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Definition df-inp 7729

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 7554	. 2
2		vl	. . . . . . . 8
3	2	cv 1397	. . . . . . 7
4		cnq 7543	. . . . . . 7
5	3, 4	wss 3201	. . . . . 6
6		vu	. . . . . . . 8
7	6	cv 1397	. . . . . . 7
8	7, 4	wss 3201	. . . . . 6
9	5, 8	wa 104	. . . . 5 $/\$
10		vq	. . . . . . . 8
11	10, 2	wel 2203	. . . . . . 7
12	11, 10, 4	wrex 2512	. . . . . 6
13		vr	. . . . . . . 8
14	13, 6	wel 2203	. . . . . . 7
15	14, 13, 4	wrex 2512	. . . . . 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 7548	. . . . . . . . . . 11
21	18, 19, 20	wbr 4093	. . . . . . . . . 10
22	13, 2	wel 2203	. . . . . . . . . 10
23	21, 22	wa 104	. . . . . . . . 9 $/\$
24	23, 13, 4	wrex 2512	. . . . . . . 8 $/\$
25	11, 24	wb 105	. . . . . . 7 $/\$
26	25, 10, 4	wral 2511	. . . . . 6 $/\$
27	10, 6	wel 2203	. . . . . . . . . 10
28	21, 27	wa 104	. . . . . . . . 9 $/\$
29	28, 10, 4	wrex 2512	. . . . . . . 8 $/\$
30	14, 29	wb 105	. . . . . . 7 $/\$
31	30, 13, 4	wral 2511	. . . . . 6 $/\$
32	26, 31	wa 104	. . . . 5 $/\$ $/\$ $/\$
33	11, 27	wa 104	. . . . . . 7 $/\$
34	33	wn 3	. . . . . 6 $/\$
35	34, 10, 4	wral 2511	. . . . 5 $/\$
36	11, 14	wo 716	. . . . . . . 8 $\/$
37	21, 36	wi 4	. . . . . . 7 $\/$
38	37, 13, 4	wral 2511	. . . . . 6 $\/$
39	38, 10, 4	wral 2511	. . . . 5 $\/$
40	32, 35, 39	w3a 1005	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
41	17, 40	wa 104	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
42	41, 2, 6	copab 4154	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
43	1, 42	wceq 1398	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

This definition is referenced by: npsspw 7734 elinp 7737

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