Definition df-sqrt 15284

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30486).

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 30486. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15410 for its closure, sqrtval 15286 for its value, sqrtth 15413 and sqsqrti 15424 for its relationship to squares, and sqrt11i 15433 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 15282	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11182	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1536	. . . . . . 7 class 𝑦
6		c2 12348	. . . . . . 7 class 2
7		cexp 14112	. . . . . . 7 class ↑
8	5, 6, 7	co 7448	. . . . . 6 class (𝑦↑2)
9	2	cv 1536	. . . . . 6 class 𝑥
10	8, 9	wceq 1537	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11184	. . . . . 6 class 0
12		cre 15146	. . . . . . 7 class ℜ
13	5, 12	cfv 6573	. . . . . 6 class (ℜ‘𝑦)
14		cle 11325	. . . . . 6 class ≤
15	11, 13, 14	wbr 5166	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11186	. . . . . . 7 class i
17		cmul 11189	. . . . . . 7 class ·
18	16, 5, 17	co 7448	. . . . . 6 class (i · 𝑦)
19		crp 13057	. . . . . 6 class ℝ+
20	18, 19	wnel 3052	. . . . 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 7403	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5249	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1537	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15286  sqrtf  15412  cphsscph  25304