Definition df-ap 7346

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 7420 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 7369), symmetry (apsym 7370), and cotransitivity (apcotr 7371). Apartness implies negated equality, as seen at apne 7387, 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 7386).

(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 7345	. 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 6693	. . . . . . . . . . . 12 class i
7		vs	. . . . . . . . . . . . 13 setvar 𝑠
8	7	cv 1241	. . . . . . . . . . . 12 class 𝑠
9		cmul 6696	. . . . . . . . . . . 12 class ·
10	6, 8, 9	co 5455	. . . . . . . . . . 11 class (i · 𝑠)
11		caddc 6694	. . . . . . . . . . 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 7338	. . . . . . . . . 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 6690	. . . . . . 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  7351  apreim  7367