ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-cj GIF version

Definition df-cj 10171
Description: Define the complex conjugate function. See cjcli 10242 for its closure and cjval 10174 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
Assertion
Ref Expression
df-cj ∗ = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ)))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-cj
StepHypRef Expression
1 ccj 10168 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 7292 . . 3 class
42cv 1286 . . . . . . 7 class 𝑥
5 vy . . . . . . . 8 setvar 𝑦
65cv 1286 . . . . . . 7 class 𝑦
7 caddc 7297 . . . . . . 7 class +
84, 6, 7co 5613 . . . . . 6 class (𝑥 + 𝑦)
9 cr 7293 . . . . . 6 class
108, 9wcel 1436 . . . . 5 wff (𝑥 + 𝑦) ∈ ℝ
11 ci 7296 . . . . . . 7 class i
12 cmin 7597 . . . . . . . 8 class
134, 6, 12co 5613 . . . . . . 7 class (𝑥𝑦)
14 cmul 7299 . . . . . . 7 class ·
1511, 13, 14co 5613 . . . . . 6 class (i · (𝑥𝑦))
1615, 9wcel 1436 . . . . 5 wff (i · (𝑥𝑦)) ∈ ℝ
1710, 16wa 102 . . . 4 wff ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ)
1817, 5, 3crio 5568 . . 3 class (𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ))
192, 3, 18cmpt 3874 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ)))
201, 19wceq 1287 1 wff ∗ = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ)))
Colors of variables: wff set class
This definition is referenced by:  cjval  10174  cjf  10176
  Copyright terms: Public domain W3C validator