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Theorem bcth 23859
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 𝑀𝑘 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 23858 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)‘(𝑀𝑘)) ≠ ∅)
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.
StepHypRef Expression
1 bcth.2 . . . . . 6 𝐽 = (MetOpen‘𝐷)
2 simpll 763 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅) → 𝐷 ∈ (CMet‘𝑋))
3 eleq1w 2892 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥𝑋𝑦𝑋))
4 eleq1w 2892 . . . . . . . . . . 11 (𝑟 = 𝑚 → (𝑟 ∈ ℝ+𝑚 ∈ ℝ+))
53, 4bi2anan9 635 . . . . . . . . . 10 ((𝑥 = 𝑦𝑟 = 𝑚) → ((𝑥𝑋𝑟 ∈ ℝ+) ↔ (𝑦𝑋𝑚 ∈ ℝ+)))
6 simpr 485 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑟 = 𝑚) → 𝑟 = 𝑚)
76breq1d 5067 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑟 = 𝑚) → (𝑟 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑘)))
8 oveq12 7154 . . . . . . . . . . . . 13 ((𝑥 = 𝑦𝑟 = 𝑚) → (𝑥(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)𝑚))
98fveq2d 6667 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑟 = 𝑚) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) = ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)))
109sseq1d 3995 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑟 = 𝑚) → (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ↔ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))
117, 10anbi12d 630 . . . . . . . . . 10 ((𝑥 = 𝑦𝑟 = 𝑚) → ((𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) ↔ (𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))))
125, 11anbi12d 630 . . . . . . . . 9 ((𝑥 = 𝑦𝑟 = 𝑚) → (((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))) ↔ ((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))))
1312cbvopabv 5129 . . . . . . . 8 {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} = {⟨𝑦, 𝑚⟩ ∣ ((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))}
14 oveq2 7153 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
1514breq2d 5069 . . . . . . . . . . 11 (𝑘 = 𝑛 → (𝑚 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑛)))
16 fveq2 6663 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑀𝑘) = (𝑀𝑛))
1716difeq2d 4096 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛)))
1817sseq2d 3996 . . . . . . . . . . 11 (𝑘 = 𝑛 → (((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ↔ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛))))
1915, 18anbi12d 630 . . . . . . . . . 10 (𝑘 = 𝑛 → ((𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) ↔ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛)))))
2019anbi2d 628 . . . . . . . . 9 (𝑘 = 𝑛 → (((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))) ↔ ((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛))))))
2120opabbidv 5123 . . . . . . . 8 (𝑘 = 𝑛 → {⟨𝑦, 𝑚⟩ ∣ ((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} = {⟨𝑦, 𝑚⟩ ∣ ((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛))))})
2213, 21syl5eq 2865 . . . . . . 7 (𝑘 = 𝑛 → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} = {⟨𝑦, 𝑚⟩ ∣ ((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛))))})
23 fveq2 6663 . . . . . . . . . . . 12 (𝑧 = 𝑔 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝑔))
2423difeq1d 4095 . . . . . . . . . . 11 (𝑧 = 𝑔 → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛)) = (((ball‘𝐷)‘𝑔) ∖ (𝑀𝑛)))
2524sseq2d 3996 . . . . . . . . . 10 (𝑧 = 𝑔 → (((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛)) ↔ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀𝑛))))
2625anbi2d 628 . . . . . . . . 9 (𝑧 = 𝑔 → ((𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛))) ↔ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀𝑛)))))
2726anbi2d 628 . . . . . . . 8 (𝑧 = 𝑔 → (((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛)))) ↔ ((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀𝑛))))))
2827opabbidv 5123 . . . . . . 7 (𝑧 = 𝑔 → {⟨𝑦, 𝑚⟩ ∣ ((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛))))} = {⟨𝑦, 𝑚⟩ ∣ ((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀𝑛))))})
2922, 28cbvmpov 7238 . . . . . 6 (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))}) = (𝑛 ∈ ℕ, 𝑔 ∈ (𝑋 × ℝ+) ↦ {⟨𝑦, 𝑚⟩ ∣ ((𝑦𝑋𝑚 ∈ ℝ+) ∧ (𝑚 < (1 / 𝑛) ∧ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀𝑛))))})
30 simplr 765 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅) → 𝑀:ℕ⟶(Clsd‘𝐽))
31 simpr 485 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅) → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
3216fveqeq2d 6671 . . . . . . . 8 (𝑘 = 𝑛 → (((int‘𝐽)‘(𝑀𝑘)) = ∅ ↔ ((int‘𝐽)‘(𝑀𝑛)) = ∅))
3332cbvralvw 3447 . . . . . . 7 (∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅ ↔ ∀𝑛 ∈ ℕ ((int‘𝐽)‘(𝑀𝑛)) = ∅)
3431, 33sylib 219 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅) → ∀𝑛 ∈ ℕ ((int‘𝐽)‘(𝑀𝑛)) = ∅)
351, 2, 29, 30, 34bcthlem5 23858 . . . . 5 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅) → ((int‘𝐽)‘ ran 𝑀) = ∅)
3635ex 413 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅ → ((int‘𝐽)‘ ran 𝑀) = ∅))
3736necon3ad 3026 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (((int‘𝐽)‘ ran 𝑀) ≠ ∅ → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅))
38373impia 1109 . 2 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘ ran 𝑀) ≠ ∅) → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
39 df-ne 3014 . . . 4 (((int‘𝐽)‘(𝑀𝑘)) ≠ ∅ ↔ ¬ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
4039rexbii 3244 . . 3 (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅ ↔ ∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
41 rexnal 3235 . . 3 (∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀𝑘)) = ∅ ↔ ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
4240, 41bitri 276 . 2 (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅ ↔ ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
4338, 42sylibr 235 1 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘ ran 𝑀) ≠ ∅) → ∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  cdif 3930  wss 3933  c0 4288   cuni 4830   class class class wbr 5057  {copab 5119   × cxp 5546  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  1c1 10526   < clt 10663   / cdiv 11285  cn 11626  +crp 12377  ballcbl 20460  MetOpencmopn 20463  Clsdccld 21552  intcnt 21553  clsccl 21554  CMetccmet 23784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-dc 9856  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  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 16684  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-top 21430  df-topon 21447  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-lm 21765  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-cfil 23785  df-cau 23786  df-cmet 23787
This theorem is referenced by:  bcth2  23860  bcth3  23861
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