Definition df-sqrt 15326

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30942).

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 30942. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15453 for its closure, sqrtval 15328 for its value, sqrtth 15456 and sqsqrti 15467 for its relationship to squares, and sqrt11i 15476 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 15324	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11126	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1569	. . . . . . 7 class 𝑦
6		c2 12323	. . . . . . 7 class 2
7		cexp 14129	. . . . . . 7 class ↑
8	5, 6, 7	co 7417	. . . . . 6 class (𝑦↑2)
9	2	cv 1569	. . . . . 6 class 𝑥
10	8, 9	wceq 1570	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11128	. . . . . 6 class 0
12		cre 15188	. . . . . . 7 class ℜ
13	5, 12	cfv 6537	. . . . . 6 class (ℜ‘𝑦)
14		cle 11272	. . . . . 6 class ≤
15	11, 13, 14	wbr 5107	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11130	. . . . . . 7 class i
17		cmul 11133	. . . . . . 7 class ·
18	16, 5, 17	co 7417	. . . . . 6 class (i · 𝑦)
19		crp 13046	. . . . . 6 class ℝ+
20	18, 19	wnel 3063	. . . . 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 7373	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5190	. 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  15328  sqrtf  15455  cphsscph  25485