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Definition df-sqr 11720
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 sqrcl 11845 for its closure, sqrval 11722 for its value, sqrth 11848 and sqsqri 11859 for its relationship to squares, and sqr11i 11868 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.)

Assertion
Ref Expression
df-sqr  |-  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-sqr
StepHypRef Expression
1 csqr 11718 . 2  class  sqr
2 vx . . 3  set  x
3 cc 8735 . . 3  class  CC
4 vy . . . . . . . 8  set  y
54cv 1622 . . . . . . 7  class  y
6 c2 9795 . . . . . . 7  class  2
7 cexp 11104 . . . . . . 7  class  ^
85, 6, 7co 5858 . . . . . 6  class  ( y ^ 2 )
92cv 1622 . . . . . 6  class  x
108, 9wceq 1623 . . . . 5  wff  ( y ^ 2 )  =  x
11 cc0 8737 . . . . . 6  class  0
12 cre 11582 . . . . . . 7  class  Re
135, 12cfv 5255 . . . . . 6  class  ( Re
`  y )
14 cle 8868 . . . . . 6  class  <_
1511, 13, 14wbr 4023 . . . . 5  wff  0  <_  ( Re `  y
)
16 ci 8739 . . . . . . 7  class  _i
17 cmul 8742 . . . . . . 7  class  x.
1816, 5, 17co 5858 . . . . . 6  class  ( _i  x.  y )
19 crp 10354 . . . . . 6  class  RR+
2018, 19wnel 2447 . . . . 5  wff  ( _i  x.  y )  e/  RR+
2110, 15, 20w3a 934 . . . 4  wff  ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
2221, 4, 3crio 6297 . . 3  class  ( iota_ y  e.  CC ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
)
232, 3, 22cmpt 4077 . 2  class  ( x  e.  CC  |->  ( iota_ y  e.  CC ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y
)  /\  ( _i  x.  y )  e/  RR+ )
) )
241, 23wceq 1623 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 set class
This definition is referenced by:  sqrval  11722  sqrf  11847
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