Theorem ballotlemieq

Description: If two countings share the same first tie, they also have the same swap function. (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 ) ) )
Assertion	ballotlemieq	⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐷 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) ) → ( 𝑆 ‘ 𝐶 ) = ( 𝑆 ‘ 𝐷 ) )

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		simpl	⊢ ( ( ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) ∧ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) )
11	10	breq2d	⊢ ( ( ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) ∧ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → ( 𝑖 ≤ ( 𝐼 ‘ 𝐶 ) ↔ 𝑖 ≤ ( 𝐼 ‘ 𝐷 ) ) )
12	10	oveq1d	⊢ ( ( ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) ∧ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝐼 ‘ 𝐶 ) + 1 ) = ( ( 𝐼 ‘ 𝐷 ) + 1 ) )
13	12	oveq1d	⊢ ( ( ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) ∧ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝑖 ) = ( ( ( 𝐼 ‘ 𝐷 ) + 1 ) − 𝑖 ) )
14	11 13	ifbieq1d	⊢ ( ( ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) ∧ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ) → if ( 𝑖 ≤ ( 𝐼 ‘ 𝐶 ) , ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝑖 ) , 𝑖 ) = if ( 𝑖 ≤ ( 𝐼 ‘ 𝐷 ) , ( ( ( 𝐼 ‘ 𝐷 ) + 1 ) − 𝑖 ) , 𝑖 ) )
15	14	mpteq2dva	⊢ ( ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) → ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↦ if ( 𝑖 ≤ ( 𝐼 ‘ 𝐶 ) , ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝑖 ) , 𝑖 ) ) = ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↦ if ( 𝑖 ≤ ( 𝐼 ‘ 𝐷 ) , ( ( ( 𝐼 ‘ 𝐷 ) + 1 ) − 𝑖 ) , 𝑖 ) ) )
16	15	3ad2ant3	⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐷 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) ) → ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↦ if ( 𝑖 ≤ ( 𝐼 ‘ 𝐶 ) , ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝑖 ) , 𝑖 ) ) = ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↦ if ( 𝑖 ≤ ( 𝐼 ‘ 𝐷 ) , ( ( ( 𝐼 ‘ 𝐷 ) + 1 ) − 𝑖 ) , 𝑖 ) ) )
17	1 2 3 4 5 6 7 8 9	ballotlemsval	⊢ ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑆 ‘ 𝐶 ) = ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↦ if ( 𝑖 ≤ ( 𝐼 ‘ 𝐶 ) , ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝑖 ) , 𝑖 ) ) )
18	17	3ad2ant1	⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐷 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) ) → ( 𝑆 ‘ 𝐶 ) = ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↦ if ( 𝑖 ≤ ( 𝐼 ‘ 𝐶 ) , ( ( ( 𝐼 ‘ 𝐶 ) + 1 ) − 𝑖 ) , 𝑖 ) ) )
19	1 2 3 4 5 6 7 8 9	ballotlemsval	⊢ ( 𝐷 ∈ ( 𝑂 ∖ 𝐸 ) → ( 𝑆 ‘ 𝐷 ) = ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↦ if ( 𝑖 ≤ ( 𝐼 ‘ 𝐷 ) , ( ( ( 𝐼 ‘ 𝐷 ) + 1 ) − 𝑖 ) , 𝑖 ) ) )
20	19	3ad2ant2	⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐷 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) ) → ( 𝑆 ‘ 𝐷 ) = ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↦ if ( 𝑖 ≤ ( 𝐼 ‘ 𝐷 ) , ( ( ( 𝐼 ‘ 𝐷 ) + 1 ) − 𝑖 ) , 𝑖 ) ) )
21	16 18 20	3eqtr4d	⊢ ( ( 𝐶 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝐷 ∈ ( 𝑂 ∖ 𝐸 ) ∧ ( 𝐼 ‘ 𝐶 ) = ( 𝐼 ‘ 𝐷 ) ) → ( 𝑆 ‘ 𝐶 ) = ( 𝑆 ‘ 𝐷 ) )