Theorem bcth 25363

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 25362 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 767	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → 𝐷 ∈ (CMet‘𝑋))
3		eleq1w 2824	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑋 ↔ 𝑦 ∈ 𝑋))
4		eleq1w 2824	. . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → 𝑟 = 𝑚)
7	6	breq1d 5153	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → (𝑟 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑘)))
8		oveq12 7440	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → (𝑥(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)𝑚))
9	8	fveq2d 6910	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) = ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)))
10	9	sseq1d 4015	. . . . . . . . . . 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 632	. . . . . . . . . 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 632	. . . . . . . . 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 5216	. . . . . . . 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 7439	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
15	14	breq2d 5155	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝑚 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑛)))
16		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑀‘𝑘) = (𝑀‘𝑛))
17	16	difeq2d 4126	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)))
18	17	sseq2d 4016	. . . . . . . . . . 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 632	. . . . . . . . . 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 630	. . . . . . . . 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 5209	. . . . . . . 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 2789	. . . . . . 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 6906	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑔 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝑔))
24	23	difeq1d 4125	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑔 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)) = (((ball‘𝐷)‘𝑔) ∖ (𝑀‘𝑛)))
25	24	sseq2d 4016	. . . . . . . . . 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 630	. . . . . . . . 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 630	. . . . . . . 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 5209	. . . . . . 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 7528	. . . . . 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 769	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → 𝑀:ℕ⟶(Clsd‘𝐽))
31		simpr 484	. . . . . . 7 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
32	16	fveqeq2d 6914	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ↔ ((int‘𝐽)‘(𝑀‘𝑛)) = ∅))
33	32	cbvralvw 3237	. . . . . . 7 ⊢ (∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ↔ ∀𝑛 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑛)) = ∅)
34	31, 33	sylib 218	. . . . . 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 25362	. . . . 5 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → ((int‘𝐽)‘∪ ran 𝑀) = ∅)
36	35	ex 412	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ → ((int‘𝐽)‘∪ ran 𝑀) = ∅))
37	36	necon3ad 2953	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (((int‘𝐽)‘∪ ran 𝑀) ≠ ∅ → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅))
38	37	3impia 1118	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘∪ ran 𝑀) ≠ ∅) → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
39		df-ne 2941	. . . 4 ⊢ (((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
40	39	rexbii 3094	. . 3 ⊢ (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
41		rexnal 3100	. . 3 ⊢ (∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ↔ ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
42	40, 41	bitri 275	. 2 ⊢ (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
43	38, 42	sylibr 234	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 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  ∪ cuni 4907   class class class wbr 5143  {copab 5205   × cxp 5683  ran crn 5686  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  1c1 11156   < clt 11295   / cdiv 11920  ℕcn 12266  ℝ⁺crp 13034  ballcbl 21351  MetOpencmopn 21354  Clsdccld 23024  intcnt 23025  clsccl 23026  CMetccmet 25288

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-inf2 9681  ax-dc 10486  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-iin 4994  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-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-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ico 13393  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lm 23237  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-cfil 25289  df-cau 25290  df-cmet 25291

This theorem is referenced by:  bcth2  25364  bcth3  25365