Theorem atandm

Description: Since the property is a little lengthy, we abbreviate A e. CC /\ A =/= -ui /\ A =/= i as A e. dom arctan . This is the necessary precondition for the definition of arctan to make sense. (Contributed by Mario Carneiro, 31-Mar-2015)

	Ref	Expression
Assertion	atandm	|- ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )

Proof

Step	Hyp	Ref	Expression
1		eldif	|- ( A e. ( CC \ { -u _i , _i } ) <-> ( A e. CC /\ -. A e. { -u _i , _i } ) )
2		elprg	|- ( A e. CC -> ( A e. { -u _i , _i } <-> ( A = -u _i \/ A = _i ) ) )
3	2	notbid	|- ( A e. CC -> ( -. A e. { -u _i , _i } <-> -. ( A = -u _i \/ A = _i ) ) )
4		neanior	|- ( ( A =/= -u _i /\ A =/= _i ) <-> -. ( A = -u _i \/ A = _i ) )
5	3 4	bitr4di	|- ( A e. CC -> ( -. A e. { -u _i , _i } <-> ( A =/= -u _i /\ A =/= _i ) ) )
6	5	pm5.32i	|- ( ( A e. CC /\ -. A e. { -u _i , _i } ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
7	1 6	bitri	|- ( A e. ( CC \ { -u _i , _i } ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
8		ovex	|- ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) e. _V
9		df-atan	|- arctan = ( x e. ( CC \ { -u _i , _i } ) |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) )
10	8 9	dmmpti	|- dom arctan = ( CC \ { -u _i , _i } )
11	10	eleq2i	|- ( A e. dom arctan <-> A e. ( CC \ { -u _i , _i } ) )
12		3anass	|- ( ( A e. CC /\ A =/= -u _i /\ A =/= _i ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
13	7 11 12	3bitr4i	|- ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )