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 ) |