Metamath Proof Explorer


Theorem bcneg1

Description: The binomial coefficent over negative one is zero. (Contributed by Scott Fenton, 29-May-2020)

Ref Expression
Assertion bcneg1 ( 𝑁 ∈ ℕ0 → ( 𝑁 C - 1 ) = 0 )

Proof

Step Hyp Ref Expression
1 neg1z - 1 ∈ ℤ
2 bcval ( ( 𝑁 ∈ ℕ0 ∧ - 1 ∈ ℤ ) → ( 𝑁 C - 1 ) = if ( - 1 ∈ ( 0 ... 𝑁 ) , ( ( ! ‘ 𝑁 ) / ( ( ! ‘ ( 𝑁 − - 1 ) ) · ( ! ‘ - 1 ) ) ) , 0 ) )
3 1 2 mpan2 ( 𝑁 ∈ ℕ0 → ( 𝑁 C - 1 ) = if ( - 1 ∈ ( 0 ... 𝑁 ) , ( ( ! ‘ 𝑁 ) / ( ( ! ‘ ( 𝑁 − - 1 ) ) · ( ! ‘ - 1 ) ) ) , 0 ) )
4 neg1lt0 - 1 < 0
5 neg1rr - 1 ∈ ℝ
6 0re 0 ∈ ℝ
7 5 6 ltnlei ( - 1 < 0 ↔ ¬ 0 ≤ - 1 )
8 4 7 mpbi ¬ 0 ≤ - 1
9 8 intnanr ¬ ( 0 ≤ - 1 ∧ - 1 ≤ 𝑁 )
10 nn0z ( 𝑁 ∈ ℕ0𝑁 ∈ ℤ )
11 0z 0 ∈ ℤ
12 elfz ( ( - 1 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( - 1 ∈ ( 0 ... 𝑁 ) ↔ ( 0 ≤ - 1 ∧ - 1 ≤ 𝑁 ) ) )
13 1 11 12 mp3an12 ( 𝑁 ∈ ℤ → ( - 1 ∈ ( 0 ... 𝑁 ) ↔ ( 0 ≤ - 1 ∧ - 1 ≤ 𝑁 ) ) )
14 10 13 syl ( 𝑁 ∈ ℕ0 → ( - 1 ∈ ( 0 ... 𝑁 ) ↔ ( 0 ≤ - 1 ∧ - 1 ≤ 𝑁 ) ) )
15 9 14 mtbiri ( 𝑁 ∈ ℕ0 → ¬ - 1 ∈ ( 0 ... 𝑁 ) )
16 15 iffalsed ( 𝑁 ∈ ℕ0 → if ( - 1 ∈ ( 0 ... 𝑁 ) , ( ( ! ‘ 𝑁 ) / ( ( ! ‘ ( 𝑁 − - 1 ) ) · ( ! ‘ - 1 ) ) ) , 0 ) = 0 )
17 3 16 eqtrd ( 𝑁 ∈ ℕ0 → ( 𝑁 C - 1 ) = 0 )