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Definition df-sqrt 13215
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 13341 for its closure, sqrtval 13217 for its value, sqrtth 13344 and sqsqrti 13355 for its relationship to squares, and sqrt11i 13364 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 13213 . 2  class  sqr
2 vx . . 3  setvar  x
3 cc 9519 . . 3  class  CC
4 vy . . . . . . . 8  setvar  y
54cv 1404 . . . . . . 7  class  y
6 c2 10625 . . . . . . 7  class  2
7 cexp 12208 . . . . . . 7  class  ^
85, 6, 7co 6277 . . . . . 6  class  ( y ^ 2 )
92cv 1404 . . . . . 6  class  x
108, 9wceq 1405 . . . . 5  wff  ( y ^ 2 )  =  x
11 cc0 9521 . . . . . 6  class  0
12 cre 13077 . . . . . . 7  class  Re
135, 12cfv 5568 . . . . . 6  class  ( Re
`  y )
14 cle 9658 . . . . . 6  class  <_
1511, 13, 14wbr 4394 . . . . 5  wff  0  <_  ( Re `  y
)
16 ci 9523 . . . . . . 7  class  _i
17 cmul 9526 . . . . . . 7  class  x.
1816, 5, 17co 6277 . . . . . 6  class  ( _i  x.  y )
19 crp 11264 . . . . . 6  class  RR+
2018, 19wnel 2599 . . . . 5  wff  ( _i  x.  y )  e/  RR+
2110, 15, 20w3a 974 . . . 4  wff  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
2221, 4, 3crio 6238 . . 3  class  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
)
232, 3, 22cmpt 4452 . 2  class  ( x  e.  CC  |->  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
) )
241, 23wceq 1405 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  13217  sqrtf  13343
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