Definition df-sqrt 13048

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 13174 for its closure, sqrtval 13050 for its value, sqrtth 13177 and sqsqrti 13188 for its relationship to squares, and sqrt11i 13197 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 13046	. 2 class sqr
2		vx	. . 3 setvar x
3		cc 9502	. . 3 class CC
4		vy	. . . . . . . 8 setvar y
5	4	cv 1378	. . . . . . 7 class y
6		c2 10597	. . . . . . 7 class 2
7		cexp 12146	. . . . . . 7 class ^
8	5, 6, 7	co 6295	. . . . . 6 class ( y ^ 2 )
9	2	cv 1378	. . . . . 6 class x
10	8, 9	wceq 1379	. . . . 5 wff ( y ^ 2 ) = x
11		cc0 9504	. . . . . 6 class 0
12		cre 12910	. . . . . . 7 class Re
13	5, 12	cfv 5594	. . . . . 6 class ( Re ` y )
14		cle 9641	. . . . . 6 class <_
15	11, 13, 14	wbr 4453	. . . . 5 wff 0 <_ ( Re ` y )
16		ci 9506	. . . . . . 7 class _i
17		cmul 9509	. . . . . . 7 class x.
18	16, 5, 17	co 6295	. . . . . 6 class ( _i x. y )
19		crp 11232	. . . . . 6 class RR+
20	18, 19	wnel 2663	. . . . 5 wff ( _i x. y ) e/ RR+
21	10, 15, 20	w3a 973	. . . 4 wff ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ )
22	21, 4, 3	crio 6255	. . 3 class ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) )
23	2, 3, 22	cmpt 4511	. 2 class ( x e. CC |-> ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) ) )
24	1, 23	wceq 1379	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  13050  sqrtf  13176