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Theorem ballotlemfval 30679
Description: The value of F. (Contributed by Thierry Arnoux, 23-Nov-2016.)
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)))))
ballotlemfval.c (𝜑𝐶𝑂)
ballotlemfval.j (𝜑𝐽 ∈ ℤ)
Assertion
Ref Expression
ballotlemfval (𝜑 → ((𝐹𝐶)‘𝐽) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))
Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂,𝑐   𝐹,𝑐,𝑖   𝐶,𝑖   𝑖,𝐽   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑥,𝑐)   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑐)   𝐹(𝑥)   𝐽(𝑥,𝑐)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemfval
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 ballotlemfval.c . . 3 (𝜑𝐶𝑂)
2 simpl 472 . . . . . . . 8 ((𝑏 = 𝐶𝑖 ∈ ℤ) → 𝑏 = 𝐶)
32ineq2d 3847 . . . . . . 7 ((𝑏 = 𝐶𝑖 ∈ ℤ) → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝐶))
43fveq2d 6233 . . . . . 6 ((𝑏 = 𝐶𝑖 ∈ ℤ) → (#‘((1...𝑖) ∩ 𝑏)) = (#‘((1...𝑖) ∩ 𝐶)))
52difeq2d 3761 . . . . . . 7 ((𝑏 = 𝐶𝑖 ∈ ℤ) → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝐶))
65fveq2d 6233 . . . . . 6 ((𝑏 = 𝐶𝑖 ∈ ℤ) → (#‘((1...𝑖) ∖ 𝑏)) = (#‘((1...𝑖) ∖ 𝐶)))
74, 6oveq12d 6708 . . . . 5 ((𝑏 = 𝐶𝑖 ∈ ℤ) → ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))) = ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶))))
87mpteq2dva 4777 . . . 4 (𝑏 = 𝐶 → (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶)))))
9 ballotth.f . . . . 5 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
10 ineq2 3841 . . . . . . . . 9 (𝑏 = 𝑐 → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝑐))
1110fveq2d 6233 . . . . . . . 8 (𝑏 = 𝑐 → (#‘((1...𝑖) ∩ 𝑏)) = (#‘((1...𝑖) ∩ 𝑐)))
12 difeq2 3755 . . . . . . . . 9 (𝑏 = 𝑐 → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝑐))
1312fveq2d 6233 . . . . . . . 8 (𝑏 = 𝑐 → (#‘((1...𝑖) ∖ 𝑏)) = (#‘((1...𝑖) ∖ 𝑐)))
1411, 13oveq12d 6708 . . . . . . 7 (𝑏 = 𝑐 → ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))) = ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐))))
1514mpteq2dv 4778 . . . . . 6 (𝑏 = 𝑐 → (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
1615cbvmptv 4783 . . . . 5 (𝑏𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))))) = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
179, 16eqtr4i 2676 . . . 4 𝐹 = (𝑏𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏)))))
18 zex 11424 . . . . 5 ℤ ∈ V
1918mptex 6527 . . . 4 (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶)))) ∈ V
208, 17, 19fvmpt 6321 . . 3 (𝐶𝑂 → (𝐹𝐶) = (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶)))))
211, 20syl 17 . 2 (𝜑 → (𝐹𝐶) = (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶)))))
22 oveq2 6698 . . . . . 6 (𝑖 = 𝐽 → (1...𝑖) = (1...𝐽))
2322ineq1d 3846 . . . . 5 (𝑖 = 𝐽 → ((1...𝑖) ∩ 𝐶) = ((1...𝐽) ∩ 𝐶))
2423fveq2d 6233 . . . 4 (𝑖 = 𝐽 → (#‘((1...𝑖) ∩ 𝐶)) = (#‘((1...𝐽) ∩ 𝐶)))
2522difeq1d 3760 . . . . 5 (𝑖 = 𝐽 → ((1...𝑖) ∖ 𝐶) = ((1...𝐽) ∖ 𝐶))
2625fveq2d 6233 . . . 4 (𝑖 = 𝐽 → (#‘((1...𝑖) ∖ 𝐶)) = (#‘((1...𝐽) ∖ 𝐶)))
2724, 26oveq12d 6708 . . 3 (𝑖 = 𝐽 → ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶))) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))
2827adantl 481 . 2 ((𝜑𝑖 = 𝐽) → ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶))) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))
29 ballotlemfval.j . 2 (𝜑𝐽 ∈ ℤ)
30 ovexd 6720 . 2 (𝜑 → ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))) ∈ V)
3121, 28, 29, 30fvmptd 6327 1 (𝜑 → ((𝐹𝐶)‘𝐽) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  cdif 3604  cin 3606  𝒫 cpw 4191  cmpt 4762  cfv 5926  (class class class)co 6690  1c1 9975   + caddc 9977  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-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  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-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  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-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-neg 10307  df-z 11416
This theorem is referenced by:  ballotlemfelz  30680  ballotlemfp1  30681  ballotlemfmpn  30684  ballotlemfval0  30685  ballotlemfg  30715  ballotlemfrc  30716
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