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Definition df-sqrt 14458
Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 28014).

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 14585 for its closure, sqrtval 14460 for its value, sqrtth 14588 and sqsqrti 14599 for its relationship to squares, and sqrt11i 14608 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 14456 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 10335 . . 3 class
4 vy . . . . . . . 8 setvar 𝑦
54cv 1506 . . . . . . 7 class 𝑦
6 c2 11498 . . . . . . 7 class 2
7 cexp 13247 . . . . . . 7 class
85, 6, 7co 6978 . . . . . 6 class (𝑦↑2)
92cv 1506 . . . . . 6 class 𝑥
108, 9wceq 1507 . . . . 5 wff (𝑦↑2) = 𝑥
11 cc0 10337 . . . . . 6 class 0
12 cre 14320 . . . . . . 7 class
135, 12cfv 6190 . . . . . 6 class (ℜ‘𝑦)
14 cle 10477 . . . . . 6 class
1511, 13, 14wbr 4930 . . . . 5 wff 0 ≤ (ℜ‘𝑦)
16 ci 10339 . . . . . . 7 class i
17 cmul 10342 . . . . . . 7 class ·
1816, 5, 17co 6978 . . . . . 6 class (i · 𝑦)
19 crp 12207 . . . . . 6 class +
2018, 19wnel 3073 . . . . 5 wff (i · 𝑦) ∉ ℝ+
2110, 15, 20w3a 1068 . . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
2221, 4, 3crio 6938 . . 3 class (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
232, 3, 22cmpt 5009 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
241, 23wceq 1507 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  14460  sqrtf  14587  cphsscph  23560
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