Definition df-inp 7526

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 7351	. 2 class P
2		vl	. . . . . . . 8 setvar 𝑙
3	2	cv 1363	. . . . . . 7 class 𝑙
4		cnq 7340	. . . . . . 7 class Q
5	3, 4	wss 3153	. . . . . 6 wff 𝑙 ⊆ Q
6		vu	. . . . . . . 8 setvar 𝑢
7	6	cv 1363	. . . . . . 7 class 𝑢
8	7, 4	wss 3153	. . . . . 6 wff 𝑢 ⊆ Q
9	5, 8	wa 104	. . . . 5 wff (𝑙 ⊆ Q ∧ 𝑢 ⊆ Q)
10		vq	. . . . . . . 8 setvar 𝑞
11	10, 2	wel 2165	. . . . . . 7 wff 𝑞 ∈ 𝑙
12	11, 10, 4	wrex 2473	. . . . . 6 wff ∃𝑞 ∈ Q 𝑞 ∈ 𝑙
13		vr	. . . . . . . 8 setvar 𝑟
14	13, 6	wel 2165	. . . . . . 7 wff 𝑟 ∈ 𝑢
15	14, 13, 4	wrex 2473	. . . . . 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 7345	. . . . . . . . . . 11 class <Q
21	18, 19, 20	wbr 4029	. . . . . . . . . 10 wff 𝑞 <Q 𝑟
22	13, 2	wel 2165	. . . . . . . . . 10 wff 𝑟 ∈ 𝑙
23	21, 22	wa 104	. . . . . . . . 9 wff (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)
24	23, 13, 4	wrex 2473	. . . . . . . 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 2472	. . . . . 6 wff ∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙))
27	10, 6	wel 2165	. . . . . . . . . 10 wff 𝑞 ∈ 𝑢
28	21, 27	wa 104	. . . . . . . . 9 wff (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢)
29	28, 10, 4	wrex 2473	. . . . . . . 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 2472	. . . . . 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 2472	. . . . 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 2472	. . . . . 6 wff ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))
39	38, 10, 4	wral 2472	. . . . 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 4089	. 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  7531  elinp  7534