Definition df-ap 7326

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 7400 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 7349), symmetry (apsym 7350), and cotransitivity (apcotr 7351). Apartness implies negated equality, as seen at apne 7367, 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 7366).

(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 7325	. 2 class #
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1241	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . . 12 setvar 𝑟
5	4	cv 1241	. . . . . . . . . . 11 class 𝑟
6		ci 6673	. . . . . . . . . . . 12 class i
7		vs	. . . . . . . . . . . . 13 setvar 𝑠
8	7	cv 1241	. . . . . . . . . . . 12 class 𝑠
9		cmul 6676	. . . . . . . . . . . 12 class ·
10	6, 8, 9	co 5455	. . . . . . . . . . 11 class (i · 𝑠)
11		caddc 6674	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5455	. . . . . . . . . 10 class (𝑟 + (i · 𝑠))
13	3, 12	wceq 1242	. . . . . . . . 9 wff x = (𝑟 + (i · 𝑠))
14		vy	. . . . . . . . . . 11 setvar y
15	14	cv 1241	. . . . . . . . . 10 class y
16		vt	. . . . . . . . . . . 12 setvar 𝑡
17	16	cv 1241	. . . . . . . . . . 11 class 𝑡
18		vu	. . . . . . . . . . . . 13 setvar u
19	18	cv 1241	. . . . . . . . . . . 12 class u
20	6, 19, 9	co 5455	. . . . . . . . . . 11 class (i · u)
21	17, 20, 11	co 5455	. . . . . . . . . 10 class (𝑡 + (i · u))
22	15, 21	wceq 1242	. . . . . . . . 9 wff y = (𝑡 + (i · u))
23	13, 22	wa 97	. . . . . . . 8 wff (x = (𝑟 + (i · 𝑠)) ∧ y = (𝑡 + (i · u)))
24		creap 7318	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 3755	. . . . . . . . 9 wff 𝑟 #ℝ 𝑡
26	8, 19, 24	wbr 3755	. . . . . . . . 9 wff 𝑠 #ℝ u
27	25, 26	wo 628	. . . . . . . 8 wff (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ u)
28	23, 27	wa 97	. . . . . . 7 wff ((x = (𝑟 + (i · 𝑠)) ∧ y = (𝑡 + (i · u))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ u))
29		cr 6670	. . . . . . 7 class ℝ
30	28, 18, 29	wrex 2301	. . . . . 6 wff ∃u ∈ ℝ ((x = (𝑟 + (i · 𝑠)) ∧ y = (𝑡 + (i · u))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ u))
31	30, 16, 29	wrex 2301	. . . . 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 2301	. . . 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 2301	. . 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 3808	. 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 1242	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  7331  apreim  7347