Step |
Hyp |
Ref |
Expression |
1 |
|
ballotth.m |
⊢ 𝑀 ∈ ℕ |
2 |
|
ballotth.n |
⊢ 𝑁 ∈ ℕ |
3 |
|
ballotth.o |
⊢ 𝑂 = { 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 } |
4 |
3
|
fveq2i |
⊢ ( ♯ ‘ 𝑂 ) = ( ♯ ‘ { 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 } ) |
5 |
|
fzfi |
⊢ ( 1 ... ( 𝑀 + 𝑁 ) ) ∈ Fin |
6 |
1
|
nnzi |
⊢ 𝑀 ∈ ℤ |
7 |
|
hashbc |
⊢ ( ( ( 1 ... ( 𝑀 + 𝑁 ) ) ∈ Fin ∧ 𝑀 ∈ ℤ ) → ( ( ♯ ‘ ( 1 ... ( 𝑀 + 𝑁 ) ) ) C 𝑀 ) = ( ♯ ‘ { 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 } ) ) |
8 |
5 6 7
|
mp2an |
⊢ ( ( ♯ ‘ ( 1 ... ( 𝑀 + 𝑁 ) ) ) C 𝑀 ) = ( ♯ ‘ { 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 } ) |
9 |
1 2
|
pm3.2i |
⊢ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) |
10 |
|
nnaddcl |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝑀 + 𝑁 ) ∈ ℕ ) |
11 |
|
nnnn0 |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ℕ → ( 𝑀 + 𝑁 ) ∈ ℕ0 ) |
12 |
9 10 11
|
mp2b |
⊢ ( 𝑀 + 𝑁 ) ∈ ℕ0 |
13 |
|
hashfz1 |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ℕ0 → ( ♯ ‘ ( 1 ... ( 𝑀 + 𝑁 ) ) ) = ( 𝑀 + 𝑁 ) ) |
14 |
12 13
|
ax-mp |
⊢ ( ♯ ‘ ( 1 ... ( 𝑀 + 𝑁 ) ) ) = ( 𝑀 + 𝑁 ) |
15 |
14
|
oveq1i |
⊢ ( ( ♯ ‘ ( 1 ... ( 𝑀 + 𝑁 ) ) ) C 𝑀 ) = ( ( 𝑀 + 𝑁 ) C 𝑀 ) |
16 |
4 8 15
|
3eqtr2i |
⊢ ( ♯ ‘ 𝑂 ) = ( ( 𝑀 + 𝑁 ) C 𝑀 ) |