Metamath Proof Explorer

Theorem fallfac0

Description: The value of the falling factorial when N = 0 . (Contributed by Scott Fenton, 5-Jan-2018)

Ref Expression
Assertion fallfac0 
|- ( A e. CC -> ( A FallFac 0 ) = 1 )

Proof

Step Hyp Ref Expression
1 0nn0 
 |-  0 e. NN0
2 fallrisefac 
 |-  ( ( A e. CC /\ 0 e. NN0 ) -> ( A FallFac 0 ) = ( ( -u 1 ^ 0 ) x. ( -u A RiseFac 0 ) ) )
3 1 2 mpan2 
 |-  ( A e. CC -> ( A FallFac 0 ) = ( ( -u 1 ^ 0 ) x. ( -u A RiseFac 0 ) ) )
4 neg1cn 
 |-  -u 1 e. CC
5 exp0 
 |-  ( -u 1 e. CC -> ( -u 1 ^ 0 ) = 1 )
6 4 5 mp1i 
 |-  ( A e. CC -> ( -u 1 ^ 0 ) = 1 )
7 negcl 
 |-  ( A e. CC -> -u A e. CC )
8 risefac0 
 |-  ( -u A e. CC -> ( -u A RiseFac 0 ) = 1 )
9 7 8 syl 
 |-  ( A e. CC -> ( -u A RiseFac 0 ) = 1 )
10 6 9 oveq12d 
 |-  ( A e. CC -> ( ( -u 1 ^ 0 ) x. ( -u A RiseFac 0 ) ) = ( 1 x. 1 ) )
11 1t1e1 
 |-  ( 1 x. 1 ) = 1
12 10 11 eqtrdi 
 |-  ( A e. CC -> ( ( -u 1 ^ 0 ) x. ( -u A RiseFac 0 ) ) = 1 )
13 3 12 eqtrd 
 |-  ( A e. CC -> ( A FallFac 0 ) = 1 )