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Theorem ballotlemgval 31101
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 4007 . . . 4 (𝑢 = 𝑈 → (𝑣𝑢) = (𝑣𝑈))
21fveq2d 6416 . . 3 (𝑢 = 𝑈 → (♯‘(𝑣𝑢)) = (♯‘(𝑣𝑈)))
3 difeq2 3921 . . . 4 (𝑢 = 𝑈 → (𝑣𝑢) = (𝑣𝑈))
43fveq2d 6416 . . 3 (𝑢 = 𝑈 → (♯‘(𝑣𝑢)) = (♯‘(𝑣𝑈)))
52, 4oveq12d 6897 . 2 (𝑢 = 𝑈 → ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))) = ((♯‘(𝑣𝑈)) − (♯‘(𝑣𝑈))))
6 ineq1 4006 . . . 4 (𝑣 = 𝑉 → (𝑣𝑈) = (𝑉𝑈))
76fveq2d 6416 . . 3 (𝑣 = 𝑉 → (♯‘(𝑣𝑈)) = (♯‘(𝑉𝑈)))
8 difeq1 3920 . . . 4 (𝑣 = 𝑉 → (𝑣𝑈) = (𝑉𝑈))
98fveq2d 6416 . . 3 (𝑣 = 𝑉 → (♯‘(𝑣𝑈)) = (♯‘(𝑉𝑈)))
107, 9oveq12d 6897 . 2 (𝑣 = 𝑉 → ((♯‘(𝑣𝑈)) − (♯‘(𝑣𝑈))) = ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))))
11 ballotlemg . 2 = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
12 ovex 6911 . 2 ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))) ∈ V
135, 10, 11, 12ovmpt2 7031 1 ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 𝑉) = ((♯‘(𝑉𝑈)) − (♯‘(𝑉𝑈))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3090  {crab 3094  cdif 3767  cin 3769  ifcif 4278  𝒫 cpw 4350   class class class wbr 4844  cmpt 4923  cima 5316  cfv 6102  (class class class)co 6879  cmpt2 6881  Fincfn 8196  infcinf 8590  cr 10224  0cc0 10225  1c1 10226   + caddc 10228   < clt 10364  cle 10365  cmin 10557   / cdiv 10977  cn 11313  cz 11665  ...cfz 12579  chash 13369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884
This theorem is referenced by:  ballotlemgun  31102  ballotlemfg  31103  ballotlemfrc  31104
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