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Theorem ballotlemirc 30923
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 30922 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ (𝑂𝐸))
121, 2, 3, 4, 5, 6, 7, 8ballotlemi 30892 . . 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 10330 . . . 4 < Or ℝ
1514a1i 11 . . 3 (𝐶 ∈ (𝑂𝐸) → < Or ℝ)
161, 2, 3, 4, 5, 6, 7, 8ballotlemiex 30893 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1716simpld 477 . . . . 5 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
18 elfzelz 12555 . . . . 5 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
1917, 18syl 17 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℤ)
2019zred 11694 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℝ)
21 eqid 2760 . . . . 5 (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢)))) = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
221, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21ballotlemfrci 30919 . . . 4 (𝐶 ∈ (𝑂𝐸) → ((𝐹‘(𝑅𝐶))‘(𝐼𝐶)) = 0)
23 fveq2 6353 . . . . . 6 (𝑘 = (𝐼𝐶) → ((𝐹‘(𝑅𝐶))‘𝑘) = ((𝐹‘(𝑅𝐶))‘(𝐼𝐶)))
2423eqeq1d 2762 . . . . 5 (𝑘 = (𝐼𝐶) → (((𝐹‘(𝑅𝐶))‘𝑘) = 0 ↔ ((𝐹‘(𝑅𝐶))‘(𝐼𝐶)) = 0))
2524elrab 3504 . . . 4 ((𝐼𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0} ↔ ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘(𝑅𝐶))‘(𝐼𝐶)) = 0))
2617, 22, 25sylanbrc 701 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0})
27 elrabi 3499 . . . . 5 (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0} → 𝑦 ∈ (1...(𝑀 + 𝑁)))
2827anim2i 594 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}) → (𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))))
29 simpr 479 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼𝐶)) → 𝑦 < (𝐼𝐶))
301, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemfrcn0 30921 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼𝐶)) → ((𝐹‘(𝑅𝐶))‘𝑦) ≠ 0)
3130neneqd 2937 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼𝐶)) → ¬ ((𝐹‘(𝑅𝐶))‘𝑦) = 0)
32 fveq2 6353 . . . . . . . . . . . . 13 (𝑘 = 𝑦 → ((𝐹‘(𝑅𝐶))‘𝑘) = ((𝐹‘(𝑅𝐶))‘𝑦))
3332eqeq1d 2762 . . . . . . . . . . . 12 (𝑘 = 𝑦 → (((𝐹‘(𝑅𝐶))‘𝑘) = 0 ↔ ((𝐹‘(𝑅𝐶))‘𝑦) = 0))
3433elrab 3504 . . . . . . . . . . 11 (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0} ↔ (𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘(𝑅𝐶))‘𝑦) = 0))
3534simprbi 483 . . . . . . . . . 10 (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0} → ((𝐹‘(𝑅𝐶))‘𝑦) = 0)
3631, 35nsyl 135 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0})
37363expa 1112 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0})
3829, 37syldan 488 . . . . . . 7 (((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0})
3938ex 449 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → (𝑦 < (𝐼𝐶) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}))
4039con2d 129 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0} → ¬ 𝑦 < (𝐼𝐶)))
4140imp 444 . . . 4 (((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}) → ¬ 𝑦 < (𝐼𝐶))
4228, 41sylancom 704 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}) → ¬ 𝑦 < (𝐼𝐶))
4315, 20, 26, 42infmin 8567 . 2 (𝐶 ∈ (𝑂𝐸) → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅𝐶))‘𝑘) = 0}, ℝ, < ) = (𝐼𝐶))
4413, 43eqtrd 2794 1 (𝐶 ∈ (𝑂𝐸) → (𝐼‘(𝑅𝐶)) = (𝐼𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  {crab 3054  cdif 3712  cin 3714  ifcif 4230  𝒫 cpw 4302   class class class wbr 4804  cmpt 4881   Or wor 5186  cima 5269  cfv 6049  (class class class)co 6814  cmpt2 6816  Fincfn 8123  infcinf 8514  cr 10147  0cc0 10148  1c1 10149   + caddc 10151   < clt 10286  cle 10287  cmin 10478   / cdiv 10896  cn 11232  cz 11589  ...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-rp 12046  df-fz 12540  df-hash 13332
This theorem is referenced by:  ballotlemrinv0  30924
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