Definition df-sqrt 14955

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 28827).

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 28827. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15082 for its closure, sqrtval 14957 for its value, sqrtth 15085 and sqsqrti 15096 for its relationship to squares, and sqrt11i 15105 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

Step	Hyp	Ref	Expression
1		csqrt 14953	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 10878	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1538	. . . . . . 7 class 𝑦
6		c2 12037	. . . . . . 7 class 2
7		cexp 13791	. . . . . . 7 class ↑
8	5, 6, 7	co 7284	. . . . . 6 class (𝑦↑2)
9	2	cv 1538	. . . . . 6 class 𝑥
10	8, 9	wceq 1539	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 10880	. . . . . 6 class 0
12		cre 14817	. . . . . . 7 class ℜ
13	5, 12	cfv 6437	. . . . . 6 class (ℜ‘𝑦)
14		cle 11019	. . . . . 6 class ≤
15	11, 13, 14	wbr 5075	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 10882	. . . . . . 7 class i
17		cmul 10885	. . . . . . 7 class ·
18	16, 5, 17	co 7284	. . . . . 6 class (i · 𝑦)
19		crp 12739	. . . . . 6 class ℝ+
20	18, 19	wnel 3050	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1086	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7240	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5158	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1539	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  14957  sqrtf  15084  cphsscph  24424