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 |
|
nnaddcl |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝑀 + 𝑁 ) ∈ ℕ ) |
8 |
1 2 7
|
mp2an |
⊢ ( 𝑀 + 𝑁 ) ∈ ℕ |
9 |
|
elnnuz |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ℕ ↔ ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 1 ) ) |
10 |
8 9
|
mpbi |
⊢ ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 1 ) |
11 |
|
eluzfz1 |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) |
12 |
10 11
|
ax-mp |
⊢ 1 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) |
13 |
|
0le1 |
⊢ 0 ≤ 1 |
14 |
|
0re |
⊢ 0 ∈ ℝ |
15 |
|
1re |
⊢ 1 ∈ ℝ |
16 |
14 15
|
lenlti |
⊢ ( 0 ≤ 1 ↔ ¬ 1 < 0 ) |
17 |
13 16
|
mpbi |
⊢ ¬ 1 < 0 |
18 |
|
ltsub13 |
⊢ ( ( 0 ∈ ℝ ∧ 0 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 0 < ( 0 − 1 ) ↔ 1 < ( 0 − 0 ) ) ) |
19 |
14 14 15 18
|
mp3an |
⊢ ( 0 < ( 0 − 1 ) ↔ 1 < ( 0 − 0 ) ) |
20 |
|
0m0e0 |
⊢ ( 0 − 0 ) = 0 |
21 |
20
|
breq2i |
⊢ ( 1 < ( 0 − 0 ) ↔ 1 < 0 ) |
22 |
19 21
|
bitri |
⊢ ( 0 < ( 0 − 1 ) ↔ 1 < 0 ) |
23 |
17 22
|
mtbir |
⊢ ¬ 0 < ( 0 − 1 ) |
24 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
25 |
24
|
fveq2i |
⊢ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) = ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) |
26 |
1 2 3 4 5
|
ballotlemfval0 |
⊢ ( 𝐶 ∈ 𝑂 → ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) = 0 ) |
27 |
25 26
|
syl5eq |
⊢ ( 𝐶 ∈ 𝑂 → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) = 0 ) |
28 |
27
|
oveq1d |
⊢ ( 𝐶 ∈ 𝑂 → ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) = ( 0 − 1 ) ) |
29 |
28
|
breq2d |
⊢ ( 𝐶 ∈ 𝑂 → ( 0 < ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ↔ 0 < ( 0 − 1 ) ) ) |
30 |
23 29
|
mtbiri |
⊢ ( 𝐶 ∈ 𝑂 → ¬ 0 < ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) |
31 |
30
|
adantr |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 1 ∈ 𝐶 ) → ¬ 0 < ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) |
32 |
|
simpl |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 1 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → 𝐶 ∈ 𝑂 ) |
33 |
|
1nn |
⊢ 1 ∈ ℕ |
34 |
33
|
a1i |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 1 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → 1 ∈ ℕ ) |
35 |
1 2 3 4 5 32 34
|
ballotlemfp1 |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 1 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → ( ( ¬ 1 ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) ∧ ( 1 ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) + 1 ) ) ) ) |
36 |
35
|
simpld |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 1 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → ( ¬ 1 ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) ) |
37 |
12 36
|
mpan2 |
⊢ ( 𝐶 ∈ 𝑂 → ( ¬ 1 ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) ) |
38 |
37
|
imp |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 1 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) |
39 |
38
|
breq2d |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 1 ∈ 𝐶 ) → ( 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ↔ 0 < ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) ) |
40 |
31 39
|
mtbird |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 1 ∈ 𝐶 ) → ¬ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ) |
41 |
|
fveq2 |
⊢ ( 𝑖 = 1 → ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) = ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ) |
42 |
41
|
breq2d |
⊢ ( 𝑖 = 1 → ( 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ↔ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ) ) |
43 |
42
|
notbid |
⊢ ( 𝑖 = 1 → ( ¬ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ↔ ¬ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ) ) |
44 |
43
|
rspcev |
⊢ ( ( 1 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ) → ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ¬ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) |
45 |
12 40 44
|
sylancr |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 1 ∈ 𝐶 ) → ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ¬ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) |
46 |
|
rexnal |
⊢ ( ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ¬ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ↔ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) |
47 |
45 46
|
sylib |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 1 ∈ 𝐶 ) → ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) |
48 |
1 2 3 4 5 6
|
ballotleme |
⊢ ( 𝐶 ∈ 𝐸 ↔ ( 𝐶 ∈ 𝑂 ∧ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) |
49 |
48
|
simprbi |
⊢ ( 𝐶 ∈ 𝐸 → ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) |
50 |
47 49
|
nsyl |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 1 ∈ 𝐶 ) → ¬ 𝐶 ∈ 𝐸 ) |
51 |
50
|
ex |
⊢ ( 𝐶 ∈ 𝑂 → ( ¬ 1 ∈ 𝐶 → ¬ 𝐶 ∈ 𝐸 ) ) |