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Definition df-inp 7780

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 and an upper set which is inhabited ( $/\$ ), rounded ( $/\$ and likewise for ), disjoint ( $/\$ ) and located ( $\/$ ). 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	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Distinct variable group:

**Detailed syntax breakdown of Definition df-inp**
Step	Hyp	Ref	Expression
1		cnp 7605	. 2
2		vl	. . . . . . . 8
3	2	cv 1397	. . . . . . 7
4		cnq 7594	. . . . . . 7
5	3, 4	wss 3210	. . . . . 6
6		vu	. . . . . . . 8
7	6	cv 1397	. . . . . . 7
8	7, 4	wss 3210	. . . . . 6
9	5, 8	wa 104	. . . . 5 $/\$
10		vq	. . . . . . . 8
11	10, 2	wel 2204	. . . . . . 7
12	11, 10, 4	wrex 2521	. . . . . 6
13		vr	. . . . . . . 8
14	13, 6	wel 2204	. . . . . . 7
15	14, 13, 4	wrex 2521	. . . . . 6
16	12, 15	wa 104	. . . . 5 $/\$
17	9, 16	wa 104	. . . 4 $/\$ $/\$ $/\$
18	10	cv 1397	. . . . . . . . . . 11
19	13	cv 1397	. . . . . . . . . . 11
20		cltq 7599	. . . . . . . . . . 11
21	18, 19, 20	wbr 4108	. . . . . . . . . 10
22	13, 2	wel 2204	. . . . . . . . . 10
23	21, 22	wa 104	. . . . . . . . 9 $/\$
24	23, 13, 4	wrex 2521	. . . . . . . 8 $/\$
25	11, 24	wb 105	. . . . . . 7 $/\$
26	25, 10, 4	wral 2520	. . . . . 6 $/\$
27	10, 6	wel 2204	. . . . . . . . . 10
28	21, 27	wa 104	. . . . . . . . 9 $/\$
29	28, 10, 4	wrex 2521	. . . . . . . 8 $/\$
30	14, 29	wb 105	. . . . . . 7 $/\$
31	30, 13, 4	wral 2520	. . . . . 6 $/\$
32	26, 31	wa 104	. . . . 5 $/\$ $/\$ $/\$
33	11, 27	wa 104	. . . . . . 7 $/\$
34	33	wn 3	. . . . . 6 $/\$
35	34, 10, 4	wral 2520	. . . . 5 $/\$
36	11, 14	wo 716	. . . . . . . 8 $\/$
37	21, 36	wi 4	. . . . . . 7 $\/$
38	37, 13, 4	wral 2520	. . . . . 6 $\/$
39	38, 10, 4	wral 2520	. . . . 5 $\/$
40	32, 35, 39	w3a 1005	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
41	17, 40	wa 104	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
42	41, 2, 6	copab 4169	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
43	1, 42	wceq 1398	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

This definition is referenced by: npsspw 7785 elinp 7788

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