Theorem bcval3

Description: Value of the binomial coefficient, N choose K , outside of its standard domain. Remark in Gleason p. 295. (Contributed by NM, 14-Jul-2005) (Revised by Mario Carneiro, 8-Nov-2013)

	Ref	Expression
Assertion	bcval3	|- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		bcval	|- ( ( N e. NN0 /\ K e. ZZ ) -> ( N _C K ) = if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) )
2	1	3adant3	|- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) )
3		iffalse	|- ( -. K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = 0 )
4	3	3ad2ant3	|- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = 0 )
5	2 4	eqtrd	|- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = 0 )