Theorem bcth 25281

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 25280 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 2817	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑋 ↔ 𝑦 ∈ 𝑋))
4		eleq1w 2817	. . . . . . . . . . 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 5129	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → (𝑟 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑘)))
8		oveq12 7414	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → (𝑥(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)𝑚))
9	8	fveq2d 6880	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑟 = 𝑚) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) = ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)))
10	9	sseq1d 3990	. . . . . . . . . . 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 5192	. . . . . . . 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 7413	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
15	14	breq2d 5131	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝑚 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑛)))
16		fveq2 6876	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑀‘𝑘) = (𝑀‘𝑛))
17	16	difeq2d 4101	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)))
18	17	sseq2d 3991	. . . . . . . . . . 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 5185	. . . . . . . 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 2782	. . . . . . 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 6876	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑔 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝑔))
24	23	difeq1d 4100	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑔 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑛)) = (((ball‘𝐷)‘𝑔) ∖ (𝑀‘𝑛)))
25	24	sseq2d 3991	. . . . . . . . . 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 5185	. . . . . . 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 7502	. . . . . 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 484	. . . . . . 7 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
32	16	fveqeq2d 6884	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ↔ ((int‘𝐽)‘(𝑀‘𝑛)) = ∅))
33	32	cbvralvw 3220	. . . . . . 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 25280	. . . . 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 2945	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (((int‘𝐽)‘∪ ran 𝑀) ≠ ∅ → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅))
38	37	3impia 1117	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘∪ ran 𝑀) ≠ ∅) → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
39		df-ne 2933	. . . 4 ⊢ (((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
40	39	rexbii 3083	. . 3 ⊢ (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅ ↔ ∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
41		rexnal 3089	. . 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 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  ∪ cuni 4883   class class class wbr 5119  {copab 5181   × cxp 5652  ran crn 5655  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  1c1 11130   < clt 11269   / cdiv 11894  ℕcn 12240  ℝ⁺crp 13008  ballcbl 21302  MetOpencmopn 21305  Clsdccld 22954  intcnt 22955  clsccl 22956  CMetccmet 25206

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655  ax-dc 10460  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-pm 8843  df-en 8960  df-dom 8961  df-sdom 8962  df-sup 9454  df-inf 9455  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-n0 12502  df-z 12589  df-uz 12853  df-q 12965  df-rp 13009  df-xneg 13128  df-xadd 13129  df-xmul 13130  df-ico 13368  df-rest 17436  df-topgen 17457  df-psmet 21307  df-xmet 21308  df-met 21309  df-bl 21310  df-mopn 21311  df-fbas 21312  df-fg 21313  df-top 22832  df-topon 22849  df-bases 22884  df-cld 22957  df-ntr 22958  df-cls 22959  df-nei 23036  df-lm 23167  df-fil 23784  df-fm 23876  df-flim 23877  df-flf 23878  df-cfil 25207  df-cau 25208  df-cmet 25209

This theorem is referenced by:  bcth2  25282  bcth3  25283