Step |
Hyp |
Ref |
Expression |
1 |
|
elnn0 |
⊢ ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) |
2 |
|
1eluzge0 |
⊢ 1 ∈ ( ℤ≥ ‘ 0 ) |
3 |
2
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 1 ∈ ( ℤ≥ ‘ 0 ) ) |
4 |
|
elnnuz |
⊢ ( 𝑁 ∈ ℕ ↔ 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) |
5 |
4
|
biimpi |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) |
6 |
|
elfzuzb |
⊢ ( 1 ∈ ( 0 ... 𝑁 ) ↔ ( 1 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) ) |
7 |
3 5 6
|
sylanbrc |
⊢ ( 𝑁 ∈ ℕ → 1 ∈ ( 0 ... 𝑁 ) ) |
8 |
|
bcval2 |
⊢ ( 1 ∈ ( 0 ... 𝑁 ) → ( 𝑁 C 1 ) = ( ( ! ‘ 𝑁 ) / ( ( ! ‘ ( 𝑁 − 1 ) ) · ( ! ‘ 1 ) ) ) ) |
9 |
7 8
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 C 1 ) = ( ( ! ‘ 𝑁 ) / ( ( ! ‘ ( 𝑁 − 1 ) ) · ( ! ‘ 1 ) ) ) ) |
10 |
|
facnn2 |
⊢ ( 𝑁 ∈ ℕ → ( ! ‘ 𝑁 ) = ( ( ! ‘ ( 𝑁 − 1 ) ) · 𝑁 ) ) |
11 |
|
fac1 |
⊢ ( ! ‘ 1 ) = 1 |
12 |
11
|
oveq2i |
⊢ ( ( ! ‘ ( 𝑁 − 1 ) ) · ( ! ‘ 1 ) ) = ( ( ! ‘ ( 𝑁 − 1 ) ) · 1 ) |
13 |
|
nnm1nn0 |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 − 1 ) ∈ ℕ0 ) |
14 |
13
|
faccld |
⊢ ( 𝑁 ∈ ℕ → ( ! ‘ ( 𝑁 − 1 ) ) ∈ ℕ ) |
15 |
14
|
nncnd |
⊢ ( 𝑁 ∈ ℕ → ( ! ‘ ( 𝑁 − 1 ) ) ∈ ℂ ) |
16 |
15
|
mulid1d |
⊢ ( 𝑁 ∈ ℕ → ( ( ! ‘ ( 𝑁 − 1 ) ) · 1 ) = ( ! ‘ ( 𝑁 − 1 ) ) ) |
17 |
12 16
|
eqtrid |
⊢ ( 𝑁 ∈ ℕ → ( ( ! ‘ ( 𝑁 − 1 ) ) · ( ! ‘ 1 ) ) = ( ! ‘ ( 𝑁 − 1 ) ) ) |
18 |
10 17
|
oveq12d |
⊢ ( 𝑁 ∈ ℕ → ( ( ! ‘ 𝑁 ) / ( ( ! ‘ ( 𝑁 − 1 ) ) · ( ! ‘ 1 ) ) ) = ( ( ( ! ‘ ( 𝑁 − 1 ) ) · 𝑁 ) / ( ! ‘ ( 𝑁 − 1 ) ) ) ) |
19 |
|
nncn |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℂ ) |
20 |
14
|
nnne0d |
⊢ ( 𝑁 ∈ ℕ → ( ! ‘ ( 𝑁 − 1 ) ) ≠ 0 ) |
21 |
19 15 20
|
divcan3d |
⊢ ( 𝑁 ∈ ℕ → ( ( ( ! ‘ ( 𝑁 − 1 ) ) · 𝑁 ) / ( ! ‘ ( 𝑁 − 1 ) ) ) = 𝑁 ) |
22 |
9 18 21
|
3eqtrd |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 C 1 ) = 𝑁 ) |
23 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
24 |
|
1z |
⊢ 1 ∈ ℤ |
25 |
|
0lt1 |
⊢ 0 < 1 |
26 |
25
|
olci |
⊢ ( 1 < 0 ∨ 0 < 1 ) |
27 |
|
bcval4 |
⊢ ( ( 0 ∈ ℕ0 ∧ 1 ∈ ℤ ∧ ( 1 < 0 ∨ 0 < 1 ) ) → ( 0 C 1 ) = 0 ) |
28 |
23 24 26 27
|
mp3an |
⊢ ( 0 C 1 ) = 0 |
29 |
|
oveq1 |
⊢ ( 𝑁 = 0 → ( 𝑁 C 1 ) = ( 0 C 1 ) ) |
30 |
|
eqeq12 |
⊢ ( ( ( 𝑁 C 1 ) = ( 0 C 1 ) ∧ 𝑁 = 0 ) → ( ( 𝑁 C 1 ) = 𝑁 ↔ ( 0 C 1 ) = 0 ) ) |
31 |
29 30
|
mpancom |
⊢ ( 𝑁 = 0 → ( ( 𝑁 C 1 ) = 𝑁 ↔ ( 0 C 1 ) = 0 ) ) |
32 |
28 31
|
mpbiri |
⊢ ( 𝑁 = 0 → ( 𝑁 C 1 ) = 𝑁 ) |
33 |
22 32
|
jaoi |
⊢ ( ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) → ( 𝑁 C 1 ) = 𝑁 ) |
34 |
1 33
|
sylbi |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 C 1 ) = 𝑁 ) |