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Theorem ballotlemrval 30402
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 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
ballotth.e 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
ballotth.mgtn 𝑁 < 𝑀
ballotth.i 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
Assertion
Ref Expression
ballotlemrval (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) = ((𝑆𝐶) “ 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.
StepHypRef Expression
1 fveq2 6158 . . 3 (𝑑 = 𝐶 → (𝑆𝑑) = (𝑆𝐶))
2 id 22 . . 3 (𝑑 = 𝐶𝑑 = 𝐶)
31, 2imaeq12d 5436 . 2 (𝑑 = 𝐶 → ((𝑆𝑑) “ 𝑑) = ((𝑆𝐶) “ 𝐶))
4 ballotth.r . . 3 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
5 fveq2 6158 . . . . 5 (𝑐 = 𝑑 → (𝑆𝑐) = (𝑆𝑑))
6 id 22 . . . . 5 (𝑐 = 𝑑𝑐 = 𝑑)
75, 6imaeq12d 5436 . . . 4 (𝑐 = 𝑑 → ((𝑆𝑐) “ 𝑐) = ((𝑆𝑑) “ 𝑑))
87cbvmptv 4720 . . 3 (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐)) = (𝑑 ∈ (𝑂𝐸) ↦ ((𝑆𝑑) “ 𝑑))
94, 8eqtri 2643 . 2 𝑅 = (𝑑 ∈ (𝑂𝐸) ↦ ((𝑆𝑑) “ 𝑑))
10 fvex 6168 . . 3 (𝑆𝐶) ∈ V
11 imaexg 7065 . . 3 ((𝑆𝐶) ∈ V → ((𝑆𝐶) “ 𝐶) ∈ V)
1210, 11ax-mp 5 . 2 ((𝑆𝐶) “ 𝐶) ∈ V
133, 9, 12fvmpt 6249 1 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) = ((𝑆𝐶) “ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2908  {crab 2912  Vcvv 3190  cdif 3557  cin 3559  ifcif 4064  𝒫 cpw 4136   class class class wbr 4623  cmpt 4683  cima 5087  cfv 5857  (class class class)co 6615  infcinf 8307  cr 9895  0cc0 9896  1c1 9897   + caddc 9899   < clt 10034  cle 10035  cmin 10226   / cdiv 10644  cn 10980  cz 11337  ...cfz 12284  #chash 13073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865
This theorem is referenced by:  ballotlemscr  30403  ballotlemrv  30404  ballotlemro  30407  ballotlemrinv0  30417
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