Metamath Proof Explorer


Theorem bcnm1

Description: The binomial coefficent 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 )