Description: Two positive integers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014) (Revised by Mario Carneiro, 19-Apr-2014) (Proof shortened by AV, 16-Sep-2026)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dvdssqnn | |- ( ( M e. NN /\ N e. NN ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2nn | |- 2 e. NN |
|
| 2 | dvdsexpnn | |- ( ( M e. NN /\ N e. NN /\ 2 e. NN ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) ) |
|
| 3 | 1 2 | mp3an3 | |- ( ( M e. NN /\ N e. NN ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) ) |