Theorem ballotlemrval 30579

Description: Value of 𝑅. (Contributed by Thierry Arnoux, 14-Apr-2017.)

Hypotheses

Ref	Expression
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

Ref	Expression
ballotlemrval	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶))

Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂   𝑘,𝑀   𝑘,𝑁   𝑘,𝑂   𝑖,𝑐,𝐹,𝑘   𝐶,𝑖,𝑘   𝑖,𝐸,𝑘   𝐶,𝑘   𝑘,𝐼,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐   𝑆,𝑘,𝑖,𝑐

Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑖,𝑘,𝑐)   𝑆(𝑥)   𝐸(𝑥)   𝐹(𝑥)   𝐼(𝑥)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemrval

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 ⊢ (𝑑 = 𝐶 → (𝑆‘𝑑) = (𝑆‘𝐶))
2		id 22	. . 3 ⊢ (𝑑 = 𝐶 → 𝑑 = 𝐶)
3	1, 2	imaeq12d 5467	. 2 ⊢ (𝑑 = 𝐶 → ((𝑆‘𝑑) “ 𝑑) = ((𝑆‘𝐶) “ 𝐶))
4		ballotth.r	. . 3 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
5		fveq2 6191	. . . . 5 ⊢ (𝑐 = 𝑑 → (𝑆‘𝑐) = (𝑆‘𝑑))
6		id 22	. . . . 5 ⊢ (𝑐 = 𝑑 → 𝑐 = 𝑑)
7	5, 6	imaeq12d 5467	. . . 4 ⊢ (𝑐 = 𝑑 → ((𝑆‘𝑐) “ 𝑐) = ((𝑆‘𝑑) “ 𝑑))
8	7	cbvmptv 4750	. . 3 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑑) “ 𝑑))
9	4, 8	eqtri 2644	. 2 ⊢ 𝑅 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑑) “ 𝑑))
10		fvex 6201	. . 3 ⊢ (𝑆‘𝐶) ∈ V
11		imaexg 7103	. . 3 ⊢ ((𝑆‘𝐶) ∈ V → ((𝑆‘𝐶) “ 𝐶) ∈ V)
12	10, 11	ax-mp 5	. 2 ⊢ ((𝑆‘𝐶) “ 𝐶) ∈ V
13	3, 9, 12	fvmpt 6282	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573  ifcif 4086  𝒫 cpw 4158   class class class wbr 4653   ↦ cmpt 4729   “ cima 5117  ‘cfv 5888  (class class class)co 6650  infcinf 8347  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266   / cdiv 10684  ℕcn 11020  ℤcz 11377  ...cfz 12326  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  ballotlemscr  30580  ballotlemrv  30581  ballotlemro  30584  ballotlemrinv0  30594