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Theorem erdsze 31522
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 5529 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
4 isoeq1 6710 . . . . . . . . . 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 6713 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝐹𝑦) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑤))))
7 imaeq2 5603 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
8 isoeq5 6714 . . . . . . . . . 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 294 . . . . . . . 8 (𝑤 = 𝑦 → ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦))))
11 elequ2 2159 . . . . . . . 8 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
1210, 11anbi12d 616 . . . . . . 7 (𝑤 = 𝑦 → (((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)))
1312cbvrabv 3349 . . . . . 6 {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)}
14 oveq2 6801 . . . . . . . 8 (𝑧 = 𝑥 → (1...𝑧) = (1...𝑥))
1514pweqd 4302 . . . . . . 7 (𝑧 = 𝑥 → 𝒫 (1...𝑧) = 𝒫 (1...𝑥))
16 elequ1 2152 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
1716anbi2d 614 . . . . . . 7 (𝑧 = 𝑥 → (((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)))
1815, 17rabeqbidv 3345 . . . . . 6 (𝑧 = 𝑥 → {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
1913, 18syl5eq 2817 . . . . 5 (𝑧 = 𝑥 → {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
2019imaeq2d 5607 . . . 4 (𝑧 = 𝑥 → (♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}) = (♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}))
2120supeq1d 8508 . . 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 4884 . 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 6710 . . . . . . . . . 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 6713 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝐹𝑦) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑤))))
26 isoeq5 6714 . . . . . . . . . 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 294 . . . . . . . 8 (𝑤 = 𝑦 → ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦))))
2928, 11anbi12d 616 . . . . . . 7 (𝑤 = 𝑦 → (((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)))
3029cbvrabv 3349 . . . . . 6 {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)}
3116anbi2d 614 . . . . . . 7 (𝑧 = 𝑥 → (((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)))
3215, 31rabeqbidv 3345 . . . . . 6 (𝑧 = 𝑥 → {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
3330, 32syl5eq 2817 . . . . 5 (𝑧 = 𝑥 → {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
3433imaeq2d 5607 . . . 4 (𝑧 = 𝑥 → (♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}) = (♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}))
3534supeq1d 8508 . . 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 4884 . 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 2771 . 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 31521 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 196  wa 382  wo 836   = wceq 1631  wcel 2145  wrex 3062  {crab 3065  𝒫 cpw 4297  cop 4322   class class class wbr 4786  cmpt 4863  ccnv 5248  cres 5251  cima 5252  1-1wf1 6028  cfv 6031   Isom wiso 6032  (class class class)co 6793  supcsup 8502  cr 10137  1c1 10139   · cmul 10143   < clt 10276  cle 10277  cmin 10468  cn 11222  ...cfz 12533  chash 13321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-hash 13322
This theorem is referenced by:  erdsze2lem2  31524
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