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Definition df-sqrt 15222
Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30284).

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 30284. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15348 for its closure, sqrtval 15224 for its value, sqrtth 15351 and sqsqrti 15362 for its relationship to squares, and sqrt11i 15371 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 15220 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 11144 . . 3 class
4 vy . . . . . . . 8 setvar 𝑦
54cv 1532 . . . . . . 7 class 𝑦
6 c2 12305 . . . . . . 7 class 2
7 cexp 14066 . . . . . . 7 class
85, 6, 7co 7426 . . . . . 6 class (𝑦↑2)
92cv 1532 . . . . . 6 class 𝑥
108, 9wceq 1533 . . . . 5 wff (𝑦↑2) = 𝑥
11 cc0 11146 . . . . . 6 class 0
12 cre 15084 . . . . . . 7 class
135, 12cfv 6553 . . . . . 6 class (ℜ‘𝑦)
14 cle 11287 . . . . . 6 class
1511, 13, 14wbr 5152 . . . . 5 wff 0 ≤ (ℜ‘𝑦)
16 ci 11148 . . . . . . 7 class i
17 cmul 11151 . . . . . . 7 class ·
1816, 5, 17co 7426 . . . . . 6 class (i · 𝑦)
19 crp 13014 . . . . . 6 class +
2018, 19wnel 3043 . . . . 5 wff (i · 𝑦) ∉ ℝ+
2110, 15, 20w3a 1084 . . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
2221, 4, 3crio 7381 . . 3 class (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
232, 3, 22cmpt 5235 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
241, 23wceq 1533 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  15224  sqrtf  15350  cphsscph  25199
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