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Definition df-sqrt 13292
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 13418 for its closure, sqrtval 13294 for its value, sqrtth 13421 and sqsqrti 13432 for its relationship to squares, and sqrt11i 13441 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 13290 . 2  class  sqr
2 vx . . 3  setvar  x
3 cc 9539 . . 3  class  CC
4 vy . . . . . . . 8  setvar  y
54cv 1437 . . . . . . 7  class  y
6 c2 10661 . . . . . . 7  class  2
7 cexp 12273 . . . . . . 7  class  ^
85, 6, 7co 6303 . . . . . 6  class  ( y ^ 2 )
92cv 1437 . . . . . 6  class  x
108, 9wceq 1438 . . . . 5  wff  ( y ^ 2 )  =  x
11 cc0 9541 . . . . . 6  class  0
12 cre 13154 . . . . . . 7  class  Re
135, 12cfv 5599 . . . . . 6  class  ( Re
`  y )
14 cle 9678 . . . . . 6  class  <_
1511, 13, 14wbr 4421 . . . . 5  wff  0  <_  ( Re `  y
)
16 ci 9543 . . . . . . 7  class  _i
17 cmul 9546 . . . . . . 7  class  x.
1816, 5, 17co 6303 . . . . . 6  class  ( _i  x.  y )
19 crp 11304 . . . . . 6  class  RR+
2018, 19wnel 2620 . . . . 5  wff  ( _i  x.  y )  e/  RR+
2110, 15, 20w3a 983 . . . 4  wff  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
2221, 4, 3crio 6264 . . 3  class  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
)
232, 3, 22cmpt 4480 . 2  class  ( x  e.  CC  |->  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
) )
241, 23wceq 1438 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  13294  sqrtf  13420
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