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Theorem ballotlemgval 33163
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 4171 . . . 4 (𝑢 = 𝑈 → (𝑣𝑢) = (𝑣𝑈))
21fveq2d 6851 . . 3 (𝑢 = 𝑈 → (♯‘(𝑣𝑢)) = (♯‘(𝑣𝑈)))
3 difeq2 4081 . . . 4 (𝑢 = 𝑈 → (𝑣𝑢) = (𝑣𝑈))
43fveq2d 6851 . . 3 (𝑢 = 𝑈 → (♯‘(𝑣𝑢)) = (♯‘(𝑣𝑈)))
52, 4oveq12d 7380 . 2 (𝑢 = 𝑈 → ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))) = ((♯‘(𝑣𝑈)) − (♯‘(𝑣𝑈))))
6 ineq1 4170 . . . 4 (𝑣 = 𝑉 → (𝑣𝑈) = (𝑉𝑈))
76fveq2d 6851 . . 3 (𝑣 = 𝑉 → (♯‘(𝑣𝑈)) = (♯‘(𝑉𝑈)))
8 difeq1 4080 . . . 4 (𝑣 = 𝑉 → (𝑣𝑈) = (𝑉𝑈))
98fveq2d 6851 . . 3 (𝑣 = 𝑉 → (♯‘(𝑣𝑈)) = (♯‘(𝑉𝑈)))
107, 9oveq12d 7380 . 2 (𝑣 = 𝑉 → ((♯‘(𝑣𝑈)) − (♯‘(𝑣𝑈))) = ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))))
11 ballotlemg . 2 = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
12 ovex 7395 . 2 ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))) ∈ V
135, 10, 11, 12ovmpo 7520 1 ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 𝑉) = ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  {crab 3410  cdif 3912  cin 3914  ifcif 4491  𝒫 cpw 4565   class class class wbr 5110  cmpt 5193  cima 5641  cfv 6501  (class class class)co 7362  cmpo 7364  Fincfn 8890  infcinf 9384  cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196  cle 11197  cmin 11392   / cdiv 11819  cn 12160  cz 12506  ...cfz 13431  chash 14237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367
This theorem is referenced by:  ballotlemgun  33164  ballotlemfg  33165  ballotlemfrc  33166
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