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Theorem ballotlemrval 34521
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 6905 . . 3 (𝑑 = 𝐶 → (𝑆𝑑) = (𝑆𝐶))
2 id 22 . . 3 (𝑑 = 𝐶𝑑 = 𝐶)
31, 2imaeq12d 6078 . 2 (𝑑 = 𝐶 → ((𝑆𝑑) “ 𝑑) = ((𝑆𝐶) “ 𝐶))
4 ballotth.r . . 3 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
5 fveq2 6905 . . . . 5 (𝑐 = 𝑑 → (𝑆𝑐) = (𝑆𝑑))
6 id 22 . . . . 5 (𝑐 = 𝑑𝑐 = 𝑑)
75, 6imaeq12d 6078 . . . 4 (𝑐 = 𝑑 → ((𝑆𝑐) “ 𝑐) = ((𝑆𝑑) “ 𝑑))
87cbvmptv 5254 . . 3 (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐)) = (𝑑 ∈ (𝑂𝐸) ↦ ((𝑆𝑑) “ 𝑑))
94, 8eqtri 2764 . 2 𝑅 = (𝑑 ∈ (𝑂𝐸) ↦ ((𝑆𝑑) “ 𝑑))
10 fvex 6918 . . 3 (𝑆𝐶) ∈ V
11 imaexg 7936 . . 3 ((𝑆𝐶) ∈ V → ((𝑆𝐶) “ 𝐶) ∈ V)
1210, 11ax-mp 5 . 2 ((𝑆𝐶) “ 𝐶) ∈ V
133, 9, 12fvmpt 7015 1 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) = ((𝑆𝐶) “ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  wral 3060  {crab 3435  Vcvv 3479  cdif 3947  cin 3949  ifcif 4524  𝒫 cpw 4599   class class class wbr 5142  cmpt 5224  cima 5687  cfv 6560  (class class class)co 7432  infcinf 9482  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   < clt 11296  cle 11297  cmin 11493   / cdiv 11921  cn 12267  cz 12615  ...cfz 13548  chash 14370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568
This theorem is referenced by:  ballotlemscr  34522  ballotlemrv  34523  ballotlemro  34526  ballotlemrinv0  34536
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