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Theorem ballotlemodife 31984
 Description: Elements of (𝑂 ∖ 𝐸). (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 3869 . 2 (𝐶 ∈ (𝑂𝐸) ↔ (𝐶𝑂 ∧ ¬ 𝐶𝐸))
2 df-or 846 . . . 4 (((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ (¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂) → (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
3 pm3.24 407 . . . . 5 ¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂)
43a1bi 367 . . . 4 ((𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)) ↔ (¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂) → (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
52, 4bitr4i 281 . . 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 31983 . . . . . 6 (𝐶𝐸 ↔ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1312notbii 324 . . . . 5 𝐶𝐸 ↔ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1413anbi2i 626 . . . 4 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ (𝐶𝑂 ∧ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
15 ianor 980 . . . . 5 (¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)) ↔ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1615anbi2i 626 . . . 4 ((𝐶𝑂 ∧ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ (𝐶𝑂 ∧ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
17 andi 1006 . . . 4 ((𝐶𝑂 ∧ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ ((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
1814, 16, 173bitri 301 . . 3 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ ((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
19 fz1ssfz0 13053 . . . . . . . . . . 11 (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
2019a1i 11 . . . . . . . . . 10 (𝐶𝑂 → (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁)))
2120sseld 3892 . . . . . . . . 9 (𝐶𝑂 → (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ (0...(𝑀 + 𝑁))))
2221imdistani 573 . . . . . . . 8 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
23 simpl 487 . . . . . . . . . . . 12 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝐶𝑂)
24 elfzelz 12957 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
2524adantl 486 . . . . . . . . . . . 12 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝑗 ∈ ℤ)
266, 7, 8, 9, 10, 23, 25ballotlemfelz 31977 . . . . . . . . . . 11 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑗) ∈ ℤ)
2726zred 12127 . . . . . . . . . 10 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑗) ∈ ℝ)
2827sbimi 2080 . . . . . . . . 9 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → [𝑖 / 𝑗]((𝐹𝐶)‘𝑗) ∈ ℝ)
29 sban 2086 . . . . . . . . . 10 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ ([𝑖 / 𝑗]𝐶𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))))
30 sbv 2096 . . . . . . . . . . 11 ([𝑖 / 𝑗]𝐶𝑂𝐶𝑂)
31 clelsb3 2880 . . . . . . . . . . 11 ([𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑖 ∈ (0...(𝑀 + 𝑁)))
3230, 31anbi12i 630 . . . . . . . . . 10 (([𝑖 / 𝑗]𝐶𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
3329, 32bitri 278 . . . . . . . . 9 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
34 nfv 1916 . . . . . . . . . 10 𝑗((𝐹𝐶)‘𝑖) ∈ ℝ
35 fveq2 6659 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝐹𝐶)‘𝑗) = ((𝐹𝐶)‘𝑖))
3635eleq1d 2837 . . . . . . . . . 10 (𝑗 = 𝑖 → (((𝐹𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹𝐶)‘𝑖) ∈ ℝ))
3734, 36sbiev 2323 . . . . . . . . 9 ([𝑖 / 𝑗]((𝐹𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹𝐶)‘𝑖) ∈ ℝ)
3828, 33, 373imtr3i 295 . . . . . . . 8 ((𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑖) ∈ ℝ)
3922, 38syl 17 . . . . . . 7 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑖) ∈ ℝ)
40 0red 10683 . . . . . . 7 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → 0 ∈ ℝ)
4139, 40lenltd 10825 . . . . . 6 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ¬ 0 < ((𝐹𝐶)‘𝑖)))
4241rexbidva 3221 . . . . 5 (𝐶𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹𝐶)‘𝑖)))
43 rexnal 3166 . . . . 5 (∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹𝐶)‘𝑖) ↔ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))
4442, 43bitrdi 290 . . . 4 (𝐶𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0 ↔ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
4544pm5.32i 579 . . 3 ((𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0) ↔ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
465, 18, 453bitr4i 307 . 2 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ (𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0))
471, 46bitri 278 1 (𝐶 ∈ (𝑂𝐸) ↔ (𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845   = wceq 1539  [wsb 2070   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  {crab 3075   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4495   class class class wbr 5033   ↦ cmpt 5113  ‘cfv 6336  (class class class)co 7151  ℝcr 10575  0cc0 10576  1c1 10577   + caddc 10579   < clt 10714   ≤ cle 10715   − cmin 10909   / cdiv 11336  ℕcn 11675  ℤcz 12021  ...cfz 12940  ♯chash 13741 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10632  ax-resscn 10633  ax-1cn 10634  ax-icn 10635  ax-addcl 10636  ax-addrcl 10637  ax-mulcl 10638  ax-mulrcl 10639  ax-mulcom 10640  ax-addass 10641  ax-mulass 10642  ax-distr 10643  ax-i2m1 10644  ax-1ne0 10645  ax-1rid 10646  ax-rnegex 10647  ax-rrecex 10648  ax-cnre 10649  ax-pre-lttri 10650  ax-pre-lttrn 10651  ax-pre-ltadd 10652  ax-pre-mulgt0 10653 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-1o 8113  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-card 9402  df-pnf 10716  df-mnf 10717  df-xr 10718  df-ltxr 10719  df-le 10720  df-sub 10911  df-neg 10912  df-nn 11676  df-n0 11936  df-z 12022  df-uz 12284  df-fz 12941  df-hash 13742 This theorem is referenced by:  ballotlem5  31986  ballotlemrc  32017
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