Metamath Proof Explorer

Theorem ballotlemfrci

Description: Reverse counting preserves a tie at the first tie. (Contributed by Thierry Arnoux, 21-Apr-2017)

Ref Expression
Hypotheses ballotth.m ⊢ 𝑀 ∈ ℕ
ballotth.n ⊢ 𝑁 ∈ ℕ
ballotth.o ⊢ 𝑂 = { 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 }
ballotth.p ⊢ 𝑃 = ( 𝑥 ∈ 𝒫 𝑂 ↦ ( ( ♯ ‘ 𝑥 ) / ( ♯ ‘ 𝑂 ) ) )
ballotth.f ⊢ 𝐹 = ( 𝑐 ∈ 𝑂 ↦ ( 𝑖 ∈ ℤ ↦ ( ( ♯ ‘ ( ( 1 ... 𝑖 ) ∩ 𝑐 ) ) − ( ♯ ‘ ( ( 1 ... 𝑖 ) ∖ 𝑐 ) ) ) ) )
ballotth.e ⊢ 𝐸 = { 𝑐 ∈ 𝑂 ∣ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹 ‘ 𝑐 ) ‘ 𝑖 ) }
ballotth.mgtn ⊢ 𝑁 < 𝑀
ballotth.i ⊢ 𝐼 = ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹 ‘ 𝑐 ) ‘ 𝑘 ) = 0 } , ℝ , < ) )
ballotth.s ⊢ 𝑆 = ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↦ if ( 𝑖 ≤ ( 𝐼 ‘ 𝑐 ) , ( ( ( 𝐼 ‘ 𝑐 ) + 1 ) − 𝑖 ) , 𝑖 ) ) )
ballotth.r ⊢ 𝑅 = ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) )
ballotlemg ⊢ ↑ = ( 𝑢 ∈ Fin , 𝑣 ∈ Fin ↦ ( ( ♯ ‘ ( 𝑣 ∩ 𝑢 ) ) − ( ♯ ‘ ( 𝑣 ∖ 𝑢 ) ) ) )
Assertion ballotlemfrci ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 )

Proof

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 ballotth.r ⊢ 𝑅 = ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) )
11 ballotlemg ⊢ ↑ = ( 𝑢 ∈ Fin , 𝑣 ∈ Fin ↦ ( ( ♯ ‘ ( 𝑣 ∩ 𝑢 ) ) − ( ♯ ‘ ( 𝑣 ∖ 𝑢 ) ) ) )
12 1 2 3 4 5 6 7 8 ballotlemiex ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) )
13 12 simpld ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) )
14 elfzuz ⊢ ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) → ( 𝐼 ‘ 𝐶 ) ∈ ( ℤ≥ ‘ 1 ) )
15 eluzfz2 ⊢ ( ( 𝐼 ‘ 𝐶 ) ∈ ( ℤ≥ ‘ 1 ) → ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝐼 ‘ 𝐶 ) ) )
16 13 14 15 3syl ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝐼 ‘ 𝐶 ) ) )
17 1 2 3 4 5 6 7 8 9 10 11 ballotlemfrc ⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝐼 ‘ 𝐶 ) ) ) → ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ ( 𝐼 ‘ 𝐶 ) ) = ( 𝐶 ↑ ( ( ( 𝑆 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) ... ( 𝐼 ‘ 𝐶 ) ) ) )
18 16 17 mpdan ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ ( 𝐼 ‘ 𝐶 ) ) = ( 𝐶 ↑ ( ( ( 𝑆 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) ... ( 𝐼 ‘ 𝐶 ) ) ) )
19 1 2 3 4 5 6 7 8 9 ballotlemsi ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝑆 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 1 )
20 19 oveq1d ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( ( 𝑆 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) ... ( 𝐼 ‘ 𝐶 ) ) = ( 1 ... ( 𝐼 ‘ 𝐶 ) ) )
21 20 oveq2d ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐶 ↑ ( ( ( 𝑆 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) ... ( 𝐼 ‘ 𝐶 ) ) ) = ( 𝐶 ↑ ( 1 ... ( 𝐼 ‘ 𝐶 ) ) ) )
22 18 21 eqtrd ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ ( 𝐼 ‘ 𝐶 ) ) = ( 𝐶 ↑ ( 1 ... ( 𝐼 ‘ 𝐶 ) ) ) )
23 fz1ssfz0 ⊢ ( 1 ... ( 𝑀 + 𝑁 ) ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) )
24 23 13 sselid ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) )
25 1 2 3 4 5 6 7 8 9 10 11 ballotlemfg ⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ( 𝐼 ‘ 𝐶 ) ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = ( 𝐶 ↑ ( 1 ... ( 𝐼 ‘ 𝐶 ) ) ) )
26 24 25 mpdan ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = ( 𝐶 ↑ ( 1 ... ( 𝐼 ‘ 𝐶 ) ) ) )
27 12 simprd ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 )
28 22 26 27 3eqtr2d ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 )