Definition df-sqrt 15222

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30284).

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 30284. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15348 for its closure, sqrtval 15224 for its value, sqrtth 15351 and sqsqrti 15362 for its relationship to squares, and sqrt11i 15371 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 15220	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11144	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1532	. . . . . . 7 class 𝑦
6		c2 12305	. . . . . . 7 class 2
7		cexp 14066	. . . . . . 7 class ↑
8	5, 6, 7	co 7426	. . . . . 6 class (𝑦↑2)
9	2	cv 1532	. . . . . 6 class 𝑥
10	8, 9	wceq 1533	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11146	. . . . . 6 class 0
12		cre 15084	. . . . . . 7 class ℜ
13	5, 12	cfv 6553	. . . . . 6 class (ℜ‘𝑦)
14		cle 11287	. . . . . 6 class ≤
15	11, 13, 14	wbr 5152	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11148	. . . . . . 7 class i
17		cmul 11151	. . . . . . 7 class ·
18	16, 5, 17	co 7426	. . . . . 6 class (i · 𝑦)
19		crp 13014	. . . . . 6 class ℝ+
20	18, 19	wnel 3043	. . . . 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 7381	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5235	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1533	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15224  sqrtf  15350  cphsscph  25199