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Definition df-sqrt 14584
Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 28161).

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 28161. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 14711 for its closure, sqrtval 14586 for its value, sqrtth 14714 and sqsqrti 14725 for its relationship to squares, and sqrt11i 14734 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 14582 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 10524 . . 3 class
4 vy . . . . . . . 8 setvar 𝑦
54cv 1527 . . . . . . 7 class 𝑦
6 c2 11681 . . . . . . 7 class 2
7 cexp 13419 . . . . . . 7 class
85, 6, 7co 7145 . . . . . 6 class (𝑦↑2)
92cv 1527 . . . . . 6 class 𝑥
108, 9wceq 1528 . . . . 5 wff (𝑦↑2) = 𝑥
11 cc0 10526 . . . . . 6 class 0
12 cre 14446 . . . . . . 7 class
135, 12cfv 6349 . . . . . 6 class (ℜ‘𝑦)
14 cle 10665 . . . . . 6 class
1511, 13, 14wbr 5058 . . . . 5 wff 0 ≤ (ℜ‘𝑦)
16 ci 10528 . . . . . . 7 class i
17 cmul 10531 . . . . . . 7 class ·
1816, 5, 17co 7145 . . . . . 6 class (i · 𝑦)
19 crp 12379 . . . . . 6 class +
2018, 19wnel 3123 . . . . 5 wff (i · 𝑦) ∉ ℝ+
2110, 15, 20w3a 1079 . . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
2221, 4, 3crio 7102 . . 3 class (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
232, 3, 22cmpt 5138 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
241, 23wceq 1528 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  14586  sqrtf  14713  cphsscph  23783
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