ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-inp GIF version

Definition df-inp 6562
Description: Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other.

Here we follow the definition of a Dedekind cut from Definition 11.2.1 of [HoTT], p. (varies) with the one exception that we define it over positive rational numbers rather than all rational numbers.

A Dedekind cut is an ordered pair of a lower set 𝑙 and an upper set 𝑢 which is inhabited (𝑞Q𝑞𝑙 ∧ ∃𝑟Q𝑟𝑢), rounded (𝑞Q(𝑞𝑙 ↔ ∃𝑟Q(𝑞 <Q 𝑟𝑟𝑙)) and likewise for 𝑢), disjoint (𝑞Q¬ (𝑞𝑙𝑞𝑢)) and located (𝑞Q𝑟Q(𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))). See HoTT for more discussion of those terms and different ways of defining Dedekind cuts.

(Note: This is a "temporary" definition used in the construction of complex numbers, and is intended to be used only by the construction.) (Contributed by Jim Kingdon, 25-Sep-2019.)

Assertion
Ref Expression
df-inp P = {⟨𝑙, 𝑢⟩ ∣ (((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)) ∧ ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))))}
Distinct variable group:   𝑢,𝑙,𝑞,𝑟

Detailed syntax breakdown of Definition df-inp
StepHypRef Expression
1 cnp 6387 . 2 class P
2 vl . . . . . . . 8 setvar 𝑙
32cv 1242 . . . . . . 7 class 𝑙
4 cnq 6376 . . . . . . 7 class Q
53, 4wss 2917 . . . . . 6 wff 𝑙Q
6 vu . . . . . . . 8 setvar 𝑢
76cv 1242 . . . . . . 7 class 𝑢
87, 4wss 2917 . . . . . 6 wff 𝑢Q
95, 8wa 97 . . . . 5 wff (𝑙Q𝑢Q)
10 vq . . . . . . . 8 setvar 𝑞
1110, 2wel 1394 . . . . . . 7 wff 𝑞𝑙
1211, 10, 4wrex 2307 . . . . . 6 wff 𝑞Q 𝑞𝑙
13 vr . . . . . . . 8 setvar 𝑟
1413, 6wel 1394 . . . . . . 7 wff 𝑟𝑢
1514, 13, 4wrex 2307 . . . . . 6 wff 𝑟Q 𝑟𝑢
1612, 15wa 97 . . . . 5 wff (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)
179, 16wa 97 . . . 4 wff ((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢))
1810cv 1242 . . . . . . . . . . 11 class 𝑞
1913cv 1242 . . . . . . . . . . 11 class 𝑟
20 cltq 6381 . . . . . . . . . . 11 class <Q
2118, 19, 20wbr 3764 . . . . . . . . . 10 wff 𝑞 <Q 𝑟
2213, 2wel 1394 . . . . . . . . . 10 wff 𝑟𝑙
2321, 22wa 97 . . . . . . . . 9 wff (𝑞 <Q 𝑟𝑟𝑙)
2423, 13, 4wrex 2307 . . . . . . . 8 wff 𝑟Q (𝑞 <Q 𝑟𝑟𝑙)
2511, 24wb 98 . . . . . . 7 wff (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙))
2625, 10, 4wral 2306 . . . . . 6 wff 𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙))
2710, 6wel 1394 . . . . . . . . . 10 wff 𝑞𝑢
2821, 27wa 97 . . . . . . . . 9 wff (𝑞 <Q 𝑟𝑞𝑢)
2928, 10, 4wrex 2307 . . . . . . . 8 wff 𝑞Q (𝑞 <Q 𝑟𝑞𝑢)
3014, 29wb 98 . . . . . . 7 wff (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))
3130, 13, 4wral 2306 . . . . . 6 wff 𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))
3226, 31wa 97 . . . . 5 wff (∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢)))
3311, 27wa 97 . . . . . . 7 wff (𝑞𝑙𝑞𝑢)
3433wn 3 . . . . . 6 wff ¬ (𝑞𝑙𝑞𝑢)
3534, 10, 4wral 2306 . . . . 5 wff 𝑞Q ¬ (𝑞𝑙𝑞𝑢)
3611, 14wo 629 . . . . . . . 8 wff (𝑞𝑙𝑟𝑢)
3721, 36wi 4 . . . . . . 7 wff (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))
3837, 13, 4wral 2306 . . . . . 6 wff 𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))
3938, 10, 4wral 2306 . . . . 5 wff 𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))
4032, 35, 39w3a 885 . . . 4 wff ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢)))
4117, 40wa 97 . . 3 wff (((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)) ∧ ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))))
4241, 2, 6copab 3817 . 2 class {⟨𝑙, 𝑢⟩ ∣ (((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)) ∧ ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))))}
431, 42wceq 1243 1 wff P = {⟨𝑙, 𝑢⟩ ∣ (((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)) ∧ ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))))}
Colors of variables: wff set class
This definition is referenced by:  npsspw  6567  elinp  6570
  Copyright terms: Public domain W3C validator