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Definition df-inp 7175

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 7000	. 2
2		vl	. . . . . . . 8
3	2	cv 1298	. . . . . . 7
4		cnq 6989	. . . . . . 7
5	3, 4	wss 3021	. . . . . 6
6		vu	. . . . . . . 8
7	6	cv 1298	. . . . . . 7
8	7, 4	wss 3021	. . . . . 6
9	5, 8	wa 103	. . . . 5 $/\$
10		vq	. . . . . . . 8
11	10, 2	wel 1449	. . . . . . 7
12	11, 10, 4	wrex 2376	. . . . . 6
13		vr	. . . . . . . 8
14	13, 6	wel 1449	. . . . . . 7
15	14, 13, 4	wrex 2376	. . . . . 6
16	12, 15	wa 103	. . . . 5 $/\$
17	9, 16	wa 103	. . . 4 $/\$ $/\$ $/\$
18	10	cv 1298	. . . . . . . . . . 11
19	13	cv 1298	. . . . . . . . . . 11
20		cltq 6994	. . . . . . . . . . 11
21	18, 19, 20	wbr 3875	. . . . . . . . . 10
22	13, 2	wel 1449	. . . . . . . . . 10
23	21, 22	wa 103	. . . . . . . . 9 $/\$
24	23, 13, 4	wrex 2376	. . . . . . . 8 $/\$
25	11, 24	wb 104	. . . . . . 7 $/\$
26	25, 10, 4	wral 2375	. . . . . 6 $/\$
27	10, 6	wel 1449	. . . . . . . . . 10
28	21, 27	wa 103	. . . . . . . . 9 $/\$
29	28, 10, 4	wrex 2376	. . . . . . . 8 $/\$
30	14, 29	wb 104	. . . . . . 7 $/\$
31	30, 13, 4	wral 2375	. . . . . 6 $/\$
32	26, 31	wa 103	. . . . 5 $/\$ $/\$ $/\$
33	11, 27	wa 103	. . . . . . 7 $/\$
34	33	wn 3	. . . . . 6 $/\$
35	34, 10, 4	wral 2375	. . . . 5 $/\$
36	11, 14	wo 670	. . . . . . . 8 $\/$
37	21, 36	wi 4	. . . . . . 7 $\/$
38	37, 13, 4	wral 2375	. . . . . 6 $\/$
39	38, 10, 4	wral 2375	. . . . 5 $\/$
40	32, 35, 39	w3a 930	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
41	17, 40	wa 103	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
42	41, 2, 6	copab 3928	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
43	1, 42	wceq 1299	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

This definition is referenced by: npsspw 7180 elinp 7183

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