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Description: The mapping G is a one-to-one mapping from _om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number C (normally 0 for the upper integers NN0 or 1 for the upper integers NN ), 1 maps to C + 1, etc. This theorem shows the value of G at ordinal natural number zero. (This series of theorems generalizes an earlier series for NN0 contributed by Raph Levien, 10-Apr-2004.) (Contributed by NM, 3-Oct-2004) (Revised by Mario Carneiro, 13-Sep-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | om2uz.1 | |- C e. ZZ |
|
| om2uz.2 | |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` _om ) |
||
| Assertion | om2uz0i | |- ( G ` (/) ) = C |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | om2uz.1 | |- C e. ZZ |
|
| 2 | om2uz.2 | |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` _om ) |
|
| 3 | 2 | fveq1i | |- ( G ` (/) ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` _om ) ` (/) ) |
| 4 | fr0g | |- ( C e. ZZ -> ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` _om ) ` (/) ) = C ) |
|
| 5 | 1 4 | ax-mp | |- ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` _om ) ` (/) ) = C |
| 6 | 3 5 | eqtri | |- ( G ` (/) ) = C |