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Theorem ballotlemodife 34500
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 3961 . 2 (𝐶 ∈ (𝑂𝐸) ↔ (𝐶𝑂 ∧ ¬ 𝐶𝐸))
2 df-or 849 . . . 4 (((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ (¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂) → (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
3 pm3.24 402 . . . . 5 ¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂)
43a1bi 362 . . . 4 ((𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)) ↔ (¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂) → (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
52, 4bitr4i 278 . . 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 34499 . . . . . 6 (𝐶𝐸 ↔ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1312notbii 320 . . . . 5 𝐶𝐸 ↔ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1413anbi2i 623 . . . 4 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ (𝐶𝑂 ∧ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
15 ianor 984 . . . . 5 (¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)) ↔ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1615anbi2i 623 . . . 4 ((𝐶𝑂 ∧ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ (𝐶𝑂 ∧ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
17 andi 1010 . . . 4 ((𝐶𝑂 ∧ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ ((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
1814, 16, 173bitri 297 . . 3 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ ((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
19 fz1ssfz0 13663 . . . . . . . . . . 11 (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
2019a1i 11 . . . . . . . . . 10 (𝐶𝑂 → (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁)))
2120sseld 3982 . . . . . . . . 9 (𝐶𝑂 → (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ (0...(𝑀 + 𝑁))))
2221imdistani 568 . . . . . . . 8 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
23 simpl 482 . . . . . . . . . . . 12 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝐶𝑂)
24 elfzelz 13564 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
2524adantl 481 . . . . . . . . . . . 12 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝑗 ∈ ℤ)
266, 7, 8, 9, 10, 23, 25ballotlemfelz 34493 . . . . . . . . . . 11 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑗) ∈ ℤ)
2726zred 12722 . . . . . . . . . 10 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑗) ∈ ℝ)
2827sbimi 2074 . . . . . . . . 9 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → [𝑖 / 𝑗]((𝐹𝐶)‘𝑗) ∈ ℝ)
29 sban 2080 . . . . . . . . . 10 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ ([𝑖 / 𝑗]𝐶𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))))
30 sbv 2088 . . . . . . . . . . 11 ([𝑖 / 𝑗]𝐶𝑂𝐶𝑂)
31 clelsb1 2868 . . . . . . . . . . 11 ([𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑖 ∈ (0...(𝑀 + 𝑁)))
3230, 31anbi12i 628 . . . . . . . . . 10 (([𝑖 / 𝑗]𝐶𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
3329, 32bitri 275 . . . . . . . . 9 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
34 nfv 1914 . . . . . . . . . 10 𝑗((𝐹𝐶)‘𝑖) ∈ ℝ
35 fveq2 6906 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝐹𝐶)‘𝑗) = ((𝐹𝐶)‘𝑖))
3635eleq1d 2826 . . . . . . . . . 10 (𝑗 = 𝑖 → (((𝐹𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹𝐶)‘𝑖) ∈ ℝ))
3734, 36sbiev 2314 . . . . . . . . 9 ([𝑖 / 𝑗]((𝐹𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹𝐶)‘𝑖) ∈ ℝ)
3828, 33, 373imtr3i 291 . . . . . . . 8 ((𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑖) ∈ ℝ)
3922, 38syl 17 . . . . . . 7 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑖) ∈ ℝ)
40 0red 11264 . . . . . . 7 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → 0 ∈ ℝ)
4139, 40lenltd 11407 . . . . . 6 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ¬ 0 < ((𝐹𝐶)‘𝑖)))
4241rexbidva 3177 . . . . 5 (𝐶𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹𝐶)‘𝑖)))
43 rexnal 3100 . . . . 5 (∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹𝐶)‘𝑖) ↔ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))
4442, 43bitrdi 287 . . . 4 (𝐶𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0 ↔ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
4544pm5.32i 574 . . 3 ((𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0) ↔ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
465, 18, 453bitr4i 303 . 2 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ (𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0))
471, 46bitri 275 1 (𝐶 ∈ (𝑂𝐸) ↔ (𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  [wsb 2064  wcel 2108  wral 3061  wrex 3070  {crab 3436  cdif 3948  cin 3950  wss 3951  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  cn 12266  cz 12613  ...cfz 13547  chash 14369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370
This theorem is referenced by:  ballotlem5  34502  ballotlemrc  34533
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