| Step |
Hyp |
Ref |
Expression |
| 1 |
|
recn |
|- ( A e. RR -> A e. CC ) |
| 2 |
|
efival |
|- ( A e. CC -> ( exp ` ( _i x. A ) ) = ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) |
| 3 |
1 2
|
syl |
|- ( A e. RR -> ( exp ` ( _i x. A ) ) = ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) |
| 4 |
3
|
fveq2d |
|- ( A e. RR -> ( abs ` ( exp ` ( _i x. A ) ) ) = ( abs ` ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) ) |
| 5 |
|
recoscl |
|- ( A e. RR -> ( cos ` A ) e. RR ) |
| 6 |
|
resincl |
|- ( A e. RR -> ( sin ` A ) e. RR ) |
| 7 |
|
absreim |
|- ( ( ( cos ` A ) e. RR /\ ( sin ` A ) e. RR ) -> ( abs ` ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( sqrt ` ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) ) ) |
| 8 |
5 6 7
|
syl2anc |
|- ( A e. RR -> ( abs ` ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( sqrt ` ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) ) ) |
| 9 |
5
|
resqcld |
|- ( A e. RR -> ( ( cos ` A ) ^ 2 ) e. RR ) |
| 10 |
9
|
recnd |
|- ( A e. RR -> ( ( cos ` A ) ^ 2 ) e. CC ) |
| 11 |
6
|
resqcld |
|- ( A e. RR -> ( ( sin ` A ) ^ 2 ) e. RR ) |
| 12 |
11
|
recnd |
|- ( A e. RR -> ( ( sin ` A ) ^ 2 ) e. CC ) |
| 13 |
10 12
|
addcomd |
|- ( A e. RR -> ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) = ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) ) |
| 14 |
|
sincossq |
|- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 ) |
| 15 |
1 14
|
syl |
|- ( A e. RR -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 ) |
| 16 |
13 15
|
eqtrd |
|- ( A e. RR -> ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) = 1 ) |
| 17 |
16
|
fveq2d |
|- ( A e. RR -> ( sqrt ` ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) ) = ( sqrt ` 1 ) ) |
| 18 |
|
sqrt1 |
|- ( sqrt ` 1 ) = 1 |
| 19 |
17 18
|
eqtrdi |
|- ( A e. RR -> ( sqrt ` ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) ) = 1 ) |
| 20 |
8 19
|
eqtrd |
|- ( A e. RR -> ( abs ` ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = 1 ) |
| 21 |
4 20
|
eqtrd |
|- ( A e. RR -> ( abs ` ( exp ` ( _i x. A ) ) ) = 1 ) |