 ↑ 2 ) ) = ( 1 + ( 𝐴 ↑ 2 ) ) ) |
| 29 |
28
|
fveq2d |
⊢ ( 𝐴 ∈ dom arctan → ( √ ‘ ( 1 + ( - 𝐴 ↑ 2 ) ) ) = ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) |
| 30 |
25 29
|
oveq12d |
⊢ ( 𝐴 ∈ dom arctan → ( ( 1 + ( i · - 𝐴 ) ) / ( √ ‘ ( 1 + ( - 𝐴 ↑ 2 ) ) ) ) = ( ( 1 − ( i · 𝐴 ) ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) ) |
| 31 |
11 14 30
|
3eqtrd |
⊢ ( 𝐴 ∈ dom arctan → ( exp ‘ ( - i · ( arctan ‘ 𝐴 ) ) ) = ( ( 1 − ( i · 𝐴 ) ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) ) |
| 32 |
4 31
|
oveq12d |
⊢ ( 𝐴 ∈ dom arctan → ( ( exp ‘ ( i · ( arctan ‘ 𝐴 ) ) ) + ( exp ‘ ( - i · ( arctan ‘ 𝐴 ) ) ) ) = ( ( ( 1 + ( i · 𝐴 ) ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) + ( ( 1 − ( i · 𝐴 ) ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) ) ) |
| 33 |
|
addcl |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( 1 + ( i · 𝐴 ) ) ∈ ℂ ) |
| 34 |
20 22 33
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( 1 + ( i · 𝐴 ) ) ∈ ℂ ) |
| 35 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( 1 − ( i · 𝐴 ) ) ∈ ℂ ) |
| 36 |
20 22 35
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( 1 − ( i · 𝐴 ) ) ∈ ℂ ) |
| 37 |
16
|
sqcld |
⊢ ( 𝐴 ∈ dom arctan → ( 𝐴 ↑ 2 ) ∈ ℂ ) |
| 38 |
|
addcl |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝐴 ↑ 2 ) ∈ ℂ ) → ( 1 + ( 𝐴 ↑ 2 ) ) ∈ ℂ ) |
| 39 |
20 37 38
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( 1 + ( 𝐴 ↑ 2 ) ) ∈ ℂ ) |
| 40 |
39
|
sqrtcld |
⊢ ( 𝐴 ∈ dom arctan → ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ∈ ℂ ) |
| 41 |
39
|
sqsqrtd |
⊢ ( 𝐴 ∈ dom arctan → ( ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ↑ 2 ) = ( 1 + ( 𝐴 ↑ 2 ) ) ) |
| 42 |
15
|
simprbi |
⊢ ( 𝐴 ∈ dom arctan → ( 1 + ( 𝐴 ↑ 2 ) ) ≠ 0 ) |
| 43 |
41 42
|
eqnetrd |
⊢ ( 𝐴 ∈ dom arctan → ( ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ↑ 2 ) ≠ 0 ) |
| 44 |
|
sqne0 |
⊢ ( ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ∈ ℂ → ( ( ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ↑ 2 ) ≠ 0 ↔ ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ≠ 0 ) ) |
| 45 |
40 44
|
syl |
⊢ ( 𝐴 ∈ dom arctan → ( ( ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ↑ 2 ) ≠ 0 ↔ ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ≠ 0 ) ) |
| 46 |
43 45
|
mpbid |
⊢ ( 𝐴 ∈ dom arctan → ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ≠ 0 ) |
| 47 |
34 36 40 46
|
divdird |
⊢ ( 𝐴 ∈ dom arctan → ( ( ( 1 + ( i · 𝐴 ) ) + ( 1 − ( i · 𝐴 ) ) ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) = ( ( ( 1 + ( i · 𝐴 ) ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) + ( ( 1 − ( i · 𝐴 ) ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) ) ) |
| 48 |
20
|
a1i |
⊢ ( 𝐴 ∈ dom arctan → 1 ∈ ℂ ) |
| 49 |
48 22 48
|
ppncand |
⊢ ( 𝐴 ∈ dom arctan → ( ( 1 + ( i · 𝐴 ) ) + ( 1 − ( i · 𝐴 ) ) ) = ( 1 + 1 ) ) |
| 50 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
| 51 |
49 50
|
eqtr4di |
⊢ ( 𝐴 ∈ dom arctan → ( ( 1 + ( i · 𝐴 ) ) + ( 1 − ( i · 𝐴 ) ) ) = 2 ) |
| 52 |
51
|
oveq1d |
⊢ ( 𝐴 ∈ dom arctan → ( ( ( 1 + ( i · 𝐴 ) ) + ( 1 − ( i · 𝐴 ) ) ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) = ( 2 / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) ) |
| 53 |
32 47 52
|
3eqtr2d |
⊢ ( 𝐴 ∈ dom arctan → ( ( exp ‘ ( i · ( arctan ‘ 𝐴 ) ) ) + ( exp ‘ ( - i · ( arctan ‘ 𝐴 ) ) ) ) = ( 2 / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) ) |
| 54 |
53
|
oveq1d |
⊢ ( 𝐴 ∈ dom arctan → ( ( ( exp ‘ ( i · ( arctan ‘ 𝐴 ) ) ) + ( exp ‘ ( - i · ( arctan ‘ 𝐴 ) ) ) ) / 2 ) = ( ( 2 / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) / 2 ) ) |
| 55 |
|
2cnd |
⊢ ( 𝐴 ∈ dom arctan → 2 ∈ ℂ ) |
| 56 |
|
2ne0 |
⊢ 2 ≠ 0 |
| 57 |
56
|
a1i |
⊢ ( 𝐴 ∈ dom arctan → 2 ≠ 0 ) |
| 58 |
55 40 55 46 57
|
divdiv32d |
⊢ ( 𝐴 ∈ dom arctan → ( ( 2 / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) / 2 ) = ( ( 2 / 2 ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) ) |
| 59 |
|
2div2e1 |
⊢ ( 2 / 2 ) = 1 |
| 60 |
59
|
oveq1i |
⊢ ( ( 2 / 2 ) / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) = ( 1 / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) |
| 61 |
58 60
|
eqtrdi |
⊢ ( 𝐴 ∈ dom arctan → ( ( 2 / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) / 2 ) = ( 1 / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) ) |
| 62 |
3 54 61
|
3eqtrd |
⊢ ( 𝐴 ∈ dom arctan → ( cos ‘ ( arctan ‘ 𝐴 ) ) = ( 1 / ( √ ‘ ( 1 + ( 𝐴 ↑ 2 ) ) ) ) ) |