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 ∈ ℕ0 → 𝑁 ∈ 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 ) |