Definition df-sqrt 13375

Description: Define a function whose value is the square root of a complex number. Since ( y ^ 2 ) = x iff ( -u y ^ 2 ) = x, we ensure uniqueness by restricting the range to numbers with positive real part, or numbers with 0 real part and nonnegative imaginary part. A description can be found under "Principal square root of a complex number" at http://en.wikipedia.org/wiki/Square_root. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 13501 for its closure, sqrtval 13377 for its value, sqrtth 13504 and sqsqrti 13515 for its relationship to squares, and sqrt11i 13524 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.)

Assertion

Ref	Expression
df-sqrt	|- sqr = ( x e. CC |-> ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) ) )

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-sqrt

Step	Hyp	Ref	Expression
1		csqrt 13373	. 2 class sqr
2		vx	. . 3 setvar x
3		cc 9555	. . 3 class CC
4		vy	. . . . . . . 8 setvar y
5	4	cv 1451	. . . . . . 7 class y
6		c2 10681	. . . . . . 7 class 2
7		cexp 12310	. . . . . . 7 class ^
8	5, 6, 7	co 6308	. . . . . 6 class ( y ^ 2 )
9	2	cv 1451	. . . . . 6 class x
10	8, 9	wceq 1452	. . . . 5 wff ( y ^ 2 ) = x
11		cc0 9557	. . . . . 6 class 0
12		cre 13237	. . . . . . 7 class Re
13	5, 12	cfv 5589	. . . . . 6 class ( Re ` y )
14		cle 9694	. . . . . 6 class <_
15	11, 13, 14	wbr 4395	. . . . 5 wff 0 <_ ( Re ` y )
16		ci 9559	. . . . . . 7 class _i
17		cmul 9562	. . . . . . 7 class x.
18	16, 5, 17	co 6308	. . . . . 6 class ( _i x. y )
19		crp 11325	. . . . . 6 class RR+
20	18, 19	wnel 2642	. . . . 5 wff ( _i x. y ) e/ RR+
21	10, 15, 20	w3a 1007	. . . 4 wff ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ )
22	21, 4, 3	crio 6269	. . 3 class ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) )
23	2, 3, 22	cmpt 4454	. 2 class ( x e. CC |-> ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) ) )
24	1, 23	wceq 1452	1 wff sqr = ( x e. CC |-> ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) ) )

This definition is referenced by:  sqrtval  13377  sqrtf  13503