Metamath Proof Explorer

Theorem ballotlemrc

Description: Range of R . (Contributed by Thierry Arnoux, 19-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 ) )
Assertion ballotlemrc ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑅 ‘ 𝐶 ) ∈ ( 𝑂 ∖ 𝐸 ) )

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 1 2 3 4 5 6 7 8 9 10 ballotlemro ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑅 ‘ 𝐶 ) ∈ 𝑂 )
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 eqid ⊢ ( 𝑢 ∈ Fin , 𝑣 ∈ Fin ↦ ( ( ♯ ‘ ( 𝑣 ∩ 𝑢 ) ) − ( ♯ ‘ ( 𝑣 ∖ 𝑢 ) ) ) ) = ( 𝑢 ∈ Fin , 𝑣 ∈ Fin ↦ ( ( ♯ ‘ ( 𝑣 ∩ 𝑢 ) ) − ( ♯ ‘ ( 𝑣 ∖ 𝑢 ) ) ) )
15 1 2 3 4 5 6 7 8 9 10 14 ballotlemfrci ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 )
16 0le0 ⊢ 0 ≤ 0
17 15 16 eqbrtrdi ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ ( 𝐼 ‘ 𝐶 ) ) ≤ 0 )
18 fveq2 ⊢ ( 𝑖 = ( 𝐼 ‘ 𝐶 ) → ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ 𝑖 ) = ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ ( 𝐼 ‘ 𝐶 ) ) )
19 18 breq1d ⊢ ( 𝑖 = ( 𝐼 ‘ 𝐶 ) → ( ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ 𝑖 ) ≤ 0 ↔ ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ ( 𝐼 ‘ 𝐶 ) ) ≤ 0 ) )
20 19 rspcev ⊢ ( ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ ( 𝐼 ‘ 𝐶 ) ) ≤ 0 ) → ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ 𝑖 ) ≤ 0 )
21 13 17 20 syl2anc ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ 𝑖 ) ≤ 0 )
22 1 2 3 4 5 6 ballotlemodife ⊢ ( ( 𝑅 ‘ 𝐶 ) ∈ ( 𝑂 ∖ 𝐸 ) ↔ ( ( 𝑅 ‘ 𝐶 ) ∈ 𝑂 ∧ ∃ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹 ‘ ( 𝑅 ‘ 𝐶 ) ) ‘ 𝑖 ) ≤ 0 ) )
23 11 21 22 sylanbrc ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑅 ‘ 𝐶 ) ∈ ( 𝑂 ∖ 𝐸 ) )