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

Definition df-ap 8311
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 8406 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8334), symmetry (apsym 8335), and cotransitivity (apcotr 8336). Apartness implies negated equality, as seen at apne 8352, 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 8351).

(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 8310 . 2 class #
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1315 . . . . . . . . . 10 class 𝑥
4 vr . . . . . . . . . . . 12 setvar 𝑟
54cv 1315 . . . . . . . . . . 11 class 𝑟
6 ci 7590 . . . . . . . . . . . 12 class i
7 vs . . . . . . . . . . . . 13 setvar 𝑠
87cv 1315 . . . . . . . . . . . 12 class 𝑠
9 cmul 7593 . . . . . . . . . . . 12 class ·
106, 8, 9co 5742 . . . . . . . . . . 11 class (i · 𝑠)
11 caddc 7591 . . . . . . . . . . 11 class +
125, 10, 11co 5742 . . . . . . . . . 10 class (𝑟 + (i · 𝑠))
133, 12wceq 1316 . . . . . . . . 9 wff 𝑥 = (𝑟 + (i · 𝑠))
14 vy . . . . . . . . . . 11 setvar 𝑦
1514cv 1315 . . . . . . . . . 10 class 𝑦
16 vt . . . . . . . . . . . 12 setvar 𝑡
1716cv 1315 . . . . . . . . . . 11 class 𝑡
18 vu . . . . . . . . . . . . 13 setvar 𝑢
1918cv 1315 . . . . . . . . . . . 12 class 𝑢
206, 19, 9co 5742 . . . . . . . . . . 11 class (i · 𝑢)
2117, 20, 11co 5742 . . . . . . . . . 10 class (𝑡 + (i · 𝑢))
2215, 21wceq 1316 . . . . . . . . 9 wff 𝑦 = (𝑡 + (i · 𝑢))
2313, 22wa 103 . . . . . . . 8 wff (𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢)))
24 creap 8303 . . . . . . . . . 10 class #
255, 17, 24wbr 3899 . . . . . . . . 9 wff 𝑟 # 𝑡
268, 19, 24wbr 3899 . . . . . . . . 9 wff 𝑠 # 𝑢
2725, 26wo 682 . . . . . . . 8 wff (𝑟 # 𝑡𝑠 # 𝑢)
2823, 27wa 103 . . . . . . 7 wff ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
29 cr 7587 . . . . . . 7 class
3028, 18, 29wrex 2394 . . . . . 6 wff 𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3130, 16, 29wrex 2394 . . . . 5 wff 𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3231, 7, 29wrex 2394 . . . 4 wff 𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3332, 4, 29wrex 2394 . . 3 wff 𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3433, 2, 14copab 3958 . 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 1316 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  8316  apreim  8332  aprcl  8375
  Copyright terms: Public domain W3C validator