Definition df-ap 8538

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 8635 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8561), symmetry (apsym 8562), and cotransitivity (apcotr 8563). Apartness implies negated equality, as seen at apne 8579, 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 8578).

(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 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))}

Distinct variable group:   𝑠,𝑟,𝑡,𝑢,𝑥,𝑦

Detailed syntax breakdown of Definition df-ap

Step	Hyp	Ref	Expression
1		cap 8537	. 2 class #
2		vx	. . . . . . . . . . 11 setvar 𝑥
3	2	cv 1352	. . . . . . . . . 10 class 𝑥
4		vr	. . . . . . . . . . . 12 setvar 𝑟
5	4	cv 1352	. . . . . . . . . . 11 class 𝑟
6		ci 7812	. . . . . . . . . . . 12 class i
7		vs	. . . . . . . . . . . . 13 setvar 𝑠
8	7	cv 1352	. . . . . . . . . . . 12 class 𝑠
9		cmul 7815	. . . . . . . . . . . 12 class ·
10	6, 8, 9	co 5874	. . . . . . . . . . 11 class (i · 𝑠)
11		caddc 7813	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5874	. . . . . . . . . 10 class (𝑟 + (i · 𝑠))
13	3, 12	wceq 1353	. . . . . . . . 9 wff 𝑥 = (𝑟 + (i · 𝑠))
14		vy	. . . . . . . . . . 11 setvar 𝑦
15	14	cv 1352	. . . . . . . . . 10 class 𝑦
16		vt	. . . . . . . . . . . 12 setvar 𝑡
17	16	cv 1352	. . . . . . . . . . 11 class 𝑡
18		vu	. . . . . . . . . . . . 13 setvar 𝑢
19	18	cv 1352	. . . . . . . . . . . 12 class 𝑢
20	6, 19, 9	co 5874	. . . . . . . . . . 11 class (i · 𝑢)
21	17, 20, 11	co 5874	. . . . . . . . . 10 class (𝑡 + (i · 𝑢))
22	15, 21	wceq 1353	. . . . . . . . 9 wff 𝑦 = (𝑡 + (i · 𝑢))
23	13, 22	wa 104	. . . . . . . 8 wff (𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢)))
24		creap 8530	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 4003	. . . . . . . . 9 wff 𝑟 #ℝ 𝑡
26	8, 19, 24	wbr 4003	. . . . . . . . 9 wff 𝑠 #ℝ 𝑢
27	25, 26	wo 708	. . . . . . . 8 wff (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)
28	23, 27	wa 104	. . . . . . 7 wff ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))
29		cr 7809	. . . . . . 7 class ℝ
30	28, 18, 29	wrex 2456	. . . . . 6 wff ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))
31	30, 16, 29	wrex 2456	. . . . 5 wff ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))
32	31, 7, 29	wrex 2456	. . . 4 wff ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))
33	32, 4, 29	wrex 2456	. . 3 wff ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))
34	33, 2, 14	copab 4063	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))}
35	1, 34	wceq 1353	1 wff # = {⟨𝑥, 𝑦⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))}

This definition is referenced by:  apreap  8543  apreim  8559  aprcl  8602  aptap  8606