Step |
Hyp |
Ref |
Expression |
1 |
|
cosargd.1 |
⊢ ( 𝜑 → 𝑋 ∈ ℂ ) |
2 |
|
cosargd.2 |
⊢ ( 𝜑 → 𝑋 ≠ 0 ) |
3 |
1
|
cjcld |
⊢ ( 𝜑 → ( ∗ ‘ 𝑋 ) ∈ ℂ ) |
4 |
1 3
|
addcld |
⊢ ( 𝜑 → ( 𝑋 + ( ∗ ‘ 𝑋 ) ) ∈ ℂ ) |
5 |
1
|
abscld |
⊢ ( 𝜑 → ( abs ‘ 𝑋 ) ∈ ℝ ) |
6 |
5
|
recnd |
⊢ ( 𝜑 → ( abs ‘ 𝑋 ) ∈ ℂ ) |
7 |
|
2cnd |
⊢ ( 𝜑 → 2 ∈ ℂ ) |
8 |
1 2
|
absne0d |
⊢ ( 𝜑 → ( abs ‘ 𝑋 ) ≠ 0 ) |
9 |
|
2ne0 |
⊢ 2 ≠ 0 |
10 |
9
|
a1i |
⊢ ( 𝜑 → 2 ≠ 0 ) |
11 |
4 6 7 8 10
|
divdiv32d |
⊢ ( 𝜑 → ( ( ( 𝑋 + ( ∗ ‘ 𝑋 ) ) / ( abs ‘ 𝑋 ) ) / 2 ) = ( ( ( 𝑋 + ( ∗ ‘ 𝑋 ) ) / 2 ) / ( abs ‘ 𝑋 ) ) ) |
12 |
1 2
|
logcld |
⊢ ( 𝜑 → ( log ‘ 𝑋 ) ∈ ℂ ) |
13 |
12
|
imcld |
⊢ ( 𝜑 → ( ℑ ‘ ( log ‘ 𝑋 ) ) ∈ ℝ ) |
14 |
13
|
recnd |
⊢ ( 𝜑 → ( ℑ ‘ ( log ‘ 𝑋 ) ) ∈ ℂ ) |
15 |
|
cosval |
⊢ ( ( ℑ ‘ ( log ‘ 𝑋 ) ) ∈ ℂ → ( cos ‘ ( ℑ ‘ ( log ‘ 𝑋 ) ) ) = ( ( ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) + ( exp ‘ ( - i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) / 2 ) ) |
16 |
14 15
|
syl |
⊢ ( 𝜑 → ( cos ‘ ( ℑ ‘ ( log ‘ 𝑋 ) ) ) = ( ( ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) + ( exp ‘ ( - i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) / 2 ) ) |
17 |
|
efiarg |
⊢ ( ( 𝑋 ∈ ℂ ∧ 𝑋 ≠ 0 ) → ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) = ( 𝑋 / ( abs ‘ 𝑋 ) ) ) |
18 |
1 2 17
|
syl2anc |
⊢ ( 𝜑 → ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) = ( 𝑋 / ( abs ‘ 𝑋 ) ) ) |
19 |
|
ax-icn |
⊢ i ∈ ℂ |
20 |
19
|
a1i |
⊢ ( 𝜑 → i ∈ ℂ ) |
21 |
20 14
|
mulcld |
⊢ ( 𝜑 → ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ∈ ℂ ) |
22 |
|
efcj |
⊢ ( ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ∈ ℂ → ( exp ‘ ( ∗ ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) = ( ∗ ‘ ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) ) |
23 |
21 22
|
syl |
⊢ ( 𝜑 → ( exp ‘ ( ∗ ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) = ( ∗ ‘ ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) ) |
24 |
20 14
|
cjmuld |
⊢ ( 𝜑 → ( ∗ ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) = ( ( ∗ ‘ i ) · ( ∗ ‘ ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) |
25 |
|
cji |
⊢ ( ∗ ‘ i ) = - i |
26 |
25
|
a1i |
⊢ ( 𝜑 → ( ∗ ‘ i ) = - i ) |
27 |
13
|
cjred |
⊢ ( 𝜑 → ( ∗ ‘ ( ℑ ‘ ( log ‘ 𝑋 ) ) ) = ( ℑ ‘ ( log ‘ 𝑋 ) ) ) |
28 |
26 27
|
oveq12d |
⊢ ( 𝜑 → ( ( ∗ ‘ i ) · ( ∗ ‘ ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) = ( - i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) |
29 |
24 28
|
eqtrd |
⊢ ( 𝜑 → ( ∗ ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) = ( - i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) |
30 |
29
|
fveq2d |
⊢ ( 𝜑 → ( exp ‘ ( ∗ ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) = ( exp ‘ ( - i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) |
31 |
18
|
fveq2d |
⊢ ( 𝜑 → ( ∗ ‘ ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) = ( ∗ ‘ ( 𝑋 / ( abs ‘ 𝑋 ) ) ) ) |
32 |
23 30 31
|
3eqtr3d |
⊢ ( 𝜑 → ( exp ‘ ( - i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) = ( ∗ ‘ ( 𝑋 / ( abs ‘ 𝑋 ) ) ) ) |
33 |
1 6 8
|
cjdivd |
⊢ ( 𝜑 → ( ∗ ‘ ( 𝑋 / ( abs ‘ 𝑋 ) ) ) = ( ( ∗ ‘ 𝑋 ) / ( ∗ ‘ ( abs ‘ 𝑋 ) ) ) ) |
34 |
5
|
cjred |
⊢ ( 𝜑 → ( ∗ ‘ ( abs ‘ 𝑋 ) ) = ( abs ‘ 𝑋 ) ) |
35 |
34
|
oveq2d |
⊢ ( 𝜑 → ( ( ∗ ‘ 𝑋 ) / ( ∗ ‘ ( abs ‘ 𝑋 ) ) ) = ( ( ∗ ‘ 𝑋 ) / ( abs ‘ 𝑋 ) ) ) |
36 |
32 33 35
|
3eqtrd |
⊢ ( 𝜑 → ( exp ‘ ( - i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) = ( ( ∗ ‘ 𝑋 ) / ( abs ‘ 𝑋 ) ) ) |
37 |
18 36
|
oveq12d |
⊢ ( 𝜑 → ( ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) + ( exp ‘ ( - i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) = ( ( 𝑋 / ( abs ‘ 𝑋 ) ) + ( ( ∗ ‘ 𝑋 ) / ( abs ‘ 𝑋 ) ) ) ) |
38 |
1 3 6 8
|
divdird |
⊢ ( 𝜑 → ( ( 𝑋 + ( ∗ ‘ 𝑋 ) ) / ( abs ‘ 𝑋 ) ) = ( ( 𝑋 / ( abs ‘ 𝑋 ) ) + ( ( ∗ ‘ 𝑋 ) / ( abs ‘ 𝑋 ) ) ) ) |
39 |
37 38
|
eqtr4d |
⊢ ( 𝜑 → ( ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) + ( exp ‘ ( - i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) = ( ( 𝑋 + ( ∗ ‘ 𝑋 ) ) / ( abs ‘ 𝑋 ) ) ) |
40 |
39
|
oveq1d |
⊢ ( 𝜑 → ( ( ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) + ( exp ‘ ( - i · ( ℑ ‘ ( log ‘ 𝑋 ) ) ) ) ) / 2 ) = ( ( ( 𝑋 + ( ∗ ‘ 𝑋 ) ) / ( abs ‘ 𝑋 ) ) / 2 ) ) |
41 |
16 40
|
eqtrd |
⊢ ( 𝜑 → ( cos ‘ ( ℑ ‘ ( log ‘ 𝑋 ) ) ) = ( ( ( 𝑋 + ( ∗ ‘ 𝑋 ) ) / ( abs ‘ 𝑋 ) ) / 2 ) ) |
42 |
|
reval |
⊢ ( 𝑋 ∈ ℂ → ( ℜ ‘ 𝑋 ) = ( ( 𝑋 + ( ∗ ‘ 𝑋 ) ) / 2 ) ) |
43 |
1 42
|
syl |
⊢ ( 𝜑 → ( ℜ ‘ 𝑋 ) = ( ( 𝑋 + ( ∗ ‘ 𝑋 ) ) / 2 ) ) |
44 |
43
|
oveq1d |
⊢ ( 𝜑 → ( ( ℜ ‘ 𝑋 ) / ( abs ‘ 𝑋 ) ) = ( ( ( 𝑋 + ( ∗ ‘ 𝑋 ) ) / 2 ) / ( abs ‘ 𝑋 ) ) ) |
45 |
11 41 44
|
3eqtr4d |
⊢ ( 𝜑 → ( cos ‘ ( ℑ ‘ ( log ‘ 𝑋 ) ) ) = ( ( ℜ ‘ 𝑋 ) / ( abs ‘ 𝑋 ) ) ) |