Metamath Proof Explorer


Theorem fallfaccl

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

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

Proof

Step Hyp Ref Expression
1 ssid
 |-  CC C_ CC
2 ax-1cn
 |-  1 e. CC
3 mulcl
 |-  ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
4 nn0cn
 |-  ( k e. NN0 -> k e. CC )
5 subcl
 |-  ( ( A e. CC /\ k e. CC ) -> ( A - k ) e. CC )
6 4 5 sylan2
 |-  ( ( A e. CC /\ k e. NN0 ) -> ( A - k ) e. CC )
7 1 2 3 6 fallfaccllem
 |-  ( ( A e. CC /\ N e. NN0 ) -> ( A FallFac N ) e. CC )