Theorem erdsze 35177

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 5929	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
4		isoeq1 7258	. . . . . . . . . 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 7261	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑤))))
7		imaeq2 6011	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹 “ 𝑤) = (𝐹 “ 𝑦))
8		isoeq5 7262	. . . . . . . . . 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 305	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦))))
11		elequ2 2124	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
12	10, 11	anbi12d 632	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ 𝑤) Isom < , < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤) ↔ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)))
13	12	cbvrabv 3407	. . . . . 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 7361	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (1...𝑧) = (1...𝑥))
15	14	pweqd 4570	. . . . . . 7 ⊢ (𝑧 = 𝑥 → 𝒫 (1...𝑧) = 𝒫 (1...𝑥))
16		elequ1 2116	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
17	16	anbi2d 630	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦) ↔ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)))
18	15, 17	rabeqbidv 3415	. . . . . 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 2776	. . . . 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 6015	. . . 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 9355	. . 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 5199	. 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 7258	. . . . . . . . . 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 7261	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑤))))
26		isoeq5 7262	. . . . . . . . . 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 305	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦))))
29	28, 11	anbi12d 632	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ 𝑤) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ∧ 𝑧 ∈ 𝑤) ↔ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)))
30	29	cbvrabv 3407	. . . . . 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 630	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦) ↔ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)))
32	15, 31	rabeqbidv 3415	. . . . . 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 2776	. . . . 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 6015	. . . 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 9355	. . 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 5199	. 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 2729	. 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 35176	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 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3396  𝒫 cpw 4553  ⟨cop 4585   class class class wbr 5095   ↦ cmpt 5176  ^◡ccnv 5622   ↾ cres 5625   “ cima 5626  –1-1→wf1 6483  ‘cfv 6486   Isom wiso 6487  (class class class)co 7353  supcsup 9349  ℝcr 11027  1c1 11029   · cmul 11033   < clt 11168   ≤ cle 11169   − cmin 11365  ℕcn 12146  ...cfz 13428  ♯chash 14255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-xnn0 12476  df-z 12490  df-uz 12754  df-fz 13429  df-hash 14256

This theorem is referenced by:  erdsze2lem2  35179