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Theorem ballotlemirc 34074
Description: Applying 𝑅 does not change first ties. (Contributed by Thierry Arnoux, 19-Apr-2017.) (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, < ))
ballotth.s 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
Assertion
Ref Expression
ballotlemirc (𝐶 ∈ (𝑂𝐸) → (𝐼‘(𝑅𝐶)) = (𝐼𝐶))
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,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐   𝑆,𝑘,𝑖,𝑐   𝑅,𝑖,𝑘   𝑥,𝑘,𝐶   𝑥,𝐹   𝑥,𝑀   𝑥,𝑁
Allowed substitution hints:   𝐶(𝑐)   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑐)   𝑆(𝑥)   𝐸(𝑥)   𝐼(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemirc
Dummy variables 𝑦 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ballotth.m . . . 4 𝑀 ∈ ℕ
2 ballotth.n . . . 4 𝑁 ∈ ℕ
3 ballotth.o . . . 4 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4 ballotth.p . . . 4 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
5 ballotth.f . . . 4 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6 ballotth.e . . . 4 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
7 ballotth.mgtn . . . 4 𝑁 < 𝑀
8 ballotth.i . . . 4 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
9 ballotth.s . . . 4 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
10 ballotth.r . . . 4 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemrc 34073 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ (𝑂𝐸))
121, 2, 3, 4, 5, 6, 7, 8ballotlemi 34043 . . 3 ((𝑅𝐶) ∈ (𝑂𝐸) → (𝐼‘(𝑅𝐶)) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}, ℝ, < ))
1311, 12syl 17 . 2 (𝐶 ∈ (𝑂𝐸) → (𝐼‘(𝑅𝐶)) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}, ℝ, < ))
14 ltso 11310 . . . 4 < Or ℝ
1514a1i 11 . . 3 (𝐶 ∈ (𝑂𝐸) → < Or ℝ)
161, 2, 3, 4, 5, 6, 7, 8ballotlemiex 34044 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1716simpld 494 . . . . 5 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
1817elfzelzd 13520 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℤ)
1918zred 12682 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℝ)
20 eqid 2727 . . . . 5 (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢)))) = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
211, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20ballotlemfrci 34070 . . . 4 (𝐶 ∈ (𝑂𝐸) → ((𝐹‘(𝑅𝐶))‘(𝐼𝐶)) = 0)
22 fveqeq2 6900 . . . . 5 (𝑘 = (𝐼𝐶) → (((𝐹‘(𝑅𝐶))‘𝑘) = 0 ↔ ((𝐹‘(𝑅𝐶))‘(𝐼𝐶)) = 0))
2322elrab 3680 . . . 4 ((𝐼𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0} ↔ ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘(𝑅𝐶))‘(𝐼𝐶)) = 0))
2417, 21, 23sylanbrc 582 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0})
25 elrabi 3674 . . . . 5 (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0} → 𝑦 ∈ (1...(𝑀 + 𝑁)))
2625anim2i 616 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}) → (𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))))
27 simpr 484 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼𝐶)) → 𝑦 < (𝐼𝐶))
281, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemfrcn0 34072 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼𝐶)) → ((𝐹‘(𝑅𝐶))‘𝑦) ≠ 0)
2928neneqd 2940 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼𝐶)) → ¬ ((𝐹‘(𝑅𝐶))‘𝑦) = 0)
30 fveqeq2 6900 . . . . . . . . . . . 12 (𝑘 = 𝑦 → (((𝐹‘(𝑅𝐶))‘𝑘) = 0 ↔ ((𝐹‘(𝑅𝐶))‘𝑦) = 0))
3130elrab 3680 . . . . . . . . . . 11 (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0} ↔ (𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘(𝑅𝐶))‘𝑦) = 0))
3231simprbi 496 . . . . . . . . . 10 (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0} → ((𝐹‘(𝑅𝐶))‘𝑦) = 0)
3329, 32nsyl 140 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0})
34333expa 1116 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0})
3527, 34syldan 590 . . . . . . 7 (((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0})
3635ex 412 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → (𝑦 < (𝐼𝐶) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}))
3736con2d 134 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0} → ¬ 𝑦 < (𝐼𝐶)))
3837imp 406 . . . 4 (((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}) → ¬ 𝑦 < (𝐼𝐶))
3926, 38sylancom 587 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}) → ¬ 𝑦 < (𝐼𝐶))
4015, 19, 24, 39infmin 9503 . 2 (𝐶 ∈ (𝑂𝐸) → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}, ℝ, < ) = (𝐼𝐶))
4113, 40eqtrd 2767 1 (𝐶 ∈ (𝑂𝐸) → (𝐼‘(𝑅𝐶)) = (𝐼𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3056  {crab 3427  cdif 3941  cin 3943  ifcif 4524  𝒫 cpw 4598   class class class wbr 5142  cmpt 5225   Or wor 5583  cima 5675  cfv 6542  (class class class)co 7414  cmpo 7416  Fincfn 8953  infcinf 9450  cr 11123  0cc0 11124  1c1 11125   + caddc 11127   < clt 11264  cle 11265  cmin 11460   / cdiv 11887  cn 12228  cz 12574  ...cfz 13502  chash 14307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-oadd 8482  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9451  df-inf 9452  df-dju 9910  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-n0 12489  df-z 12575  df-uz 12839  df-rp 12993  df-fz 13503  df-hash 14308
This theorem is referenced by:  ballotlemrinv0  34075
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