Metamath Proof Explorer

Theorem zfallfaccl

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

Ref Expression
Assertion zfallfaccl 
|- ( ( A e. ZZ /\ N e. NN0 ) -> ( A FallFac N ) e. ZZ )

Proof

Step Hyp Ref Expression
1 zsscn 
 |-  ZZ C_ CC
2 1z 
 |-  1 e. ZZ
3 zmulcl 
 |-  ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
4 nn0z 
 |-  ( k e. NN0 -> k e. ZZ )
5 zsubcl 
 |-  ( ( A e. ZZ /\ k e. ZZ ) -> ( A - k ) e. ZZ )
6 4 5 sylan2 
 |-  ( ( A e. ZZ /\ k e. NN0 ) -> ( A - k ) e. ZZ )
7 1 2 3 6 fallfaccllem 
 |-  ( ( A e. ZZ /\ N e. NN0 ) -> ( A FallFac N ) e. ZZ )