Theorem bcval2

Description: Value of the binomial coefficient, N choose K , in its standard domain. (Contributed by NM, 9-Jun-2005) (Revised by Mario Carneiro, 7-Nov-2013)

	Ref	Expression
Assertion	bcval2	|- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		elfz3nn0	|- ( K e. ( 0 ... N ) -> N e. NN0 )
2		elfzelz	|- ( K e. ( 0 ... N ) -> K e. ZZ )
3		bcval	|- ( ( N e. NN0 /\ K e. ZZ ) -> ( N _C K ) = if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) )
4	1 2 3	syl2anc	|- ( K e. ( 0 ... N ) -> ( N _C K ) = if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) )
5		iftrue	|- ( K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
6	4 5	eqtrd	|- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )