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

Definition df-ap 8356
Description: Define complex apartness. Definition 6.1 of [Geuvers], p. 17.

Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 8451 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8379), symmetry (apsym 8380), and cotransitivity (apcotr 8381). Apartness implies negated equality, as seen at apne 8397, and the converse would also follow if we assumed excluded middle.

In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 8396).

(Contributed by Jim Kingdon, 26-Jan-2020.)

Assertion
Ref Expression
df-ap # = {⟨𝑥, 𝑦⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))}
Distinct variable group:   𝑠,𝑟,𝑡,𝑢,𝑥,𝑦

Detailed syntax breakdown of Definition df-ap
StepHypRef Expression
1 cap 8355 . 2 class #
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1330 . . . . . . . . . 10 class 𝑥
4 vr . . . . . . . . . . . 12 setvar 𝑟
54cv 1330 . . . . . . . . . . 11 class 𝑟
6 ci 7634 . . . . . . . . . . . 12 class i
7 vs . . . . . . . . . . . . 13 setvar 𝑠
87cv 1330 . . . . . . . . . . . 12 class 𝑠
9 cmul 7637 . . . . . . . . . . . 12 class ·
106, 8, 9co 5774 . . . . . . . . . . 11 class (i · 𝑠)
11 caddc 7635 . . . . . . . . . . 11 class +
125, 10, 11co 5774 . . . . . . . . . 10 class (𝑟 + (i · 𝑠))
133, 12wceq 1331 . . . . . . . . 9 wff 𝑥 = (𝑟 + (i · 𝑠))
14 vy . . . . . . . . . . 11 setvar 𝑦
1514cv 1330 . . . . . . . . . 10 class 𝑦
16 vt . . . . . . . . . . . 12 setvar 𝑡
1716cv 1330 . . . . . . . . . . 11 class 𝑡
18 vu . . . . . . . . . . . . 13 setvar 𝑢
1918cv 1330 . . . . . . . . . . . 12 class 𝑢
206, 19, 9co 5774 . . . . . . . . . . 11 class (i · 𝑢)
2117, 20, 11co 5774 . . . . . . . . . 10 class (𝑡 + (i · 𝑢))
2215, 21wceq 1331 . . . . . . . . 9 wff 𝑦 = (𝑡 + (i · 𝑢))
2313, 22wa 103 . . . . . . . 8 wff (𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢)))
24 creap 8348 . . . . . . . . . 10 class #
255, 17, 24wbr 3929 . . . . . . . . 9 wff 𝑟 # 𝑡
268, 19, 24wbr 3929 . . . . . . . . 9 wff 𝑠 # 𝑢
2725, 26wo 697 . . . . . . . 8 wff (𝑟 # 𝑡𝑠 # 𝑢)
2823, 27wa 103 . . . . . . 7 wff ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
29 cr 7631 . . . . . . 7 class
3028, 18, 29wrex 2417 . . . . . 6 wff 𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3130, 16, 29wrex 2417 . . . . 5 wff 𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3231, 7, 29wrex 2417 . . . 4 wff 𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3332, 4, 29wrex 2417 . . 3 wff 𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3433, 2, 14copab 3988 . 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))}
351, 34wceq 1331 1 wff # = {⟨𝑥, 𝑦⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))}
Colors of variables: wff set class
This definition is referenced by:  apreap  8361  apreim  8377  aprcl  8420
  Copyright terms: Public domain W3C validator