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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 𝑠 𝑡 u ℝ ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))}
Distinct variable group:   𝑠,𝑟,𝑡,u,x,y

Detailed syntax breakdown of Definition df-ap
StepHypRef Expression
1 cap 7317 . 2 class #
2 vx . . . . . . . . . . 11 setvar x
32cv 1241 . . . . . . . . . 10 class x
4 vr . . . . . . . . . . . 12 setvar 𝑟
54cv 1241 . . . . . . . . . . 11 class 𝑟
6 ci 6665 . . . . . . . . . . . 12 class i
7 vs . . . . . . . . . . . . 13 setvar 𝑠
87cv 1241 . . . . . . . . . . . 12 class 𝑠
9 cmul 6668 . . . . . . . . . . . 12 class ·
106, 8, 9co 5452 . . . . . . . . . . 11 class (i · 𝑠)
11 caddc 6666 . . . . . . . . . . 11 class +
125, 10, 11co 5452 . . . . . . . . . 10 class (𝑟 + (i · 𝑠))
133, 12wceq 1242 . . . . . . . . 9 wff x = (𝑟 + (i · 𝑠))
14 vy . . . . . . . . . . 11 setvar y
1514cv 1241 . . . . . . . . . 10 class y
16 vt . . . . . . . . . . . 12 setvar 𝑡
1716cv 1241 . . . . . . . . . . 11 class 𝑡
18 vu . . . . . . . . . . . . 13 setvar u
1918cv 1241 . . . . . . . . . . . 12 class u
206, 19, 9co 5452 . . . . . . . . . . 11 class (i · u)
2117, 20, 11co 5452 . . . . . . . . . 10 class (𝑡 + (i · u))
2215, 21wceq 1242 . . . . . . . . 9 wff y = (𝑡 + (i · u))
2313, 22wa 97 . . . . . . . 8 wff (x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u)))
24 creap 7310 . . . . . . . . . 10 class #
255, 17, 24wbr 3754 . . . . . . . . 9 wff 𝑟 # 𝑡
268, 19, 24wbr 3754 . . . . . . . . 9 wff 𝑠 # u
2725, 26wo 628 . . . . . . . 8 wff (𝑟 # 𝑡 𝑠 # u)
2823, 27wa 97 . . . . . . 7 wff ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))
29 cr 6662 . . . . . . 7 class
3028, 18, 29wrex 2301 . . . . . 6 wff u ℝ ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))
3130, 16, 29wrex 2301 . . . . 5 wff 𝑡 u ℝ ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))
3231, 7, 29wrex 2301 . . . 4 wff 𝑠 𝑡 u ℝ ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))
3332, 4, 29wrex 2301 . . 3 wff 𝑟 𝑠 𝑡 u ℝ ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))
3433, 2, 14copab 3807 . 2 class {⟨x, y⟩ ∣ 𝑟 𝑠 𝑡 u ℝ ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))}
351, 34wceq 1242 1 wff # = {⟨x, y⟩ ∣ 𝑟 𝑠 𝑡 u ℝ ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))}
Colors of variables: wff set class
This definition is referenced by:  apreap  7323  apreim  7339
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