Step | Hyp | Ref
| Expression |
1 | | 1zzd 11408 |
. . . . 5
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ∈ ℤ) |
2 | | ballotth.m |
. . . . . . . 8
⊢ 𝑀 ∈ ℕ |
3 | | ballotth.n |
. . . . . . . 8
⊢ 𝑁 ∈ ℕ |
4 | | nnaddcl 11042 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) |
5 | 2, 3, 4 | mp2an 708 |
. . . . . . 7
⊢ (𝑀 + 𝑁) ∈ ℕ |
6 | 5 | nnzi 11401 |
. . . . . 6
⊢ (𝑀 + 𝑁) ∈ ℤ |
7 | 6 | a1i 11 |
. . . . 5
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝑀 + 𝑁) ∈ ℤ) |
8 | | ballotth.o |
. . . . . . . . 9
⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀} |
9 | | ballotth.p |
. . . . . . . . 9
⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂))) |
10 | | ballotth.f |
. . . . . . . . 9
⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐))))) |
11 | | ballotth.e |
. . . . . . . . 9
⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} |
12 | | ballotth.mgtn |
. . . . . . . . 9
⊢ 𝑁 < 𝑀 |
13 | | ballotth.i |
. . . . . . . . 9
⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) |
14 | | ballotth.s |
. . . . . . . . 9
⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) |
15 | 2, 3, 8, 9, 10, 11, 12, 13, 14 | ballotlemsdom 30573 |
. . . . . . . 8
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁))) |
16 | | elfzelz 12342 |
. . . . . . . 8
⊢ (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ) |
17 | 15, 16 | syl 17 |
. . . . . . 7
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ) |
18 | 17 | 3adant3 1081 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ) |
19 | 18, 1 | zsubcld 11487 |
. . . . 5
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈
ℤ) |
20 | 2, 3, 8, 9, 10, 11, 12, 13, 14 | ballotlemsgt1 30572 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 < ((𝑆‘𝐶)‘𝐽)) |
21 | | zltlem1 11430 |
. . . . . . 7
⊢ ((1
∈ ℤ ∧ ((𝑆‘𝐶)‘𝐽) ∈ ℤ) → (1 < ((𝑆‘𝐶)‘𝐽) ↔ 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1))) |
22 | 21 | biimpa 501 |
. . . . . 6
⊢ (((1
∈ ℤ ∧ ((𝑆‘𝐶)‘𝐽) ∈ ℤ) ∧ 1 < ((𝑆‘𝐶)‘𝐽)) → 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1)) |
23 | 1, 18, 20, 22 | syl21anc 1325 |
. . . . 5
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1)) |
24 | 18 | zred 11482 |
. . . . . . 7
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ ℝ) |
25 | | 1red 10055 |
. . . . . . 7
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ∈ ℝ) |
26 | 24, 25 | resubcld 10458 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈
ℝ) |
27 | | simp1 1061 |
. . . . . . . 8
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐶 ∈ (𝑂 ∖ 𝐸)) |
28 | 2, 3, 8, 9, 10, 11, 12, 13 | ballotlemiex 30563 |
. . . . . . . . 9
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)) |
29 | 28 | simpld 475 |
. . . . . . . 8
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁))) |
30 | | elfzelz 12342 |
. . . . . . . 8
⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ) |
31 | 27, 29, 30 | 3syl 18 |
. . . . . . 7
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℤ) |
32 | 31 | zred 11482 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℝ) |
33 | 7 | zred 11482 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝑀 + 𝑁) ∈ ℝ) |
34 | | elfzelz 12342 |
. . . . . . . . . . . 12
⊢ (𝐽 ∈ (1...(𝑀 + 𝑁)) → 𝐽 ∈ ℤ) |
35 | 34 | 3ad2ant2 1083 |
. . . . . . . . . . 11
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ ℤ) |
36 | | elfzle1 12344 |
. . . . . . . . . . . 12
⊢ (𝐽 ∈ (1...(𝑀 + 𝑁)) → 1 ≤ 𝐽) |
37 | 36 | 3ad2ant2 1083 |
. . . . . . . . . . 11
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ≤ 𝐽) |
38 | 35 | zred 11482 |
. . . . . . . . . . . 12
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ ℝ) |
39 | | simp3 1063 |
. . . . . . . . . . . 12
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 < (𝐼‘𝐶)) |
40 | 38, 32, 39 | ltled 10185 |
. . . . . . . . . . 11
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ≤ (𝐼‘𝐶)) |
41 | | elfz4 12335 |
. . . . . . . . . . 11
⊢ (((1
∈ ℤ ∧ (𝐼‘𝐶) ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (1 ≤ 𝐽 ∧ 𝐽 ≤ (𝐼‘𝐶))) → 𝐽 ∈ (1...(𝐼‘𝐶))) |
42 | 1, 31, 35, 37, 40, 41 | syl32anc 1334 |
. . . . . . . . . 10
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ (1...(𝐼‘𝐶))) |
43 | 2, 3, 8, 9, 10, 11, 12, 13, 14 | ballotlemsel1i 30574 |
. . . . . . . . . 10
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶))) |
44 | 27, 42, 43 | syl2anc 693 |
. . . . . . . . 9
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶))) |
45 | | elfzle2 12345 |
. . . . . . . . 9
⊢ (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶)) |
46 | 44, 45 | syl 17 |
. . . . . . . 8
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶)) |
47 | | zlem1lt 11429 |
. . . . . . . . 9
⊢ ((((𝑆‘𝐶)‘𝐽) ∈ ℤ ∧ (𝐼‘𝐶) ∈ ℤ) → (((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶))) |
48 | 18, 31, 47 | syl2anc 693 |
. . . . . . . 8
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶))) |
49 | 46, 48 | mpbid 222 |
. . . . . . 7
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶)) |
50 | 26, 32, 49 | ltled 10185 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ≤ (𝐼‘𝐶)) |
51 | | elfzle2 12345 |
. . . . . . 7
⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁)) |
52 | 27, 29, 51 | 3syl 18 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁)) |
53 | 26, 32, 33, 50, 52 | letrd 10194 |
. . . . 5
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ≤ (𝑀 + 𝑁)) |
54 | | elfz4 12335 |
. . . . 5
⊢ (((1
∈ ℤ ∧ (𝑀 +
𝑁) ∈ ℤ ∧
(((𝑆‘𝐶)‘𝐽) − 1) ∈ ℤ) ∧ (1 ≤
(((𝑆‘𝐶)‘𝐽) − 1) ∧ (((𝑆‘𝐶)‘𝐽) − 1) ≤ (𝑀 + 𝑁))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁))) |
55 | 1, 7, 19, 23, 53, 54 | syl32anc 1334 |
. . . 4
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁))) |
56 | | biid 251 |
. . . . . . . . 9
⊢ ((((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶)) |
57 | 49, 56 | sylibr 224 |
. . . . . . . 8
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶)) |
58 | 2, 3, 8, 9, 10, 11, 12, 13 | ballotlemi 30562 |
. . . . . . . . . 10
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )) |
59 | 58 | breq2d 4665 |
. . . . . . . . 9
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))) |
60 | 59 | 3ad2ant1 1082 |
. . . . . . . 8
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))) |
61 | 57, 60 | mpbid 222 |
. . . . . . 7
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )) |
62 | | ltso 10118 |
. . . . . . . . . 10
⊢ < Or
ℝ |
63 | 62 | a1i 11 |
. . . . . . . . 9
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → < Or ℝ) |
64 | 2, 3, 8, 9, 10, 11, 12, 13 | ballotlemsup 30566 |
. . . . . . . . 9
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}𝑦 < 𝑤))) |
65 | 63, 64 | inflb 8395 |
. . . . . . . 8
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((((𝑆‘𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} → ¬ (((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))) |
66 | 65 | con2d 129 |
. . . . . . 7
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ) → ¬ (((𝑆‘𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})) |
67 | 27, 61, 66 | sylc 65 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ¬ (((𝑆‘𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}) |
68 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑘 = (((𝑆‘𝐶)‘𝐽) − 1) → ((𝐹‘𝐶)‘𝑘) = ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1))) |
69 | 68 | eqeq1d 2624 |
. . . . . . 7
⊢ (𝑘 = (((𝑆‘𝐶)‘𝐽) − 1) → (((𝐹‘𝐶)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0)) |
70 | 69 | elrab 3363 |
. . . . . 6
⊢ ((((𝑆‘𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ↔ ((((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0)) |
71 | 67, 70 | sylnib 318 |
. . . . 5
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ¬ ((((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0)) |
72 | | imnan 438 |
. . . . 5
⊢
(((((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) → ¬ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0) ↔ ¬ ((((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0)) |
73 | 71, 72 | sylibr 224 |
. . . 4
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) → ¬ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0)) |
74 | 55, 73 | mpd 15 |
. . 3
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ¬ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0) |
75 | 74 | neqned 2801 |
. 2
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ≠ 0) |
76 | | ballotth.r |
. . . . . . . . . 10
⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) |
77 | 2, 3, 8, 9, 10, 11, 12, 13, 14, 76 | ballotlemro 30584 |
. . . . . . . . 9
⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂) |
78 | 77 | adantr 481 |
. . . . . . . 8
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂) |
79 | | elfzelz 12342 |
. . . . . . . . 9
⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ) |
80 | 79 | adantl 482 |
. . . . . . . 8
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ) |
81 | 2, 3, 8, 9, 10, 78, 80 | ballotlemfelz 30552 |
. . . . . . 7
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℤ) |
82 | 81 | zcnd 11483 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℂ) |
83 | 82 | negeq0d 10384 |
. . . . 5
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) = 0 ↔ -((𝐹‘(𝑅‘𝐶))‘𝐽) = 0)) |
84 | | eqid 2622 |
. . . . . . 7
⊢ (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣 ∩ 𝑢)) − (#‘(𝑣 ∖ 𝑢)))) = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣 ∩ 𝑢)) − (#‘(𝑣 ∖ 𝑢)))) |
85 | 2, 3, 8, 9, 10, 11, 12, 13, 14, 76, 84 | ballotlemfrceq 30590 |
. . . . . 6
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽)) |
86 | 85 | eqeq1d 2624 |
. . . . 5
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0 ↔ -((𝐹‘(𝑅‘𝐶))‘𝐽) = 0)) |
87 | 83, 86 | bitr4d 271 |
. . . 4
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) = 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0)) |
88 | 87 | necon3bid 2838 |
. . 3
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ≠ 0)) |
89 | 27, 42, 88 | syl2anc 693 |
. 2
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ≠ 0)) |
90 | 75, 89 | mpbird 247 |
1
⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0) |