Definition df-ap 7366

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 7440 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 7389), symmetry (apsym 7390), and cotransitivity (apcotr 7391). Apartness implies negated equality, as seen at apne 7407, 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 7406).

(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 7365	. 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 6713	. . . . . . . . . . . 12 class i
7		vs	. . . . . . . . . . . . 13 setvar 𝑠
8	7	cv 1241	. . . . . . . . . . . 12 class 𝑠
9		cmul 6716	. . . . . . . . . . . 12 class ·
10	6, 8, 9	co 5455	. . . . . . . . . . 11 class (i · 𝑠)
11		caddc 6714	. . . . . . . . . . 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 7358	. . . . . . . . . 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 6710	. . . . . . 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  7371  apreim  7387