MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-sqrt Structured version   Visualization version   GIF version

Definition df-sqrt 14944
Description: Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 28814).

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 28814. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrtcl 15071 for its closure, sqrtval 14946 for its value, sqrtth 15074 and sqsqrti 15085 for its relationship to squares, and sqrt11i 15094 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
StepHypRef Expression
1 csqrt 14942 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 10870 . . 3 class
4 vy . . . . . . . 8 setvar 𝑦
54cv 1541 . . . . . . 7 class 𝑦
6 c2 12028 . . . . . . 7 class 2
7 cexp 13780 . . . . . . 7 class
85, 6, 7co 7271 . . . . . 6 class (𝑦↑2)
92cv 1541 . . . . . 6 class 𝑥
108, 9wceq 1542 . . . . 5 wff (𝑦↑2) = 𝑥
11 cc0 10872 . . . . . 6 class 0
12 cre 14806 . . . . . . 7 class
135, 12cfv 6432 . . . . . 6 class (ℜ‘𝑦)
14 cle 11011 . . . . . 6 class
1511, 13, 14wbr 5079 . . . . 5 wff 0 ≤ (ℜ‘𝑦)
16 ci 10874 . . . . . . 7 class i
17 cmul 10877 . . . . . . 7 class ·
1816, 5, 17co 7271 . . . . . 6 class (i · 𝑦)
19 crp 12729 . . . . . 6 class +
2018, 19wnel 3051 . . . . 5 wff (i · 𝑦) ∉ ℝ+
2110, 15, 20w3a 1086 . . . 4 wff ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)
2221, 4, 3crio 7227 . . 3 class (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))
232, 3, 22cmpt 5162 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
241, 23wceq 1542 1 wff √ = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
Colors of variables: wff setvar class
This definition is referenced by:  sqrtval  14946  sqrtf  15073  cphsscph  24413
  Copyright terms: Public domain W3C validator