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Theorem erdsze 32444
Description: The Erdős-Szekeres theorem. For any injective sequence 𝐹 on the reals of length at least (𝑅 − 1) · (𝑆 − 1) + 1, there is either a subsequence of length at least 𝑅 on which 𝐹 is increasing (i.e. a < , < order isomorphism) or a subsequence of length at least 𝑆 on which 𝐹 is decreasing (i.e. a < , < order isomorphism, recalling that < is the "greater than" relation). This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n (𝜑𝑁 ∈ ℕ)
erdsze.f (𝜑𝐹:(1...𝑁)–1-1→ℝ)
erdsze.r (𝜑𝑅 ∈ ℕ)
erdsze.s (𝜑𝑆 ∈ ℕ)
erdsze.l (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)
Assertion
Ref Expression
erdsze (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠)))))
Distinct variable groups:   𝐹,𝑠   𝑅,𝑠   𝑁,𝑠   𝜑,𝑠   𝑆,𝑠

Proof of Theorem erdsze
Dummy variables 𝑤 𝑥 𝑦 𝑧 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 erdsze.n . 2 (𝜑𝑁 ∈ ℕ)
2 erdsze.f . 2 (𝜑𝐹:(1...𝑁)–1-1→ℝ)
3 reseq2 5842 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
4 isoeq1 7064 . . . . . . . . . 10 ((𝐹𝑤) = (𝐹𝑦) → ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑤, (𝐹𝑤))))
53, 4syl 17 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑤, (𝐹𝑤))))
6 isoeq4 7067 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝐹𝑦) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑤))))
7 imaeq2 5919 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
8 isoeq5 7068 . . . . . . . . . 10 ((𝐹𝑤) = (𝐹𝑦) → ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦))))
97, 8syl 17 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦))))
105, 6, 93bitrd 307 . . . . . . . 8 (𝑤 = 𝑦 → ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦))))
11 elequ2 2125 . . . . . . . 8 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
1210, 11anbi12d 632 . . . . . . 7 (𝑤 = 𝑦 → (((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)))
1312cbvrabv 3491 . . . . . 6 {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)}
14 oveq2 7158 . . . . . . . 8 (𝑧 = 𝑥 → (1...𝑧) = (1...𝑥))
1514pweqd 4543 . . . . . . 7 (𝑧 = 𝑥 → 𝒫 (1...𝑧) = 𝒫 (1...𝑥))
16 elequ1 2117 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
1716anbi2d 630 . . . . . . 7 (𝑧 = 𝑥 → (((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)))
1815, 17rabeqbidv 3485 . . . . . 6 (𝑧 = 𝑥 → {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
1913, 18syl5eq 2868 . . . . 5 (𝑧 = 𝑥 → {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
2019imaeq2d 5923 . . . 4 (𝑧 = 𝑥 → (♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}) = (♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}))
2120supeq1d 8904 . . 3 (𝑧 = 𝑥 → sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}), ℝ, < ) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
2221cbvmptv 5161 . 2 (𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}), ℝ, < )) = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
23 isoeq1 7064 . . . . . . . . . 10 ((𝐹𝑤) = (𝐹𝑦) → ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑤, (𝐹𝑤))))
243, 23syl 17 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑤, (𝐹𝑤))))
25 isoeq4 7067 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝐹𝑦) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑤))))
26 isoeq5 7068 . . . . . . . . . 10 ((𝐹𝑤) = (𝐹𝑦) → ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦))))
277, 26syl 17 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦))))
2824, 25, 273bitrd 307 . . . . . . . 8 (𝑤 = 𝑦 → ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦))))
2928, 11anbi12d 632 . . . . . . 7 (𝑤 = 𝑦 → (((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)))
3029cbvrabv 3491 . . . . . 6 {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)}
3116anbi2d 630 . . . . . . 7 (𝑧 = 𝑥 → (((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)))
3215, 31rabeqbidv 3485 . . . . . 6 (𝑧 = 𝑥 → {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
3330, 32syl5eq 2868 . . . . 5 (𝑧 = 𝑥 → {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
3433imaeq2d 5923 . . . 4 (𝑧 = 𝑥 → (♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}) = (♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}))
3534supeq1d 8904 . . 3 (𝑧 = 𝑥 → sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}), ℝ, < ) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
3635cbvmptv 5161 . 2 (𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}), ℝ, < )) = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
37 eqid 2821 . 2 (𝑛 ∈ (1...𝑁) ↦ ⟨((𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}), ℝ, < ))‘𝑛), ((𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}), ℝ, < ))‘𝑛)⟩) = (𝑛 ∈ (1...𝑁) ↦ ⟨((𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}), ℝ, < ))‘𝑛), ((𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}), ℝ, < ))‘𝑛)⟩)
38 erdsze.r . 2 (𝜑𝑅 ∈ ℕ)
39 erdsze.s . 2 (𝜑𝑆 ∈ ℕ)
40 erdsze.l . 2 (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)
411, 2, 22, 36, 37, 38, 39, 40erdszelem11 32443 1 (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843   = wceq 1533  wcel 2110  wrex 3139  {crab 3142  𝒫 cpw 4538  cop 4566   class class class wbr 5058  cmpt 5138  ccnv 5548  cres 5551  cima 5552  1-1wf1 6346  cfv 6349   Isom wiso 6350  (class class class)co 7150  supcsup 8898  cr 10530  1c1 10532   · cmul 10536   < clt 10669  cle 10670  cmin 10864  cn 11632  ...cfz 12886  chash 13684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-fz 12887  df-hash 13685
This theorem is referenced by:  erdsze2lem2  32446
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