Definition df-sqrt 15286

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30771).

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 30771. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15413 for its closure, sqrtval 15288 for its value, sqrtth 15416 and sqsqrti 15427 for its relationship to squares, and sqrt11i 15436 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 15284	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11097	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1567	. . . . . . 7 class 𝑦
6		c2 12294	. . . . . . 7 class 2
7		cexp 14097	. . . . . . 7 class ↑
8	5, 6, 7	co 7410	. . . . . 6 class (𝑦↑2)
9	2	cv 1567	. . . . . 6 class 𝑥
10	8, 9	wceq 1568	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11099	. . . . . 6 class 0
12		cre 15148	. . . . . . 7 class ℜ
13	5, 12	cfv 6536	. . . . . 6 class (ℜ‘𝑦)
14		cle 11243	. . . . . 6 class ≤
15	11, 13, 14	wbr 5108	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11101	. . . . . . 7 class i
17		cmul 11104	. . . . . . 7 class ·
18	16, 5, 17	co 7410	. . . . . 6 class (i · 𝑦)
19		crp 13015	. . . . . 6 class ℝ+
20	18, 19	wnel 3062	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1101	. . . 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 5191	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1568	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15288  sqrtf  15415  cphsscph  25389