Theorem erdsze 35207

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 5992	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
4		isoeq1 7337	. . . . . . . . . 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 7340	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑤))))
7		imaeq2 6074	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹 “ 𝑤) = (𝐹 “ 𝑦))
8		isoeq5 7341	. . . . . . . . . 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 2123	. . . . . . . 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 3447	. . . . . 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 7439	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (1...𝑧) = (1...𝑥))
15	14	pweqd 4617	. . . . . . 7 ⊢ (𝑧 = 𝑥 → 𝒫 (1...𝑧) = 𝒫 (1...𝑥))
16		elequ1 2115	. . . . . . . 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 3455	. . . . . 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 2789	. . . . 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 6078	. . . 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 9486	. . 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 5255	. 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 7337	. . . . . . . . . 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 7340	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑤, (𝐹 “ 𝑤)) ↔ (𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑤))))
26		isoeq5 7341	. . . . . . . . . 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 3447	. . . . . 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 3455	. . . . . 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 2789	. . . . 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 6078	. . . 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 9486	. . 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 5255	. 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 2737	. 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 35206	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 848   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  {crab 3436  𝒫 cpw 4600  ⟨cop 4632   class class class wbr 5143   ↦ cmpt 5225  ^◡ccnv 5684   ↾ cres 5687   “ cima 5688  –1-1→wf1 6558  ‘cfv 6561   Isom wiso 6562  (class class class)co 7431  supcsup 9480  ℝcr 11154  1c1 11156   · cmul 11160   < clt 11295   ≤ cle 11296   − cmin 11492  ℕcn 12266  ...cfz 13547  ♯chash 14369

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370

This theorem is referenced by:  erdsze2lem2  35209