Theorem efif1o

Description: The exponential function of an imaginary number maps any open-below, closed-above interval of length 2 _pi one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008) (Revised by Mario Carneiro, 13-May-2014)

	Ref	Expression
Hypotheses	efif1o.1	|- F = ( w e. D |-> ( exp ` ( _i x. w ) ) )
	efif1o.2	|- C = ( `' abs " { 1 } )
	efif1o.3	|- D = ( A (,] ( A + ( 2 x. _pi ) ) )
Assertion	efif1o	|- ( A e. RR -> F : D -1-1-onto-> C )

Proof

Step	Hyp	Ref	Expression
1		efif1o.1	|- F = ( w e. D |-> ( exp ` ( _i x. w ) ) )
2		efif1o.2	|- C = ( `' abs " { 1 } )
3		efif1o.3	|- D = ( A (,] ( A + ( 2 x. _pi ) ) )
4		rexr	|- ( A e. RR -> A e. RR* )
5		2re	|- 2 e. RR
6		pire	|- _pi e. RR
7	5 6	remulcli	|- ( 2 x. _pi ) e. RR
8		readdcl	|- ( ( A e. RR /\ ( 2 x. _pi ) e. RR ) -> ( A + ( 2 x. _pi ) ) e. RR )
9	7 8	mpan2	|- ( A e. RR -> ( A + ( 2 x. _pi ) ) e. RR )
10		elioc2	|- ( ( A e. RR* /\ ( A + ( 2 x. _pi ) ) e. RR ) -> ( x e. ( A (,] ( A + ( 2 x. _pi ) ) ) <-> ( x e. RR /\ A < x /\ x <_ ( A + ( 2 x. _pi ) ) ) ) )
11	4 9 10	syl2anc	|- ( A e. RR -> ( x e. ( A (,] ( A + ( 2 x. _pi ) ) ) <-> ( x e. RR /\ A < x /\ x <_ ( A + ( 2 x. _pi ) ) ) ) )
12		simp1	|- ( ( x e. RR /\ A < x /\ x <_ ( A + ( 2 x. _pi ) ) ) -> x e. RR )
13	11 12	biimtrdi	|- ( A e. RR -> ( x e. ( A (,] ( A + ( 2 x. _pi ) ) ) -> x e. RR ) )
14	13	ssrdv	|- ( A e. RR -> ( A (,] ( A + ( 2 x. _pi ) ) ) C_ RR )
15	3 14	eqsstrid	|- ( A e. RR -> D C_ RR )
16	3	efif1olem1	|- ( ( A e. RR /\ ( x e. D /\ y e. D ) ) -> ( abs ` ( x - y ) ) < ( 2 x. _pi ) )
17	3	efif1olem2	|- ( ( A e. RR /\ z e. RR ) -> E. y e. D ( ( z - y ) / ( 2 x. _pi ) ) e. ZZ )
18		eqid	|- ( sin |` ( -u ( _pi / 2 ) [,] ( _pi / 2 ) ) ) = ( sin |` ( -u ( _pi / 2 ) [,] ( _pi / 2 ) ) )
19	1 2 15 16 17 18	efif1olem4	|- ( A e. RR -> F : D -1-1-onto-> C )