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 |
1 2 3 4 5 6 7 8
|
ballotlemi |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) = inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ) |
10 |
|
ltso |
⊢ < Or ℝ |
11 |
10
|
a1i |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → < Or ℝ ) |
12 |
|
fzfi |
⊢ ( 1 ... ( 𝑀 + 𝑁 ) ) ∈ Fin |
13 |
|
ssrab2 |
⊢ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) |
14 |
|
ssfi |
⊢ ( ( ( 1 ... ( 𝑀 + 𝑁 ) ) ∈ Fin ∧ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ∈ Fin ) |
15 |
12 13 14
|
mp2an |
⊢ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ∈ Fin |
16 |
15
|
a1i |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ∈ Fin ) |
17 |
1 2 3 4 5 6 7
|
ballotlem5 |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ∃ 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 ) |
18 |
|
rabn0 |
⊢ ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ≠ ∅ ↔ ∃ 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 ) |
19 |
17 18
|
sylibr |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ≠ ∅ ) |
20 |
|
fzssuz |
⊢ ( 1 ... ( 𝑀 + 𝑁 ) ) ⊆ ( ℤ≥ ‘ 1 ) |
21 |
|
uzssz |
⊢ ( ℤ≥ ‘ 1 ) ⊆ ℤ |
22 |
20 21
|
sstri |
⊢ ( 1 ... ( 𝑀 + 𝑁 ) ) ⊆ ℤ |
23 |
|
zssre |
⊢ ℤ ⊆ ℝ |
24 |
22 23
|
sstri |
⊢ ( 1 ... ( 𝑀 + 𝑁 ) ) ⊆ ℝ |
25 |
13 24
|
sstri |
⊢ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ⊆ ℝ |
26 |
25
|
a1i |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ⊆ ℝ ) |
27 |
|
fiinfcl |
⊢ ( ( < Or ℝ ∧ ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ∈ Fin ∧ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ≠ ∅ ∧ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ⊆ ℝ ) ) → inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ∈ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ) |
28 |
11 16 19 26 27
|
syl13anc |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ∈ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ) |
29 |
9 28
|
eqeltrd |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ) |
30 |
|
fveqeq2 |
⊢ ( 𝑘 = ( 𝐼 ‘ 𝐶 ) → ( ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 ↔ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) ) |
31 |
30
|
elrab |
⊢ ( ( 𝐼 ‘ 𝐶 ) ∈ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 } ↔ ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) ) |
32 |
29 31
|
sylib |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) ) |