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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
StepHypRef Expression
1 csqrt 13046 . 2  class  sqr
2 vx . . 3  setvar  x
3 cc 9502 . . 3  class  CC
4 vy . . . . . . . 8  setvar  y
54cv 1378 . . . . . . 7  class  y
6 c2 10597 . . . . . . 7  class  2
7 cexp 12146 . . . . . . 7  class  ^
85, 6, 7co 6295 . . . . . 6  class  ( y ^ 2 )
92cv 1378 . . . . . 6  class  x
108, 9wceq 1379 . . . . 5  wff  ( y ^ 2 )  =  x
11 cc0 9504 . . . . . 6  class  0
12 cre 12910 . . . . . . 7  class  Re
135, 12cfv 5594 . . . . . 6  class  ( Re
`  y )
14 cle 9641 . . . . . 6  class  <_
1511, 13, 14wbr 4453 . . . . 5  wff  0  <_  ( Re `  y
)
16 ci 9506 . . . . . . 7  class  _i
17 cmul 9509 . . . . . . 7  class  x.
1816, 5, 17co 6295 . . . . . 6  class  ( _i  x.  y )
19 crp 11232 . . . . . 6  class  RR+
2018, 19wnel 2663 . . . . 5  wff  ( _i  x.  y )  e/  RR+
2110, 15, 20w3a 973 . . . 4  wff  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
2221, 4, 3crio 6255 . . 3  class  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
)
232, 3, 22cmpt 4511 . 2  class  ( x  e.  CC  |->  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
) )
241, 23wceq 1379 1  wff  sqr  =  ( x  e.  CC  |->  ( iota_ y  e.  CC  ( ( y ^
2 )  =  x  /\  0  <_  (
Re `  y )  /\  ( _i  x.  y
)  e/  RR+ ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  sqrtval  13050  sqrtf  13176
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