Definition df-sqrt 15159

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30513).

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 30513. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15286 for its closure, sqrtval 15161 for its value, sqrtth 15289 and sqsqrti 15300 for its relationship to squares, and sqrt11i 15309 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 15157	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11025	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1541	. . . . . . 7 class 𝑦
6		c2 12201	. . . . . . 7 class 2
7		cexp 13985	. . . . . . 7 class ↑
8	5, 6, 7	co 7358	. . . . . 6 class (𝑦↑2)
9	2	cv 1541	. . . . . 6 class 𝑥
10	8, 9	wceq 1542	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11027	. . . . . 6 class 0
12		cre 15021	. . . . . . 7 class ℜ
13	5, 12	cfv 6490	. . . . . 6 class (ℜ‘𝑦)
14		cle 11168	. . . . . 6 class ≤
15	11, 13, 14	wbr 5086	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11029	. . . . . . 7 class i
17		cmul 11032	. . . . . . 7 class ·
18	16, 5, 17	co 7358	. . . . . 6 class (i · 𝑦)
19		crp 12906	. . . . . 6 class ℝ+
20	18, 19	wnel 3037	. . . . 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 7314	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5167	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1542	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15161  sqrtf  15288  cphsscph  25196