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Definition df-inp 6621

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 6446	. 2
2		vl	. . . . . . . 8
3	2	cv 1258	. . . . . . 7
4		cnq 6435	. . . . . . 7
5	3, 4	wss 2944	. . . . . 6
6		vu	. . . . . . . 8
7	6	cv 1258	. . . . . . 7
8	7, 4	wss 2944	. . . . . 6
9	5, 8	wa 101	. . . . 5 $/\$
10		vq	. . . . . . . 8
11	10, 2	wel 1410	. . . . . . 7
12	11, 10, 4	wrex 2324	. . . . . 6
13		vr	. . . . . . . 8
14	13, 6	wel 1410	. . . . . . 7
15	14, 13, 4	wrex 2324	. . . . . 6
16	12, 15	wa 101	. . . . 5 $/\$
17	9, 16	wa 101	. . . 4 $/\$ $/\$ $/\$
18	10	cv 1258	. . . . . . . . . . 11
19	13	cv 1258	. . . . . . . . . . 11
20		cltq 6440	. . . . . . . . . . 11
21	18, 19, 20	wbr 3791	. . . . . . . . . 10
22	13, 2	wel 1410	. . . . . . . . . 10
23	21, 22	wa 101	. . . . . . . . 9 $/\$
24	23, 13, 4	wrex 2324	. . . . . . . 8 $/\$
25	11, 24	wb 102	. . . . . . 7 $/\$
26	25, 10, 4	wral 2323	. . . . . 6 $/\$
27	10, 6	wel 1410	. . . . . . . . . 10
28	21, 27	wa 101	. . . . . . . . 9 $/\$
29	28, 10, 4	wrex 2324	. . . . . . . 8 $/\$
30	14, 29	wb 102	. . . . . . 7 $/\$
31	30, 13, 4	wral 2323	. . . . . 6 $/\$
32	26, 31	wa 101	. . . . 5 $/\$ $/\$ $/\$
33	11, 27	wa 101	. . . . . . 7 $/\$
34	33	wn 3	. . . . . 6 $/\$
35	34, 10, 4	wral 2323	. . . . 5 $/\$
36	11, 14	wo 639	. . . . . . . 8 $\/$
37	21, 36	wi 4	. . . . . . 7 $\/$
38	37, 13, 4	wral 2323	. . . . . 6 $\/$
39	38, 10, 4	wral 2323	. . . . 5 $\/$
40	32, 35, 39	w3a 896	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
41	17, 40	wa 101	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
42	41, 2, 6	copab 3844	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
43	1, 42	wceq 1259	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

This definition is referenced by: npsspw 6626 elinp 6629

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