Definition df-sqrt 15152

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30445).

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 30445. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15279 for its closure, sqrtval 15154 for its value, sqrtth 15282 and sqsqrti 15293 for its relationship to squares, and sqrt11i 15302 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 15150	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11014	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1540	. . . . . . 7 class 𝑦
6		c2 12190	. . . . . . 7 class 2
7		cexp 13978	. . . . . . 7 class ↑
8	5, 6, 7	co 7355	. . . . . 6 class (𝑦↑2)
9	2	cv 1540	. . . . . 6 class 𝑥
10	8, 9	wceq 1541	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11016	. . . . . 6 class 0
12		cre 15014	. . . . . . 7 class ℜ
13	5, 12	cfv 6489	. . . . . 6 class (ℜ‘𝑦)
14		cle 11157	. . . . . 6 class ≤
15	11, 13, 14	wbr 5095	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11018	. . . . . . 7 class i
17		cmul 11021	. . . . . . 7 class ·
18	16, 5, 17	co 7355	. . . . . 6 class (i · 𝑦)
19		crp 12900	. . . . . 6 class ℝ+
20	18, 19	wnel 3034	. . . . 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 7311	. . 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 1541	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15154  sqrtf  15281  cphsscph  25188