Definition df-sqrt 14874

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 28719).

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 28719. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15001 for its closure, sqrtval 14876 for its value, sqrtth 15004 and sqsqrti 15015 for its relationship to squares, and sqrt11i 15024 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 14872	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 10800	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1538	. . . . . . 7 class 𝑦
6		c2 11958	. . . . . . 7 class 2
7		cexp 13710	. . . . . . 7 class ↑
8	5, 6, 7	co 7255	. . . . . 6 class (𝑦↑2)
9	2	cv 1538	. . . . . 6 class 𝑥
10	8, 9	wceq 1539	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 10802	. . . . . 6 class 0
12		cre 14736	. . . . . . 7 class ℜ
13	5, 12	cfv 6418	. . . . . 6 class (ℜ‘𝑦)
14		cle 10941	. . . . . 6 class ≤
15	11, 13, 14	wbr 5070	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 10804	. . . . . . 7 class i
17		cmul 10807	. . . . . . 7 class ·
18	16, 5, 17	co 7255	. . . . . 6 class (i · 𝑦)
19		crp 12659	. . . . . 6 class ℝ+
20	18, 19	wnel 3048	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1085	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7211	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5153	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1539	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  14876  sqrtf  15003  cphsscph  24320