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Definition df-sqrt 14174
Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 27622).

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 14300 for its closure, sqrtval 14176 for its value, sqrtth 14303 and sqsqrti 14314 for its relationship to squares, and sqrt11i 14323 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 14172 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 10126 . . 3 class
4 vy . . . . . . . 8 setvar 𝑦
54cv 1631 . . . . . . 7 class 𝑦
6 c2 11262 . . . . . . 7 class 2
7 cexp 13054 . . . . . . 7 class
85, 6, 7co 6813 . . . . . 6 class (𝑦↑2)
92cv 1631 . . . . . 6 class 𝑥
108, 9wceq 1632 . . . . 5 wff (𝑦↑2) = 𝑥
11 cc0 10128 . . . . . 6 class 0
12 cre 14036 . . . . . . 7 class
135, 12cfv 6049 . . . . . 6 class (ℜ‘𝑦)
14 cle 10267 . . . . . 6 class
1511, 13, 14wbr 4804 . . . . 5 wff 0 ≤ (ℜ‘𝑦)
16 ci 10130 . . . . . . 7 class i
17 cmul 10133 . . . . . . 7 class ·
1816, 5, 17co 6813 . . . . . 6 class (i · 𝑦)
19 crp 12025 . . . . . 6 class +
2018, 19wnel 3035 . . . . 5 wff (i · 𝑦) ∉ ℝ+
2110, 15, 20w3a 1072 . . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
2221, 4, 3crio 6773 . . 3 class (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
232, 3, 22cmpt 4881 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
241, 23wceq 1632 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  14176  sqrtf  14302
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