Definition df-sqrt 15403

Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 31056).

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 31056. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15530 for its closure, sqrtval 15405 for its value, sqrtth 15533 and sqsqrti 15544 for its relationship to squares, and sqrt11i 15553 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 15401	. 2 class √
2		vx	. . 3 setvar 𝑥
3		cc 11199	. . 3 class ℂ
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1568	. . . . . . 7 class 𝑦
6		c2 12398	. . . . . . 7 class 2
7		cexp 14205	. . . . . . 7 class ↑
8	5, 6, 7	co 7420	. . . . . 6 class (𝑦↑2)
9	2	cv 1568	. . . . . 6 class 𝑥
10	8, 9	wceq 1569	. . . . 5 wff (𝑦↑2) = 𝑥
11		cc0 11201	. . . . . 6 class 0
12		cre 15265	. . . . . . 7 class ℜ
13	5, 12	cfv 6538	. . . . . 6 class (ℜ‘𝑦)
14		cle 11345	. . . . . 6 class ≤
15	11, 13, 14	wbr 5102	. . . . 5 wff 0 ≤ (ℜ‘𝑦)
16		ci 11203	. . . . . . 7 class i
17		cmul 11206	. . . . . . 7 class ·
18	16, 5, 17	co 7420	. . . . . 6 class (i · 𝑦)
19		crp 13121	. . . . . 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 7376	. . 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 1569	1 wff √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))

This definition is used by:  sqrtval  15405  sqrtf  15532  cphsscph  25573