Description: A mod B is zero iff A is evenly divisible by B . (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Fan Zheng, 7-Jun-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | mod0 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modval | |
|
2 | 1 | eqeq1d | |
3 | recn | |
|
4 | 3 | adantr | |
5 | rpre | |
|
6 | 5 | adantl | |
7 | refldivcl | |
|
8 | 6 7 | remulcld | |
9 | 8 | recnd | |
10 | 4 9 | subeq0ad | |
11 | 2 10 | bitrd | |
12 | 7 | recnd | |
13 | rpcnne0 | |
|
14 | 13 | adantl | |
15 | divmul2 | |
|
16 | 4 12 14 15 | syl3anc | |
17 | eqcom | |
|
18 | 16 17 | bitr3di | |
19 | 11 18 | bitrd | |
20 | rerpdivcl | |
|
21 | flidz | |
|
22 | 20 21 | syl | |
23 | 19 22 | bitrd | |