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Theorem ballotleme 30898
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 6332 . . . . 5 (𝑑 = 𝐶 → (𝐹𝑑) = (𝐹𝐶))
21fveq1d 6334 . . . 4 (𝑑 = 𝐶 → ((𝐹𝑑)‘𝑖) = ((𝐹𝐶)‘𝑖))
32breq2d 4798 . . 3 (𝑑 = 𝐶 → (0 < ((𝐹𝑑)‘𝑖) ↔ 0 < ((𝐹𝐶)‘𝑖)))
43ralbidv 3135 . 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 6332 . . . . . . 7 (𝑐 = 𝑑 → (𝐹𝑐) = (𝐹𝑑))
76fveq1d 6334 . . . . . 6 (𝑐 = 𝑑 → ((𝐹𝑐)‘𝑖) = ((𝐹𝑑)‘𝑖))
87breq2d 4798 . . . . 5 (𝑐 = 𝑑 → (0 < ((𝐹𝑐)‘𝑖) ↔ 0 < ((𝐹𝑑)‘𝑖)))
98ralbidv 3135 . . . 4 (𝑐 = 𝑑 → (∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖) ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑑)‘𝑖)))
109cbvrabv 3349 . . 3 {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)} = {𝑑𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑑)‘𝑖)}
115, 10eqtri 2793 . 2 𝐸 = {𝑑𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑑)‘𝑖)}
124, 11elrab2 3518 1 (𝐶𝐸 ↔ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  cdif 3720  cin 3722  𝒫 cpw 4297   class class class wbr 4786  cmpt 4863  cfv 6031  (class class class)co 6793  0cc0 10138  1c1 10139   + caddc 10141   < clt 10276  cmin 10468   / cdiv 10886  cn 11222  cz 11579  ...cfz 12533  chash 13321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039
This theorem is referenced by:  ballotlemodife  30899  ballotlem4  30900
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