Definition df-sqrt 14594

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 28233).

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 28233. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 14721 for its closure, sqrtval 14596 for its value, sqrtth 14724 and sqsqrti 14735 for its relationship to squares, and sqrt11i 14744 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 14592	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 10535	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1536	. . . . . . 7 class 𝑦
6		c2 11693	. . . . . . 7 class 2
7		cexp 13430	. . . . . . 7 class ↑
8	5, 6, 7	co 7156	. . . . . 6 class (𝑦↑2)
9	2	cv 1536	. . . . . 6 class 𝑥
10	8, 9	wceq 1537	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 10537	. . . . . 6 class 0
12		cre 14456	. . . . . . 7 class ℜ
13	5, 12	cfv 6355	. . . . . 6 class (ℜ‘𝑦)
14		cle 10676	. . . . . 6 class ≤
15	11, 13, 14	wbr 5066	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 10539	. . . . . . 7 class i
17		cmul 10542	. . . . . . 7 class ·
18	16, 5, 17	co 7156	. . . . . 6 class (i · 𝑦)
19		crp 12390	. . . . . 6 class ℝ+
20	18, 19	wnel 3123	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1083	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7113	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5146	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1537	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  14596  sqrtf  14723  cphsscph  23854