Theorem ballotlemrval

Description: Value of R . (Contributed by Thierry Arnoux, 14-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	ballotlemrval	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑅 ‘ 𝐶 ) = ( ( 𝑆 ‘ 𝐶 ) “ 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		fveq2	⊢ ( 𝑑 = 𝐶 → ( 𝑆 ‘ 𝑑 ) = ( 𝑆 ‘ 𝐶 ) )
12		id	⊢ ( 𝑑 = 𝐶 → 𝑑 = 𝐶 )
13	11 12	imaeq12d	⊢ ( 𝑑 = 𝐶 → ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) = ( ( 𝑆 ‘ 𝐶 ) “ 𝐶 ) )
14		fveq2	⊢ ( 𝑐 = 𝑑 → ( 𝑆 ‘ 𝑐 ) = ( 𝑆 ‘ 𝑑 ) )
15		id	⊢ ( 𝑐 = 𝑑 → 𝑐 = 𝑑 )
16	14 15	imaeq12d	⊢ ( 𝑐 = 𝑑 → ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) = ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) )
17	16	cbvmptv	⊢ ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) ) = ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) )
18	10 17	eqtri	⊢ 𝑅 = ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) )
19		fvex	⊢ ( 𝑆 ‘ 𝐶 ) ∈ V
20		imaexg	⊢ ( ( 𝑆 ‘ 𝐶 ) ∈ V → ( ( 𝑆 ‘ 𝐶 ) “ 𝐶 ) ∈ V )
21	19 20	ax-mp	⊢ ( ( 𝑆 ‘ 𝐶 ) “ 𝐶 ) ∈ V
22	13 18 21	fvmpt	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑅 ‘ 𝐶 ) = ( ( 𝑆 ‘ 𝐶 ) “ 𝐶 ) )