Definition df-sqrt 15186

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30544).

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 30544. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15313 for its closure, sqrtval 15188 for its value, sqrtth 15316 and sqsqrti 15327 for its relationship to squares, and sqrt11i 15336 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 15184	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11025	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1541	. . . . . . 7 class 𝑦
6		c2 12225	. . . . . . 7 class 2
7		cexp 14012	. . . . . . 7 class ↑
8	5, 6, 7	co 7358	. . . . . 6 class (𝑦↑2)
9	2	cv 1541	. . . . . 6 class 𝑥
10	8, 9	wceq 1542	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11027	. . . . . 6 class 0
12		cre 15048	. . . . . . 7 class ℜ
13	5, 12	cfv 6490	. . . . . 6 class (ℜ‘𝑦)
14		cle 11169	. . . . . 6 class ≤
15	11, 13, 14	wbr 5086	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11029	. . . . . . 7 class i
17		cmul 11032	. . . . . . 7 class ·
18	16, 5, 17	co 7358	. . . . . 6 class (i · 𝑦)
19		crp 12931	. . . . . 6 class ℝ+
20	18, 19	wnel 3037	. . . . 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 7314	. . 3 class (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
23	2, 3, 22	cmpt 5167	. 2 class (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
24	1, 23	wceq 1542	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is referenced by:  sqrtval  15188  sqrtf  15315  cphsscph  25227