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Definition df-sqrt 13969
Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 27295).

Since (𝑦↑2) = 𝑥 iff (-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 14095 for its closure, sqrtval 13971 for its value, sqrtth 14098 and sqsqrti 14109 for its relationship to squares, and sqrt11i 14118 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.)

Assertion
Ref Expression
df-sqrt √ = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-sqrt
StepHypRef Expression
1 csqrt 13967 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 9931 . . 3 class
4 vy . . . . . . . 8 setvar 𝑦
54cv 1481 . . . . . . 7 class 𝑦
6 c2 11067 . . . . . . 7 class 2
7 cexp 12855 . . . . . . 7 class
85, 6, 7co 6647 . . . . . 6 class (𝑦↑2)
92cv 1481 . . . . . 6 class 𝑥
108, 9wceq 1482 . . . . 5 wff (𝑦↑2) = 𝑥
11 cc0 9933 . . . . . 6 class 0
12 cre 13831 . . . . . . 7 class
135, 12cfv 5886 . . . . . 6 class (ℜ‘𝑦)
14 cle 10072 . . . . . 6 class
1511, 13, 14wbr 4651 . . . . 5 wff 0 ≤ (ℜ‘𝑦)
16 ci 9935 . . . . . . 7 class i
17 cmul 9938 . . . . . . 7 class ·
1816, 5, 17co 6647 . . . . . 6 class (i · 𝑦)
19 crp 11829 . . . . . 6 class +
2018, 19wnel 2896 . . . . 5 wff (i · 𝑦) ∉ ℝ+
2110, 15, 20w3a 1037 . . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
2221, 4, 3crio 6607 . . 3 class (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
232, 3, 22cmpt 4727 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
241, 23wceq 1482 1 wff √ = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
Colors of variables: wff setvar class
This definition is referenced by:  sqrtval  13971  sqrtf  14097
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