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Theorem ballotlemgval 34176
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 4208 . . . 4 (𝑢 = 𝑈 → (𝑣𝑢) = (𝑣𝑈))
21fveq2d 6906 . . 3 (𝑢 = 𝑈 → (♯‘(𝑣𝑢)) = (♯‘(𝑣𝑈)))
3 difeq2 4116 . . . 4 (𝑢 = 𝑈 → (𝑣𝑢) = (𝑣𝑈))
43fveq2d 6906 . . 3 (𝑢 = 𝑈 → (♯‘(𝑣𝑢)) = (♯‘(𝑣𝑈)))
52, 4oveq12d 7444 . 2 (𝑢 = 𝑈 → ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))) = ((♯‘(𝑣𝑈)) − (♯‘(𝑣𝑈))))
6 ineq1 4207 . . . 4 (𝑣 = 𝑉 → (𝑣𝑈) = (𝑉𝑈))
76fveq2d 6906 . . 3 (𝑣 = 𝑉 → (♯‘(𝑣𝑈)) = (♯‘(𝑉𝑈)))
8 difeq1 4115 . . . 4 (𝑣 = 𝑉 → (𝑣𝑈) = (𝑉𝑈))
98fveq2d 6906 . . 3 (𝑣 = 𝑉 → (♯‘(𝑣𝑈)) = (♯‘(𝑉𝑈)))
107, 9oveq12d 7444 . 2 (𝑣 = 𝑉 → ((♯‘(𝑣𝑈)) − (♯‘(𝑣𝑈))) = ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))))
11 ballotlemg . 2 = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
12 ovex 7459 . 2 ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))) ∈ V
135, 10, 11, 12ovmpo 7587 1 ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 𝑉) = ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3058  {crab 3430  cdif 3946  cin 3948  ifcif 4532  𝒫 cpw 4606   class class class wbr 5152  cmpt 5235  cima 5685  cfv 6553  (class class class)co 7426  cmpo 7428  Fincfn 8970  infcinf 9472  cr 11145  0cc0 11146  1c1 11147   + caddc 11149   < clt 11286  cle 11287  cmin 11482   / cdiv 11909  cn 12250  cz 12596  ...cfz 13524  chash 14329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431
This theorem is referenced by:  ballotlemgun  34177  ballotlemfg  34178  ballotlemfrc  34179
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