Definition df-inp 7533

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 7358	. 2 class P
2		vl	. . . . . . . 8 setvar 𝑙
3	2	cv 1363	. . . . . . 7 class 𝑙
4		cnq 7347	. . . . . . 7 class Q
5	3, 4	wss 3157	. . . . . 6 wff 𝑙 ⊆ Q
6		vu	. . . . . . . 8 setvar 𝑢
7	6	cv 1363	. . . . . . 7 class 𝑢
8	7, 4	wss 3157	. . . . . 6 wff 𝑢 ⊆ Q
9	5, 8	wa 104	. . . . 5 wff (𝑙 ⊆ Q ∧ 𝑢 ⊆ Q)
10		vq	. . . . . . . 8 setvar 𝑞
11	10, 2	wel 2168	. . . . . . 7 wff 𝑞 ∈ 𝑙
12	11, 10, 4	wrex 2476	. . . . . 6 wff ∃𝑞 ∈ Q 𝑞 ∈ 𝑙
13		vr	. . . . . . . 8 setvar 𝑟
14	13, 6	wel 2168	. . . . . . 7 wff 𝑟 ∈ 𝑢
15	14, 13, 4	wrex 2476	. . . . . 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 1363	. . . . . . . . . . 11 class 𝑞
19	13	cv 1363	. . . . . . . . . . 11 class 𝑟
20		cltq 7352	. . . . . . . . . . 11 class <Q
21	18, 19, 20	wbr 4033	. . . . . . . . . 10 wff 𝑞 <Q 𝑟
22	13, 2	wel 2168	. . . . . . . . . 10 wff 𝑟 ∈ 𝑙
23	21, 22	wa 104	. . . . . . . . 9 wff (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)
24	23, 13, 4	wrex 2476	. . . . . . . 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 2475	. . . . . 6 wff ∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙))
27	10, 6	wel 2168	. . . . . . . . . 10 wff 𝑞 ∈ 𝑢
28	21, 27	wa 104	. . . . . . . . 9 wff (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢)
29	28, 10, 4	wrex 2476	. . . . . . . 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 2475	. . . . . 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 2475	. . . . 5 wff ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢)
36	11, 14	wo 709	. . . . . . . 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 2475	. . . . . 6 wff ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))
39	38, 10, 4	wral 2475	. . . . 5 wff ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))
40	32, 35, 39	w3a 980	. . . 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 4093	. 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 1364	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  7538  elinp  7541