Definition df-inp 7464

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 7289	. 2 class P
2		vl	. . . . . . . 8 setvar 𝑙
3	2	cv 1352	. . . . . . 7 class 𝑙
4		cnq 7278	. . . . . . 7 class Q
5	3, 4	wss 3129	. . . . . 6 wff 𝑙 ⊆ Q
6		vu	. . . . . . . 8 setvar 𝑢
7	6	cv 1352	. . . . . . 7 class 𝑢
8	7, 4	wss 3129	. . . . . 6 wff 𝑢 ⊆ Q
9	5, 8	wa 104	. . . . 5 wff (𝑙 ⊆ Q ∧ 𝑢 ⊆ Q)
10		vq	. . . . . . . 8 setvar 𝑞
11	10, 2	wel 2149	. . . . . . 7 wff 𝑞 ∈ 𝑙
12	11, 10, 4	wrex 2456	. . . . . 6 wff ∃𝑞 ∈ Q 𝑞 ∈ 𝑙
13		vr	. . . . . . . 8 setvar 𝑟
14	13, 6	wel 2149	. . . . . . 7 wff 𝑟 ∈ 𝑢
15	14, 13, 4	wrex 2456	. . . . . 6 wff ∃𝑟 ∈ Q 𝑟 ∈ 𝑢
16	12, 15	wa 104	. . . . 5 wff (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)
17	9, 16	wa 104	. . . 4 wff ((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢))
18	10	cv 1352	. . . . . . . . . . 11 class 𝑞
19	13	cv 1352	. . . . . . . . . . 11 class 𝑟
20		cltq 7283	. . . . . . . . . . 11 class <Q
21	18, 19, 20	wbr 4003	. . . . . . . . . 10 wff 𝑞 <Q 𝑟
22	13, 2	wel 2149	. . . . . . . . . 10 wff 𝑟 ∈ 𝑙
23	21, 22	wa 104	. . . . . . . . 9 wff (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)
24	23, 13, 4	wrex 2456	. . . . . . . 8 wff ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)
25	11, 24	wb 105	. . . . . . 7 wff (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙))
26	25, 10, 4	wral 2455	. . . . . 6 wff ∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙))
27	10, 6	wel 2149	. . . . . . . . . 10 wff 𝑞 ∈ 𝑢
28	21, 27	wa 104	. . . . . . . . 9 wff (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢)
29	28, 10, 4	wrex 2456	. . . . . . . 8 wff ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢)
30	14, 29	wb 105	. . . . . . 7 wff (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))
31	30, 13, 4	wral 2455	. . . . . 6 wff ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))
32	26, 31	wa 104	. . . . 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 104	. . . . . . 7 wff (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢)
34	33	wn 3	. . . . . 6 wff ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢)
35	34, 10, 4	wral 2455	. . . . 5 wff ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢)
36	11, 14	wo 708	. . . . . . . 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 2455	. . . . . 6 wff ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))
39	38, 10, 4	wral 2455	. . . . 5 wff ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))
40	32, 35, 39	w3a 978	. . . 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 104	. . 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 4063	. 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 1353	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  7469  elinp  7472