Theorem bcth 24493

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 24492 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 764	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → 𝐷 ∈ (CMet‘𝑋))
3		eleq1w 2821	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑋 ↔ 𝑦 ∈ 𝑋))
4		eleq1w 2821	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑚 → (𝑟 ∈ ℝ+ ↔ 𝑚 ∈ ℝ+))
5	3, 4	bi2anan9 636	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ↔ (𝑦 ∈ 𝑋 ∧ 𝑚 ∈ ℝ+)))
6		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → 𝑟 = 𝑚)
7	6	breq1d 5084	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → (𝑟 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑘)))
8		oveq12 7284	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → (𝑥(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)𝑚))
9	8	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) = ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)))
10	9	sseq1d 3952	. . . . . . . . . . 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 631	. . . . . . . . . 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 631	. . . . . . . . 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 5147	. . . . . . . 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 7283	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
15	14	breq2d 5086	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝑚 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑛)))
16		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑀‘𝑘) = (𝑀‘𝑛))
17	16	difeq2d 4057	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)))
18	17	sseq2d 3953	. . . . . . . . . . 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 631	. . . . . . . . . 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 629	. . . . . . . . 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 5140	. . . . . . . 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	eqtrid 2790	. . . . . . 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 6774	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑔 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝑔))
24	23	difeq1d 4056	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑔 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)) = (((ball‘𝐷)‘𝑔) ∖ (𝑀‘𝑛)))
25	24	sseq2d 3953	. . . . . . . . . 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 629	. . . . . . . . 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 629	. . . . . . . 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 5140	. . . . . . 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 7370	. . . . . 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 766	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → 𝑀:ℕ⟶(Clsd‘𝐽))
31		simpr 485	. . . . . . 7 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
32	16	fveqeq2d 6782	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ↔ ((int‘𝐽)‘(𝑀‘𝑛)) = ∅))
33	32	cbvralvw 3383	. . . . . . 7 ⊢ (∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ↔ ∀𝑛 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑛)) = ∅)
34	31, 33	sylib 217	. . . . . 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 24492	. . . . 5 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → ((int‘𝐽)‘∪ ran 𝑀) = ∅)
36	35	ex 413	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ → ((int‘𝐽)‘∪ ran 𝑀) = ∅))
37	36	necon3ad 2956	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (((int‘𝐽)‘∪ ran 𝑀) ≠ ∅ → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅))
38	37	3impia 1116	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘∪ ran 𝑀) ≠ ∅) → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
39		df-ne 2944	. . . 4 ⊢ (((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
40	39	rexbii 3181	. . 3 ⊢ (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
41		rexnal 3169	. . 3 ⊢ (∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ↔ ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
42	40, 41	bitri 274	. 2 ⊢ (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
43	38, 42	sylibr 233	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 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839   class class class wbr 5074  {copab 5136   × cxp 5587  ran crn 5590  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  1c1 10872   < clt 11009   / cdiv 11632  ℕcn 11973  ℝ⁺crp 12730  ballcbl 20584  MetOpencmopn 20587  Clsdccld 22167  intcnt 22168  clsccl 22169  CMetccmet 24418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-dc 10202  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ico 13085  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-top 22043  df-topon 22060  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lm 22380  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-cfil 24419  df-cau 24420  df-cmet 24421

This theorem is referenced by:  bcth2  24494  bcth3  24495