Definition df-ap 8852

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 8949 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8875), symmetry (apsym 8876), and cotransitivity (apcotr 8877). Apartness implies negated equality, as seen at apne 8893, 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 8892).

(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 8851	. 2 class #
2		vx	. . . . . . . . . . 11 setvar 𝑥
3	2	cv 1397	. . . . . . . . . 10 class 𝑥
4		vr	. . . . . . . . . . . 12 setvar 𝑟
5	4	cv 1397	. . . . . . . . . . 11 class 𝑟
6		ci 8125	. . . . . . . . . . . 12 class i
7		vs	. . . . . . . . . . . . 13 setvar 𝑠
8	7	cv 1397	. . . . . . . . . . . 12 class 𝑠
9		cmul 8128	. . . . . . . . . . . 12 class ·
10	6, 8, 9	co 6049	. . . . . . . . . . 11 class (i · 𝑠)
11		caddc 8126	. . . . . . . . . . 11 class +
12	5, 10, 11	co 6049	. . . . . . . . . 10 class (𝑟 + (i · 𝑠))
13	3, 12	wceq 1398	. . . . . . . . 9 wff 𝑥 = (𝑟 + (i · 𝑠))
14		vy	. . . . . . . . . . 11 setvar 𝑦
15	14	cv 1397	. . . . . . . . . 10 class 𝑦
16		vt	. . . . . . . . . . . 12 setvar 𝑡
17	16	cv 1397	. . . . . . . . . . 11 class 𝑡
18		vu	. . . . . . . . . . . . 13 setvar 𝑢
19	18	cv 1397	. . . . . . . . . . . 12 class 𝑢
20	6, 19, 9	co 6049	. . . . . . . . . . 11 class (i · 𝑢)
21	17, 20, 11	co 6049	. . . . . . . . . 10 class (𝑡 + (i · 𝑢))
22	15, 21	wceq 1398	. . . . . . . . 9 wff 𝑦 = (𝑡 + (i · 𝑢))
23	13, 22	wa 104	. . . . . . . 8 wff (𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢)))
24		creap 8844	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 4108	. . . . . . . . 9 wff 𝑟 #ℝ 𝑡
26	8, 19, 24	wbr 4108	. . . . . . . . 9 wff 𝑠 #ℝ 𝑢
27	25, 26	wo 716	. . . . . . . 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 8122	. . . . . . 7 class ℝ
30	28, 18, 29	wrex 2521	. . . . . 6 wff ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))
31	30, 16, 29	wrex 2521	. . . . 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 2521	. . . 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 2521	. . 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 4169	. 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 1398	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  8857  apreim  8873  aprcl  8916  aptap  8920