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Definition df-sqrt 13298
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 13424 for its closure, sqrtval 13300 for its value, sqrtth 13427 and sqsqrti 13438 for its relationship to squares, and sqrt11i 13447 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 13296 . 2  class  sqr
2 vx . . 3  setvar  x
3 cc 9537 . . 3  class  CC
4 vy . . . . . . . 8  setvar  y
54cv 1443 . . . . . . 7  class  y
6 c2 10659 . . . . . . 7  class  2
7 cexp 12272 . . . . . . 7  class  ^
85, 6, 7co 6290 . . . . . 6  class  ( y ^ 2 )
92cv 1443 . . . . . 6  class  x
108, 9wceq 1444 . . . . 5  wff  ( y ^ 2 )  =  x
11 cc0 9539 . . . . . 6  class  0
12 cre 13160 . . . . . . 7  class  Re
135, 12cfv 5582 . . . . . 6  class  ( Re
`  y )
14 cle 9676 . . . . . 6  class  <_
1511, 13, 14wbr 4402 . . . . 5  wff  0  <_  ( Re `  y
)
16 ci 9541 . . . . . . 7  class  _i
17 cmul 9544 . . . . . . 7  class  x.
1816, 5, 17co 6290 . . . . . 6  class  ( _i  x.  y )
19 crp 11302 . . . . . 6  class  RR+
2018, 19wnel 2623 . . . . 5  wff  ( _i  x.  y )  e/  RR+
2110, 15, 20w3a 985 . . . 4  wff  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
2221, 4, 3crio 6251 . . 3  class  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
)
232, 3, 22cmpt 4461 . 2  class  ( x  e.  CC  |->  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
) )
241, 23wceq 1444 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  13300  sqrtf  13426
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