Theorem ballotlem8

Description: There are as many countings with ties starting with a ballot for A as there are starting with a ballot for B . (Contributed by Thierry Arnoux, 7-Dec-2016)

	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	ballotlem8	⊢ ( ♯ ‘ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ 1 ∈ 𝑐 } ) = ( ♯ ‘ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ ¬ 1 ∈ 𝑐 } )

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	ballotlem7	⊢ ( 𝑅 ↾ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ 1 ∈ 𝑐 } ) : { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ 1 ∈ 𝑐 } –1-1-onto→ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ ¬ 1 ∈ 𝑐 }
12	1 2 3	ballotlemoex	⊢ 𝑂 ∈ V
13		difexg	⊢ ( 𝑂 ∈ V → ( 𝑂 ∖ 𝐸 ) ∈ V )
14	12 13	ax-mp	⊢ ( 𝑂 ∖ 𝐸 ) ∈ V
15	14	rabex	⊢ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ 1 ∈ 𝑐 } ∈ V
16	15	f1oen	⊢ ( ( 𝑅 ↾ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ 1 ∈ 𝑐 } ) : { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ 1 ∈ 𝑐 } –1-1-onto→ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ ¬ 1 ∈ 𝑐 } → { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ 1 ∈ 𝑐 } ≈ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ ¬ 1 ∈ 𝑐 } )
17		hasheni	⊢ ( { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ 1 ∈ 𝑐 } ≈ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ ¬ 1 ∈ 𝑐 } → ( ♯ ‘ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ 1 ∈ 𝑐 } ) = ( ♯ ‘ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ ¬ 1 ∈ 𝑐 } ) )
18	11 16 17	mp2b	⊢ ( ♯ ‘ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ 1 ∈ 𝑐 } ) = ( ♯ ‘ { 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∣ ¬ 1 ∈ 𝑐 } )