Definition df-sqrt 15162

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30512).

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 30512. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15289 for its closure, sqrtval 15164 for its value, sqrtth 15292 and sqsqrti 15303 for its relationship to squares, and sqrt11i 15312 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 15160	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11028	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1541	. . . . . . 7 class 𝑦
6		c2 12204	. . . . . . 7 class 2
7		cexp 13988	. . . . . . 7 class ↑
8	5, 6, 7	co 7360	. . . . . 6 class (𝑦↑2)
9	2	cv 1541	. . . . . 6 class 𝑥
10	8, 9	wceq 1542	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11030	. . . . . 6 class 0
12		cre 15024	. . . . . . 7 class ℜ
13	5, 12	cfv 6493	. . . . . 6 class (ℜ‘𝑦)
14		cle 11171	. . . . . 6 class ≤
15	11, 13, 14	wbr 5099	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11032	. . . . . . 7 class i
17		cmul 11035	. . . . . . 7 class ·
18	16, 5, 17	co 7360	. . . . . 6 class (i · 𝑦)
19		crp 12909	. . . . . 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 7316	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5180	. 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  15164  sqrtf  15291  cphsscph  25211