Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballotlemirc Structured version   Visualization version   GIF version

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  ℝcr 10147  0cc0 10148  1c1 10149   + caddc 10151   < clt 10286   ≤ cle 10287   − cmin 10478   / cdiv 10896  ℕcn 11232  ℤcz 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
 Copyright terms: Public domain W3C validator