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Theorem ballotlemsval 29897
Description: Value of 𝑆. (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
ballotlemsval (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
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)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemsval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 simpl 472 . . . . . 6 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → 𝑑 = 𝐶)
21fveq2d 6107 . . . . 5 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼𝑑) = (𝐼𝐶))
32breq2d 4595 . . . 4 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼𝑑) ↔ 𝑖 ≤ (𝐼𝐶)))
42oveq1d 6564 . . . . 5 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼𝑑) + 1) = ((𝐼𝐶) + 1))
54oveq1d 6564 . . . 4 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼𝑑) + 1) − 𝑖) = (((𝐼𝐶) + 1) − 𝑖))
63, 5ifbieq1d 4059 . . 3 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖) = if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖))
76mpteq2dva 4672 . 2 (𝑑 = 𝐶 → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖)) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
8 ballotth.s . . 3 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
9 simpl 472 . . . . . . . 8 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → 𝑐 = 𝑑)
109fveq2d 6107 . . . . . . 7 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼𝑐) = (𝐼𝑑))
1110breq2d 4595 . . . . . 6 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼𝑐) ↔ 𝑖 ≤ (𝐼𝑑)))
1210oveq1d 6564 . . . . . . 7 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼𝑐) + 1) = ((𝐼𝑑) + 1))
1312oveq1d 6564 . . . . . 6 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼𝑐) + 1) − 𝑖) = (((𝐼𝑑) + 1) − 𝑖))
1411, 13ifbieq1d 4059 . . . . 5 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖) = if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖))
1514mpteq2dva 4672 . . . 4 (𝑐 = 𝑑 → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖)))
1615cbvmptv 4678 . . 3 (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖))) = (𝑑 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖)))
178, 16eqtri 2632 . 2 𝑆 = (𝑑 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖)))
18 ovex 6577 . . 3 (1...(𝑀 + 𝑁)) ∈ V
1918mptex 6390 . 2 (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)) ∈ V
207, 17, 19fvmpt 6191 1 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900  cdif 3537  cin 3539  ifcif 4036  𝒫 cpw 4108   class class class wbr 4583  cmpt 4643  cfv 5804  (class class class)co 6549  infcinf 8230  cr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953  cle 9954  cmin 10145   / cdiv 10563  cn 10897  cz 11254  ...cfz 12197  #chash 12979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552
This theorem is referenced by:  ballotlemsv  29898  ballotlemsf1o  29902  ballotlemieq  29905
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