Metamath Proof Explorer


Theorem ballotlemiex

Description: Properties of ( IC ) . (Contributed by Thierry Arnoux, 12-Dec-2016) (Revised by AV, 6-Oct-2020)

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 = ( 𝑐𝑂 ↦ ( 𝑖 ∈ ℤ ↦ ( ( ♯ ‘ ( ( 1 ... 𝑖 ) ∩ 𝑐 ) ) − ( ♯ ‘ ( ( 1 ... 𝑖 ) ∖ 𝑐 ) ) ) ) )
ballotth.e 𝐸 = { 𝑐𝑂 ∣ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹𝑐 ) ‘ 𝑖 ) }
ballotth.mgtn 𝑁 < 𝑀
ballotth.i 𝐼 = ( 𝑐 ∈ ( 𝑂𝐸 ) ↦ inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝑐 ) ‘ 𝑘 ) = 0 } , ℝ , < ) )
Assertion ballotlemiex ( 𝐶 ∈ ( 𝑂𝐸 ) → ( ( 𝐼𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹𝐶 ) ‘ ( 𝐼𝐶 ) ) = 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 = ( 𝑐𝑂 ↦ ( 𝑖 ∈ ℤ ↦ ( ( ♯ ‘ ( ( 1 ... 𝑖 ) ∩ 𝑐 ) ) − ( ♯ ‘ ( ( 1 ... 𝑖 ) ∖ 𝑐 ) ) ) ) )
6 ballotth.e 𝐸 = { 𝑐𝑂 ∣ ∀ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) 0 < ( ( 𝐹𝑐 ) ‘ 𝑖 ) }
7 ballotth.mgtn 𝑁 < 𝑀
8 ballotth.i 𝐼 = ( 𝑐 ∈ ( 𝑂𝐸 ) ↦ inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝑐 ) ‘ 𝑘 ) = 0 } , ℝ , < ) )
9 1 2 3 4 5 6 7 8 ballotlemi ( 𝐶 ∈ ( 𝑂𝐸 ) → ( 𝐼𝐶 ) = inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } , ℝ , < ) )
10 ltso < Or ℝ
11 10 a1i ( 𝐶 ∈ ( 𝑂𝐸 ) → < Or ℝ )
12 fzfi ( 1 ... ( 𝑀 + 𝑁 ) ) ∈ Fin
13 ssrab2 { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ⊆ ( 1 ... ( 𝑀 + 𝑁 ) )
14 ssfi ( ( ( 1 ... ( 𝑀 + 𝑁 ) ) ∈ Fin ∧ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ∈ Fin )
15 12 13 14 mp2an { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ∈ Fin
16 15 a1i ( 𝐶 ∈ ( 𝑂𝐸 ) → { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ∈ Fin )
17 1 2 3 4 5 6 7 ballotlem5 ( 𝐶 ∈ ( 𝑂𝐸 ) → ∃ 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 )
18 rabn0 ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ≠ ∅ ↔ ∃ 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 )
19 17 18 sylibr ( 𝐶 ∈ ( 𝑂𝐸 ) → { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ≠ ∅ )
20 fzssuz ( 1 ... ( 𝑀 + 𝑁 ) ) ⊆ ( ℤ ‘ 1 )
21 uzssz ( ℤ ‘ 1 ) ⊆ ℤ
22 20 21 sstri ( 1 ... ( 𝑀 + 𝑁 ) ) ⊆ ℤ
23 zssre ℤ ⊆ ℝ
24 22 23 sstri ( 1 ... ( 𝑀 + 𝑁 ) ) ⊆ ℝ
25 13 24 sstri { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ⊆ ℝ
26 25 a1i ( 𝐶 ∈ ( 𝑂𝐸 ) → { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ⊆ ℝ )
27 fiinfcl ( ( < Or ℝ ∧ ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ∈ Fin ∧ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ≠ ∅ ∧ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ⊆ ℝ ) ) → inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ∈ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } )
28 11 16 19 26 27 syl13anc ( 𝐶 ∈ ( 𝑂𝐸 ) → inf ( { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } , ℝ , < ) ∈ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } )
29 9 28 eqeltrd ( 𝐶 ∈ ( 𝑂𝐸 ) → ( 𝐼𝐶 ) ∈ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } )
30 fveqeq2 ( 𝑘 = ( 𝐼𝐶 ) → ( ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 ↔ ( ( 𝐹𝐶 ) ‘ ( 𝐼𝐶 ) ) = 0 ) )
31 30 elrab ( ( 𝐼𝐶 ) ∈ { 𝑘 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ( 𝐹𝐶 ) ‘ 𝑘 ) = 0 } ↔ ( ( 𝐼𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹𝐶 ) ‘ ( 𝐼𝐶 ) ) = 0 ) )
32 29 31 sylib ( 𝐶 ∈ ( 𝑂𝐸 ) → ( ( 𝐼𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹𝐶 ) ‘ ( 𝐼𝐶 ) ) = 0 ) )