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Definition df-sqrt 14585
Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 28237).

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 28237. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 14712 for its closure, sqrtval 14587 for its value, sqrtth 14715 and sqsqrti 14726 for its relationship to squares, and sqrt11i 14735 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 14583 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 10524 . . 3 class
4 vy . . . . . . . 8 setvar 𝑦
54cv 1537 . . . . . . 7 class 𝑦
6 c2 11680 . . . . . . 7 class 2
7 cexp 13425 . . . . . . 7 class
85, 6, 7co 7140 . . . . . 6 class (𝑦↑2)
92cv 1537 . . . . . 6 class 𝑥
108, 9wceq 1538 . . . . 5 wff (𝑦↑2) = 𝑥
11 cc0 10526 . . . . . 6 class 0
12 cre 14447 . . . . . . 7 class
135, 12cfv 6334 . . . . . 6 class (ℜ‘𝑦)
14 cle 10665 . . . . . 6 class
1511, 13, 14wbr 5042 . . . . 5 wff 0 ≤ (ℜ‘𝑦)
16 ci 10528 . . . . . . 7 class i
17 cmul 10531 . . . . . . 7 class ·
1816, 5, 17co 7140 . . . . . 6 class (i · 𝑦)
19 crp 12377 . . . . . 6 class +
2018, 19wnel 3115 . . . . 5 wff (i · 𝑦) ∉ ℝ+
2110, 15, 20w3a 1084 . . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
2221, 4, 3crio 7097 . . 3 class (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
232, 3, 22cmpt 5122 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
241, 23wceq 1538 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  14587  sqrtf  14714  cphsscph  23853
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