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Definition df-inp 6788

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 6613	. 2
2		vl	. . . . . . . 8
3	2	cv 1284	. . . . . . 7
4		cnq 6602	. . . . . . 7
5	3, 4	wss 2982	. . . . . 6
6		vu	. . . . . . . 8
7	6	cv 1284	. . . . . . 7
8	7, 4	wss 2982	. . . . . 6
9	5, 8	wa 102	. . . . 5 $/\$
10		vq	. . . . . . . 8
11	10, 2	wel 1435	. . . . . . 7
12	11, 10, 4	wrex 2354	. . . . . 6
13		vr	. . . . . . . 8
14	13, 6	wel 1435	. . . . . . 7
15	14, 13, 4	wrex 2354	. . . . . 6
16	12, 15	wa 102	. . . . 5 $/\$
17	9, 16	wa 102	. . . 4 $/\$ $/\$ $/\$
18	10	cv 1284	. . . . . . . . . . 11
19	13	cv 1284	. . . . . . . . . . 11
20		cltq 6607	. . . . . . . . . . 11
21	18, 19, 20	wbr 3805	. . . . . . . . . 10
22	13, 2	wel 1435	. . . . . . . . . 10
23	21, 22	wa 102	. . . . . . . . 9 $/\$
24	23, 13, 4	wrex 2354	. . . . . . . 8 $/\$
25	11, 24	wb 103	. . . . . . 7 $/\$
26	25, 10, 4	wral 2353	. . . . . 6 $/\$
27	10, 6	wel 1435	. . . . . . . . . 10
28	21, 27	wa 102	. . . . . . . . 9 $/\$
29	28, 10, 4	wrex 2354	. . . . . . . 8 $/\$
30	14, 29	wb 103	. . . . . . 7 $/\$
31	30, 13, 4	wral 2353	. . . . . 6 $/\$
32	26, 31	wa 102	. . . . 5 $/\$ $/\$ $/\$
33	11, 27	wa 102	. . . . . . 7 $/\$
34	33	wn 3	. . . . . 6 $/\$
35	34, 10, 4	wral 2353	. . . . 5 $/\$
36	11, 14	wo 662	. . . . . . . 8 $\/$
37	21, 36	wi 4	. . . . . . 7 $\/$
38	37, 13, 4	wral 2353	. . . . . 6 $\/$
39	38, 10, 4	wral 2353	. . . . 5 $\/$
40	32, 35, 39	w3a 920	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
41	17, 40	wa 102	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
42	41, 2, 6	copab 3858	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
43	1, 42	wceq 1285	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

This definition is referenced by: npsspw 6793 elinp 6796

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