Metamath Proof Explorer

Theorem ballotlemro

Description: Range of R is included in O . (Contributed by Thierry Arnoux, 17-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 ballotlemro ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑅 ‘ 𝐶 ) ∈ 𝑂 )

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 ballotlemrval ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑅 ‘ 𝐶 ) = ( ( 𝑆 ‘ 𝐶 ) “ 𝐶 ) )
12 imassrn ⊢ ( ( 𝑆 ‘ 𝐶 ) “ 𝐶 ) ⊆ ran ( 𝑆 ‘ 𝐶 )
13 1 2 3 4 5 6 7 8 9 ballotlemsf1o ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝑆 ‘ 𝐶 ) : ( 1 ... ( 𝑀 + 𝑁 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ◡ ( 𝑆 ‘ 𝐶 ) = ( 𝑆 ‘ 𝐶 ) ) )
14 13 simpld ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑆 ‘ 𝐶 ) : ( 1 ... ( 𝑀 + 𝑁 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 𝑁 ) ) )
15 f1ofo ⊢ ( ( 𝑆 ‘ 𝐶 ) : ( 1 ... ( 𝑀 + 𝑁 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 𝑁 ) ) → ( 𝑆 ‘ 𝐶 ) : ( 1 ... ( 𝑀 + 𝑁 ) ) –onto→ ( 1 ... ( 𝑀 + 𝑁 ) ) )
16 forn ⊢ ( ( 𝑆 ‘ 𝐶 ) : ( 1 ... ( 𝑀 + 𝑁 ) ) –onto→ ( 1 ... ( 𝑀 + 𝑁 ) ) → ran ( 𝑆 ‘ 𝐶 ) = ( 1 ... ( 𝑀 + 𝑁 ) ) )
17 14 15 16 3syl ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ran ( 𝑆 ‘ 𝐶 ) = ( 1 ... ( 𝑀 + 𝑁 ) ) )
18 12 17 sseqtrid ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝑆 ‘ 𝐶 ) “ 𝐶 ) ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) )
19 11 18 eqsstrd ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑅 ‘ 𝐶 ) ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) )
20 f1of1 ⊢ ( ( 𝑆 ‘ 𝐶 ) : ( 1 ... ( 𝑀 + 𝑁 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 𝑁 ) ) → ( 𝑆 ‘ 𝐶 ) : ( 1 ... ( 𝑀 + 𝑁 ) ) –1-1→ ( 1 ... ( 𝑀 + 𝑁 ) ) )
21 14 20 syl ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑆 ‘ 𝐶 ) : ( 1 ... ( 𝑀 + 𝑁 ) ) –1-1→ ( 1 ... ( 𝑀 + 𝑁 ) ) )
22 eldifi ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → 𝐶 ∈ 𝑂 )
23 1 2 3 ballotlemelo ⊢ ( 𝐶 ∈ 𝑂 ↔ ( 𝐶 ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ♯ ‘ 𝐶 ) = 𝑀 ) )
24 22 23 sylib ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐶 ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ♯ ‘ 𝐶 ) = 𝑀 ) )
25 24 simpld ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → 𝐶 ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) )
26 id ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) )
27 f1imaeng ⊢ ( ( ( 𝑆 ‘ 𝐶 ) : ( 1 ... ( 𝑀 + 𝑁 ) ) –1-1→ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐶 ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ) → ( ( 𝑆 ‘ 𝐶 ) “ 𝐶 ) ≈ 𝐶 )
28 21 25 26 27 syl3anc ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝑆 ‘ 𝐶 ) “ 𝐶 ) ≈ 𝐶 )
29 11 28 eqbrtrd ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑅 ‘ 𝐶 ) ≈ 𝐶 )
30 hasheni ⊢ ( ( 𝑅 ‘ 𝐶 ) ≈ 𝐶 → ( ♯ ‘ ( 𝑅 ‘ 𝐶 ) ) = ( ♯ ‘ 𝐶 ) )
31 29 30 syl ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ♯ ‘ ( 𝑅 ‘ 𝐶 ) ) = ( ♯ ‘ 𝐶 ) )
32 24 simprd ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ♯ ‘ 𝐶 ) = 𝑀 )
33 31 32 eqtrd ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ♯ ‘ ( 𝑅 ‘ 𝐶 ) ) = 𝑀 )
34 1 2 3 ballotlemelo ⊢ ( ( 𝑅 ‘ 𝐶 ) ∈ 𝑂 ↔ ( ( 𝑅 ‘ 𝐶 ) ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ♯ ‘ ( 𝑅 ‘ 𝐶 ) ) = 𝑀 ) )
35 19 33 34 sylanbrc ⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑅 ‘ 𝐶 ) ∈ 𝑂 )