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Theorem erdsze 32908
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 5864 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
4 isoeq1 7148 . . . . . . . . . 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 7151 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝐹𝑦) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑤))))
7 imaeq2 5943 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
8 isoeq5 7152 . . . . . . . . . 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 308 . . . . . . . 8 (𝑤 = 𝑦 → ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦))))
11 elequ2 2127 . . . . . . . 8 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
1210, 11anbi12d 634 . . . . . . 7 (𝑤 = 𝑦 → (((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)))
1312cbvrabv 3417 . . . . . 6 {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)}
14 oveq2 7243 . . . . . . . 8 (𝑧 = 𝑥 → (1...𝑧) = (1...𝑥))
1514pweqd 4549 . . . . . . 7 (𝑧 = 𝑥 → 𝒫 (1...𝑧) = 𝒫 (1...𝑥))
16 elequ1 2119 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
1716anbi2d 632 . . . . . . 7 (𝑧 = 𝑥 → (((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)))
1815, 17rabeqbidv 3411 . . . . . 6 (𝑧 = 𝑥 → {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
1913, 18syl5eq 2792 . . . . 5 (𝑧 = 𝑥 → {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
2019imaeq2d 5947 . . . 4 (𝑧 = 𝑥 → (♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}) = (♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}))
2120supeq1d 9092 . . 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 5175 . 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 7148 . . . . . . . . . 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 7151 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝐹𝑦) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑤))))
26 isoeq5 7152 . . . . . . . . . 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 308 . . . . . . . 8 (𝑤 = 𝑦 → ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ↔ (𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦))))
2928, 11anbi12d 634 . . . . . . 7 (𝑤 = 𝑦 → (((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)))
3029cbvrabv 3417 . . . . . 6 {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)}
3116anbi2d 632 . . . . . . 7 (𝑧 = 𝑥 → (((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦) ↔ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)))
3215, 31rabeqbidv 3411 . . . . . 6 (𝑧 = 𝑥 → {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑧𝑦)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
3330, 32syl5eq 2792 . . . . 5 (𝑧 = 𝑥 → {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)})
3433imaeq2d 5947 . . . 4 (𝑧 = 𝑥 → (♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹𝑤) Isom < , < (𝑤, (𝐹𝑤)) ∧ 𝑧𝑤)}) = (♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}))
3534supeq1d 9092 . . 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 5175 . 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 2739 . 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 32907 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 209  wa 399  wo 847   = wceq 1543  wcel 2112  wrex 3065  {crab 3068  𝒫 cpw 4530  cop 4564   class class class wbr 5070  cmpt 5152  ccnv 5568  cres 5571  cima 5572  1-1wf1 6398  cfv 6401   Isom wiso 6402  (class class class)co 7235  supcsup 9086  cr 10758  1c1 10760   · cmul 10764   < clt 10897  cle 10898  cmin 11092  cn 11860  ...cfz 13125  chash 13929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5196  ax-sep 5209  ax-nul 5216  ax-pow 5275  ax-pr 5339  ax-un 7545  ax-cnex 10815  ax-resscn 10816  ax-1cn 10817  ax-icn 10818  ax-addcl 10819  ax-addrcl 10820  ax-mulcl 10821  ax-mulrcl 10822  ax-mulcom 10823  ax-addass 10824  ax-mulass 10825  ax-distr 10826  ax-i2m1 10827  ax-1ne0 10828  ax-1rid 10829  ax-rnegex 10830  ax-rrecex 10831  ax-cnre 10832  ax-pre-lttri 10833  ax-pre-lttrn 10834  ax-pre-ltadd 10835  ax-pre-mulgt0 10836  ax-pre-sup 10837
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5153  df-tr 5179  df-id 5472  df-eprel 5478  df-po 5486  df-so 5487  df-fr 5527  df-we 5529  df-xp 5575  df-rel 5576  df-cnv 5577  df-co 5578  df-dm 5579  df-rn 5580  df-res 5581  df-ima 5582  df-pred 6179  df-ord 6237  df-on 6238  df-lim 6239  df-suc 6240  df-iota 6359  df-fun 6403  df-fn 6404  df-f 6405  df-f1 6406  df-fo 6407  df-f1o 6408  df-fv 6409  df-isom 6410  df-riota 7192  df-ov 7238  df-oprab 7239  df-mpo 7240  df-om 7667  df-1st 7783  df-2nd 7784  df-wrecs 8071  df-recs 8132  df-rdg 8170  df-1o 8226  df-oadd 8230  df-er 8415  df-en 8651  df-dom 8652  df-sdom 8653  df-fin 8654  df-sup 9088  df-dju 9547  df-card 9585  df-pnf 10899  df-mnf 10900  df-xr 10901  df-ltxr 10902  df-le 10903  df-sub 11094  df-neg 11095  df-nn 11861  df-n0 12121  df-xnn0 12193  df-z 12207  df-uz 12469  df-fz 13126  df-hash 13930
This theorem is referenced by:  erdsze2lem2  32910
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