Description: A nonnegative integer is not itself plus 1 modulo an integer greater than 1 and the nonnegative integer. (Contributed by AV, 6-Sep-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | p1modne | |- ( ( N e. ( ZZ>= ` 2 ) /\ A e. ( 0 ..^ N ) ) -> ( ( A + 1 ) mod N ) =/= A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzoelz | |- ( A e. ( 0 ..^ N ) -> A e. ZZ ) |
|
| 2 | zp1modne | |- ( ( N e. ( ZZ>= ` 2 ) /\ A e. ZZ ) -> ( ( A + 1 ) mod N ) =/= ( A mod N ) ) |
|
| 3 | 1 2 | sylan2 | |- ( ( N e. ( ZZ>= ` 2 ) /\ A e. ( 0 ..^ N ) ) -> ( ( A + 1 ) mod N ) =/= ( A mod N ) ) |
| 4 | zmodidfzoimp | |- ( A e. ( 0 ..^ N ) -> ( A mod N ) = A ) |
|
| 5 | 4 | adantl | |- ( ( N e. ( ZZ>= ` 2 ) /\ A e. ( 0 ..^ N ) ) -> ( A mod N ) = A ) |
| 6 | 3 5 | neeqtrd | |- ( ( N e. ( ZZ>= ` 2 ) /\ A e. ( 0 ..^ N ) ) -> ( ( A + 1 ) mod N ) =/= A ) |