Definition df-sqrt 15201

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30383).

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 30383. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15328 for its closure, sqrtval 15203 for its value, sqrtth 15331 and sqsqrti 15342 for its relationship to squares, and sqrt11i 15351 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 15199	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11066	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1539	. . . . . . 7 class 𝑦
6		c2 12241	. . . . . . 7 class 2
7		cexp 14026	. . . . . . 7 class ↑
8	5, 6, 7	co 7387	. . . . . 6 class (𝑦↑2)
9	2	cv 1539	. . . . . 6 class 𝑥
10	8, 9	wceq 1540	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11068	. . . . . 6 class 0
12		cre 15063	. . . . . . 7 class ℜ
13	5, 12	cfv 6511	. . . . . 6 class (ℜ‘𝑦)
14		cle 11209	. . . . . 6 class ≤
15	11, 13, 14	wbr 5107	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11070	. . . . . . 7 class i
17		cmul 11073	. . . . . . 7 class ·
18	16, 5, 17	co 7387	. . . . . 6 class (i · 𝑦)
19		crp 12951	. . . . . 6 class ℝ+
20	18, 19	wnel 3029	. . . . 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 7343	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5188	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1540	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15203  sqrtf  15330  cphsscph  25151