Definition df-sqrt 15186

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 29974).

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 29974. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15312 for its closure, sqrtval 15188 for its value, sqrtth 15315 and sqsqrti 15326 for its relationship to squares, and sqrt11i 15335 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 15184	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11110	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1538	. . . . . . 7 class 𝑦
6		c2 12271	. . . . . . 7 class 2
7		cexp 14031	. . . . . . 7 class ↑
8	5, 6, 7	co 7411	. . . . . 6 class (𝑦↑2)
9	2	cv 1538	. . . . . 6 class 𝑥
10	8, 9	wceq 1539	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11112	. . . . . 6 class 0
12		cre 15048	. . . . . . 7 class ℜ
13	5, 12	cfv 6542	. . . . . 6 class (ℜ‘𝑦)
14		cle 11253	. . . . . 6 class ≤
15	11, 13, 14	wbr 5147	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11114	. . . . . . 7 class i
17		cmul 11117	. . . . . . 7 class ·
18	16, 5, 17	co 7411	. . . . . 6 class (i · 𝑦)
19		crp 12978	. . . . . 6 class ℝ+
20	18, 19	wnel 3044	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1085	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7366	. . 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 1539	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15188  sqrtf  15314  cphsscph  24999