Definition df-sqrt 15208

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30390).

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 30390. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15335 for its closure, sqrtval 15210 for its value, sqrtth 15338 and sqsqrti 15349 for its relationship to squares, and sqrt11i 15358 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 15206	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11073	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1539	. . . . . . 7 class 𝑦
6		c2 12248	. . . . . . 7 class 2
7		cexp 14033	. . . . . . 7 class ↑
8	5, 6, 7	co 7390	. . . . . 6 class (𝑦↑2)
9	2	cv 1539	. . . . . 6 class 𝑥
10	8, 9	wceq 1540	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11075	. . . . . 6 class 0
12		cre 15070	. . . . . . 7 class ℜ
13	5, 12	cfv 6514	. . . . . 6 class (ℜ‘𝑦)
14		cle 11216	. . . . . 6 class ≤
15	11, 13, 14	wbr 5110	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11077	. . . . . . 7 class i
17		cmul 11080	. . . . . . 7 class ·
18	16, 5, 17	co 7390	. . . . . 6 class (i · 𝑦)
19		crp 12958	. . . . . 6 class ℝ+
20	18, 19	wnel 3030	. . . . 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 7346	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5191	. 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  15210  sqrtf  15337  cphsscph  25158