Definition df-ap 7318

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 7392 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 7341), symmetry (apsym 7342), and cotransitivity (apcotr 7343). Apartness implies negated equality, as seen at apne 7359, 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 7358).

(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 7317	. 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 6665	. . . . . . . . . . . 12 class i
7		vs	. . . . . . . . . . . . 13 setvar 𝑠
8	7	cv 1241	. . . . . . . . . . . 12 class 𝑠
9		cmul 6668	. . . . . . . . . . . 12 class ·
10	6, 8, 9	co 5452	. . . . . . . . . . 11 class (i · 𝑠)
11		caddc 6666	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5452	. . . . . . . . . 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 5452	. . . . . . . . . . 11 class (i · u)
21	17, 20, 11	co 5452	. . . . . . . . . 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 7310	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 3754	. . . . . . . . 9 wff 𝑟 #ℝ 𝑡
26	8, 19, 24	wbr 3754	. . . . . . . . 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 6662	. . . . . . 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 3807	. 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  7323  apreim  7339