Step |
Hyp |
Ref |
Expression |
1 |
|
eqsqrtd.1 |
|- ( ph -> A e. CC ) |
2 |
|
eqsqrtd.2 |
|- ( ph -> B e. CC ) |
3 |
|
eqsqrtd.3 |
|- ( ph -> ( A ^ 2 ) = B ) |
4 |
|
eqsqrt2d.4 |
|- ( ph -> 0 < ( Re ` A ) ) |
5 |
|
0re |
|- 0 e. RR |
6 |
1
|
recld |
|- ( ph -> ( Re ` A ) e. RR ) |
7 |
|
ltle |
|- ( ( 0 e. RR /\ ( Re ` A ) e. RR ) -> ( 0 < ( Re ` A ) -> 0 <_ ( Re ` A ) ) ) |
8 |
5 6 7
|
sylancr |
|- ( ph -> ( 0 < ( Re ` A ) -> 0 <_ ( Re ` A ) ) ) |
9 |
4 8
|
mpd |
|- ( ph -> 0 <_ ( Re ` A ) ) |
10 |
|
reim |
|- ( A e. CC -> ( Re ` A ) = ( Im ` ( _i x. A ) ) ) |
11 |
1 10
|
syl |
|- ( ph -> ( Re ` A ) = ( Im ` ( _i x. A ) ) ) |
12 |
4
|
gt0ne0d |
|- ( ph -> ( Re ` A ) =/= 0 ) |
13 |
11 12
|
eqnetrrd |
|- ( ph -> ( Im ` ( _i x. A ) ) =/= 0 ) |
14 |
|
rpre |
|- ( ( _i x. A ) e. RR+ -> ( _i x. A ) e. RR ) |
15 |
14
|
reim0d |
|- ( ( _i x. A ) e. RR+ -> ( Im ` ( _i x. A ) ) = 0 ) |
16 |
15
|
necon3ai |
|- ( ( Im ` ( _i x. A ) ) =/= 0 -> -. ( _i x. A ) e. RR+ ) |
17 |
13 16
|
syl |
|- ( ph -> -. ( _i x. A ) e. RR+ ) |
18 |
1 2 3 9 17
|
eqsqrtd |
|- ( ph -> A = ( sqrt ` B ) ) |