Definition df-sqrt 15270

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30482).

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 30482. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15396 for its closure, sqrtval 15272 for its value, sqrtth 15399 and sqsqrti 15410 for its relationship to squares, and sqrt11i 15419 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 15268	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11150	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1535	. . . . . . 7 class 𝑦
6		c2 12318	. . . . . . 7 class 2
7		cexp 14098	. . . . . . 7 class ↑
8	5, 6, 7	co 7430	. . . . . 6 class (𝑦↑2)
9	2	cv 1535	. . . . . 6 class 𝑥
10	8, 9	wceq 1536	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11152	. . . . . 6 class 0
12		cre 15132	. . . . . . 7 class ℜ
13	5, 12	cfv 6562	. . . . . 6 class (ℜ‘𝑦)
14		cle 11293	. . . . . 6 class ≤
15	11, 13, 14	wbr 5147	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11154	. . . . . . 7 class i
17		cmul 11157	. . . . . . 7 class ·
18	16, 5, 17	co 7430	. . . . . 6 class (i · 𝑦)
19		crp 13031	. . . . . 6 class ℝ+
20	18, 19	wnel 3043	. . . . 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 7386	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5230	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1536	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15272  sqrtf  15398  cphsscph  25298