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Mirrors > Home > ILE Home > Th. List > df-inp | Unicode version |
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 (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.) |
Ref | Expression |
---|---|
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
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42 | 41, 2, 6 | copab 3817 |
. 2
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43 | 1, 42 | wceq 1243 |
1
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Colors of variables: wff set class |
This definition is referenced by: npsspw 6569 elinp 6572 |
Copyright terms: Public domain | W3C validator |