Description: The Ackermann function at 0 (for the first argument). This is the first equation of Péter's definition of the Ackermann function. (Contributed by AV, 4-May-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ackval0val | |- ( M e. NN0 -> ( ( Ack ` 0 ) ` M ) = ( M + 1 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ackval0 | |- ( Ack ` 0 ) = ( m e. NN0 |-> ( m + 1 ) ) |
|
| 2 | 1 | a1i | |- ( M e. NN0 -> ( Ack ` 0 ) = ( m e. NN0 |-> ( m + 1 ) ) ) |
| 3 | oveq1 | |- ( m = M -> ( m + 1 ) = ( M + 1 ) ) |
|
| 4 | 3 | adantl | |- ( ( M e. NN0 /\ m = M ) -> ( m + 1 ) = ( M + 1 ) ) |
| 5 | id | |- ( M e. NN0 -> M e. NN0 ) |
|
| 6 | peano2nn0 | |- ( M e. NN0 -> ( M + 1 ) e. NN0 ) |
|
| 7 | 2 4 5 6 | fvmptd | |- ( M e. NN0 -> ( ( Ack ` 0 ) ` M ) = ( M + 1 ) ) |