Definition df-sqrt 13298

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 13424 for its closure, sqrtval 13300 for its value, sqrtth 13427 and sqsqrti 13438 for its relationship to squares, and sqrt11i 13447 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 13296	. 2 class sqr
2		vx	. . 3 setvar x
3		cc 9537	. . 3 class CC
4		vy	. . . . . . . 8 setvar y
5	4	cv 1443	. . . . . . 7 class y
6		c2 10659	. . . . . . 7 class 2
7		cexp 12272	. . . . . . 7 class ^
8	5, 6, 7	co 6290	. . . . . 6 class ( y ^ 2 )
9	2	cv 1443	. . . . . 6 class x
10	8, 9	wceq 1444	. . . . 5 wff ( y ^ 2 ) = x
11		cc0 9539	. . . . . 6 class 0
12		cre 13160	. . . . . . 7 class Re
13	5, 12	cfv 5582	. . . . . 6 class ( Re ` y )
14		cle 9676	. . . . . 6 class <_
15	11, 13, 14	wbr 4402	. . . . 5 wff 0 <_ ( Re ` y )
16		ci 9541	. . . . . . 7 class _i
17		cmul 9544	. . . . . . 7 class x.
18	16, 5, 17	co 6290	. . . . . 6 class ( _i x. y )
19		crp 11302	. . . . . 6 class RR+
20	18, 19	wnel 2623	. . . . 5 wff ( _i x. y ) e/ RR+
21	10, 15, 20	w3a 985	. . . 4 wff ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ )
22	21, 4, 3	crio 6251	. . 3 class ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) )
23	2, 3, 22	cmpt 4461	. 2 class ( x e. CC |-> ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) ) )
24	1, 23	wceq 1444	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  13300  sqrtf  13426