Description: F times the least common multiple of 1 to n is a natural number. (Contributed by metakunt, 10-May-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lcmineqlem15.1 | |
|
lcmineqlem15.2 | |
||
lcmineqlem15.3 | |
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lcmineqlem15.4 | |
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Assertion | lcmineqlem15 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmineqlem15.1 | |
|
2 | lcmineqlem15.2 | |
|
3 | lcmineqlem15.3 | |
|
4 | lcmineqlem15.4 | |
|
5 | 1 2 3 4 | lcmineqlem6 | |
6 | fz1ssnn | |
|
7 | fzfi | |
|
8 | lcmfnncl | |
|
9 | 6 7 8 | mp2an | |
10 | 9 | a1i | |
11 | 10 | nnred | |
12 | 1 3 2 4 | lcmineqlem13 | |
13 | 1red | |
|
14 | 3 | nnnn0d | |
15 | 2 14 4 | bccl2d | |
16 | 3 15 | nnmulcld | |
17 | 16 | nnred | |
18 | 16 | nnne0d | |
19 | 13 17 18 | redivcld | |
20 | 12 19 | eqeltrd | |
21 | 10 | nngt0d | |
22 | nnrecgt0 | |
|
23 | 16 22 | syl | |
24 | 23 12 | breqtrrd | |
25 | 11 20 21 24 | mulgt0d | |
26 | elnnz | |
|
27 | 5 25 26 | sylanbrc | |