Definition df-sqrt 13292

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 13418 for its closure, sqrtval 13294 for its value, sqrtth 13421 and sqsqrti 13432 for its relationship to squares, and sqrt11i 13441 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 13290	. 2 class sqr
2		vx	. . 3 setvar x
3		cc 9539	. . 3 class CC
4		vy	. . . . . . . 8 setvar y
5	4	cv 1437	. . . . . . 7 class y
6		c2 10661	. . . . . . 7 class 2
7		cexp 12273	. . . . . . 7 class ^
8	5, 6, 7	co 6303	. . . . . 6 class ( y ^ 2 )
9	2	cv 1437	. . . . . 6 class x
10	8, 9	wceq 1438	. . . . 5 wff ( y ^ 2 ) = x
11		cc0 9541	. . . . . 6 class 0
12		cre 13154	. . . . . . 7 class Re
13	5, 12	cfv 5599	. . . . . 6 class ( Re ` y )
14		cle 9678	. . . . . 6 class <_
15	11, 13, 14	wbr 4421	. . . . 5 wff 0 <_ ( Re ` y )
16		ci 9543	. . . . . . 7 class _i
17		cmul 9546	. . . . . . 7 class x.
18	16, 5, 17	co 6303	. . . . . 6 class ( _i x. y )
19		crp 11304	. . . . . 6 class RR+
20	18, 19	wnel 2620	. . . . 5 wff ( _i x. y ) e/ RR+
21	10, 15, 20	w3a 983	. . . 4 wff ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ )
22	21, 4, 3	crio 6264	. . 3 class ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) )
23	2, 3, 22	cmpt 4480	. 2 class ( x e. CC |-> ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) ) )
24	1, 23	wceq 1438	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  13294  sqrtf  13420