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Theorem ballotlemro 31101
Description: Range of 𝑅 is included in 𝑂. (Contributed by Thierry Arnoux, 17-Apr-2017.)
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
ballotlemro (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
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
Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑖,𝑘,𝑐)   𝑆(𝑥)   𝐸(𝑥)   𝐹(𝑥)   𝐼(𝑥)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemro
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, 10ballotlemrval 31096 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) = ((𝑆𝐶) “ 𝐶))
12 imassrn 5694 . . . 4 ((𝑆𝐶) “ 𝐶) ⊆ ran (𝑆𝐶)
131, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1o 31092 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑆𝐶)))
1413simpld 489 . . . . 5 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
15 f1ofo 6363 . . . . 5 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)))
16 forn 6334 . . . . 5 ((𝑆𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)) → ran (𝑆𝐶) = (1...(𝑀 + 𝑁)))
1714, 15, 163syl 18 . . . 4 (𝐶 ∈ (𝑂𝐸) → ran (𝑆𝐶) = (1...(𝑀 + 𝑁)))
1812, 17syl5sseq 3849 . . 3 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ 𝐶) ⊆ (1...(𝑀 + 𝑁)))
1911, 18eqsstrd 3835 . 2 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ⊆ (1...(𝑀 + 𝑁)))
20 f1of1 6355 . . . . . . 7 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
2114, 20syl 17 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
22 eldifi 3930 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
231, 2, 3ballotlemelo 31066 . . . . . . . 8 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
2422, 23sylib 210 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
2524simpld 489 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
26 id 22 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ∈ (𝑂𝐸))
27 f1imaeng 8255 . . . . . 6 (((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)) ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ (𝑂𝐸)) → ((𝑆𝐶) “ 𝐶) ≈ 𝐶)
2821, 25, 26, 27syl3anc 1491 . . . . 5 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ 𝐶) ≈ 𝐶)
2911, 28eqbrtrd 4865 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ≈ 𝐶)
30 hasheni 13388 . . . 4 ((𝑅𝐶) ≈ 𝐶 → (♯‘(𝑅𝐶)) = (♯‘𝐶))
3129, 30syl 17 . . 3 (𝐶 ∈ (𝑂𝐸) → (♯‘(𝑅𝐶)) = (♯‘𝐶))
3224simprd 490 . . 3 (𝐶 ∈ (𝑂𝐸) → (♯‘𝐶) = 𝑀)
3331, 32eqtrd 2833 . 2 (𝐶 ∈ (𝑂𝐸) → (♯‘(𝑅𝐶)) = 𝑀)
341, 2, 3ballotlemelo 31066 . 2 ((𝑅𝐶) ∈ 𝑂 ↔ ((𝑅𝐶) ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘(𝑅𝐶)) = 𝑀))
3519, 33, 34sylanbrc 579 1 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  {crab 3093  cdif 3766  cin 3768  wss 3769  ifcif 4277  𝒫 cpw 4349   class class class wbr 4843  cmpt 4922  ccnv 5311  ran crn 5313  cima 5315  1-1wf1 6098  ontowfo 6099  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  cen 8192  infcinf 8589  cr 10223  0cc0 10224  1c1 10225   + caddc 10227   < clt 10363  cle 10364  cmin 10556   / cdiv 10976  cn 11312  cz 11666  ...cfz 12580  chash 13370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-sup 8590  df-inf 8591  df-card 9051  df-cda 9278  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-n0 11581  df-z 11667  df-uz 11931  df-rp 12075  df-fz 12581  df-hash 13371
This theorem is referenced by:  ballotlemfrc  31105  ballotlemfrceq  31107  ballotlemfrcn0  31108  ballotlemrc  31109
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