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Theorem ballotleme 34477
Description: Elements of 𝐸. (Contributed by Thierry Arnoux, 14-Dec-2016.)
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)}
Assertion
Ref Expression
ballotleme (𝐶𝐸 ↔ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂,𝑐   𝐹,𝑐,𝑖   𝐶,𝑖
Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑐)   𝐸(𝑥,𝑖,𝑐)   𝐹(𝑥)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotleme
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6904 . . . . 5 (𝑑 = 𝐶 → (𝐹𝑑) = (𝐹𝐶))
21fveq1d 6906 . . . 4 (𝑑 = 𝐶 → ((𝐹𝑑)‘𝑖) = ((𝐹𝐶)‘𝑖))
32breq2d 5153 . . 3 (𝑑 = 𝐶 → (0 < ((𝐹𝑑)‘𝑖) ↔ 0 < ((𝐹𝐶)‘𝑖)))
43ralbidv 3177 . 2 (𝑑 = 𝐶 → (∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑑)‘𝑖) ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
5 ballotth.e . . 3 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
6 fveq2 6904 . . . . . . 7 (𝑐 = 𝑑 → (𝐹𝑐) = (𝐹𝑑))
76fveq1d 6906 . . . . . 6 (𝑐 = 𝑑 → ((𝐹𝑐)‘𝑖) = ((𝐹𝑑)‘𝑖))
87breq2d 5153 . . . . 5 (𝑐 = 𝑑 → (0 < ((𝐹𝑐)‘𝑖) ↔ 0 < ((𝐹𝑑)‘𝑖)))
98ralbidv 3177 . . . 4 (𝑐 = 𝑑 → (∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖) ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑑)‘𝑖)))
109cbvrabv 3446 . . 3 {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)} = {𝑑𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑑)‘𝑖)}
115, 10eqtri 2764 . 2 𝐸 = {𝑑𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑑)‘𝑖)}
124, 11elrab2 3694 1 (𝐶𝐸 ↔ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3060  {crab 3435  cdif 3947  cin 3949  𝒫 cpw 4598   class class class wbr 5141  cmpt 5223  cfv 6559  (class class class)co 7429  0cc0 11151  1c1 11152   + caddc 11154   < clt 11291  cmin 11488   / cdiv 11916  cn 12262  cz 12609  ...cfz 13543  chash 14365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fv 6567
This theorem is referenced by:  ballotlemodife  34478  ballotlem4  34479
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