Metamath Proof Explorer

Theorem nn0risefaccl

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

Ref Expression
Assertion nn0risefaccl 
|- ( ( A e. NN0 /\ N e. NN0 ) -> ( A RiseFac N ) e. NN0 )

Proof

Step Hyp Ref Expression
1 nn0sscn 
 |-  NN0 C_ CC
2 1nn0 
 |-  1 e. NN0
3 nn0mulcl 
 |-  ( ( x e. NN0 /\ y e. NN0 ) -> ( x x. y ) e. NN0 )
4 nn0addcl 
 |-  ( ( A e. NN0 /\ k e. NN0 ) -> ( A + k ) e. NN0 )
5 1 2 3 4 risefaccllem 
 |-  ( ( A e. NN0 /\ N e. NN0 ) -> ( A RiseFac N ) e. NN0 )