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Definition df-ap 8120
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 8207 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8143), symmetry (apsym 8144), and cotransitivity (apcotr 8145). Apartness implies negated equality, as seen at apne 8161, 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 8160).

(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 8119 . 2 class #
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1289 . . . . . . . . . 10 class 𝑥
4 vr . . . . . . . . . . . 12 setvar 𝑟
54cv 1289 . . . . . . . . . . 11 class 𝑟
6 ci 7413 . . . . . . . . . . . 12 class i
7 vs . . . . . . . . . . . . 13 setvar 𝑠
87cv 1289 . . . . . . . . . . . 12 class 𝑠
9 cmul 7416 . . . . . . . . . . . 12 class ·
106, 8, 9co 5666 . . . . . . . . . . 11 class (i · 𝑠)
11 caddc 7414 . . . . . . . . . . 11 class +
125, 10, 11co 5666 . . . . . . . . . 10 class (𝑟 + (i · 𝑠))
133, 12wceq 1290 . . . . . . . . 9 wff 𝑥 = (𝑟 + (i · 𝑠))
14 vy . . . . . . . . . . 11 setvar 𝑦
1514cv 1289 . . . . . . . . . 10 class 𝑦
16 vt . . . . . . . . . . . 12 setvar 𝑡
1716cv 1289 . . . . . . . . . . 11 class 𝑡
18 vu . . . . . . . . . . . . 13 setvar 𝑢
1918cv 1289 . . . . . . . . . . . 12 class 𝑢
206, 19, 9co 5666 . . . . . . . . . . 11 class (i · 𝑢)
2117, 20, 11co 5666 . . . . . . . . . 10 class (𝑡 + (i · 𝑢))
2215, 21wceq 1290 . . . . . . . . 9 wff 𝑦 = (𝑡 + (i · 𝑢))
2313, 22wa 103 . . . . . . . 8 wff (𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢)))
24 creap 8112 . . . . . . . . . 10 class #
255, 17, 24wbr 3851 . . . . . . . . 9 wff 𝑟 # 𝑡
268, 19, 24wbr 3851 . . . . . . . . 9 wff 𝑠 # 𝑢
2725, 26wo 665 . . . . . . . 8 wff (𝑟 # 𝑡𝑠 # 𝑢)
2823, 27wa 103 . . . . . . 7 wff ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
29 cr 7410 . . . . . . 7 class
3028, 18, 29wrex 2361 . . . . . 6 wff 𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3130, 16, 29wrex 2361 . . . . 5 wff 𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3231, 7, 29wrex 2361 . . . 4 wff 𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3332, 4, 29wrex 2361 . . 3 wff 𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3433, 2, 14copab 3904 . 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 1290 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  8125  apreim  8141
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