Step | Hyp | Ref
| Expression |
1 | | ballotth.m |
. . . 4
⊢ 𝑀 ∈ ℕ |
2 | | ballotth.n |
. . . 4
⊢ 𝑁 ∈ ℕ |
3 | | ballotth.o |
. . . 4
⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} |
4 | | ballotth.p |
. . . 4
⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂))) |
5 | | ballotth.f |
. . . 4
⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦
((♯‘((1...𝑖)
∩ 𝑐)) −
(♯‘((1...𝑖)
∖ 𝑐))))) |
6 | | ballotth.e |
. . . 4
⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} |
7 | | ballotth.mgtn |
. . . 4
⊢ 𝑁 < 𝑀 |
8 | | ballotth.i |
. . . 4
⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) |
9 | | ballotth.s |
. . . 4
⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) |
10 | | ballotth.r |
. . . 4
⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | ballotlemrc 31790 |
. . 3
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ (𝑂 ∖ 𝐸)) |
12 | 1, 2, 3, 4, 5, 6, 7, 8 | ballotlemi 31760 |
. . 3
⊢ ((𝑅‘𝐶) ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}, ℝ, < )) |
13 | 11, 12 | syl 17 |
. 2
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}, ℝ, < )) |
14 | | ltso 10723 |
. . . 4
⊢ < Or
ℝ |
15 | 14 | a1i 11 |
. . 3
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → < Or ℝ) |
16 | 1, 2, 3, 4, 5, 6, 7, 8 | ballotlemiex 31761 |
. . . . . 6
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)) |
17 | 16 | simpld 497 |
. . . . 5
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁))) |
18 | | elfzelz 12911 |
. . . . 5
⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ) |
19 | 17, 18 | syl 17 |
. . . 4
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ) |
20 | 19 | zred 12090 |
. . 3
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℝ) |
21 | | eqid 2823 |
. . . . 5
⊢ (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
21 | ballotlemfrci 31787 |
. . . 4
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘(𝑅‘𝐶))‘(𝐼‘𝐶)) = 0) |
23 | | fveqeq2 6681 |
. . . . 5
⊢ (𝑘 = (𝐼‘𝐶) → (((𝐹‘(𝑅‘𝐶))‘𝑘) = 0 ↔ ((𝐹‘(𝑅‘𝐶))‘(𝐼‘𝐶)) = 0)) |
24 | 23 | elrab 3682 |
. . . 4
⊢ ((𝐼‘𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} ↔ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘(𝑅‘𝐶))‘(𝐼‘𝐶)) = 0)) |
25 | 17, 22, 24 | sylanbrc 585 |
. . 3
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) |
26 | | elrabi 3677 |
. . . . 5
⊢ (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} → 𝑦 ∈ (1...(𝑀 + 𝑁))) |
27 | 26 | anim2i 618 |
. . . 4
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) → (𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)))) |
28 | | simpr 487 |
. . . . . . . 8
⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼‘𝐶)) → 𝑦 < (𝐼‘𝐶)) |
29 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | ballotlemfrcn0 31789 |
. . . . . . . . . . 11
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼‘𝐶)) → ((𝐹‘(𝑅‘𝐶))‘𝑦) ≠ 0) |
30 | 29 | neneqd 3023 |
. . . . . . . . . 10
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼‘𝐶)) → ¬ ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0) |
31 | | fveqeq2 6681 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑦 → (((𝐹‘(𝑅‘𝐶))‘𝑘) = 0 ↔ ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0)) |
32 | 31 | elrab 3682 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} ↔ (𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0)) |
33 | 32 | simprbi 499 |
. . . . . . . . . 10
⊢ (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} → ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0) |
34 | 30, 33 | nsyl 142 |
. . . . . . . . 9
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼‘𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) |
35 | 34 | 3expa 1114 |
. . . . . . . 8
⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼‘𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) |
36 | 28, 35 | syldan 593 |
. . . . . . 7
⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼‘𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) |
37 | 36 | ex 415 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → (𝑦 < (𝐼‘𝐶) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0})) |
38 | 37 | con2d 136 |
. . . . 5
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} → ¬ 𝑦 < (𝐼‘𝐶))) |
39 | 38 | imp 409 |
. . . 4
⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) → ¬ 𝑦 < (𝐼‘𝐶)) |
40 | 27, 39 | sylancom 590 |
. . 3
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) → ¬ 𝑦 < (𝐼‘𝐶)) |
41 | 15, 20, 25, 40 | infmin 8960 |
. 2
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}, ℝ, < ) = (𝐼‘𝐶)) |
42 | 13, 41 | eqtrd 2858 |
1
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = (𝐼‘𝐶)) |