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Theorem ballotlemfval 29878
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 3776 . . . . . . 7 ((𝑏 = 𝐶𝑖 ∈ ℤ) → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝐶))
43fveq2d 6107 . . . . . 6 ((𝑏 = 𝐶𝑖 ∈ ℤ) → (#‘((1...𝑖) ∩ 𝑏)) = (#‘((1...𝑖) ∩ 𝐶)))
52difeq2d 3690 . . . . . . 7 ((𝑏 = 𝐶𝑖 ∈ ℤ) → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝐶))
65fveq2d 6107 . . . . . 6 ((𝑏 = 𝐶𝑖 ∈ ℤ) → (#‘((1...𝑖) ∖ 𝑏)) = (#‘((1...𝑖) ∖ 𝐶)))
74, 6oveq12d 6567 . . . . 5 ((𝑏 = 𝐶𝑖 ∈ ℤ) → ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))) = ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶))))
87mpteq2dva 4672 . . . 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 3770 . . . . . . . . 9 (𝑏 = 𝑐 → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝑐))
1110fveq2d 6107 . . . . . . . 8 (𝑏 = 𝑐 → (#‘((1...𝑖) ∩ 𝑏)) = (#‘((1...𝑖) ∩ 𝑐)))
12 difeq2 3684 . . . . . . . . 9 (𝑏 = 𝑐 → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝑐))
1312fveq2d 6107 . . . . . . . 8 (𝑏 = 𝑐 → (#‘((1...𝑖) ∖ 𝑏)) = (#‘((1...𝑖) ∖ 𝑐)))
1411, 13oveq12d 6567 . . . . . . 7 (𝑏 = 𝑐 → ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))) = ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐))))
1514mpteq2dv 4673 . . . . . 6 (𝑏 = 𝑐 → (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
1615cbvmptv 4678 . . . . 5 (𝑏𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))))) = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
179, 16eqtr4i 2635 . . . 4 𝐹 = (𝑏𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏)))))
18 zex 11263 . . . . 5 ℤ ∈ V
1918mptex 6390 . . . 4 (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶)))) ∈ V
208, 17, 19fvmpt 6191 . . 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 6557 . . . . . 6 (𝑖 = 𝐽 → (1...𝑖) = (1...𝐽))
2322ineq1d 3775 . . . . 5 (𝑖 = 𝐽 → ((1...𝑖) ∩ 𝐶) = ((1...𝐽) ∩ 𝐶))
2423fveq2d 6107 . . . 4 (𝑖 = 𝐽 → (#‘((1...𝑖) ∩ 𝐶)) = (#‘((1...𝐽) ∩ 𝐶)))
2522difeq1d 3689 . . . . 5 (𝑖 = 𝐽 → ((1...𝑖) ∖ 𝐶) = ((1...𝐽) ∖ 𝐶))
2625fveq2d 6107 . . . 4 (𝑖 = 𝐽 → (#‘((1...𝑖) ∖ 𝐶)) = (#‘((1...𝐽) ∖ 𝐶)))
2724, 26oveq12d 6567 . . 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 ovex 6577 . . 3 ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))) ∈ V
3130a1i 11 . 2 (𝜑 → ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))) ∈ V)
3221, 28, 29, 31fvmptd 6197 1 (𝜑 → ((𝐹𝐶)‘𝐽) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {crab 2900  Vcvv 3173  cdif 3537  cin 3539  𝒫 cpw 4108  cmpt 4643  cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818  cmin 10145   / cdiv 10563  cn 10897  cz 11254  ...cfz 12197  #chash 12979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-cnex 9871  ax-resscn 9872
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-neg 10148  df-z 11255
This theorem is referenced by:  ballotlemfelz  29879  ballotlemfp1  29880  ballotlemfmpn  29883  ballotlemfval0  29884  ballotlemfg  29914  ballotlemfrc  29915
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