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Theorem ballotlemgval 32233
Description: Expand the value of . (Contributed by Thierry Arnoux, 21-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))
ballotlemg = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
Assertion
Ref Expression
ballotlemgval ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 𝑉) = ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))))
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   𝑖,𝐸,𝑘   𝑘,𝐼,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐   𝑆,𝑘,𝑖,𝑐   𝑅,𝑖   𝑣,𝑢,𝐼   𝑢,𝑅,𝑣   𝑢,𝑆,𝑣   𝑢,𝑈,𝑣   𝑢,𝑉,𝑣
Allowed substitution hints:   𝑃(𝑥,𝑣,𝑢,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑘,𝑐)   𝑆(𝑥)   𝑈(𝑥,𝑖,𝑘,𝑐)   𝐸(𝑥,𝑣,𝑢)   (𝑥,𝑣,𝑢,𝑖,𝑘,𝑐)   𝐹(𝑥,𝑣,𝑢)   𝐼(𝑥)   𝑀(𝑥,𝑣,𝑢)   𝑁(𝑥,𝑣,𝑢)   𝑂(𝑥,𝑣,𝑢)   𝑉(𝑥,𝑖,𝑘,𝑐)

Proof of Theorem ballotlemgval
StepHypRef Expression
1 ineq2 4137 . . . 4 (𝑢 = 𝑈 → (𝑣𝑢) = (𝑣𝑈))
21fveq2d 6742 . . 3 (𝑢 = 𝑈 → (♯‘(𝑣𝑢)) = (♯‘(𝑣𝑈)))
3 difeq2 4047 . . . 4 (𝑢 = 𝑈 → (𝑣𝑢) = (𝑣𝑈))
43fveq2d 6742 . . 3 (𝑢 = 𝑈 → (♯‘(𝑣𝑢)) = (♯‘(𝑣𝑈)))
52, 4oveq12d 7252 . 2 (𝑢 = 𝑈 → ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))) = ((♯‘(𝑣𝑈)) − (♯‘(𝑣𝑈))))
6 ineq1 4136 . . . 4 (𝑣 = 𝑉 → (𝑣𝑈) = (𝑉𝑈))
76fveq2d 6742 . . 3 (𝑣 = 𝑉 → (♯‘(𝑣𝑈)) = (♯‘(𝑉𝑈)))
8 difeq1 4046 . . . 4 (𝑣 = 𝑉 → (𝑣𝑈) = (𝑉𝑈))
98fveq2d 6742 . . 3 (𝑣 = 𝑉 → (♯‘(𝑣𝑈)) = (♯‘(𝑉𝑈)))
107, 9oveq12d 7252 . 2 (𝑣 = 𝑉 → ((♯‘(𝑣𝑈)) − (♯‘(𝑣𝑈))) = ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))))
11 ballotlemg . 2 = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
12 ovex 7267 . 2 ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))) ∈ V
135, 10, 11, 12ovmpo 7390 1 ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 𝑉) = ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2112  wral 3064  {crab 3068  cdif 3880  cin 3882  ifcif 4455  𝒫 cpw 4529   class class class wbr 5069  cmpt 5151  cima 5571  cfv 6400  (class class class)co 7234  cmpo 7236  Fincfn 8649  infcinf 9086  cr 10757  0cc0 10758  1c1 10759   + caddc 10761   < clt 10896  cle 10897  cmin 11091   / cdiv 11518  cn 11859  cz 12205  ...cfz 13124  chash 13928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-iota 6358  df-fun 6402  df-fv 6408  df-ov 7237  df-oprab 7238  df-mpo 7239
This theorem is referenced by:  ballotlemgun  32234  ballotlemfg  32235  ballotlemfrc  32236
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