Definition df-ap 8494

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 8589 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8517), symmetry (apsym 8518), and cotransitivity (apcotr 8519). Apartness implies negated equality, as seen at apne 8535, 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 8534).

(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 8493	. 2 class #
2		vx	. . . . . . . . . . 11 setvar 𝑥
3	2	cv 1347	. . . . . . . . . 10 class 𝑥
4		vr	. . . . . . . . . . . 12 setvar 𝑟
5	4	cv 1347	. . . . . . . . . . 11 class 𝑟
6		ci 7769	. . . . . . . . . . . 12 class i
7		vs	. . . . . . . . . . . . 13 setvar 𝑠
8	7	cv 1347	. . . . . . . . . . . 12 class 𝑠
9		cmul 7772	. . . . . . . . . . . 12 class ·
10	6, 8, 9	co 5851	. . . . . . . . . . 11 class (i · 𝑠)
11		caddc 7770	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5851	. . . . . . . . . 10 class (𝑟 + (i · 𝑠))
13	3, 12	wceq 1348	. . . . . . . . 9 wff 𝑥 = (𝑟 + (i · 𝑠))
14		vy	. . . . . . . . . . 11 setvar 𝑦
15	14	cv 1347	. . . . . . . . . 10 class 𝑦
16		vt	. . . . . . . . . . . 12 setvar 𝑡
17	16	cv 1347	. . . . . . . . . . 11 class 𝑡
18		vu	. . . . . . . . . . . . 13 setvar 𝑢
19	18	cv 1347	. . . . . . . . . . . 12 class 𝑢
20	6, 19, 9	co 5851	. . . . . . . . . . 11 class (i · 𝑢)
21	17, 20, 11	co 5851	. . . . . . . . . 10 class (𝑡 + (i · 𝑢))
22	15, 21	wceq 1348	. . . . . . . . 9 wff 𝑦 = (𝑡 + (i · 𝑢))
23	13, 22	wa 103	. . . . . . . 8 wff (𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢)))
24		creap 8486	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 3987	. . . . . . . . 9 wff 𝑟 #ℝ 𝑡
26	8, 19, 24	wbr 3987	. . . . . . . . 9 wff 𝑠 #ℝ 𝑢
27	25, 26	wo 703	. . . . . . . 8 wff (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)
28	23, 27	wa 103	. . . . . . 7 wff ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))
29		cr 7766	. . . . . . 7 class ℝ
30	28, 18, 29	wrex 2449	. . . . . 6 wff ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))
31	30, 16, 29	wrex 2449	. . . . 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 2449	. . . 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 2449	. . 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 4047	. 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 1348	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  8499  apreim  8515  aprcl  8558