Theorem ballotlemirc 32498

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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r	⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))

Assertion

Ref	Expression
ballotlemirc	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = (𝐼‘𝐶))

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.

Step	Hyp	Ref	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotth.e	. . . 4 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
7		ballotth.mgtn	. . . 4 ⊢ 𝑁 < 𝑀
8		ballotth.i	. . . 4 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
9		ballotth.s	. . . 4 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
10		ballotth.r	. . . 4 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemrc 32497	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ (𝑂 ∖ 𝐸))
12	1, 2, 3, 4, 5, 6, 7, 8	ballotlemi 32467	. . 3 ⊢ ((𝑅‘𝐶) ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}, ℝ, < ))
13	11, 12	syl 17	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}, ℝ, < ))
14		ltso 11055	. . . 4 ⊢ < Or ℝ
15	14	a1i 11	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → < Or ℝ)
16	1, 2, 3, 4, 5, 6, 7, 8	ballotlemiex 32468	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
17	16	simpld 495	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
18	17	elfzelzd 13257	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
19	18	zred 12426	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℝ)
20		eqid 2738	. . . . 5 ⊢ (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
21	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20	ballotlemfrci 32494	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘(𝑅‘𝐶))‘(𝐼‘𝐶)) = 0)
22		fveqeq2 6783	. . . . 5 ⊢ (𝑘 = (𝐼‘𝐶) → (((𝐹‘(𝑅‘𝐶))‘𝑘) = 0 ↔ ((𝐹‘(𝑅‘𝐶))‘(𝐼‘𝐶)) = 0))
23	22	elrab 3624	. . . 4 ⊢ ((𝐼‘𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} ↔ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘(𝑅‘𝐶))‘(𝐼‘𝐶)) = 0))
24	17, 21, 23	sylanbrc 583	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0})
25		elrabi 3618	. . . . 5 ⊢ (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} → 𝑦 ∈ (1...(𝑀 + 𝑁)))
26	25	anim2i 617	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) → (𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))))
27		simpr 485	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼‘𝐶)) → 𝑦 < (𝐼‘𝐶))
28	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemfrcn0 32496	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼‘𝐶)) → ((𝐹‘(𝑅‘𝐶))‘𝑦) ≠ 0)
29	28	neneqd 2948	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼‘𝐶)) → ¬ ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0)
30		fveqeq2 6783	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → (((𝐹‘(𝑅‘𝐶))‘𝑘) = 0 ↔ ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0))
31	30	elrab 3624	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} ↔ (𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0))
32	31	simprbi 497	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} → ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0)
33	29, 32	nsyl 140	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼‘𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0})
34	33	3expa 1117	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼‘𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0})
35	27, 34	syldan 591	. . . . . . 7 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 < (𝐼‘𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0})
36	35	ex 413	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → (𝑦 < (𝐼‘𝐶) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}))
37	36	con2d 134	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} → ¬ 𝑦 < (𝐼‘𝐶)))
38	37	imp 407	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) → ¬ 𝑦 < (𝐼‘𝐶))
39	26, 38	sylancom 588	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) → ¬ 𝑦 < (𝐼‘𝐶))
40	15, 19, 24, 39	infmin 9253	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}, ℝ, < ) = (𝐼‘𝐶))
41	13, 40	eqtrd 2778	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = (𝐼‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ∖ cdif 3884   ∩ cin 3886  ifcif 4459  𝒫 cpw 4533   class class class wbr 5074   ↦ cmpt 5157   Or wor 5502   “ cima 5592  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Fincfn 8733  infcinf 9200  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874   < clt 11009   ≤ cle 11010   − cmin 11205   / cdiv 11632  ℕcn 11973  ℤcz 12319  ...cfz 13239  ♯chash 14044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-hash 14045

This theorem is referenced by:  ballotlemrinv0  32499