Definition df-sqrt 15252

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30381).

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 30381. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15378 for its closure, sqrtval 15254 for its value, sqrtth 15381 and sqsqrti 15392 for its relationship to squares, and sqrt11i 15401 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 15250	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11125	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1539	. . . . . . 7 class 𝑦
6		c2 12293	. . . . . . 7 class 2
7		cexp 14077	. . . . . . 7 class ↑
8	5, 6, 7	co 7403	. . . . . 6 class (𝑦↑2)
9	2	cv 1539	. . . . . 6 class 𝑥
10	8, 9	wceq 1540	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11127	. . . . . 6 class 0
12		cre 15114	. . . . . . 7 class ℜ
13	5, 12	cfv 6530	. . . . . 6 class (ℜ‘𝑦)
14		cle 11268	. . . . . 6 class ≤
15	11, 13, 14	wbr 5119	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11129	. . . . . . 7 class i
17		cmul 11132	. . . . . . 7 class ·
18	16, 5, 17	co 7403	. . . . . 6 class (i · 𝑦)
19		crp 13006	. . . . . 6 class ℝ+
20	18, 19	wnel 3036	. . . . 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 7359	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5201	. 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  15254  sqrtf  15380  cphsscph  25201