Step |
Hyp |
Ref |
Expression |
1 |
|
atandm3 |
|- ( A e. dom arctan <-> ( A e. CC /\ ( A ^ 2 ) =/= -u 1 ) ) |
2 |
1
|
simplbi |
|- ( A e. dom arctan -> A e. CC ) |
3 |
2
|
cjcld |
|- ( A e. dom arctan -> ( * ` A ) e. CC ) |
4 |
|
2nn0 |
|- 2 e. NN0 |
5 |
|
cjexp |
|- ( ( A e. CC /\ 2 e. NN0 ) -> ( * ` ( A ^ 2 ) ) = ( ( * ` A ) ^ 2 ) ) |
6 |
2 4 5
|
sylancl |
|- ( A e. dom arctan -> ( * ` ( A ^ 2 ) ) = ( ( * ` A ) ^ 2 ) ) |
7 |
2
|
sqcld |
|- ( A e. dom arctan -> ( A ^ 2 ) e. CC ) |
8 |
7
|
cjcjd |
|- ( A e. dom arctan -> ( * ` ( * ` ( A ^ 2 ) ) ) = ( A ^ 2 ) ) |
9 |
1
|
simprbi |
|- ( A e. dom arctan -> ( A ^ 2 ) =/= -u 1 ) |
10 |
8 9
|
eqnetrd |
|- ( A e. dom arctan -> ( * ` ( * ` ( A ^ 2 ) ) ) =/= -u 1 ) |
11 |
|
fveq2 |
|- ( ( * ` ( A ^ 2 ) ) = -u 1 -> ( * ` ( * ` ( A ^ 2 ) ) ) = ( * ` -u 1 ) ) |
12 |
|
neg1rr |
|- -u 1 e. RR |
13 |
|
cjre |
|- ( -u 1 e. RR -> ( * ` -u 1 ) = -u 1 ) |
14 |
12 13
|
ax-mp |
|- ( * ` -u 1 ) = -u 1 |
15 |
11 14
|
eqtrdi |
|- ( ( * ` ( A ^ 2 ) ) = -u 1 -> ( * ` ( * ` ( A ^ 2 ) ) ) = -u 1 ) |
16 |
15
|
necon3i |
|- ( ( * ` ( * ` ( A ^ 2 ) ) ) =/= -u 1 -> ( * ` ( A ^ 2 ) ) =/= -u 1 ) |
17 |
10 16
|
syl |
|- ( A e. dom arctan -> ( * ` ( A ^ 2 ) ) =/= -u 1 ) |
18 |
6 17
|
eqnetrrd |
|- ( A e. dom arctan -> ( ( * ` A ) ^ 2 ) =/= -u 1 ) |
19 |
|
atandm3 |
|- ( ( * ` A ) e. dom arctan <-> ( ( * ` A ) e. CC /\ ( ( * ` A ) ^ 2 ) =/= -u 1 ) ) |
20 |
3 18 19
|
sylanbrc |
|- ( A e. dom arctan -> ( * ` A ) e. dom arctan ) |