Theorem erdsze 35707

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)–1-1→ℝ)
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)–1-1→ℝ)
3		reseq2 5975	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
4		isoeq1 7321	. . . . . . . . . 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 7324	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑤))))
7		imaeq2 6060	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹 “ 𝑤) = (𝐹 “ 𝑦))
8		isoeq5 7325	. . . . . . . . . 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 2161	. . . . . . . 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 3428	. . . . . 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 7424	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (1...𝑧) = (1...𝑥))
15	14	pweqd 4581	. . . . . . 7 ⊢ (𝑧 = 𝑥 → 𝒫 (1...𝑧) = 𝒫 (1...𝑥))
16		elequ1 2153	. . . . . . . 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 3436	. . . . . 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 2812	. . . . 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 6064	. . . 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 9409	. . 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 5217	. 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 7321	. . . . . . . . . 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 7324	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑤))))
26		isoeq5 7325	. . . . . . . . . 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 3428	. . . . . 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 3436	. . . . . 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 2812	. . . . 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 6064	. . . 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 9409	. . 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 5217	. 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 2765	. 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 35706	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 2146  ∃wrex 3091  {crab 3418  𝒫 cpw 4564  ⟨cop 4597   class class class wbr 5111   ↦ cmpt 5194  ^◡ccnv 5662   ↾ cres 5665   “ cima 5666  –1-1→wf1 6537  ‘cfv 6540   Isom wiso 6541  (class class class)co 7416  supcsup 9403  ℝcr 11110  1c1 11112   · cmul 11116   < clt 11254   ≤ cle 11255   − cmin 11452  ℕcn 12244  ...cfz 13546  ♯chash 14379

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188  ax-pre-sup 11189

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9405  df-dju 9899  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-n0 12516  df-xnn0 12589  df-z 12603  df-uz 12874  df-fz 13547  df-hash 14380

This theorem is used by:  erdsze2lem2  35709