Description: The difference between a real number and its nearest integer is less than or equal to one half. (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Mario Carneiro, 14-Sep-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | rddif | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfcn | |
|
2 | 1 | 2timesi | |
3 | 2cn | |
|
4 | 2ne0 | |
|
5 | 3 4 | recidi | |
6 | 2 5 | eqtr3i | |
7 | 6 | oveq2i | |
8 | recn | |
|
9 | 1 | a1i | |
10 | 8 9 9 | nppcan3d | |
11 | 7 10 | eqtr3id | |
12 | halfre | |
|
13 | readdcl | |
|
14 | 12 13 | mpan2 | |
15 | fllep1 | |
|
16 | 14 15 | syl | |
17 | 11 16 | eqbrtrd | |
18 | resubcl | |
|
19 | 12 18 | mpan2 | |
20 | reflcl | |
|
21 | 14 20 | syl | |
22 | 1red | |
|
23 | 19 21 22 | leadd1d | |
24 | 17 23 | mpbird | |
25 | flle | |
|
26 | 14 25 | syl | |
27 | id | |
|
28 | 12 | a1i | |
29 | absdifle | |
|
30 | 21 27 28 29 | syl3anc | |
31 | 24 26 30 | mpbir2and | |