Definition df-sqrt 14586

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 28239).

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 28239. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 14713 for its closure, sqrtval 14588 for its value, sqrtth 14716 and sqsqrti 14727 for its relationship to squares, and sqrt11i 14736 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 14584	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 10524	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1537	. . . . . . 7 class 𝑦
6		c2 11680	. . . . . . 7 class 2
7		cexp 13425	. . . . . . 7 class ↑
8	5, 6, 7	co 7135	. . . . . 6 class (𝑦↑2)
9	2	cv 1537	. . . . . 6 class 𝑥
10	8, 9	wceq 1538	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 10526	. . . . . 6 class 0
12		cre 14448	. . . . . . 7 class ℜ
13	5, 12	cfv 6324	. . . . . 6 class (ℜ‘𝑦)
14		cle 10665	. . . . . 6 class ≤
15	11, 13, 14	wbr 5030	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 10528	. . . . . . 7 class i
17		cmul 10531	. . . . . . 7 class ·
18	16, 5, 17	co 7135	. . . . . 6 class (i · 𝑦)
19		crp 12377	. . . . . 6 class ℝ+
20	18, 19	wnel 3091	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1084	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7092	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5110	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1538	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  14588  sqrtf  14715  cphsscph  23855