Definition df-sqrt 15197

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30524).

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 30524. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15324 for its closure, sqrtval 15199 for its value, sqrtth 15327 and sqsqrti 15338 for its relationship to squares, and sqrt11i 15347 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 15195	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11036	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1541	. . . . . . 7 class 𝑦
6		c2 12236	. . . . . . 7 class 2
7		cexp 14023	. . . . . . 7 class ↑
8	5, 6, 7	co 7367	. . . . . 6 class (𝑦↑2)
9	2	cv 1541	. . . . . 6 class 𝑥
10	8, 9	wceq 1542	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11038	. . . . . 6 class 0
12		cre 15059	. . . . . . 7 class ℜ
13	5, 12	cfv 6499	. . . . . 6 class (ℜ‘𝑦)
14		cle 11180	. . . . . 6 class ≤
15	11, 13, 14	wbr 5086	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11040	. . . . . . 7 class i
17		cmul 11043	. . . . . . 7 class ·
18	16, 5, 17	co 7367	. . . . . 6 class (i · 𝑦)
19		crp 12942	. . . . . 6 class ℝ+
20	18, 19	wnel 3037	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1087	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7323	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5167	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1542	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15199  sqrtf  15326  cphsscph  25218