Metamath Proof Explorer


Definition df-fallfac

Description: Define the falling factorial function. This is the function ( A x. ( A - 1 ) x. ... ( A - N ) ) for complex A and nonnegative integers N . (Contributed by Scott Fenton, 5-Jan-2018)

Ref Expression
Assertion df-fallfac
|- FallFac = ( x e. CC , n e. NN0 |-> prod_ k e. ( 0 ... ( n - 1 ) ) ( x - k ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cfallfac
 |-  FallFac
1 vx
 |-  x
2 cc
 |-  CC
3 vn
 |-  n
4 cn0
 |-  NN0
5 vk
 |-  k
6 cc0
 |-  0
7 cfz
 |-  ...
8 3 cv
 |-  n
9 cmin
 |-  -
10 c1
 |-  1
11 8 10 9 co
 |-  ( n - 1 )
12 6 11 7 co
 |-  ( 0 ... ( n - 1 ) )
13 1 cv
 |-  x
14 5 cv
 |-  k
15 13 14 9 co
 |-  ( x - k )
16 12 15 5 cprod
 |-  prod_ k e. ( 0 ... ( n - 1 ) ) ( x - k )
17 1 3 2 4 16 cmpo
 |-  ( x e. CC , n e. NN0 |-> prod_ k e. ( 0 ... ( n - 1 ) ) ( x - k ) )
18 0 17 wceq
 |-  FallFac = ( x e. CC , n e. NN0 |-> prod_ k e. ( 0 ... ( n - 1 ) ) ( x - k ) )