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Theorem ballotlemodife 30533
Description: Elements of (𝑂𝐸). (Contributed by Thierry Arnoux, 7-Dec-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)))))
ballotth.e 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
Assertion
Ref Expression
ballotlemodife (𝐶 ∈ (𝑂𝐸) ↔ (𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0))
Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂,𝑐   𝐹,𝑐,𝑖   𝐶,𝑖
Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑐)   𝐸(𝑥,𝑖,𝑐)   𝐹(𝑥)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemodife
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 eldif 3577 . 2 (𝐶 ∈ (𝑂𝐸) ↔ (𝐶𝑂 ∧ ¬ 𝐶𝐸))
2 df-or 385 . . . 4 (((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ (¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂) → (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
3 pm3.24 925 . . . . 5 ¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂)
43a1bi 352 . . . 4 ((𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)) ↔ (¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂) → (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
52, 4bitr4i 267 . . 3 (((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
6 ballotth.m . . . . . . 7 𝑀 ∈ ℕ
7 ballotth.n . . . . . . 7 𝑁 ∈ ℕ
8 ballotth.o . . . . . . 7 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}
9 ballotth.p . . . . . . 7 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))
10 ballotth.f . . . . . . 7 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
11 ballotth.e . . . . . . 7 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
126, 7, 8, 9, 10, 11ballotleme 30532 . . . . . 6 (𝐶𝐸 ↔ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1312notbii 310 . . . . 5 𝐶𝐸 ↔ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1413anbi2i 729 . . . 4 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ (𝐶𝑂 ∧ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
15 ianor 509 . . . . 5 (¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)) ↔ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1615anbi2i 729 . . . 4 ((𝐶𝑂 ∧ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ (𝐶𝑂 ∧ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
17 andi 910 . . . 4 ((𝐶𝑂 ∧ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ ((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
1814, 16, 173bitri 286 . . 3 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ ((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
19 0p1e1 11117 . . . . . . . . . . . . 13 (0 + 1) = 1
2019oveq1i 6645 . . . . . . . . . . . 12 ((0 + 1)...(𝑀 + 𝑁)) = (1...(𝑀 + 𝑁))
21 0z 11373 . . . . . . . . . . . . 13 0 ∈ ℤ
22 fzp1ss 12377 . . . . . . . . . . . . 13 (0 ∈ ℤ → ((0 + 1)...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁)))
2321, 22ax-mp 5 . . . . . . . . . . . 12 ((0 + 1)...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
2420, 23eqsstr3i 3628 . . . . . . . . . . 11 (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
2524a1i 11 . . . . . . . . . 10 (𝐶𝑂 → (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁)))
2625sseld 3594 . . . . . . . . 9 (𝐶𝑂 → (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ (0...(𝑀 + 𝑁))))
2726imdistani 725 . . . . . . . 8 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
28 simpl 473 . . . . . . . . . . . 12 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝐶𝑂)
29 elfzelz 12327 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
3029adantl 482 . . . . . . . . . . . 12 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝑗 ∈ ℤ)
316, 7, 8, 9, 10, 28, 30ballotlemfelz 30526 . . . . . . . . . . 11 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑗) ∈ ℤ)
3231zred 11467 . . . . . . . . . 10 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑗) ∈ ℝ)
3332sbimi 1884 . . . . . . . . 9 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → [𝑖 / 𝑗]((𝐹𝐶)‘𝑗) ∈ ℝ)
34 sban 2397 . . . . . . . . . 10 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ ([𝑖 / 𝑗]𝐶𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))))
35 nfv 1841 . . . . . . . . . . . 12 𝑗 𝐶𝑂
3635sbf 2378 . . . . . . . . . . 11 ([𝑖 / 𝑗]𝐶𝑂𝐶𝑂)
37 clelsb3 2727 . . . . . . . . . . 11 ([𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑖 ∈ (0...(𝑀 + 𝑁)))
3836, 37anbi12i 732 . . . . . . . . . 10 (([𝑖 / 𝑗]𝐶𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
3934, 38bitri 264 . . . . . . . . 9 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
40 nfv 1841 . . . . . . . . . 10 𝑗((𝐹𝐶)‘𝑖) ∈ ℝ
41 fveq2 6178 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝐹𝐶)‘𝑗) = ((𝐹𝐶)‘𝑖))
4241eleq1d 2684 . . . . . . . . . 10 (𝑗 = 𝑖 → (((𝐹𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹𝐶)‘𝑖) ∈ ℝ))
4340, 42sbie 2406 . . . . . . . . 9 ([𝑖 / 𝑗]((𝐹𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹𝐶)‘𝑖) ∈ ℝ)
4433, 39, 433imtr3i 280 . . . . . . . 8 ((𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑖) ∈ ℝ)
4527, 44syl 17 . . . . . . 7 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑖) ∈ ℝ)
46 0red 10026 . . . . . . 7 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → 0 ∈ ℝ)
4745, 46lenltd 10168 . . . . . 6 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ¬ 0 < ((𝐹𝐶)‘𝑖)))
4847rexbidva 3045 . . . . 5 (𝐶𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹𝐶)‘𝑖)))
49 rexnal 2992 . . . . 5 (∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹𝐶)‘𝑖) ↔ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))
5048, 49syl6bb 276 . . . 4 (𝐶𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0 ↔ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
5150pm5.32i 668 . . 3 ((𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0) ↔ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
525, 18, 513bitr4i 292 . 2 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ (𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0))
531, 52bitri 264 1 (𝐶 ∈ (𝑂𝐸) ↔ (𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1481  [wsb 1878  wcel 1988  wral 2909  wrex 2910  {crab 2913  cdif 3564  cin 3566  wss 3567  𝒫 cpw 4149   class class class wbr 4644  cmpt 4720  cfv 5876  (class class class)co 6635  cr 9920  0cc0 9921  1c1 9922   + caddc 9924   < clt 10059  cle 10060  cmin 10251   / cdiv 10669  cn 11005  cz 11362  ...cfz 12311  #chash 13100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101
This theorem is referenced by:  ballotlem5  30535  ballotlemrc  30566
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