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 |
|
1e0p1 |
⊢ 1 = ( 0 + 1 ) |
10 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
11 |
9 10
|
eqnetrri |
⊢ ( 0 + 1 ) ≠ 0 |
12 |
11
|
neii |
⊢ ¬ ( 0 + 1 ) = 0 |
13 |
|
eldifi |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → 𝐶 ∈ 𝑂 ) |
14 |
|
1nn |
⊢ 1 ∈ ℕ |
15 |
14
|
a1i |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → 1 ∈ ℕ ) |
16 |
1 2 3 4 5 13 15
|
ballotlemfp1 |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( ¬ 1 ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) − 1 ) ) ∧ ( 1 ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) + 1 ) ) ) ) |
17 |
16
|
simprd |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 1 ∈ 𝐶 → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) + 1 ) ) ) |
18 |
17
|
imp |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) + 1 ) ) |
19 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
20 |
19
|
fveq2i |
⊢ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) = ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) |
21 |
20
|
oveq1i |
⊢ ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) + 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) + 1 ) |
22 |
21
|
a1i |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) → ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 1 − 1 ) ) + 1 ) = ( ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) + 1 ) ) |
23 |
1 2 3 4 5
|
ballotlemfval0 |
⊢ ( 𝐶 ∈ 𝑂 → ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) = 0 ) |
24 |
13 23
|
syl |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) = 0 ) |
25 |
24
|
adantr |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) = 0 ) |
26 |
25
|
oveq1d |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) → ( ( ( 𝐹 ‘ 𝐶 ) ‘ 0 ) + 1 ) = ( 0 + 1 ) ) |
27 |
18 22 26
|
3eqtrrd |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) → ( 0 + 1 ) = ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) ) |
28 |
27
|
eqeq1d |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) → ( ( 0 + 1 ) = 0 ↔ ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = 0 ) ) |
29 |
12 28
|
mtbii |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) → ¬ ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = 0 ) |
30 |
1 2 3 4 5 6 7 8
|
ballotlemiex |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) ) |
31 |
30
|
simprd |
⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) |
32 |
31
|
ad2antrr |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) ∧ ( 𝐼 ‘ 𝐶 ) = 1 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) |
33 |
|
fveqeq2 |
⊢ ( ( 𝐼 ‘ 𝐶 ) = 1 → ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ↔ ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = 0 ) ) |
34 |
33
|
adantl |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) ∧ ( 𝐼 ‘ 𝐶 ) = 1 ) → ( ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ↔ ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = 0 ) ) |
35 |
32 34
|
mpbid |
⊢ ( ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) ∧ ( 𝐼 ‘ 𝐶 ) = 1 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ 1 ) = 0 ) |
36 |
29 35
|
mtand |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) → ¬ ( 𝐼 ‘ 𝐶 ) = 1 ) |
37 |
36
|
neqned |
⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ 𝐶 ) → ( 𝐼 ‘ 𝐶 ) ≠ 1 ) |