Definition df-ap 8745

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 8842 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8768), symmetry (apsym 8769), and cotransitivity (apcotr 8770). Apartness implies negated equality, as seen at apne 8786, 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 8785).

(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 8744	. 2 class #
2		vx	. . . . . . . . . . 11 setvar 𝑥
3	2	cv 1394	. . . . . . . . . 10 class 𝑥
4		vr	. . . . . . . . . . . 12 setvar 𝑟
5	4	cv 1394	. . . . . . . . . . 11 class 𝑟
6		ci 8017	. . . . . . . . . . . 12 class i
7		vs	. . . . . . . . . . . . 13 setvar 𝑠
8	7	cv 1394	. . . . . . . . . . . 12 class 𝑠
9		cmul 8020	. . . . . . . . . . . 12 class ·
10	6, 8, 9	co 6010	. . . . . . . . . . 11 class (i · 𝑠)
11		caddc 8018	. . . . . . . . . . 11 class +
12	5, 10, 11	co 6010	. . . . . . . . . 10 class (𝑟 + (i · 𝑠))
13	3, 12	wceq 1395	. . . . . . . . 9 wff 𝑥 = (𝑟 + (i · 𝑠))
14		vy	. . . . . . . . . . 11 setvar 𝑦
15	14	cv 1394	. . . . . . . . . 10 class 𝑦
16		vt	. . . . . . . . . . . 12 setvar 𝑡
17	16	cv 1394	. . . . . . . . . . 11 class 𝑡
18		vu	. . . . . . . . . . . . 13 setvar 𝑢
19	18	cv 1394	. . . . . . . . . . . 12 class 𝑢
20	6, 19, 9	co 6010	. . . . . . . . . . 11 class (i · 𝑢)
21	17, 20, 11	co 6010	. . . . . . . . . 10 class (𝑡 + (i · 𝑢))
22	15, 21	wceq 1395	. . . . . . . . 9 wff 𝑦 = (𝑡 + (i · 𝑢))
23	13, 22	wa 104	. . . . . . . 8 wff (𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢)))
24		creap 8737	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 4083	. . . . . . . . 9 wff 𝑟 #ℝ 𝑡
26	8, 19, 24	wbr 4083	. . . . . . . . 9 wff 𝑠 #ℝ 𝑢
27	25, 26	wo 713	. . . . . . . 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 8014	. . . . . . 7 class ℝ
30	28, 18, 29	wrex 2509	. . . . . 6 wff ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))
31	30, 16, 29	wrex 2509	. . . . 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 2509	. . . 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 2509	. . 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 4144	. 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 1395	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  8750  apreim  8766  aprcl  8809  aptap  8813