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Definition df-ap 7957
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 8042 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 7980), symmetry (apsym 7981), and cotransitivity (apcotr 7982). Apartness implies negated equality, as seen at apne 7998, 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 7997).

(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 7956 . 2 class #
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1284 . . . . . . . . . 10 class 𝑥
4 vr . . . . . . . . . . . 12 setvar 𝑟
54cv 1284 . . . . . . . . . . 11 class 𝑟
6 ci 7253 . . . . . . . . . . . 12 class i
7 vs . . . . . . . . . . . . 13 setvar 𝑠
87cv 1284 . . . . . . . . . . . 12 class 𝑠
9 cmul 7256 . . . . . . . . . . . 12 class ·
106, 8, 9co 5589 . . . . . . . . . . 11 class (i · 𝑠)
11 caddc 7254 . . . . . . . . . . 11 class +
125, 10, 11co 5589 . . . . . . . . . 10 class (𝑟 + (i · 𝑠))
133, 12wceq 1285 . . . . . . . . 9 wff 𝑥 = (𝑟 + (i · 𝑠))
14 vy . . . . . . . . . . 11 setvar 𝑦
1514cv 1284 . . . . . . . . . 10 class 𝑦
16 vt . . . . . . . . . . . 12 setvar 𝑡
1716cv 1284 . . . . . . . . . . 11 class 𝑡
18 vu . . . . . . . . . . . . 13 setvar 𝑢
1918cv 1284 . . . . . . . . . . . 12 class 𝑢
206, 19, 9co 5589 . . . . . . . . . . 11 class (i · 𝑢)
2117, 20, 11co 5589 . . . . . . . . . 10 class (𝑡 + (i · 𝑢))
2215, 21wceq 1285 . . . . . . . . 9 wff 𝑦 = (𝑡 + (i · 𝑢))
2313, 22wa 102 . . . . . . . 8 wff (𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢)))
24 creap 7949 . . . . . . . . . 10 class #
255, 17, 24wbr 3811 . . . . . . . . 9 wff 𝑟 # 𝑡
268, 19, 24wbr 3811 . . . . . . . . 9 wff 𝑠 # 𝑢
2725, 26wo 662 . . . . . . . 8 wff (𝑟 # 𝑡𝑠 # 𝑢)
2823, 27wa 102 . . . . . . 7 wff ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
29 cr 7250 . . . . . . 7 class
3028, 18, 29wrex 2354 . . . . . 6 wff 𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3130, 16, 29wrex 2354 . . . . 5 wff 𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3231, 7, 29wrex 2354 . . . 4 wff 𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3332, 4, 29wrex 2354 . . 3 wff 𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))
3433, 2, 14copab 3864 . 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 1285 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  7962  apreim  7978
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