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Definition df-sqrt 13765
Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 26465).

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 13891 for its closure, sqrtval 13767 for its value, sqrtth 13894 and sqsqrti 13905 for its relationship to squares, and sqrt11i 13914 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 13763 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 9786 . . 3 class
4 vy . . . . . . . 8 setvar 𝑦
54cv 1473 . . . . . . 7 class 𝑦
6 c2 10913 . . . . . . 7 class 2
7 cexp 12673 . . . . . . 7 class
85, 6, 7co 6523 . . . . . 6 class (𝑦↑2)
92cv 1473 . . . . . 6 class 𝑥
108, 9wceq 1474 . . . . 5 wff (𝑦↑2) = 𝑥
11 cc0 9788 . . . . . 6 class 0
12 cre 13627 . . . . . . 7 class
135, 12cfv 5786 . . . . . 6 class (ℜ‘𝑦)
14 cle 9927 . . . . . 6 class
1511, 13, 14wbr 4573 . . . . 5 wff 0 ≤ (ℜ‘𝑦)
16 ci 9790 . . . . . . 7 class i
17 cmul 9793 . . . . . . 7 class ·
1816, 5, 17co 6523 . . . . . 6 class (i · 𝑦)
19 crp 11660 . . . . . 6 class +
2018, 19wnel 2776 . . . . 5 wff (i · 𝑦) ∉ ℝ+
2110, 15, 20w3a 1030 . . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
2221, 4, 3crio 6484 . . 3 class (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
232, 3, 22cmpt 4633 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
241, 23wceq 1474 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  13767  sqrtf  13893
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