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Definition df-sqrt 13823
Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 26703).

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 13949 for its closure, sqrtval 13825 for its value, sqrtth 13952 and sqsqrti 13963 for its relationship to squares, and sqrt11i 13972 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 13821 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 9813 . . 3 class
4 vy . . . . . . . 8 setvar 𝑦
54cv 1474 . . . . . . 7 class 𝑦
6 c2 10947 . . . . . . 7 class 2
7 cexp 12722 . . . . . . 7 class
85, 6, 7co 6549 . . . . . 6 class (𝑦↑2)
92cv 1474 . . . . . 6 class 𝑥
108, 9wceq 1475 . . . . 5 wff (𝑦↑2) = 𝑥
11 cc0 9815 . . . . . 6 class 0
12 cre 13685 . . . . . . 7 class
135, 12cfv 5804 . . . . . 6 class (ℜ‘𝑦)
14 cle 9954 . . . . . 6 class
1511, 13, 14wbr 4583 . . . . 5 wff 0 ≤ (ℜ‘𝑦)
16 ci 9817 . . . . . . 7 class i
17 cmul 9820 . . . . . . 7 class ·
1816, 5, 17co 6549 . . . . . 6 class (i · 𝑦)
19 crp 11708 . . . . . 6 class +
2018, 19wnel 2781 . . . . 5 wff (i · 𝑦) ∉ ℝ+
2110, 15, 20w3a 1031 . . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
2221, 4, 3crio 6510 . . 3 class (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
232, 3, 22cmpt 4643 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
241, 23wceq 1475 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  13825  sqrtf  13951
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