Definition df-sqrt 13049

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 13175 for its closure, sqrtval 13051 for its value, sqrtth 13178 and sqsqrti 13189 for its relationship to squares, and sqrt11i 13198 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 13047	. 2 class sqr
2		vx	. . 3 setvar x
3		cc 9493	. . 3 class CC
4		vy	. . . . . . . 8 setvar y
5	4	cv 1382	. . . . . . 7 class y
6		c2 10592	. . . . . . 7 class 2
7		cexp 12147	. . . . . . 7 class ^
8	5, 6, 7	co 6281	. . . . . 6 class ( y ^ 2 )
9	2	cv 1382	. . . . . 6 class x
10	8, 9	wceq 1383	. . . . 5 wff ( y ^ 2 ) = x
11		cc0 9495	. . . . . 6 class 0
12		cre 12911	. . . . . . 7 class Re
13	5, 12	cfv 5578	. . . . . 6 class ( Re ` y )
14		cle 9632	. . . . . 6 class <_
15	11, 13, 14	wbr 4437	. . . . 5 wff 0 <_ ( Re ` y )
16		ci 9497	. . . . . . 7 class _i
17		cmul 9500	. . . . . . 7 class x.
18	16, 5, 17	co 6281	. . . . . 6 class ( _i x. y )
19		crp 11230	. . . . . 6 class RR+
20	18, 19	wnel 2639	. . . . 5 wff ( _i x. y ) e/ RR+
21	10, 15, 20	w3a 974	. . . 4 wff ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ )
22	21, 4, 3	crio 6241	. . 3 class ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) )
23	2, 3, 22	cmpt 4495	. 2 class ( x e. CC |-> ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) ) )
24	1, 23	wceq 1383	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  13051  sqrtf  13177