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Definition df-sqrt 13049
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 13175 for its closure, sqrtval 13051 for its value, sqrtth 13178 and sqsqrti 13189 for its relationship to squares, and sqrt11i 13198 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 13047 . 2  class  sqr
2 vx . . 3  setvar  x
3 cc 9493 . . 3  class  CC
4 vy . . . . . . . 8  setvar  y
54cv 1382 . . . . . . 7  class  y
6 c2 10592 . . . . . . 7  class  2
7 cexp 12147 . . . . . . 7  class  ^
85, 6, 7co 6281 . . . . . 6  class  ( y ^ 2 )
92cv 1382 . . . . . 6  class  x
108, 9wceq 1383 . . . . 5  wff  ( y ^ 2 )  =  x
11 cc0 9495 . . . . . 6  class  0
12 cre 12911 . . . . . . 7  class  Re
135, 12cfv 5578 . . . . . 6  class  ( Re
`  y )
14 cle 9632 . . . . . 6  class  <_
1511, 13, 14wbr 4437 . . . . 5  wff  0  <_  ( Re `  y
)
16 ci 9497 . . . . . . 7  class  _i
17 cmul 9500 . . . . . . 7  class  x.
1816, 5, 17co 6281 . . . . . 6  class  ( _i  x.  y )
19 crp 11230 . . . . . 6  class  RR+
2018, 19wnel 2639 . . . . 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 6241 . . 3  class  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
)
232, 3, 22cmpt 4495 . 2  class  ( x  e.  CC  |->  ( iota_ y  e.  CC  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
) )
241, 23wceq 1383 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  13051  sqrtf  13177
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