Metamath Proof Explorer

Theorem refallfaccl

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

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

Proof

Step Hyp Ref Expression
1 ax-resscn 
 |-  RR C_ CC
2 1re 
 |-  1 e. RR
3 remulcl 
 |-  ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
4 nn0re 
 |-  ( k e. NN0 -> k e. RR )
5 resubcl 
 |-  ( ( A e. RR /\ k e. RR ) -> ( A - k ) e. RR )
6 4 5 sylan2 
 |-  ( ( A e. RR /\ k e. NN0 ) -> ( A - k ) e. RR )
7 1 2 3 6 fallfaccllem 
 |-  ( ( A e. RR /\ N e. NN0 ) -> ( A FallFac N ) e. RR )