Metamath Proof Explorer

Theorem nnrisefaccl

Description: Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018)

Ref Expression
Assertion nnrisefaccl 
|- ( ( A e. NN /\ N e. NN0 ) -> ( A RiseFac N ) e. NN )

Proof

Step Hyp Ref Expression
1 nnsscn 
 |-  NN C_ CC
2 1nn 
 |-  1 e. NN
3 nnmulcl 
 |-  ( ( x e. NN /\ y e. NN ) -> ( x x. y ) e. NN )
4 nnnn0addcl 
 |-  ( ( A e. NN /\ k e. NN0 ) -> ( A + k ) e. NN )
5 1 2 3 4 risefaccllem 
 |-  ( ( A e. NN /\ N e. NN0 ) -> ( A RiseFac N ) e. NN )