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Theorem ballotlemro 31666
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 31661 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) = ((𝑆𝐶) “ 𝐶))
12 imassrn 5938 . . . 4 ((𝑆𝐶) “ 𝐶) ⊆ ran (𝑆𝐶)
131, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1o 31657 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑆𝐶)))
1413simpld 495 . . . . 5 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
15 f1ofo 6619 . . . . 5 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)))
16 forn 6590 . . . . 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, 17sseqtrid 4023 . . 3 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ 𝐶) ⊆ (1...(𝑀 + 𝑁)))
1911, 18eqsstrd 4009 . 2 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ⊆ (1...(𝑀 + 𝑁)))
20 f1of1 6611 . . . . . . 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 4107 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
231, 2, 3ballotlemelo 31631 . . . . . . . 8 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
2422, 23sylib 219 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
2524simpld 495 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
26 id 22 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ∈ (𝑂𝐸))
27 f1imaeng 8558 . . . . . 6 (((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)) ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ (𝑂𝐸)) → ((𝑆𝐶) “ 𝐶) ≈ 𝐶)
2821, 25, 26, 27syl3anc 1365 . . . . 5 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ 𝐶) ≈ 𝐶)
2911, 28eqbrtrd 5085 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ≈ 𝐶)
30 hasheni 13698 . . . 4 ((𝑅𝐶) ≈ 𝐶 → (♯‘(𝑅𝐶)) = (♯‘𝐶))
3129, 30syl 17 . . 3 (𝐶 ∈ (𝑂𝐸) → (♯‘(𝑅𝐶)) = (♯‘𝐶))
3224simprd 496 . . 3 (𝐶 ∈ (𝑂𝐸) → (♯‘𝐶) = 𝑀)
3331, 32eqtrd 2861 . 2 (𝐶 ∈ (𝑂𝐸) → (♯‘(𝑅𝐶)) = 𝑀)
341, 2, 3ballotlemelo 31631 . 2 ((𝑅𝐶) ∈ 𝑂 ↔ ((𝑅𝐶) ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘(𝑅𝐶)) = 𝑀))
3519, 33, 34sylanbrc 583 1 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  {crab 3147  cdif 3937  cin 3939  wss 3940  ifcif 4470  𝒫 cpw 4542   class class class wbr 5063  cmpt 5143  ccnv 5553  ran crn 5555  cima 5557  1-1wf1 6349  ontowfo 6350  1-1-ontowf1o 6351  cfv 6352  (class class class)co 7148  cen 8495  infcinf 8894  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   < clt 10664  cle 10665  cmin 10859   / cdiv 11286  cn 11627  cz 11970  ...cfz 12882  chash 13680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-hash 13681
This theorem is referenced by:  ballotlemfrc  31670  ballotlemfrceq  31672  ballotlemfrcn0  31673  ballotlemrc  31674
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