Theorem bcth 23933

Description: Baire's Category Theorem. If a nonempty metric space is complete, it is nonmeager in itself. In other words, no open set in the metric space can be the countable union of rare closed subsets (where rare means having a closure with empty interior), so some subset 𝑀‘𝑘 must have a nonempty interior. Theorem 4.7-2 of [Kreyszig] p. 247. (The terminology "meager" and "nonmeager" is used by Kreyszig to replace Baire's "of the first category" and "of the second category." The latter terms are going out of favor to avoid confusion with category theory.) See bcthlem5 23932 for an overview of the proof. (Contributed by NM, 28-Oct-2007.) (Proof shortened by Mario Carneiro, 6-Jan-2014.)

Hypothesis

Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)

Assertion

Ref	Expression
bcth	⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘∪ ran 𝑀) ≠ ∅) → ∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅)

Distinct variable groups:   𝐷,𝑘   𝑘,𝐽   𝑘,𝑀   𝑘,𝑋

Proof of Theorem bcth

Dummy variables 𝑛 𝑟 𝑥 𝑧 𝑔 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bcth.2	. . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷)
2		simpll 766	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → 𝐷 ∈ (CMet‘𝑋))
3		eleq1w 2872	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑋 ↔ 𝑦 ∈ 𝑋))
4		eleq1w 2872	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑚 → (𝑟 ∈ ℝ+ ↔ 𝑚 ∈ ℝ+))
5	3, 4	bi2anan9 638	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ↔ (𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+)))
6		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → 𝑟 = 𝑚)
7	6	breq1d 5040	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → (𝑟 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑘)))
8		oveq12 7144	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → (𝑥(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)𝑚))
9	8	fveq2d 6649	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) = ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)))
10	9	sseq1d 3946	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ↔ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))
11	7, 10	anbi12d 633	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → ((𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) ↔ (𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))))
12	5, 11	anbi12d 633	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → (((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))) ↔ ((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))))
13	12	cbvopabv 5102	. . . . . . . 8 ⊢ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} = {⟨𝑦, 𝑚⟩ ∣ ((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))}
14		oveq2 7143	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
15	14	breq2d 5042	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝑚 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑛)))
16		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑀‘𝑘) = (𝑀‘𝑛))
17	16	difeq2d 4050	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)))
18	17	sseq2d 3947	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ↔ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛))))
19	15, 18	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → ((𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) ↔ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)))))
20	19	anbi2d 631	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))) ↔ ((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛))))))
21	20	opabbidv 5096	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → {⟨𝑦, 𝑚⟩ ∣ ((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} = {⟨𝑦, 𝑚⟩ ∣ ((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛))))})
22	13, 21	syl5eq 2845	. . . . . . 7 ⊢ (𝑘 = 𝑛 → {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} = {⟨𝑦, 𝑚⟩ ∣ ((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛))))})
23		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑔 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝑔))
24	23	difeq1d 4049	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑔 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)) = (((ball‘𝐷)‘𝑔) ∖ (𝑀‘𝑛)))
25	24	sseq2d 3947	. . . . . . . . . 10 ⊢ (𝑧 = 𝑔 → (((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)) ↔ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀‘𝑛))))
26	25	anbi2d 631	. . . . . . . . 9 ⊢ (𝑧 = 𝑔 → ((𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛))) ↔ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀‘𝑛)))))
27	26	anbi2d 631	. . . . . . . 8 ⊢ (𝑧 = 𝑔 → (((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)))) ↔ ((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀‘𝑛))))))
28	27	opabbidv 5096	. . . . . . 7 ⊢ (𝑧 = 𝑔 → {⟨𝑦, 𝑚⟩ ∣ ((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛))))} = {⟨𝑦, 𝑚⟩ ∣ ((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀‘𝑛))))})
29	22, 28	cbvmpov 7228	. . . . . 6 ⊢ (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))}) = (𝑛 ∈ ℕ, 𝑔 ∈ (𝑋 × ℝ+) ↦ {⟨𝑦, 𝑚⟩ ∣ ((𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀‘𝑛))))})
30		simplr 768	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → 𝑀:ℕ⟶(Clsd‘𝐽))
31		simpr 488	. . . . . . 7 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
32	16	fveqeq2d 6653	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ↔ ((int‘𝐽)‘(𝑀‘𝑛)) = ∅))
33	32	cbvralvw 3396	. . . . . . 7 ⊢ (∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ↔ ∀𝑛 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑛)) = ∅)
34	31, 33	sylib 221	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → ∀𝑛 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑛)) = ∅)
35	1, 2, 29, 30, 34	bcthlem5 23932	. . . . 5 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → ((int‘𝐽)‘∪ ran 𝑀) = ∅)
36	35	ex 416	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ → ((int‘𝐽)‘∪ ran 𝑀) = ∅))
37	36	necon3ad 3000	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (((int‘𝐽)‘∪ ran 𝑀) ≠ ∅ → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅))
38	37	3impia 1114	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘∪ ran 𝑀) ≠ ∅) → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
39		df-ne 2988	. . . 4 ⊢ (((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
40	39	rexbii 3210	. . 3 ⊢ (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
41		rexnal 3201	. . 3 ⊢ (∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ↔ ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
42	40, 41	bitri 278	. 2 ⊢ (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
43	38, 42	sylibr 237	1 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘∪ ran 𝑀) ≠ ∅) → ∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  ∪ cuni 4800   class class class wbr 5030  {copab 5092   × cxp 5517  ran crn 5520  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1c1 10527   < clt 10664   / cdiv 11286  ℕcn 11625  ℝ⁺crp 12377  ballcbl 20078  MetOpencmopn 20081  Clsdccld 21621  intcnt 21622  clsccl 21623  CMetccmet 23858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-dc 9857  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ico 12732  df-rest 16688  df-topgen 16709  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-fbas 20088  df-fg 20089  df-top 21499  df-topon 21516  df-bases 21551  df-cld 21624  df-ntr 21625  df-cls 21626  df-nei 21703  df-lm 21834  df-fil 22451  df-fm 22543  df-flim 22544  df-flf 22545  df-cfil 23859  df-cau 23860  df-cmet 23861

This theorem is referenced by:  bcth2  23934  bcth3  23935