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 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ ( 0 ... 𝑁 ) ) → ( 𝑁 C 𝐾 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		bcval	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ) → ( 𝑁 C 𝐾 ) = if ( 𝐾 ∈ ( 0 ... 𝑁 ) , ( ( ! ‘ 𝑁 ) / ( ( ! ‘ ( 𝑁 − 𝐾 ) ) · ( ! ‘ 𝐾 ) ) ) , 0 ) )
2	1	3adant3	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ ( 0 ... 𝑁 ) ) → ( 𝑁 C 𝐾 ) = if ( 𝐾 ∈ ( 0 ... 𝑁 ) , ( ( ! ‘ 𝑁 ) / ( ( ! ‘ ( 𝑁 − 𝐾 ) ) · ( ! ‘ 𝐾 ) ) ) , 0 ) )
3		iffalse	⊢ ( ¬ 𝐾 ∈ ( 0 ... 𝑁 ) → if ( 𝐾 ∈ ( 0 ... 𝑁 ) , ( ( ! ‘ 𝑁 ) / ( ( ! ‘ ( 𝑁 − 𝐾 ) ) · ( ! ‘ 𝐾 ) ) ) , 0 ) = 0 )
4	3	3ad2ant3	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ ( 0 ... 𝑁 ) ) → if ( 𝐾 ∈ ( 0 ... 𝑁 ) , ( ( ! ‘ 𝑁 ) / ( ( ! ‘ ( 𝑁 − 𝐾 ) ) · ( ! ‘ 𝐾 ) ) ) , 0 ) = 0 )
5	2 4	eqtrd	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ ( 0 ... 𝑁 ) ) → ( 𝑁 C 𝐾 ) = 0 )