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Theorem ballotlemodife 33792
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 3959 . 2 (𝐶 ∈ (𝑂𝐸) ↔ (𝐶𝑂 ∧ ¬ 𝐶𝐸))
2 df-or 844 . . . 4 (((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ (¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂) → (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
3 pm3.24 401 . . . . 5 ¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂)
43a1bi 361 . . . 4 ((𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)) ↔ (¬ (𝐶𝑂 ∧ ¬ 𝐶𝑂) → (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
52, 4bitr4i 277 . . 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 33791 . . . . . 6 (𝐶𝐸 ↔ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1312notbii 319 . . . . 5 𝐶𝐸 ↔ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1413anbi2i 621 . . . 4 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ (𝐶𝑂 ∧ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
15 ianor 978 . . . . 5 (¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)) ↔ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
1615anbi2i 621 . . . 4 ((𝐶𝑂 ∧ ¬ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ (𝐶𝑂 ∧ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
17 andi 1004 . . . 4 ((𝐶𝑂 ∧ (¬ 𝐶𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))) ↔ ((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
1814, 16, 173bitri 296 . . 3 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ ((𝐶𝑂 ∧ ¬ 𝐶𝑂) ∨ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))))
19 fz1ssfz0 13603 . . . . . . . . . . 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 567 . . . . . . . 8 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
23 simpl 481 . . . . . . . . . . . 12 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝐶𝑂)
24 elfzelz 13507 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
2524adantl 480 . . . . . . . . . . . 12 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝑗 ∈ ℤ)
266, 7, 8, 9, 10, 23, 25ballotlemfelz 33785 . . . . . . . . . . 11 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑗) ∈ ℤ)
2726zred 12672 . . . . . . . . . 10 ((𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑗) ∈ ℝ)
2827sbimi 2075 . . . . . . . . 9 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) → [𝑖 / 𝑗]((𝐹𝐶)‘𝑗) ∈ ℝ)
29 sban 2081 . . . . . . . . . 10 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ ([𝑖 / 𝑗]𝐶𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))))
30 sbv 2089 . . . . . . . . . . 11 ([𝑖 / 𝑗]𝐶𝑂𝐶𝑂)
31 clelsb1 2858 . . . . . . . . . . 11 ([𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑖 ∈ (0...(𝑀 + 𝑁)))
3230, 31anbi12i 625 . . . . . . . . . 10 (([𝑖 / 𝑗]𝐶𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
3329, 32bitri 274 . . . . . . . . 9 ([𝑖 / 𝑗](𝐶𝑂𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))))
34 nfv 1915 . . . . . . . . . 10 𝑗((𝐹𝐶)‘𝑖) ∈ ℝ
35 fveq2 6892 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝐹𝐶)‘𝑗) = ((𝐹𝐶)‘𝑖))
3635eleq1d 2816 . . . . . . . . . 10 (𝑗 = 𝑖 → (((𝐹𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹𝐶)‘𝑖) ∈ ℝ))
3734, 36sbiev 2306 . . . . . . . . 9 ([𝑖 / 𝑗]((𝐹𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹𝐶)‘𝑖) ∈ ℝ)
3828, 33, 373imtr3i 290 . . . . . . . 8 ((𝐶𝑂𝑖 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑖) ∈ ℝ)
3922, 38syl 17 . . . . . . 7 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝑖) ∈ ℝ)
40 0red 11223 . . . . . . 7 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → 0 ∈ ℝ)
4139, 40lenltd 11366 . . . . . 6 ((𝐶𝑂𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ¬ 0 < ((𝐹𝐶)‘𝑖)))
4241rexbidva 3174 . . . . 5 (𝐶𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹𝐶)‘𝑖)))
43 rexnal 3098 . . . . 5 (∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹𝐶)‘𝑖) ↔ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖))
4442, 43bitrdi 286 . . . 4 (𝐶𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0 ↔ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
4544pm5.32i 573 . . 3 ((𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0) ↔ (𝐶𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))
465, 18, 453bitr4i 302 . 2 ((𝐶𝑂 ∧ ¬ 𝐶𝐸) ↔ (𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0))
471, 46bitri 274 1 (𝐶 ∈ (𝑂𝐸) ↔ (𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 843   = wceq 1539  [wsb 2065  wcel 2104  wral 3059  wrex 3068  {crab 3430  cdif 3946  cin 3948  wss 3949  𝒫 cpw 4603   class class class wbr 5149  cmpt 5232  cfv 6544  (class class class)co 7413  cr 11113  0cc0 11114  1c1 11115   + caddc 11117   < clt 11254  cle 11255  cmin 11450   / cdiv 11877  cn 12218  cz 12564  ...cfz 13490  chash 14296
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9938  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-nn 12219  df-n0 12479  df-z 12565  df-uz 12829  df-fz 13491  df-hash 14297
This theorem is referenced by:  ballotlem5  33794  ballotlemrc  33825
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