Definition df-inp 7407

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 (∃𝑞 ∈ Q𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q𝑟 ∈ 𝑢), rounded (∀𝑞 ∈ Q(𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q(𝑞 <_Q 𝑟 ∧ 𝑟 ∈ 𝑙)) and likewise for 𝑢), disjoint (∀𝑞 ∈ Q¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢)) and located (∀𝑞 ∈ Q∀𝑟 ∈ Q(𝑞 <_Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))). 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⟩ ∣ (((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))))}

Distinct variable group:   𝑢,𝑙,𝑞,𝑟

Detailed syntax breakdown of Definition df-inp

Step	Hyp	Ref	Expression
1		cnp 7232	. 2 class P
2		vl	. . . . . . . 8 setvar 𝑙
3	2	cv 1342	. . . . . . 7 class 𝑙
4		cnq 7221	. . . . . . 7 class Q
5	3, 4	wss 3116	. . . . . 6 wff 𝑙 ⊆ Q
6		vu	. . . . . . . 8 setvar 𝑢
7	6	cv 1342	. . . . . . 7 class 𝑢
8	7, 4	wss 3116	. . . . . 6 wff 𝑢 ⊆ Q
9	5, 8	wa 103	. . . . 5 wff (𝑙 ⊆ Q ∧ 𝑢 ⊆ Q)
10		vq	. . . . . . . 8 setvar 𝑞
11	10, 2	wel 2137	. . . . . . 7 wff 𝑞 ∈ 𝑙
12	11, 10, 4	wrex 2445	. . . . . 6 wff ∃𝑞 ∈ Q 𝑞 ∈ 𝑙
13		vr	. . . . . . . 8 setvar 𝑟
14	13, 6	wel 2137	. . . . . . 7 wff 𝑟 ∈ 𝑢
15	14, 13, 4	wrex 2445	. . . . . 6 wff ∃𝑟 ∈ Q 𝑟 ∈ 𝑢
16	12, 15	wa 103	. . . . 5 wff (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)
17	9, 16	wa 103	. . . 4 wff ((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢))
18	10	cv 1342	. . . . . . . . . . 11 class 𝑞
19	13	cv 1342	. . . . . . . . . . 11 class 𝑟
20		cltq 7226	. . . . . . . . . . 11 class <Q
21	18, 19, 20	wbr 3982	. . . . . . . . . 10 wff 𝑞 <Q 𝑟
22	13, 2	wel 2137	. . . . . . . . . 10 wff 𝑟 ∈ 𝑙
23	21, 22	wa 103	. . . . . . . . 9 wff (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)
24	23, 13, 4	wrex 2445	. . . . . . . 8 wff ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)
25	11, 24	wb 104	. . . . . . 7 wff (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙))
26	25, 10, 4	wral 2444	. . . . . 6 wff ∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙))
27	10, 6	wel 2137	. . . . . . . . . 10 wff 𝑞 ∈ 𝑢
28	21, 27	wa 103	. . . . . . . . 9 wff (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢)
29	28, 10, 4	wrex 2445	. . . . . . . 8 wff ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢)
30	14, 29	wb 104	. . . . . . 7 wff (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))
31	30, 13, 4	wral 2444	. . . . . 6 wff ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))
32	26, 31	wa 103	. . . . 5 wff (∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢)))
33	11, 27	wa 103	. . . . . . 7 wff (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢)
34	33	wn 3	. . . . . 6 wff ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢)
35	34, 10, 4	wral 2444	. . . . 5 wff ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢)
36	11, 14	wo 698	. . . . . . . 8 wff (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢)
37	21, 36	wi 4	. . . . . . 7 wff (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))
38	37, 13, 4	wral 2444	. . . . . 6 wff ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))
39	38, 10, 4	wral 2444	. . . . 5 wff ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))
40	32, 35, 39	w3a 968	. . . 4 wff ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢)))
41	17, 40	wa 103	. . 3 wff (((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))))
42	41, 2, 6	copab 4042	. 2 class {⟨𝑙, 𝑢⟩ ∣ (((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))))}
43	1, 42	wceq 1343	1 wff P = {⟨𝑙, 𝑢⟩ ∣ (((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))))}

This definition is referenced by:  npsspw  7412  elinp  7415