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Definition df-inp 7298

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 7123	. 2
2		vl	. . . . . . . 8
3	2	cv 1331	. . . . . . 7
4		cnq 7112	. . . . . . 7
5	3, 4	wss 3076	. . . . . 6
6		vu	. . . . . . . 8
7	6	cv 1331	. . . . . . 7
8	7, 4	wss 3076	. . . . . 6
9	5, 8	wa 103	. . . . 5 $/\$
10		vq	. . . . . . . 8
11	10, 2	wel 1482	. . . . . . 7
12	11, 10, 4	wrex 2418	. . . . . 6
13		vr	. . . . . . . 8
14	13, 6	wel 1482	. . . . . . 7
15	14, 13, 4	wrex 2418	. . . . . 6
16	12, 15	wa 103	. . . . 5 $/\$
17	9, 16	wa 103	. . . 4 $/\$ $/\$ $/\$
18	10	cv 1331	. . . . . . . . . . 11
19	13	cv 1331	. . . . . . . . . . 11
20		cltq 7117	. . . . . . . . . . 11
21	18, 19, 20	wbr 3937	. . . . . . . . . 10
22	13, 2	wel 1482	. . . . . . . . . 10
23	21, 22	wa 103	. . . . . . . . 9 $/\$
24	23, 13, 4	wrex 2418	. . . . . . . 8 $/\$
25	11, 24	wb 104	. . . . . . 7 $/\$
26	25, 10, 4	wral 2417	. . . . . 6 $/\$
27	10, 6	wel 1482	. . . . . . . . . 10
28	21, 27	wa 103	. . . . . . . . 9 $/\$
29	28, 10, 4	wrex 2418	. . . . . . . 8 $/\$
30	14, 29	wb 104	. . . . . . 7 $/\$
31	30, 13, 4	wral 2417	. . . . . 6 $/\$
32	26, 31	wa 103	. . . . 5 $/\$ $/\$ $/\$
33	11, 27	wa 103	. . . . . . 7 $/\$
34	33	wn 3	. . . . . 6 $/\$
35	34, 10, 4	wral 2417	. . . . 5 $/\$
36	11, 14	wo 698	. . . . . . . 8 $\/$
37	21, 36	wi 4	. . . . . . 7 $\/$
38	37, 13, 4	wral 2417	. . . . . 6 $\/$
39	38, 10, 4	wral 2417	. . . . 5 $\/$
40	32, 35, 39	w3a 963	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
41	17, 40	wa 103	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
42	41, 2, 6	copab 3996	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
43	1, 42	wceq 1332	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

This definition is referenced by: npsspw 7303 elinp 7306

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