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 |
|
eldifi |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → 𝐶 ∈ 𝑂 ) |
10 |
9
|
ad2antrr |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → 𝐶 ∈ 𝑂 ) |
11 |
1 2 3 4 5 6 7 8
|
ballotlemiex |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) ) |
12 |
11
|
simpld |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) |
13 |
|
elfznn |
⊢ ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) → ( 𝐼 ‘ 𝐶 ) ∈ ℕ ) |
14 |
12 13
|
syl |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ℕ ) |
15 |
14
|
adantr |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( 𝐼 ‘ 𝐶 ) ∈ ℕ ) |
16 |
1 2 3 4 5 6 7 8
|
ballotlemi1 |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( 𝐼 ‘ 𝐶 ) ≠ 1 ) |
17 |
|
eluz2b3 |
⊢ ( ( 𝐼 ‘ 𝐶 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( ( 𝐼 ‘ 𝐶 ) ∈ ℕ ∧ ( 𝐼 ‘ 𝐶 ) ≠ 1 ) ) |
18 |
15 16 17
|
sylanbrc |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( 𝐼 ‘ 𝐶 ) ∈ ( ℤ≥ ‘ 2 ) ) |
19 |
|
uz2m1nn |
⊢ ( ( 𝐼 ‘ 𝐶 ) ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝐼 ‘ 𝐶 ) − 1 ) ∈ ℕ ) |
20 |
18 19
|
syl |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( ( 𝐼 ‘ 𝐶 ) − 1 ) ∈ ℕ ) |
21 |
20
|
adantr |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( ( 𝐼 ‘ 𝐶 ) − 1 ) ∈ ℕ ) |
22 |
|
elnnuz |
⊢ ( ( ( 𝐼 ‘ 𝐶 ) − 1 ) ∈ ℕ ↔ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
23 |
22
|
biimpi |
⊢ ( ( ( 𝐼 ‘ 𝐶 ) − 1 ) ∈ ℕ → ( ( 𝐼 ‘ 𝐶 ) − 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
24 |
|
eluzfz1 |
⊢ ( ( ( 𝐼 ‘ 𝐶 ) − 1 ) ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ) |
25 |
20 23 24
|
3syl |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → 1 ∈ ( 1 ... ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ) |
26 |
25
|
adantr |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → 1 ∈ ( 1 ... ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ) |
27 |
|
1nn |
⊢ 1 ∈ ℕ |
28 |
27
|
a1i |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → 1 ∈ ℕ ) |
29 |
1 2 3 4 5 9 28
|
ballotlemfp1 |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( ¬ 1 ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) ∧ ( 1 ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) + 1 ) ) ) ) |
30 |
29
|
simpld |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ¬ 1 ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) ) |
31 |
30
|
imp |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) |
32 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
33 |
32
|
fveq2i |
⊢ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) = ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) |
34 |
33
|
oveq1i |
⊢ ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) − 1 ) |
35 |
34
|
a1i |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) − 1 ) ) |
36 |
1 2 3 4 5
|
ballotlemfval0 |
⊢ ( 𝐶 ∈ 𝑂 → ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) = 0 ) |
37 |
9 36
|
syl |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) = 0 ) |
38 |
37
|
adantr |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) = 0 ) |
39 |
38
|
oveq1d |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) − 1 ) = ( 0 − 1 ) ) |
40 |
31 35 39
|
3eqtrrd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( 0 − 1 ) = ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ) |
41 |
|
0le1 |
⊢ 0 ≤ 1 |
42 |
|
0re |
⊢ 0 ∈ ℝ |
43 |
|
1re |
⊢ 1 ∈ ℝ |
44 |
|
suble0 |
⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 0 − 1 ) ≤ 0 ↔ 0 ≤ 1 ) ) |
45 |
42 43 44
|
mp2an |
⊢ ( ( 0 − 1 ) ≤ 0 ↔ 0 ≤ 1 ) |
46 |
41 45
|
mpbir |
⊢ ( 0 − 1 ) ≤ 0 |
47 |
40 46
|
eqbrtrrdi |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ≤ 0 ) |
48 |
47
|
adantr |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ≤ 0 ) |
49 |
|
fveq2 |
⊢ ( 𝑖 = 1 → ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) = ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ) |
50 |
49
|
breq1d |
⊢ ( 𝑖 = 1 → ( ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ≤ 0 ↔ ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ≤ 0 ) ) |
51 |
50
|
rspcev |
⊢ ( ( 1 ∈ ( 1 ... ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ∧ ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ≤ 0 ) → ∃ 𝑖 ∈ ( 1 ... ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ≤ 0 ) |
52 |
26 48 51
|
syl2anc |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ∃ 𝑖 ∈ ( 1 ... ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ≤ 0 ) |
53 |
|
0lt1 |
⊢ 0 < 1 |
54 |
|
1p0e1 |
⊢ ( 1 + 0 ) = 1 |
55 |
1 2 3 4 5 9 14
|
ballotlemfp1 |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) − 1 ) ) ∧ ( ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) + 1 ) ) ) ) |
56 |
55
|
simpld |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) − 1 ) ) ) |
57 |
56
|
imp |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) − 1 ) ) |
58 |
11
|
simprd |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) |
59 |
58
|
adantr |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) |
60 |
57 59
|
eqtr3d |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) − 1 ) = 0 ) |
61 |
9
|
adantr |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → 𝐶 ∈ 𝑂 ) |
62 |
14
|
nnzd |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ℤ ) |
63 |
62
|
adantr |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( 𝐼 ‘ 𝐶 ) ∈ ℤ ) |
64 |
|
1zzd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → 1 ∈ ℤ ) |
65 |
63 64
|
zsubcld |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( ( 𝐼 ‘ 𝐶 ) − 1 ) ∈ ℤ ) |
66 |
1 2 3 4 5 61 65
|
ballotlemfelz |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ∈ ℤ ) |
67 |
66
|
zcnd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ∈ ℂ ) |
68 |
|
1cnd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → 1 ∈ ℂ ) |
69 |
|
0cnd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → 0 ∈ ℂ ) |
70 |
67 68 69
|
subaddd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) − 1 ) = 0 ↔ ( 1 + 0 ) = ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ) ) |
71 |
60 70
|
mpbid |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ( 1 + 0 ) = ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ) |
72 |
54 71
|
eqtr3id |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → 1 = ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ) |
73 |
53 72
|
breqtrid |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ) |
74 |
73
|
adantlr |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ) |
75 |
1 2 3 4 5 10 21 52 74
|
ballotlemfc0 |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ∃ 𝑘 ∈ ( 1 ... ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 ) |
76 |
1 2 3 4 5 6 7 8
|
ballotlemimin |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ¬ ∃ 𝑘 ∈ ( 1 ... ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 ) |
77 |
76
|
ad2antrr |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) ∧ ¬ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) → ¬ ∃ 𝑘 ∈ ( 1 ... ( ( 𝐼 ‘ 𝐶 ) − 1 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑘 ) = 0 ) |
78 |
75 77
|
condan |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ¬ 1 ∈ 𝐶 ) → ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) |