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Definition df-sqrt 14205
Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 27652).

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 14331 for its closure, sqrtval 14207 for its value, sqrtth 14334 and sqsqrti 14345 for its relationship to squares, and sqrt11i 14354 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 14203 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 10226 . . 3 class
4 vy . . . . . . . 8 setvar 𝑦
54cv 1636 . . . . . . 7 class 𝑦
6 c2 11363 . . . . . . 7 class 2
7 cexp 13090 . . . . . . 7 class
85, 6, 7co 6881 . . . . . 6 class (𝑦↑2)
92cv 1636 . . . . . 6 class 𝑥
108, 9wceq 1637 . . . . 5 wff (𝑦↑2) = 𝑥
11 cc0 10228 . . . . . 6 class 0
12 cre 14067 . . . . . . 7 class
135, 12cfv 6108 . . . . . 6 class (ℜ‘𝑦)
14 cle 10367 . . . . . 6 class
1511, 13, 14wbr 4855 . . . . 5 wff 0 ≤ (ℜ‘𝑦)
16 ci 10230 . . . . . . 7 class i
17 cmul 10233 . . . . . . 7 class ·
1816, 5, 17co 6881 . . . . . 6 class (i · 𝑦)
19 crp 12053 . . . . . 6 class +
2018, 19wnel 3092 . . . . 5 wff (i · 𝑦) ∉ ℝ+
2110, 15, 20w3a 1100 . . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
2221, 4, 3crio 6841 . . 3 class (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
232, 3, 22cmpt 4934 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
241, 23wceq 1637 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  14207  sqrtf  14333  cphsscph  23270
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