Step |
Hyp |
Ref |
Expression |
1 |
|
ax-icn |
|- _i e. CC |
2 |
|
mulcl |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC ) |
3 |
1 2
|
mpan |
|- ( A e. CC -> ( _i x. A ) e. CC ) |
4 |
|
ax-1cn |
|- 1 e. CC |
5 |
|
sqcl |
|- ( A e. CC -> ( A ^ 2 ) e. CC ) |
6 |
|
subcl |
|- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 - ( A ^ 2 ) ) e. CC ) |
7 |
4 5 6
|
sylancr |
|- ( A e. CC -> ( 1 - ( A ^ 2 ) ) e. CC ) |
8 |
7
|
sqrtcld |
|- ( A e. CC -> ( sqrt ` ( 1 - ( A ^ 2 ) ) ) e. CC ) |
9 |
3 8
|
addcomd |
|- ( A e. CC -> ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) = ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) ) |
10 |
|
mulneg2 |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. -u A ) = -u ( _i x. A ) ) |
11 |
1 10
|
mpan |
|- ( A e. CC -> ( _i x. -u A ) = -u ( _i x. A ) ) |
12 |
|
sqneg |
|- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) ) |
13 |
12
|
oveq2d |
|- ( A e. CC -> ( 1 - ( -u A ^ 2 ) ) = ( 1 - ( A ^ 2 ) ) ) |
14 |
13
|
fveq2d |
|- ( A e. CC -> ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) = ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) |
15 |
11 14
|
oveq12d |
|- ( A e. CC -> ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) = ( -u ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) ) |
16 |
3
|
negcld |
|- ( A e. CC -> -u ( _i x. A ) e. CC ) |
17 |
16 8
|
addcomd |
|- ( A e. CC -> ( -u ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) = ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + -u ( _i x. A ) ) ) |
18 |
8 3
|
negsubd |
|- ( A e. CC -> ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + -u ( _i x. A ) ) = ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) |
19 |
15 17 18
|
3eqtrd |
|- ( A e. CC -> ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) = ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) |
20 |
9 19
|
oveq12d |
|- ( A e. CC -> ( ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) x. ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) ) = ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) ) |
21 |
7
|
sqsqrtd |
|- ( A e. CC -> ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( 1 - ( A ^ 2 ) ) ) |
22 |
|
sqmul |
|- ( ( _i e. CC /\ A e. CC ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) ) |
23 |
1 22
|
mpan |
|- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) ) |
24 |
|
i2 |
|- ( _i ^ 2 ) = -u 1 |
25 |
24
|
oveq1i |
|- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) ) |
26 |
5
|
mulm1d |
|- ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) ) |
27 |
25 26
|
syl5eq |
|- ( A e. CC -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = -u ( A ^ 2 ) ) |
28 |
23 27
|
eqtrd |
|- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) ) |
29 |
21 28
|
oveq12d |
|- ( A e. CC -> ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( 1 - ( A ^ 2 ) ) - -u ( A ^ 2 ) ) ) |
30 |
|
subsq |
|- ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) e. CC /\ ( _i x. A ) e. CC ) -> ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) ) |
31 |
8 3 30
|
syl2anc |
|- ( A e. CC -> ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) ) |
32 |
7 5
|
subnegd |
|- ( A e. CC -> ( ( 1 - ( A ^ 2 ) ) - -u ( A ^ 2 ) ) = ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) ) |
33 |
29 31 32
|
3eqtr3d |
|- ( A e. CC -> ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) = ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) ) |
34 |
|
npcan |
|- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) = 1 ) |
35 |
4 5 34
|
sylancr |
|- ( A e. CC -> ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) = 1 ) |
36 |
20 33 35
|
3eqtrd |
|- ( A e. CC -> ( ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) x. ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) ) = 1 ) |