Theorem erdsze 36168

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	⊢ (𝜑 → 𝑁 ∈ ℕ)
erdsze.f	⊢ (𝜑 → 𝐹:(1...𝑁)⟶1-1ℝ)
erdsze.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
erdsze.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
erdsze.l	⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)

Assertion

Ref	Expression
erdsze	⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))

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.

Step	Hyp	Ref	Expression
1		erdsze.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		erdsze.f	. 2 ⊢ (𝜑 → 𝐹:(1...𝑁)⟶1-1ℝ)
3		reseq2 5979	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
4		isoeq1 7354	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦) → ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑤, (𝐹 “ 𝑤))))
5	3, 4	syl 18	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑤, (𝐹 “ 𝑤))))
6		isoeq4 7357	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑤))))
7		imaeq2 6064	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹 “ 𝑤) = (𝐹 “ 𝑦))
8		isoeq5 7358	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑤) = (𝐹 “ 𝑦) → ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦))))
9	7, 8	syl 18	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦))))
10	5, 6, 9	3bitrd 308	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦))))
11		elequ2 2162	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
12	10, 11	anbi12d 644	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤) ↔ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)))
13	12	cbvrabv 3426	. . . . . 6 ⊢ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)} = {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)}
14		oveq2 7457	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (1...𝑧) = (1...𝑥))
15	14	pweqd 4577	. . . . . . 7 ⊢ (𝑧 = 𝑥 → 𝒫 (1...𝑧) = 𝒫 (1...𝑥))
16		elequ1 2154	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
17	16	anbi2d 642	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦) ↔ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)))
18	15, 17	rabeqbidv 3433	. . . . . 6 ⊢ (𝑧 = 𝑥 → {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)})
19	13, 18	eqtrid 2810	. . . . 5 ⊢ (𝑧 = 𝑥 → {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)})
20	19	imaeq2d 6068	. . . 4 ⊢ (𝑧 = 𝑥 → (♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)}) = (♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}))
21	20	supeq1d 9473	. . 3 ⊢ (𝑧 = 𝑥 → sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)}), ℝ, < ) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
22	21	cbvmptv 5213	. 2 ⊢ (𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)}), ℝ, < )) = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
23		isoeq1 7354	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦) → ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤))))
24	3, 23	syl 18	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤))))
25		isoeq4 7357	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑤))))
26		isoeq5 7358	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑤) = (𝐹 “ 𝑦) → ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦))))
27	7, 26	syl 18	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦))))
28	24, 25, 27	3bitrd 308	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦))))
29	28, 11	anbi12d 644	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤) ↔ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)))
30	29	cbvrabv 3426	. . . . . 6 ⊢ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)} = {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)}
31	16	anbi2d 642	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦) ↔ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)))
32	15, 31	rabeqbidv 3433	. . . . . 6 ⊢ (𝑧 = 𝑥 → {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)})
33	30, 32	eqtrid 2810	. . . . 5 ⊢ (𝑧 = 𝑥 → {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)} = {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)})
34	33	imaeq2d 6068	. . . 4 ⊢ (𝑧 = 𝑥 → (♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)}) = (♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}))
35	34	supeq1d 9473	. . 3 ⊢ (𝑧 = 𝑥 → sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)}), ℝ, < ) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
36	35	cbvmptv 5213	. 2 ⊢ (𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)}), ℝ, < )) = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
37		eqid 2763	. 2 ⊢ (𝑛 ∈ (1...𝑁) ↦ ⟨((𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)}), ℝ, < ))‘𝑛), ((𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)}), ℝ, < ))‘𝑛)⟩) = (𝑛 ∈ (1...𝑁) ↦ ⟨((𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)}), ℝ, < ))‘𝑛), ((𝑧 ∈ (1...𝑁) ↦ sup((♯ “ {𝑤 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤)}), ℝ, < ))‘𝑛)⟩)
38		erdsze.r	. 2 ⊢ (𝜑 → 𝑅 ∈ ℕ)
39		erdsze.s	. 2 ⊢ (𝜑 → 𝑆 ∈ ℕ)
40		erdsze.l	. 2 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)
41	1, 2, 22, 36, 37, 38, 39, 40	erdszelem11 36167	1 ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416  𝒫 cpw 4560  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5190  ^◡ccnv 5655   ↾ cres 5658   “ cima 5659  ⟶1-1wf1 6561  ‘cfv 6564   Isom wiso 6565  (class class class)co 7449  supcsup 9467  ℝcr 11236  1c1 11238   · cmul 11242   < clt 11380   ≤ cle 11381   − cmin 11578  ℕcn 12373  ...cfz 13677  ♯chash 14513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314  ax-pre-sup 11315

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-isom 6573  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-oadd 8516  df-er 8753  df-en 9010  df-dom 9011  df-sdom 9012  df-fin 9013  df-sup 9469  df-dju 10017  df-card 10055  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-nn 12374  df-n0 12645  df-xnn0 12718  df-z 12732  df-uz 13004  df-fz 13678  df-hash 14514

This theorem is used by:  erdsze2lem2  36170