Definition df-ap 8737

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 8834 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8760), symmetry (apsym 8761), and cotransitivity (apcotr 8762). Apartness implies negated equality, as seen at apne 8778, 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 8777).

(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 8736	. 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 8009	. . . . . . . . . . . 12 class i
7		vs	. . . . . . . . . . . . 13 setvar 𝑠
8	7	cv 1394	. . . . . . . . . . . 12 class 𝑠
9		cmul 8012	. . . . . . . . . . . 12 class ·
10	6, 8, 9	co 6007	. . . . . . . . . . 11 class (i · 𝑠)
11		caddc 8010	. . . . . . . . . . 11 class +
12	5, 10, 11	co 6007	. . . . . . . . . 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 6007	. . . . . . . . . . 11 class (i · 𝑢)
21	17, 20, 11	co 6007	. . . . . . . . . 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 8729	. . . . . . . . . 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 8006	. . . . . . 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  8742  apreim  8758  aprcl  8801  aptap  8805