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Definition df-inp 6656

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 6481	. 2
2		vl	. . . . . . . 8
3	2	cv 1283	. . . . . . 7
4		cnq 6470	. . . . . . 7
5	3, 4	wss 2973	. . . . . 6
6		vu	. . . . . . . 8
7	6	cv 1283	. . . . . . 7
8	7, 4	wss 2973	. . . . . 6
9	5, 8	wa 102	. . . . 5 $/\$
10		vq	. . . . . . . 8
11	10, 2	wel 1434	. . . . . . 7
12	11, 10, 4	wrex 2349	. . . . . 6
13		vr	. . . . . . . 8
14	13, 6	wel 1434	. . . . . . 7
15	14, 13, 4	wrex 2349	. . . . . 6
16	12, 15	wa 102	. . . . 5 $/\$
17	9, 16	wa 102	. . . 4 $/\$ $/\$ $/\$
18	10	cv 1283	. . . . . . . . . . 11
19	13	cv 1283	. . . . . . . . . . 11
20		cltq 6475	. . . . . . . . . . 11
21	18, 19, 20	wbr 3785	. . . . . . . . . 10
22	13, 2	wel 1434	. . . . . . . . . 10
23	21, 22	wa 102	. . . . . . . . 9 $/\$
24	23, 13, 4	wrex 2349	. . . . . . . 8 $/\$
25	11, 24	wb 103	. . . . . . 7 $/\$
26	25, 10, 4	wral 2348	. . . . . 6 $/\$
27	10, 6	wel 1434	. . . . . . . . . 10
28	21, 27	wa 102	. . . . . . . . 9 $/\$
29	28, 10, 4	wrex 2349	. . . . . . . 8 $/\$
30	14, 29	wb 103	. . . . . . 7 $/\$
31	30, 13, 4	wral 2348	. . . . . 6 $/\$
32	26, 31	wa 102	. . . . 5 $/\$ $/\$ $/\$
33	11, 27	wa 102	. . . . . . 7 $/\$
34	33	wn 3	. . . . . 6 $/\$
35	34, 10, 4	wral 2348	. . . . 5 $/\$
36	11, 14	wo 661	. . . . . . . 8 $\/$
37	21, 36	wi 4	. . . . . . 7 $\/$
38	37, 13, 4	wral 2348	. . . . . 6 $\/$
39	38, 10, 4	wral 2348	. . . . 5 $\/$
40	32, 35, 39	w3a 919	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
41	17, 40	wa 102	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
42	41, 2, 6	copab 3838	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
43	1, 42	wceq 1284	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

This definition is referenced by: npsspw 6661 elinp 6664

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