Definition df-sqrt 14944

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 28814).

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 28814. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15071 for its closure, sqrtval 14946 for its value, sqrtth 15074 and sqsqrti 15085 for its relationship to squares, and sqrt11i 15094 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 14942	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 10870	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1541	. . . . . . 7 class 𝑦
6		c2 12028	. . . . . . 7 class 2
7		cexp 13780	. . . . . . 7 class ↑
8	5, 6, 7	co 7271	. . . . . 6 class (𝑦↑2)
9	2	cv 1541	. . . . . 6 class 𝑥
10	8, 9	wceq 1542	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 10872	. . . . . 6 class 0
12		cre 14806	. . . . . . 7 class ℜ
13	5, 12	cfv 6432	. . . . . 6 class (ℜ‘𝑦)
14		cle 11011	. . . . . 6 class ≤
15	11, 13, 14	wbr 5079	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 10874	. . . . . . 7 class i
17		cmul 10877	. . . . . . 7 class ·
18	16, 5, 17	co 7271	. . . . . 6 class (i · 𝑦)
19		crp 12729	. . . . . 6 class ℝ+
20	18, 19	wnel 3051	. . . . 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 7227	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5162	. 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  14946  sqrtf  15073  cphsscph  24413