Definition df-sqrt 15370

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30989).

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 30989. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15497 for its closure, sqrtval 15372 for its value, sqrtth 15500 and sqsqrti 15511 for its relationship to squares, and sqrt11i 15520 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 15368	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11170	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1569	. . . . . . 7 class 𝑦
6		c2 12367	. . . . . . 7 class 2
7		cexp 14173	. . . . . . 7 class ↑
8	5, 6, 7	co 7408	. . . . . 6 class (𝑦↑2)
9	2	cv 1569	. . . . . 6 class 𝑥
10	8, 9	wceq 1570	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11172	. . . . . 6 class 0
12		cre 15232	. . . . . . 7 class ℜ
13	5, 12	cfv 6527	. . . . . 6 class (ℜ‘𝑦)
14		cle 11316	. . . . . 6 class ≤
15	11, 13, 14	wbr 5102	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11174	. . . . . . 7 class i
17		cmul 11177	. . . . . . 7 class ·
18	16, 5, 17	co 7408	. . . . . 6 class (i · 𝑦)
19		crp 13090	. . . . . 6 class ℝ+
20	18, 19	wnel 3061	. . . . 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 7364	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5185	. 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  15372  sqrtf  15499  cphsscph  25534