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Definition df-cj 10626
Description: Define the complex conjugate function. See cjcli 10697 for its closure and cjval 10629 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 10623 . 2 class
2 vx . . 3 setvar 𝑥
3 cc 7630 . . 3 class
42cv 1330 . . . . . . 7 class 𝑥
5 vy . . . . . . . 8 setvar 𝑦
65cv 1330 . . . . . . 7 class 𝑦
7 caddc 7635 . . . . . . 7 class +
84, 6, 7co 5774 . . . . . 6 class (𝑥 + 𝑦)
9 cr 7631 . . . . . 6 class
108, 9wcel 1480 . . . . 5 wff (𝑥 + 𝑦) ∈ ℝ
11 ci 7634 . . . . . . 7 class i
12 cmin 7945 . . . . . . . 8 class
134, 6, 12co 5774 . . . . . . 7 class (𝑥𝑦)
14 cmul 7637 . . . . . . 7 class ·
1511, 13, 14co 5774 . . . . . 6 class (i · (𝑥𝑦))
1615, 9wcel 1480 . . . . 5 wff (i · (𝑥𝑦)) ∈ ℝ
1710, 16wa 103 . . . 4 wff ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ)
1817, 5, 3crio 5729 . . 3 class (𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ))
192, 3, 18cmpt 3989 . 2 class (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ)))
201, 19wceq 1331 1 wff ∗ = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ)))
Colors of variables: wff set class
This definition is referenced by:  cjval  10629  cjf  10631
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