Theorem erdsze 35513

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 5956	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
4		isoeq1 7296	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦) → ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑤, (𝐹 “ 𝑤))))
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑤, (𝐹 “ 𝑤))))
6		isoeq4 7299	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑤))))
7		imaeq2 6041	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹 “ 𝑤) = (𝐹 “ 𝑦))
8		isoeq5 7300	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑤) = (𝐹 “ 𝑦) → ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦))))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦))))
10	5, 6, 9	3bitrd 307	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦))))
11		elequ2 2156	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
12	10, 11	anbi12d 641	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤) ↔ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)))
13	12	cbvrabv 3423	. . . . . 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 7399	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (1...𝑧) = (1...𝑥))
15	14	pweqd 4569	. . . . . . 7 ⊢ (𝑧 = 𝑥 → 𝒫 (1...𝑧) = 𝒫 (1...𝑥))
16		elequ1 2148	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
17	16	anbi2d 639	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦) ↔ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)))
18	15, 17	rabeqbidv 3431	. . . . . 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 2808	. . . . 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 6045	. . . 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 9386	. . 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 5201	. 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 7296	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦) → ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤))))
24	3, 23	syl 17	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤))))
25		isoeq4 7299	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑤))))
26		isoeq5 7300	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑤) = (𝐹 “ 𝑦) → ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦))))
27	7, 26	syl 17	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦))))
28	24, 25, 27	3bitrd 307	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦))))
29	28, 11	anbi12d 641	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤) ↔ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)))
30	29	cbvrabv 3423	. . . . . 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 639	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦) ↔ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)))
32	15, 31	rabeqbidv 3431	. . . . . 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 2808	. . . . 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 6045	. . . 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 9386	. . 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 5201	. 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 2761	. 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 35512	1 ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  {crab 3413  𝒫 cpw 4552  ⟨cop 4585   class class class wbr 5097   ↦ cmpt 5178  ^◡ccnv 5642   ↾ cres 5645   “ cima 5646  –1-1→wf1 6513  ‘cfv 6516   Isom wiso 6517  (class class class)co 7391  supcsup 9380  ℝcr 11066  1c1 11068   · cmul 11072   < clt 11210   ≤ cle 11211   − cmin 11408  ℕcn 12204  ...cfz 13506  ♯chash 14337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144  ax-pre-sup 11145

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-oadd 8435  df-er 8672  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9382  df-dju 9853  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-n0 12476  df-xnn0 12549  df-z 12563  df-uz 12834  df-fz 13507  df-hash 14338

This theorem is referenced by:  erdsze2lem2  35515