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
|
subnegd |
|- ( A e. CC -> ( ( _i x. A ) - -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) = ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) ) |
10 |
8
|
negcld |
|- ( A e. CC -> -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) e. CC ) |
11 |
|
0ne1 |
|- 0 =/= 1 |
12 |
|
0cnd |
|- ( A e. CC -> 0 e. CC ) |
13 |
|
1cnd |
|- ( A e. CC -> 1 e. CC ) |
14 |
|
subcan2 |
|- ( ( 0 e. CC /\ 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( ( 0 - ( A ^ 2 ) ) = ( 1 - ( A ^ 2 ) ) <-> 0 = 1 ) ) |
15 |
14
|
necon3bid |
|- ( ( 0 e. CC /\ 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( ( 0 - ( A ^ 2 ) ) =/= ( 1 - ( A ^ 2 ) ) <-> 0 =/= 1 ) ) |
16 |
12 13 5 15
|
syl3anc |
|- ( A e. CC -> ( ( 0 - ( A ^ 2 ) ) =/= ( 1 - ( A ^ 2 ) ) <-> 0 =/= 1 ) ) |
17 |
11 16
|
mpbiri |
|- ( A e. CC -> ( 0 - ( A ^ 2 ) ) =/= ( 1 - ( A ^ 2 ) ) ) |
18 |
|
sqmul |
|- ( ( _i e. CC /\ A e. CC ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) ) |
19 |
1 18
|
mpan |
|- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) ) |
20 |
|
i2 |
|- ( _i ^ 2 ) = -u 1 |
21 |
20
|
oveq1i |
|- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) ) |
22 |
5
|
mulm1d |
|- ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) ) |
23 |
21 22
|
syl5eq |
|- ( A e. CC -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = -u ( A ^ 2 ) ) |
24 |
19 23
|
eqtrd |
|- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) ) |
25 |
|
df-neg |
|- -u ( A ^ 2 ) = ( 0 - ( A ^ 2 ) ) |
26 |
24 25
|
eqtrdi |
|- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( 0 - ( A ^ 2 ) ) ) |
27 |
|
sqneg |
|- ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) e. CC -> ( -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) ) |
28 |
8 27
|
syl |
|- ( A e. CC -> ( -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) ) |
29 |
7
|
sqsqrtd |
|- ( A e. CC -> ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( 1 - ( A ^ 2 ) ) ) |
30 |
28 29
|
eqtrd |
|- ( A e. CC -> ( -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( 1 - ( A ^ 2 ) ) ) |
31 |
17 26 30
|
3netr4d |
|- ( A e. CC -> ( ( _i x. A ) ^ 2 ) =/= ( -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) ) |
32 |
|
oveq1 |
|- ( ( _i x. A ) = -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) -> ( ( _i x. A ) ^ 2 ) = ( -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) ) |
33 |
32
|
necon3i |
|- ( ( ( _i x. A ) ^ 2 ) =/= ( -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) -> ( _i x. A ) =/= -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) |
34 |
31 33
|
syl |
|- ( A e. CC -> ( _i x. A ) =/= -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) |
35 |
3 10 34
|
subne0d |
|- ( A e. CC -> ( ( _i x. A ) - -u ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) =/= 0 ) |
36 |
9 35
|
eqnetrrd |
|- ( A e. CC -> ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) =/= 0 ) |