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 |
|
eldif |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ↔ ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸 ) ) |
8 |
|
df-or |
⊢ ( ( ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂 ) ∨ ( 𝐶 ∈ 𝑂 ∧ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) ↔ ( ¬ ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂 ) → ( 𝐶 ∈ 𝑂 ∧ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) ) |
9 |
|
pm3.24 |
⊢ ¬ ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂 ) |
10 |
9
|
a1bi |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ↔ ( ¬ ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂 ) → ( 𝐶 ∈ 𝑂 ∧ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) ) |
11 |
8 10
|
bitr4i |
⊢ ( ( ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂 ) ∨ ( 𝐶 ∈ 𝑂 ∧ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) ↔ ( 𝐶 ∈ 𝑂 ∧ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) |
12 |
1 2 3 4 5 6
|
ballotleme |
⊢ ( 𝐶 ∈ 𝐸 ↔ ( 𝐶 ∈ 𝑂 ∧ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) |
13 |
12
|
notbii |
⊢ ( ¬ 𝐶 ∈ 𝐸 ↔ ¬ ( 𝐶 ∈ 𝑂 ∧ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) |
14 |
13
|
anbi2i |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸 ) ↔ ( 𝐶 ∈ 𝑂 ∧ ¬ ( 𝐶 ∈ 𝑂 ∧ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) ) |
15 |
|
ianor |
⊢ ( ¬ ( 𝐶 ∈ 𝑂 ∧ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ↔ ( ¬ 𝐶 ∈ 𝑂 ∨ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) |
16 |
15
|
anbi2i |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ ( 𝐶 ∈ 𝑂 ∧ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) ↔ ( 𝐶 ∈ 𝑂 ∧ ( ¬ 𝐶 ∈ 𝑂 ∨ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) ) |
17 |
|
andi |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ( ¬ 𝐶 ∈ 𝑂 ∨ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) ↔ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂 ) ∨ ( 𝐶 ∈ 𝑂 ∧ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) ) |
18 |
14 16 17
|
3bitri |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸 ) ↔ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂 ) ∨ ( 𝐶 ∈ 𝑂 ∧ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) ) |
19 |
|
fz1ssfz0 |
⊢ ( 1 ... ( 𝑀 + 𝑁 ) ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) |
20 |
19
|
a1i |
⊢ ( 𝐶 ∈ 𝑂 → ( 1 ... ( 𝑀 + 𝑁 ) ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
21 |
20
|
sseld |
⊢ ( 𝐶 ∈ 𝑂 → ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) → 𝑖 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ) |
22 |
21
|
imdistani |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → ( 𝐶 ∈ 𝑂 ∧ 𝑖 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ) |
23 |
|
simpl |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → 𝐶 ∈ 𝑂 ) |
24 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) → 𝑗 ∈ ℤ ) |
25 |
24
|
adantl |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → 𝑗 ∈ ℤ ) |
26 |
1 2 3 4 5 23 25
|
ballotlemfelz |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑗 ) ∈ ℤ ) |
27 |
26
|
zred |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑗 ) ∈ ℝ ) |
28 |
27
|
sbimi |
⊢ ( [ 𝑖 / 𝑗 ] ( 𝐶 ∈ 𝑂 ∧ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → [ 𝑖 / 𝑗 ] ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑗 ) ∈ ℝ ) |
29 |
|
sban |
⊢ ( [ 𝑖 / 𝑗 ] ( 𝐶 ∈ 𝑂 ∧ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ↔ ( [ 𝑖 / 𝑗 ] 𝐶 ∈ 𝑂 ∧ [ 𝑖 / 𝑗 ] 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ) |
30 |
|
sbv |
⊢ ( [ 𝑖 / 𝑗 ] 𝐶 ∈ 𝑂 ↔ 𝐶 ∈ 𝑂 ) |
31 |
|
clelsb3 |
⊢ ( [ 𝑖 / 𝑗 ] 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ↔ 𝑖 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
32 |
30 31
|
anbi12i |
⊢ ( ( [ 𝑖 / 𝑗 ] 𝐶 ∈ 𝑂 ∧ [ 𝑖 / 𝑗 ] 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ↔ ( 𝐶 ∈ 𝑂 ∧ 𝑖 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ) |
33 |
29 32
|
bitri |
⊢ ( [ 𝑖 / 𝑗 ] ( 𝐶 ∈ 𝑂 ∧ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ↔ ( 𝐶 ∈ 𝑂 ∧ 𝑖 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ) |
34 |
|
nfv |
⊢ Ⅎ 𝑗 ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ∈ ℝ |
35 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑗 ) = ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) |
36 |
35
|
eleq1d |
⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑗 ) ∈ ℝ ↔ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ∈ ℝ ) ) |
37 |
34 36
|
sbiev |
⊢ ( [ 𝑖 / 𝑗 ] ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑗 ) ∈ ℝ ↔ ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ∈ ℝ ) |
38 |
28 33 37
|
3imtr3i |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 𝑖 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ∈ ℝ ) |
39 |
22 38
|
syl |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ∈ ℝ ) |
40 |
|
0red |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → 0 ∈ ℝ ) |
41 |
39 40
|
lenltd |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → ( ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ≤ 0 ↔ ¬ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) |
42 |
41
|
rexbidva |
⊢ ( 𝐶 ∈ 𝑂 → ( ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ≤ 0 ↔ ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ¬ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) |
43 |
|
rexnal |
⊢ ( ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ¬ 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ↔ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) |
44 |
42 43
|
bitrdi |
⊢ ( 𝐶 ∈ 𝑂 → ( ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ≤ 0 ↔ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) |
45 |
44
|
pm5.32i |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ≤ 0 ) ↔ ( 𝐶 ∈ 𝑂 ∧ ¬ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ) ) |
46 |
11 18 45
|
3bitr4i |
⊢ ( ( 𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸 ) ↔ ( 𝐶 ∈ 𝑂 ∧ ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ≤ 0 ) ) |
47 |
7 46
|
bitri |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ↔ ( 𝐶 ∈ 𝑂 ∧ ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹 ‘ 𝐶 ) ‘ 𝑖 ) ≤ 0 ) ) |