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Definition df-sqrt 13375
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 13501 for its closure, sqrtval 13377 for its value, sqrtth 13504 and sqsqrti 13515 for its relationship to squares, and sqrt11i 13524 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 13373 . 2  class  sqr
2 vx . . 3  setvar  x
3 cc 9555 . . 3  class  CC
4 vy . . . . . . . 8  setvar  y
54cv 1451 . . . . . . 7  class  y
6 c2 10681 . . . . . . 7  class  2
7 cexp 12310 . . . . . . 7  class  ^
85, 6, 7co 6308 . . . . . 6  class  ( y ^ 2 )
92cv 1451 . . . . . 6  class  x
108, 9wceq 1452 . . . . 5  wff  ( y ^ 2 )  =  x
11 cc0 9557 . . . . . 6  class  0
12 cre 13237 . . . . . . 7  class  Re
135, 12cfv 5589 . . . . . 6  class  ( Re
`  y )
14 cle 9694 . . . . . 6  class  <_
1511, 13, 14wbr 4395 . . . . 5  wff  0  <_  ( Re `  y
)
16 ci 9559 . . . . . . 7  class  _i
17 cmul 9562 . . . . . . 7  class  x.
1816, 5, 17co 6308 . . . . . 6  class  ( _i  x.  y )
19 crp 11325 . . . . . 6  class  RR+
2018, 19wnel 2642 . . . . 5  wff  ( _i  x.  y )  e/  RR+
2110, 15, 20w3a 1007 . . . 4  wff  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
2221, 4, 3crio 6269 . . 3  class  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
)
232, 3, 22cmpt 4454 . 2  class  ( x  e.  CC  |->  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
) )
241, 23wceq 1452 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  13377  sqrtf  13503
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