Definition df-sqrt 15134

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30424).

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 30424. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15261 for its closure, sqrtval 15136 for its value, sqrtth 15264 and sqsqrti 15275 for its relationship to squares, and sqrt11i 15284 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 15132	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 10996	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1540	. . . . . . 7 class 𝑦
6		c2 12172	. . . . . . 7 class 2
7		cexp 13960	. . . . . . 7 class ↑
8	5, 6, 7	co 7341	. . . . . 6 class (𝑦↑2)
9	2	cv 1540	. . . . . 6 class 𝑥
10	8, 9	wceq 1541	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 10998	. . . . . 6 class 0
12		cre 14996	. . . . . . 7 class ℜ
13	5, 12	cfv 6477	. . . . . 6 class (ℜ‘𝑦)
14		cle 11139	. . . . . 6 class ≤
15	11, 13, 14	wbr 5089	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11000	. . . . . . 7 class i
17		cmul 11003	. . . . . . 7 class ·
18	16, 5, 17	co 7341	. . . . . 6 class (i · 𝑦)
19		crp 12882	. . . . . 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 7297	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5170	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1541	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15136  sqrtf  15263  cphsscph  25171