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Definition df-fallfac 15895
Description: Define the falling factorial function. This is the function (๐ด ยท (๐ด โˆ’ 1) ยท ...(๐ด โˆ’ ๐‘)) for complex ๐ด and nonnegative integers ๐‘. (Contributed by Scott Fenton, 5-Jan-2018.)
Assertion
Ref Expression
df-fallfac FallFac = (๐‘ฅ โˆˆ โ„‚, ๐‘› โˆˆ โ„•0 โ†ฆ โˆ๐‘˜ โˆˆ (0...(๐‘› โˆ’ 1))(๐‘ฅ โˆ’ ๐‘˜))
Distinct variable group:   ๐‘ฅ,๐‘›,๐‘˜

Detailed syntax breakdown of Definition df-fallfac
StepHypRef Expression
1 cfallfac 15892 . 2 class FallFac
2 vx . . 3 setvar ๐‘ฅ
3 vn . . 3 setvar ๐‘›
4 cc 11054 . . 3 class โ„‚
5 cn0 12418 . . 3 class โ„•0
6 cc0 11056 . . . . 5 class 0
73cv 1541 . . . . . 6 class ๐‘›
8 c1 11057 . . . . . 6 class 1
9 cmin 11390 . . . . . 6 class โˆ’
107, 8, 9co 7358 . . . . 5 class (๐‘› โˆ’ 1)
11 cfz 13430 . . . . 5 class ...
126, 10, 11co 7358 . . . 4 class (0...(๐‘› โˆ’ 1))
132cv 1541 . . . . 5 class ๐‘ฅ
14 vk . . . . . 6 setvar ๐‘˜
1514cv 1541 . . . . 5 class ๐‘˜
1613, 15, 9co 7358 . . . 4 class (๐‘ฅ โˆ’ ๐‘˜)
1712, 16, 14cprod 15793 . . 3 class โˆ๐‘˜ โˆˆ (0...(๐‘› โˆ’ 1))(๐‘ฅ โˆ’ ๐‘˜)
182, 3, 4, 5, 17cmpo 7360 . 2 class (๐‘ฅ โˆˆ โ„‚, ๐‘› โˆˆ โ„•0 โ†ฆ โˆ๐‘˜ โˆˆ (0...(๐‘› โˆ’ 1))(๐‘ฅ โˆ’ ๐‘˜))
191, 18wceq 1542 1 wff FallFac = (๐‘ฅ โˆˆ โ„‚, ๐‘› โˆˆ โ„•0 โ†ฆ โˆ๐‘˜ โˆˆ (0...(๐‘› โˆ’ 1))(๐‘ฅ โˆ’ ๐‘˜))
Colors of variables: wff setvar class
This definition is referenced by:  fallfacval  15897
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