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 |
|
ballotth.p |
⊢ 𝑃 = ( 𝑥 ∈ 𝒫 𝑂 ↦ ( ( ♯ ‘ 𝑥 ) / ( ♯ ‘ 𝑂 ) ) ) |
5 |
|
ballotth.f |
⊢ 𝐹 = ( 𝑐 ∈ 𝑂 ↦ ( 𝑖 ∈ ℤ ↦ ( ( ♯ ‘ ( ( 1 ... 𝑖 ) ∩ 𝑐 ) ) − ( ♯ ‘ ( ( 1 ... 𝑖 ) ∖ 𝑐 ) ) ) ) ) |
6 |
|
ballotth.e |
⊢ 𝐸 = { 𝑐 ∈ 𝑂 ∣ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝑐 ) ‘ 𝑖 ) } |
7 |
|
ballotth.mgtn |
⊢ 𝑁 < 𝑀 |
8 |
|
ballotth.i |
⊢ 𝐼 = ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑐 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ) |
9 |
|
fveq2 |
⊢ ( 𝑑 = 𝐶 → ( 𝐹 ‘ 𝑑 ) = ( 𝐹 ‘ 𝐶 ) ) |
10 |
9
|
fveq1d |
⊢ ( 𝑑 = 𝐶 → ( ( 𝐹 ‘ 𝑑 ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) ) |
11 |
10
|
eqeq1d |
⊢ ( 𝑑 = 𝐶 → ( ( ( 𝐹 ‘ 𝑑 ) ‘ 𝑘 ) = 0 ↔ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 ) ) |
12 |
11
|
rabbidv |
⊢ ( 𝑑 = 𝐶 → { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑑 ) ‘ 𝑘 ) = 0 } = { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ) |
13 |
12
|
infeq1d |
⊢ ( 𝑑 = 𝐶 → inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑑 ) ‘ 𝑘 ) = 0 } , ℝ , < ) = inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ) |
14 |
|
fveq2 |
⊢ ( 𝑐 = 𝑑 → ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑑 ) ) |
15 |
14
|
fveq1d |
⊢ ( 𝑐 = 𝑑 → ( ( 𝐹 ‘ 𝑐 ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑑 ) ‘ 𝑘 ) ) |
16 |
15
|
eqeq1d |
⊢ ( 𝑐 = 𝑑 → ( ( ( 𝐹 ‘ 𝑐 ) ‘ 𝑘 ) = 0 ↔ ( ( 𝐹 ‘ 𝑑 ) ‘ 𝑘 ) = 0 ) ) |
17 |
16
|
rabbidv |
⊢ ( 𝑐 = 𝑑 → { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑐 ) ‘ 𝑘 ) = 0 } = { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑑 ) ‘ 𝑘 ) = 0 } ) |
18 |
17
|
infeq1d |
⊢ ( 𝑐 = 𝑑 → inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑐 ) ‘ 𝑘 ) = 0 } , ℝ , < ) = inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑑 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ) |
19 |
18
|
cbvmptv |
⊢ ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑐 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ) = ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ↦ inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑑 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ) |
20 |
8 19
|
eqtri |
⊢ 𝐼 = ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ↦ inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑑 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ) |
21 |
|
ltso |
⊢ < Or ℝ |
22 |
21
|
infex |
⊢ inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ∈ V |
23 |
13 20 22
|
fvmpt |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) = inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ) |