Definition df-sqrt 15431

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 31085).

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 31085. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15558 for its closure, sqrtval 15433 for its value, sqrtth 15561 and sqsqrti 15572 for its relationship to squares, and sqrt11i 15581 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 15429	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11225	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1569	. . . . . . 7 class 𝑦
6		c2 12424	. . . . . . 7 class 2
7		cexp 14231	. . . . . . 7 class ↑
8	5, 6, 7	co 7443	. . . . . 6 class (𝑦↑2)
9	2	cv 1569	. . . . . 6 class 𝑥
10	8, 9	wceq 1570	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11227	. . . . . 6 class 0
12		cre 15293	. . . . . . 7 class ℜ
13	5, 12	cfv 6558	. . . . . 6 class (ℜ‘𝑦)
14		cle 11371	. . . . . 6 class ≤
15	11, 13, 14	wbr 5106	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11229	. . . . . . 7 class i
17		cmul 11232	. . . . . . 7 class ·
18	16, 5, 17	co 7443	. . . . . 6 class (i · 𝑦)
19		crp 13147	. . . . . 6 class ℝ+
20	18, 19	wnel 3065	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1103	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7399	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5189	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1570	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is used by:  sqrtval  15433  sqrtf  15560  cphsscph  25602