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Theorem ballotlemfval 30344
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 473 . . . . . . . 8 ((𝑏 = 𝐶𝑖 ∈ ℤ) → 𝑏 = 𝐶)
32ineq2d 3794 . . . . . . 7 ((𝑏 = 𝐶𝑖 ∈ ℤ) → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝐶))
43fveq2d 6154 . . . . . 6 ((𝑏 = 𝐶𝑖 ∈ ℤ) → (#‘((1...𝑖) ∩ 𝑏)) = (#‘((1...𝑖) ∩ 𝐶)))
52difeq2d 3708 . . . . . . 7 ((𝑏 = 𝐶𝑖 ∈ ℤ) → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝐶))
65fveq2d 6154 . . . . . 6 ((𝑏 = 𝐶𝑖 ∈ ℤ) → (#‘((1...𝑖) ∖ 𝑏)) = (#‘((1...𝑖) ∖ 𝐶)))
74, 6oveq12d 6625 . . . . 5 ((𝑏 = 𝐶𝑖 ∈ ℤ) → ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))) = ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶))))
87mpteq2dva 4706 . . . 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 3788 . . . . . . . . 9 (𝑏 = 𝑐 → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝑐))
1110fveq2d 6154 . . . . . . . 8 (𝑏 = 𝑐 → (#‘((1...𝑖) ∩ 𝑏)) = (#‘((1...𝑖) ∩ 𝑐)))
12 difeq2 3702 . . . . . . . . 9 (𝑏 = 𝑐 → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝑐))
1312fveq2d 6154 . . . . . . . 8 (𝑏 = 𝑐 → (#‘((1...𝑖) ∖ 𝑏)) = (#‘((1...𝑖) ∖ 𝑐)))
1411, 13oveq12d 6625 . . . . . . 7 (𝑏 = 𝑐 → ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))) = ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐))))
1514mpteq2dv 4707 . . . . . 6 (𝑏 = 𝑐 → (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
1615cbvmptv 4712 . . . . 5 (𝑏𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))))) = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
179, 16eqtr4i 2646 . . . 4 𝐹 = (𝑏𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏)))))
18 zex 11333 . . . . 5 ℤ ∈ V
1918mptex 6443 . . . 4 (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶)))) ∈ V
208, 17, 19fvmpt 6241 . . 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 6615 . . . . . 6 (𝑖 = 𝐽 → (1...𝑖) = (1...𝐽))
2322ineq1d 3793 . . . . 5 (𝑖 = 𝐽 → ((1...𝑖) ∩ 𝐶) = ((1...𝐽) ∩ 𝐶))
2423fveq2d 6154 . . . 4 (𝑖 = 𝐽 → (#‘((1...𝑖) ∩ 𝐶)) = (#‘((1...𝐽) ∩ 𝐶)))
2522difeq1d 3707 . . . . 5 (𝑖 = 𝐽 → ((1...𝑖) ∖ 𝐶) = ((1...𝐽) ∖ 𝐶))
2625fveq2d 6154 . . . 4 (𝑖 = 𝐽 → (#‘((1...𝑖) ∖ 𝐶)) = (#‘((1...𝐽) ∖ 𝐶)))
2724, 26oveq12d 6625 . . 3 (𝑖 = 𝐽 → ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶))) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))
2827adantl 482 . 2 ((𝜑𝑖 = 𝐽) → ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶))) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))
29 ballotlemfval.j . 2 (𝜑𝐽 ∈ ℤ)
30 ovex 6635 . . 3 ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))) ∈ V
3130a1i 11 . 2 (𝜑 → ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))) ∈ V)
3221, 28, 29, 31fvmptd 6247 1 (𝜑 → ((𝐹𝐶)‘𝐽) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  cdif 3553  cin 3555  𝒫 cpw 4132  cmpt 4675  cfv 5849  (class class class)co 6607  1c1 9884   + caddc 9886  cmin 10213   / cdiv 10631  cn 10967  cz 11324  ...cfz 12271  #chash 13060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-cnex 9939  ax-resscn 9940
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-neg 10216  df-z 11325
This theorem is referenced by:  ballotlemfelz  30345  ballotlemfp1  30346  ballotlemfmpn  30349  ballotlemfval0  30350  ballotlemfg  30380  ballotlemfrc  30381
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