Theorem bcnm1

Description: The binomial coefficient of ( N - 1 ) is N . (Contributed by Scott Fenton, 16-May-2014)

	Ref	Expression
Assertion	bcnm1	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 C ( 𝑁 − 1 ) ) = 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		1z	⊢ 1 ∈ ℤ
2		bccmpl	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 1 ∈ ℤ ) → ( 𝑁 C 1 ) = ( 𝑁 C ( 𝑁 − 1 ) ) )
3	1 2	mpan2	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 C 1 ) = ( 𝑁 C ( 𝑁 − 1 ) ) )
4		bcn1	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 C 1 ) = 𝑁 )
5	3 4	eqtr3d	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 C ( 𝑁 − 1 ) ) = 𝑁 )