Definition df-sqrt 15179

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 29697).

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 29697. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15305 for its closure, sqrtval 15181 for its value, sqrtth 15308 and sqsqrti 15319 for its relationship to squares, and sqrt11i 15328 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 15177	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11105	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1541	. . . . . . 7 class 𝑦
6		c2 12264	. . . . . . 7 class 2
7		cexp 14024	. . . . . . 7 class ↑
8	5, 6, 7	co 7406	. . . . . 6 class (𝑦↑2)
9	2	cv 1541	. . . . . 6 class 𝑥
10	8, 9	wceq 1542	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11107	. . . . . 6 class 0
12		cre 15041	. . . . . . 7 class ℜ
13	5, 12	cfv 6541	. . . . . 6 class (ℜ‘𝑦)
14		cle 11246	. . . . . 6 class ≤
15	11, 13, 14	wbr 5148	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11109	. . . . . . 7 class i
17		cmul 11112	. . . . . . 7 class ·
18	16, 5, 17	co 7406	. . . . . 6 class (i · 𝑦)
19		crp 12971	. . . . . 6 class ℝ+
20	18, 19	wnel 3047	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1088	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7361	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5231	. 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  15181  sqrtf  15307  cphsscph  24760