Theorem ballotlemrinv

Description: R is its own inverse : it is an involution. (Contributed by Thierry Arnoux, 10-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	ballotlemrinv	⊢ ◡ 𝑅 = 𝑅

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	ballotlemrinv0	⊢ ( ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝑑 = ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) ) → ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝑐 = ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) ) )
12	1 2 3 4 5 6 7 8 9 10	ballotlemrinv0	⊢ ( ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝑐 = ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) ) → ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝑑 = ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) ) )
13	11 12	impbii	⊢ ( ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝑑 = ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) ) ↔ ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝑐 = ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) ) )
14	13	a1i	⊢ ( ⊤ → ( ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝑑 = ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) ) ↔ ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ∧ 𝑐 = ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) ) ) )
15	14	mptcnv	⊢ ( ⊤ → ◡ ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) ) = ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) ) )
16	15	mptru	⊢ ◡ ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) ) = ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) )
17		fveq2	⊢ ( 𝑑 = 𝑐 → ( 𝑆 ‘ 𝑑 ) = ( 𝑆 ‘ 𝑐 ) )
18		id	⊢ ( 𝑑 = 𝑐 → 𝑑 = 𝑐 )
19	17 18	imaeq12d	⊢ ( 𝑑 = 𝑐 → ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) = ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) )
20	19	cbvmptv	⊢ ( 𝑑 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑑 ) “ 𝑑 ) ) = ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) )
21	16 20	eqtri	⊢ ◡ ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) ) = ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) )
22	10	cnveqi	⊢ ◡ 𝑅 = ◡ ( 𝑐 ∈ ( 𝑂 ∖ 𝐸 ) ↦ ( ( 𝑆 ‘ 𝑐 ) “ 𝑐 ) )
23	21 22 10	3eqtr4i	⊢ ◡ 𝑅 = 𝑅