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Theorem ballotlemgval 30713
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 3841 . . . 4 (𝑢 = 𝑈 → (𝑣𝑢) = (𝑣𝑈))
21fveq2d 6233 . . 3 (𝑢 = 𝑈 → (#‘(𝑣𝑢)) = (#‘(𝑣𝑈)))
3 difeq2 3755 . . . 4 (𝑢 = 𝑈 → (𝑣𝑢) = (𝑣𝑈))
43fveq2d 6233 . . 3 (𝑢 = 𝑈 → (#‘(𝑣𝑢)) = (#‘(𝑣𝑈)))
52, 4oveq12d 6708 . 2 (𝑢 = 𝑈 → ((#‘(𝑣𝑢)) − (#‘(𝑣𝑢))) = ((#‘(𝑣𝑈)) − (#‘(𝑣𝑈))))
6 ineq1 3840 . . . 4 (𝑣 = 𝑉 → (𝑣𝑈) = (𝑉𝑈))
76fveq2d 6233 . . 3 (𝑣 = 𝑉 → (#‘(𝑣𝑈)) = (#‘(𝑉𝑈)))
8 difeq1 3754 . . . 4 (𝑣 = 𝑉 → (𝑣𝑈) = (𝑉𝑈))
98fveq2d 6233 . . 3 (𝑣 = 𝑉 → (#‘(𝑣𝑈)) = (#‘(𝑉𝑈)))
107, 9oveq12d 6708 . 2 (𝑣 = 𝑉 → ((#‘(𝑣𝑈)) − (#‘(𝑣𝑈))) = ((#‘(𝑉𝑈)) − (#‘(𝑉𝑈))))
11 ballotlemg . 2 = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣𝑢)) − (#‘(𝑣𝑢))))
12 ovex 6718 . 2 ((#‘(𝑉𝑈)) − (#‘(𝑉𝑈))) ∈ V
135, 10, 11, 12ovmpt2 6838 1 ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 𝑉) = ((#‘(𝑉𝑈)) − (#‘(𝑉𝑈))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cdif 3604  cin 3606  ifcif 4119  𝒫 cpw 4191   class class class wbr 4685  cmpt 4762  cima 5146  cfv 5926  (class class class)co 6690  cmpt2 6692  Fincfn 7997  infcinf 8388  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  cz 11415  ...cfz 12364  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  ballotlemgun  30714  ballotlemfg  30715  ballotlemfrc  30716
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