Definition df-sqrt 15195

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30549).

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 30549. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15322 for its closure, sqrtval 15197 for its value, sqrtth 15325 and sqsqrti 15336 for its relationship to squares, and sqrt11i 15345 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 15193	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11034	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1546	. . . . . . 7 class 𝑦
6		c2 12234	. . . . . . 7 class 2
7		cexp 14021	. . . . . . 7 class ↑
8	5, 6, 7	co 7363	. . . . . 6 class (𝑦↑2)
9	2	cv 1546	. . . . . 6 class 𝑥
10	8, 9	wceq 1547	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11036	. . . . . 6 class 0
12		cre 15057	. . . . . . 7 class ℜ
13	5, 12	cfv 6492	. . . . . 6 class (ℜ‘𝑦)
14		cle 11178	. . . . . 6 class ≤
15	11, 13, 14	wbr 5079	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11038	. . . . . . 7 class i
17		cmul 11041	. . . . . . 7 class ·
18	16, 5, 17	co 7363	. . . . . 6 class (i · 𝑦)
19		crp 12940	. . . . . 6 class ℝ+
20	18, 19	wnel 3039	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1092	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7319	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5160	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1547	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15197  sqrtf  15324  cphsscph  25243