Definition df-sqrt 15160

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30416).

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 30416. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15287 for its closure, sqrtval 15162 for its value, sqrtth 15290 and sqsqrti 15301 for its relationship to squares, and sqrt11i 15310 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 15158	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11026	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1539	. . . . . . 7 class 𝑦
6		c2 12201	. . . . . . 7 class 2
7		cexp 13986	. . . . . . 7 class ↑
8	5, 6, 7	co 7353	. . . . . 6 class (𝑦↑2)
9	2	cv 1539	. . . . . 6 class 𝑥
10	8, 9	wceq 1540	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11028	. . . . . 6 class 0
12		cre 15022	. . . . . . 7 class ℜ
13	5, 12	cfv 6486	. . . . . 6 class (ℜ‘𝑦)
14		cle 11169	. . . . . 6 class ≤
15	11, 13, 14	wbr 5095	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11030	. . . . . . 7 class i
17		cmul 11033	. . . . . . 7 class ·
18	16, 5, 17	co 7353	. . . . . 6 class (i · 𝑦)
19		crp 12911	. . . . . 6 class ℝ+
20	18, 19	wnel 3029	. . . . 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 7309	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5176	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1540	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15162  sqrtf  15289  cphsscph  25167