| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lcmdvds |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) |
| 2 |
|
dvdslcm |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) ) |
| 3 |
2
|
simpld |
|- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M lcm N ) ) |
| 4 |
3
|
3adant1 |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M || ( M lcm N ) ) |
| 5 |
|
simp2 |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ ) |
| 6 |
|
lcmcl |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 ) |
| 7 |
6
|
nn0zd |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. ZZ ) |
| 8 |
7
|
3adant1 |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. ZZ ) |
| 9 |
|
simp1 |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ ) |
| 10 |
|
dvdstr |
|- ( ( M e. ZZ /\ ( M lcm N ) e. ZZ /\ K e. ZZ ) -> ( ( M || ( M lcm N ) /\ ( M lcm N ) || K ) -> M || K ) ) |
| 11 |
5 8 9 10
|
syl3anc |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || ( M lcm N ) /\ ( M lcm N ) || K ) -> M || K ) ) |
| 12 |
4 11
|
mpand |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) || K -> M || K ) ) |
| 13 |
2
|
simprd |
|- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( M lcm N ) ) |
| 14 |
13
|
3adant1 |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N || ( M lcm N ) ) |
| 15 |
|
dvdstr |
|- ( ( N e. ZZ /\ ( M lcm N ) e. ZZ /\ K e. ZZ ) -> ( ( N || ( M lcm N ) /\ ( M lcm N ) || K ) -> N || K ) ) |
| 16 |
15
|
3com13 |
|- ( ( K e. ZZ /\ ( M lcm N ) e. ZZ /\ N e. ZZ ) -> ( ( N || ( M lcm N ) /\ ( M lcm N ) || K ) -> N || K ) ) |
| 17 |
8 16
|
syld3an2 |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( N || ( M lcm N ) /\ ( M lcm N ) || K ) -> N || K ) ) |
| 18 |
14 17
|
mpand |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) || K -> N || K ) ) |
| 19 |
12 18
|
jcad |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) || K -> ( M || K /\ N || K ) ) ) |
| 20 |
1 19
|
impbid |
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || K /\ N || K ) <-> ( M lcm N ) || K ) ) |