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Definition df-inp 7440

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 7265	. 2
2		vl	. . . . . . . 8
3	2	cv 1352	. . . . . . 7
4		cnq 7254	. . . . . . 7
5	3, 4	wss 3127	. . . . . 6
6		vu	. . . . . . . 8
7	6	cv 1352	. . . . . . 7
8	7, 4	wss 3127	. . . . . 6
9	5, 8	wa 104	. . . . 5 $/\$
10		vq	. . . . . . . 8
11	10, 2	wel 2147	. . . . . . 7
12	11, 10, 4	wrex 2454	. . . . . 6
13		vr	. . . . . . . 8
14	13, 6	wel 2147	. . . . . . 7
15	14, 13, 4	wrex 2454	. . . . . 6
16	12, 15	wa 104	. . . . 5 $/\$
17	9, 16	wa 104	. . . 4 $/\$ $/\$ $/\$
18	10	cv 1352	. . . . . . . . . . 11
19	13	cv 1352	. . . . . . . . . . 11
20		cltq 7259	. . . . . . . . . . 11
21	18, 19, 20	wbr 3998	. . . . . . . . . 10
22	13, 2	wel 2147	. . . . . . . . . 10
23	21, 22	wa 104	. . . . . . . . 9 $/\$
24	23, 13, 4	wrex 2454	. . . . . . . 8 $/\$
25	11, 24	wb 105	. . . . . . 7 $/\$
26	25, 10, 4	wral 2453	. . . . . 6 $/\$
27	10, 6	wel 2147	. . . . . . . . . 10
28	21, 27	wa 104	. . . . . . . . 9 $/\$
29	28, 10, 4	wrex 2454	. . . . . . . 8 $/\$
30	14, 29	wb 105	. . . . . . 7 $/\$
31	30, 13, 4	wral 2453	. . . . . 6 $/\$
32	26, 31	wa 104	. . . . 5 $/\$ $/\$ $/\$
33	11, 27	wa 104	. . . . . . 7 $/\$
34	33	wn 3	. . . . . 6 $/\$
35	34, 10, 4	wral 2453	. . . . 5 $/\$
36	11, 14	wo 708	. . . . . . . 8 $\/$
37	21, 36	wi 4	. . . . . . 7 $\/$
38	37, 13, 4	wral 2453	. . . . . 6 $\/$
39	38, 10, 4	wral 2453	. . . . 5 $\/$
40	32, 35, 39	w3a 978	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
41	17, 40	wa 104	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
42	41, 2, 6	copab 4058	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
43	1, 42	wceq 1353	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

This definition is referenced by: npsspw 7445 elinp 7448

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