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Theorem ballotlemimin 30897
Description: (𝐼𝐶) is the first tie. (Contributed by Thierry Arnoux, 1-Dec-2016.) (Revised by AV, 6-Oct-2020.)
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)}
ballotth.mgtn 𝑁 < 𝑀
ballotth.i 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
Assertion
Ref Expression
ballotlemimin (𝐶 ∈ (𝑂𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼𝐶) − 1))((𝐹𝐶)‘𝑘) = 0)
Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂   𝑘,𝑀   𝑘,𝑁   𝑘,𝑂   𝑖,𝑐,𝐹,𝑘   𝐶,𝑖,𝑘   𝑖,𝐸,𝑘   𝐶,𝑘   𝑘,𝐼   𝑘,𝑐,𝐸   𝑖,𝐼
Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑘,𝑐)   𝐸(𝑥)   𝐹(𝑥)   𝐼(𝑥,𝑐)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemimin
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfzle2 12558 . . . . . 6 (𝑘 ∈ (1...((𝐼𝐶) − 1)) → 𝑘 ≤ ((𝐼𝐶) − 1))
21adantl 473 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑘 ∈ (1...((𝐼𝐶) − 1))) → 𝑘 ≤ ((𝐼𝐶) − 1))
3 elfzelz 12555 . . . . . 6 (𝑘 ∈ (1...((𝐼𝐶) − 1)) → 𝑘 ∈ ℤ)
4 ballotth.m . . . . . . . . . 10 𝑀 ∈ ℕ
5 ballotth.n . . . . . . . . . 10 𝑁 ∈ ℕ
6 ballotth.o . . . . . . . . . 10 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
7 ballotth.p . . . . . . . . . 10 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
8 ballotth.f . . . . . . . . . 10 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
9 ballotth.e . . . . . . . . . 10 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
10 ballotth.mgtn . . . . . . . . . 10 𝑁 < 𝑀
11 ballotth.i . . . . . . . . . 10 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
124, 5, 6, 7, 8, 9, 10, 11ballotlemiex 30893 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1312simpld 477 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
14 elfznn 12583 . . . . . . . 8 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℕ)
1513, 14syl 17 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℕ)
1615nnzd 11693 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℤ)
17 zltlem1 11642 . . . . . 6 ((𝑘 ∈ ℤ ∧ (𝐼𝐶) ∈ ℤ) → (𝑘 < (𝐼𝐶) ↔ 𝑘 ≤ ((𝐼𝐶) − 1)))
183, 16, 17syl2anr 496 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑘 ∈ (1...((𝐼𝐶) − 1))) → (𝑘 < (𝐼𝐶) ↔ 𝑘 ≤ ((𝐼𝐶) − 1)))
192, 18mpbird 247 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑘 ∈ (1...((𝐼𝐶) − 1))) → 𝑘 < (𝐼𝐶))
2019adantr 472 . . 3 (((𝐶 ∈ (𝑂𝐸) ∧ 𝑘 ∈ (1...((𝐼𝐶) − 1))) ∧ ((𝐹𝐶)‘𝑘) = 0) → 𝑘 < (𝐼𝐶))
21 1zzd 11620 . . . . . . . . . . . . 13 (𝐶 ∈ (𝑂𝐸) → 1 ∈ ℤ)
2216, 21zsubcld 11699 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℤ)
2322zred 11694 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℝ)
24 nnaddcl 11254 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
254, 5, 24mp2an 710 . . . . . . . . . . . . 13 (𝑀 + 𝑁) ∈ ℕ
2625a1i 11 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℕ)
2726nnred 11247 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℝ)
28 elfzle2 12558 . . . . . . . . . . . . 13 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
2913, 28syl 17 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
3026nnzd 11693 . . . . . . . . . . . . 13 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℤ)
31 zlem1lt 11641 . . . . . . . . . . . . 13 (((𝐼𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
3216, 30, 31syl2anc 696 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
3329, 32mpbid 222 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) < (𝑀 + 𝑁))
3423, 27, 33ltled 10397 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁))
35 eluz 11913 . . . . . . . . . . 11 ((((𝐼𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)) ↔ ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
3622, 30, 35syl2anc 696 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)) ↔ ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
3734, 36mpbird 247 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)))
38 fzss2 12594 . . . . . . . . 9 ((𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)) → (1...((𝐼𝐶) − 1)) ⊆ (1...(𝑀 + 𝑁)))
3937, 38syl 17 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → (1...((𝐼𝐶) − 1)) ⊆ (1...(𝑀 + 𝑁)))
4039sseld 3743 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → (𝑘 ∈ (1...((𝐼𝐶) − 1)) → 𝑘 ∈ (1...(𝑀 + 𝑁))))
41 rabid 3254 . . . . . . . 8 (𝑘 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} ↔ (𝑘 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘𝑘) = 0))
424, 5, 6, 7, 8, 9, 10, 11ballotlemsup 30896 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}𝑦 < 𝑤)))
43 ltso 10330 . . . . . . . . . . . 12 < Or ℝ
4443a1i 11 . . . . . . . . . . 11 (∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}𝑦 < 𝑤)) → < Or ℝ)
45 id 22 . . . . . . . . . . 11 (∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}𝑦 < 𝑤)) → ∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}𝑦 < 𝑤)))
4644, 45inflb 8562 . . . . . . . . . 10 (∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}𝑦 < 𝑤)) → (𝑘 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} → ¬ 𝑘 < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < )))
4742, 46syl 17 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝑘 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} → ¬ 𝑘 < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < )))
484, 5, 6, 7, 8, 9, 10, 11ballotlemi 30892 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < ))
4948breq2d 4816 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝑘 < (𝐼𝐶) ↔ 𝑘 < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < )))
5049notbid 307 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (¬ 𝑘 < (𝐼𝐶) ↔ ¬ 𝑘 < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < )))
5147, 50sylibrd 249 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → (𝑘 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} → ¬ 𝑘 < (𝐼𝐶)))
5241, 51syl5bir 233 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → ((𝑘 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘𝑘) = 0) → ¬ 𝑘 < (𝐼𝐶)))
5340, 52syland 499 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → ((𝑘 ∈ (1...((𝐼𝐶) − 1)) ∧ ((𝐹𝐶)‘𝑘) = 0) → ¬ 𝑘 < (𝐼𝐶)))
5453imp 444 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑘 ∈ (1...((𝐼𝐶) − 1)) ∧ ((𝐹𝐶)‘𝑘) = 0)) → ¬ 𝑘 < (𝐼𝐶))
55 biid 251 . . . . 5 (𝑘 < (𝐼𝐶) ↔ 𝑘 < (𝐼𝐶))
5654, 55sylnib 317 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑘 ∈ (1...((𝐼𝐶) − 1)) ∧ ((𝐹𝐶)‘𝑘) = 0)) → ¬ 𝑘 < (𝐼𝐶))
5756anassrs 683 . . 3 (((𝐶 ∈ (𝑂𝐸) ∧ 𝑘 ∈ (1...((𝐼𝐶) − 1))) ∧ ((𝐹𝐶)‘𝑘) = 0) → ¬ 𝑘 < (𝐼𝐶))
5820, 57pm2.65da 601 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑘 ∈ (1...((𝐼𝐶) − 1))) → ¬ ((𝐹𝐶)‘𝑘) = 0)
5958nrexdv 3139 1 (𝐶 ∈ (𝑂𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼𝐶) − 1))((𝐹𝐶)‘𝑘) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  cdif 3712  cin 3714  wss 3715  𝒫 cpw 4302   class class class wbr 4804  cmpt 4881   Or wor 5186  cfv 6049  (class class class)co 6814  infcinf 8514  cr 10147  0cc0 10148  1c1 10149   + caddc 10151   < clt 10286  cle 10287  cmin 10478   / cdiv 10896  cn 11232  cz 11589  cuz 11899  ...cfz 12539  chash 13331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-hash 13332
This theorem is referenced by:  ballotlemic  30898  ballotlem1c  30899
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