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Description: Define complex apartness.
Definition 6.1 of [Geuvers], p. 17.
Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 8406 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 8334), symmetry (apsym 8335), and cotransitivity (apcotr 8336). Apartness implies negated equality, as seen at apne 8352, and the converse would also follow if we assumed excluded middle. In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 8351). (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
df-ap | # #ℝ #ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cap 8310 | . 2 # | |
2 | vx | . . . . . . . . . . 11 | |
3 | 2 | cv 1315 | . . . . . . . . . 10 |
4 | vr | . . . . . . . . . . . 12 | |
5 | 4 | cv 1315 | . . . . . . . . . . 11 |
6 | ci 7590 | . . . . . . . . . . . 12 | |
7 | vs | . . . . . . . . . . . . 13 | |
8 | 7 | cv 1315 | . . . . . . . . . . . 12 |
9 | cmul 7593 | . . . . . . . . . . . 12 | |
10 | 6, 8, 9 | co 5742 | . . . . . . . . . . 11 |
11 | caddc 7591 | . . . . . . . . . . 11 | |
12 | 5, 10, 11 | co 5742 | . . . . . . . . . 10 |
13 | 3, 12 | wceq 1316 | . . . . . . . . 9 |
14 | vy | . . . . . . . . . . 11 | |
15 | 14 | cv 1315 | . . . . . . . . . 10 |
16 | vt | . . . . . . . . . . . 12 | |
17 | 16 | cv 1315 | . . . . . . . . . . 11 |
18 | vu | . . . . . . . . . . . . 13 | |
19 | 18 | cv 1315 | . . . . . . . . . . . 12 |
20 | 6, 19, 9 | co 5742 | . . . . . . . . . . 11 |
21 | 17, 20, 11 | co 5742 | . . . . . . . . . 10 |
22 | 15, 21 | wceq 1316 | . . . . . . . . 9 |
23 | 13, 22 | wa 103 | . . . . . . . 8 |
24 | creap 8303 | . . . . . . . . . 10 #ℝ | |
25 | 5, 17, 24 | wbr 3899 | . . . . . . . . 9 #ℝ |
26 | 8, 19, 24 | wbr 3899 | . . . . . . . . 9 #ℝ |
27 | 25, 26 | wo 682 | . . . . . . . 8 #ℝ #ℝ |
28 | 23, 27 | wa 103 | . . . . . . 7 #ℝ #ℝ |
29 | cr 7587 | . . . . . . 7 | |
30 | 28, 18, 29 | wrex 2394 | . . . . . 6 #ℝ #ℝ |
31 | 30, 16, 29 | wrex 2394 | . . . . 5 #ℝ #ℝ |
32 | 31, 7, 29 | wrex 2394 | . . . 4 #ℝ #ℝ |
33 | 32, 4, 29 | wrex 2394 | . . 3 #ℝ #ℝ |
34 | 33, 2, 14 | copab 3958 | . 2 #ℝ #ℝ |
35 | 1, 34 | wceq 1316 | 1 # #ℝ #ℝ |
Colors of variables: wff set class |
This definition is referenced by: apreap 8316 apreim 8332 aprcl 8375 |
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