Definition df-sqrt 15312

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30835).

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 30835. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15439 for its closure, sqrtval 15314 for its value, sqrtth 15442 and sqsqrti 15453 for its relationship to squares, and sqrt11i 15462 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 15310	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11116	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1569	. . . . . . 7 class 𝑦
6		c2 12313	. . . . . . 7 class 2
7		cexp 14117	. . . . . . 7 class ↑
8	5, 6, 7	co 7423	. . . . . 6 class (𝑦↑2)
9	2	cv 1569	. . . . . 6 class 𝑥
10	8, 9	wceq 1570	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11118	. . . . . 6 class 0
12		cre 15174	. . . . . . 7 class ℜ
13	5, 12	cfv 6543	. . . . . 6 class (ℜ‘𝑦)
14		cle 11262	. . . . . 6 class ≤
15	11, 13, 14	wbr 5114	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11120	. . . . . . 7 class i
17		cmul 11123	. . . . . . 7 class ·
18	16, 5, 17	co 7423	. . . . . 6 class (i · 𝑦)
19		crp 13034	. . . . . 6 class ℝ+
20	18, 19	wnel 3067	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1103	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7379	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5197	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1570	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is used by:  sqrtval  15314  sqrtf  15441  cphsscph  25440