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Metamath Proof Explorer

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 )