Metamath Proof Explorer


Theorem atandmcj

Description: The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015)

Ref Expression
Assertion atandmcj
|- ( A e. dom arctan -> ( * ` A ) e. dom arctan )

Proof

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 )