Definition df-sqrt 15276

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30714).

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 30714. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15403 for its closure, sqrtval 15278 for its value, sqrtth 15406 and sqsqrti 15417 for its relationship to squares, and sqrt11i 15426 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 15274	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11086	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1562	. . . . . . 7 class 𝑦
6		c2 12286	. . . . . . 7 class 2
7		cexp 14088	. . . . . . 7 class ↑
8	5, 6, 7	co 7400	. . . . . 6 class (𝑦↑2)
9	2	cv 1562	. . . . . 6 class 𝑥
10	8, 9	wceq 1563	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11088	. . . . . 6 class 0
12		cre 15138	. . . . . . 7 class ℜ
13	5, 12	cfv 6525	. . . . . 6 class (ℜ‘𝑦)
14		cle 11232	. . . . . 6 class ≤
15	11, 13, 14	wbr 5105	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11090	. . . . . . 7 class i
17		cmul 11093	. . . . . . 7 class ·
18	16, 5, 17	co 7400	. . . . . 6 class (i · 𝑦)
19		crp 13007	. . . . . 6 class ℝ+
20	18, 19	wnel 3064	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1101	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7356	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5186	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1563	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15278  sqrtf  15405  cphsscph  25371