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 |
|
ballotth.s |
⊢ 𝑆 = ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↦ if ( 𝑖 ≤ ( 𝐼 ‘ 𝑐 ) , ( ( ( 𝐼 ‘ 𝑐 ) + 1 ) − 𝑖 ) , 𝑖 ) ) ) |
10 |
|
elfzelz |
⊢ ( 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) → 𝐽 ∈ ℤ ) |
11 |
10
|
3ad2ant2 |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → 𝐽 ∈ ℤ ) |
12 |
11
|
zred |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → 𝐽 ∈ ℝ ) |
13 |
1 2 3 4 5 6 7 8
|
ballotlemiex |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) ) |
14 |
13
|
simpld |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) |
15 |
14
|
elfzelzd |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ℤ ) |
16 |
15
|
3ad2ant1 |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → ( 𝐼 ‘ 𝐶 ) ∈ ℤ ) |
17 |
16
|
zred |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → ( 𝐼 ‘ 𝐶 ) ∈ ℝ ) |
18 |
|
1red |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → 1 ∈ ℝ ) |
19 |
17 18
|
readdcld |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → ( ( 𝐼 ‘ 𝐶 ) + 1 ) ∈ ℝ ) |
20 |
|
simp3 |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → 𝐽 < ( 𝐼 ‘ 𝐶 ) ) |
21 |
16
|
zcnd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → ( 𝐼 ‘ 𝐶 ) ∈ ℂ ) |
22 |
|
1cnd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → 1 ∈ ℂ ) |
23 |
21 22
|
pncand |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 1 ) = ( 𝐼 ‘ 𝐶 ) ) |
24 |
20 23
|
breqtrrd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → 𝐽 < ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 1 ) ) |
25 |
12 19 18 24
|
ltsub13d |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → 1 < ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝐽 ) ) |
26 |
1 2 3 4 5 6 7 8 9
|
ballotlemsv |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝑆 ‘ 𝐶 ) ‘ 𝐽 ) = if ( 𝐽 ≤ ( 𝐼 ‘ 𝐶 ) , ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝐽 ) , 𝐽 ) ) |
27 |
26
|
3adant3 |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → ( ( 𝑆 ‘ 𝐶 ) ‘ 𝐽 ) = if ( 𝐽 ≤ ( 𝐼 ‘ 𝐶 ) , ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝐽 ) , 𝐽 ) ) |
28 |
12 17 20
|
ltled |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → 𝐽 ≤ ( 𝐼 ‘ 𝐶 ) ) |
29 |
28
|
iftrued |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → if ( 𝐽 ≤ ( 𝐼 ‘ 𝐶 ) , ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝐽 ) , 𝐽 ) = ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝐽 ) ) |
30 |
27 29
|
eqtrd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → ( ( 𝑆 ‘ 𝐶 ) ‘ 𝐽 ) = ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝐽 ) ) |
31 |
25 30
|
breqtrrd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐽 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐽 < ( 𝐼 ‘ 𝐶 ) ) → 1 < ( ( 𝑆 ‘ 𝐶 ) ‘ 𝐽 ) ) |