Definition df-sqrt 15274

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30473).

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 30473. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15400 for its closure, sqrtval 15276 for its value, sqrtth 15403 and sqsqrti 15414 for its relationship to squares, and sqrt11i 15423 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 15272	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11153	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1539	. . . . . . 7 class 𝑦
6		c2 12321	. . . . . . 7 class 2
7		cexp 14102	. . . . . . 7 class ↑
8	5, 6, 7	co 7431	. . . . . 6 class (𝑦↑2)
9	2	cv 1539	. . . . . 6 class 𝑥
10	8, 9	wceq 1540	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11155	. . . . . 6 class 0
12		cre 15136	. . . . . . 7 class ℜ
13	5, 12	cfv 6561	. . . . . 6 class (ℜ‘𝑦)
14		cle 11296	. . . . . 6 class ≤
15	11, 13, 14	wbr 5143	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11157	. . . . . . 7 class i
17		cmul 11160	. . . . . . 7 class ·
18	16, 5, 17	co 7431	. . . . . 6 class (i · 𝑦)
19		crp 13034	. . . . . 6 class ℝ+
20	18, 19	wnel 3046	. . . . 5 wff (i · 𝑦) ∉ ℝ+
21	10, 15, 20	w3a 1087	. . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
22	21, 4, 3	crio 7387	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5225	. 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  15276  sqrtf  15402  cphsscph  25285