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Theorem ballotlemsv 31170
Description: Value of 𝑆 evaluated at 𝐽 for a given counting 𝐶. (Contributed by Thierry Arnoux, 12-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)))
Assertion
Ref Expression
ballotlemsv ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) = if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽))
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
Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑆(𝑥,𝑖,𝑘,𝑐)   𝐸(𝑥)   𝐹(𝑥)   𝐼(𝑥)   𝐽(𝑥,𝑖,𝑘,𝑐)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemsv
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 ballotth.m . . . . 5 𝑀 ∈ ℕ
2 ballotth.n . . . . 5 𝑁 ∈ ℕ
3 ballotth.o . . . . 5 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4 ballotth.p . . . . 5 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
5 ballotth.f . . . . 5 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6 ballotth.e . . . . 5 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
7 ballotth.mgtn . . . . 5 𝑁 < 𝑀
8 ballotth.i . . . . 5 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
9 ballotth.s . . . . 5 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
101, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsval 31169 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
11 breq1 4889 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ≤ (𝐼𝐶) ↔ 𝑗 ≤ (𝐼𝐶)))
12 oveq2 6930 . . . . . 6 (𝑖 = 𝑗 → (((𝐼𝐶) + 1) − 𝑖) = (((𝐼𝐶) + 1) − 𝑗))
13 id 22 . . . . . 6 (𝑖 = 𝑗𝑖 = 𝑗)
1411, 12, 13ifbieq12d 4333 . . . . 5 (𝑖 = 𝑗 → if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖) = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
1514cbvmptv 4985 . . . 4 (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
1610, 15syl6eq 2829 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗)))
1716adantr 474 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → (𝑆𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗)))
18 simpr 479 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
1918breq1d 4896 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 = 𝐽) → (𝑗 ≤ (𝐼𝐶) ↔ 𝐽 ≤ (𝐼𝐶)))
2018oveq2d 6938 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 = 𝐽) → (((𝐼𝐶) + 1) − 𝑗) = (((𝐼𝐶) + 1) − 𝐽))
2119, 20, 18ifbieq12d 4333 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 = 𝐽) → if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗) = if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽))
2221adantlr 705 . 2 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑗 = 𝐽) → if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗) = if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽))
23 simpr 479 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
24 ovexd 6956 . . 3 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) ∧ 𝐽 ≤ (𝐼𝐶)) → (((𝐼𝐶) + 1) − 𝐽) ∈ V)
25 elex 3413 . . . 4 (𝐽 ∈ (1...(𝑀 + 𝑁)) → 𝐽 ∈ V)
2625ad2antlr 717 . . 3 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) ∧ ¬ 𝐽 ≤ (𝐼𝐶)) → 𝐽 ∈ V)
2724, 26ifclda 4340 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽) ∈ V)
2817, 22, 23, 27fvmptd 6548 1 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) = if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1601  wcel 2106  wral 3089  {crab 3093  Vcvv 3397  cdif 3788  cin 3790  ifcif 4306  𝒫 cpw 4378   class class class wbr 4886  cmpt 4965  cfv 6135  (class class class)co 6922  infcinf 8635  cr 10271  0cc0 10272  1c1 10273   + caddc 10275   < clt 10411  cle 10412  cmin 10606   / cdiv 11032  cn 11374  cz 11728  ...cfz 12643  chash 13435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925
This theorem is referenced by:  ballotlemsgt1  31171  ballotlemsdom  31172  ballotlemsel1i  31173  ballotlemsf1o  31174  ballotlemsi  31175  ballotlemsima  31176  ballotlemrv  31180
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