Theorem ballotlem1ri

Description: When the vote on the first tie is for A, the first vote is also for A on the reverse counting. (Contributed by Thierry Arnoux, 18-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	ballotlem1ri	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 1 ∈ ( 𝑅 ‘ 𝐶 ) ↔ ( 𝐼 ‘ 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		nnaddcl	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝑀 + 𝑁 ) ∈ ℕ )
12	1 2 11	mp2an	⊢ ( 𝑀 + 𝑁 ) ∈ ℕ
13		nnuz	⊢ ℕ = ( ℤ≥ ‘ 1 )
14	12 13	eleqtri	⊢ ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 1 )
15		eluzfz1	⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) )
16	14 15	mp1i	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → 1 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) )
17	1 2 3 4 5 6 7 8	ballotlemiex	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ( 𝐹 ‘ 𝐶 ) ‘ ( 𝐼 ‘ 𝐶 ) ) = 0 ) )
18	17	simpld	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) )
19		elfzle1	⊢ ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) → 1 ≤ ( 𝐼 ‘ 𝐶 ) )
20	18 19	syl	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → 1 ≤ ( 𝐼 ‘ 𝐶 ) )
21	1 2 3 4 5 6 7 8 9 10	ballotlemrv1	⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 1 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 1 ≤ ( 𝐼 ‘ 𝐶 ) ) → ( 1 ∈ ( 𝑅 ‘ 𝐶 ) ↔ ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 1 ) ∈ 𝐶 ) )
22	16 20 21	mpd3an23	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 1 ∈ ( 𝑅 ‘ 𝐶 ) ↔ ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 1 ) ∈ 𝐶 ) )
23		elfzelz	⊢ ( ( 𝐼 ‘ 𝐶 ) ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) → ( 𝐼 ‘ 𝐶 ) ∈ ℤ )
24	18 23	syl	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ℤ )
25	24	zcnd	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝐼 ‘ 𝐶 ) ∈ ℂ )
26		1cnd	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → 1 ∈ ℂ )
27	25 26	pncand	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 1 ) = ( 𝐼 ‘ 𝐶 ) )
28	27	eleq1d	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 1 ) ∈ 𝐶 ↔ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) )
29	22 28	bitrd	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 1 ∈ ( 𝑅 ‘ 𝐶 ) ↔ ( 𝐼 ‘ 𝐶 ) ∈ 𝐶 ) )