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Definition df-inp 6564

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 6389	. 2
2		vl	. . . . . . . 8
3	2	cv 1242	. . . . . . 7
4		cnq 6378	. . . . . . 7
5	3, 4	wss 2917	. . . . . 6
6		vu	. . . . . . . 8
7	6	cv 1242	. . . . . . 7
8	7, 4	wss 2917	. . . . . 6
9	5, 8	wa 97	. . . . 5 $/\$
10		vq	. . . . . . . 8
11	10, 2	wel 1394	. . . . . . 7
12	11, 10, 4	wrex 2307	. . . . . 6
13		vr	. . . . . . . 8
14	13, 6	wel 1394	. . . . . . 7
15	14, 13, 4	wrex 2307	. . . . . 6
16	12, 15	wa 97	. . . . 5 $/\$
17	9, 16	wa 97	. . . 4 $/\$ $/\$ $/\$
18	10	cv 1242	. . . . . . . . . . 11
19	13	cv 1242	. . . . . . . . . . 11
20		cltq 6383	. . . . . . . . . . 11
21	18, 19, 20	wbr 3764	. . . . . . . . . 10
22	13, 2	wel 1394	. . . . . . . . . 10
23	21, 22	wa 97	. . . . . . . . 9 $/\$
24	23, 13, 4	wrex 2307	. . . . . . . 8 $/\$
25	11, 24	wb 98	. . . . . . 7 $/\$
26	25, 10, 4	wral 2306	. . . . . 6 $/\$
27	10, 6	wel 1394	. . . . . . . . . 10
28	21, 27	wa 97	. . . . . . . . 9 $/\$
29	28, 10, 4	wrex 2307	. . . . . . . 8 $/\$
30	14, 29	wb 98	. . . . . . 7 $/\$
31	30, 13, 4	wral 2306	. . . . . 6 $/\$
32	26, 31	wa 97	. . . . 5 $/\$ $/\$ $/\$
33	11, 27	wa 97	. . . . . . 7 $/\$
34	33	wn 3	. . . . . 6 $/\$
35	34, 10, 4	wral 2306	. . . . 5 $/\$
36	11, 14	wo 629	. . . . . . . 8 $\/$
37	21, 36	wi 4	. . . . . . 7 $\/$
38	37, 13, 4	wral 2306	. . . . . 6 $\/$
39	38, 10, 4	wral 2306	. . . . 5 $\/$
40	32, 35, 39	w3a 885	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
41	17, 40	wa 97	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
42	41, 2, 6	copab 3817	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
43	1, 42	wceq 1243	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

This definition is referenced by: npsspw 6569 elinp 6572

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