Description: An integer is odd iff its successor divided by 2 is an integer. This is a representation of odd numbers without using the divides relation, see zeo and zeo2 . (Contributed by AV, 22-Jun-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | oddp1d2 | |- ( N e. ZZ -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. ZZ ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oddp1even | |- ( N e. ZZ -> ( -. 2 || N <-> 2 || ( N + 1 ) ) ) |
|
| 2 | 2z | |- 2 e. ZZ |
|
| 3 | 2ne0 | |- 2 =/= 0 |
|
| 4 | peano2z | |- ( N e. ZZ -> ( N + 1 ) e. ZZ ) |
|
| 5 | dvdsval2 | |- ( ( 2 e. ZZ /\ 2 =/= 0 /\ ( N + 1 ) e. ZZ ) -> ( 2 || ( N + 1 ) <-> ( ( N + 1 ) / 2 ) e. ZZ ) ) |
|
| 6 | 2 3 4 5 | mp3an12i | |- ( N e. ZZ -> ( 2 || ( N + 1 ) <-> ( ( N + 1 ) / 2 ) e. ZZ ) ) |
| 7 | 1 6 | bitrd | |- ( N e. ZZ -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. ZZ ) ) |