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Theorem bcth 22851
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 22850 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 785 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅) → 𝐷 ∈ (CMet‘𝑋))
3 eleq1 2675 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥𝑋𝑦𝑋))
4 eleq1 2675 . . . . . . . . . . 11 (𝑟 = 𝑚 → (𝑟 ∈ ℝ+𝑚 ∈ ℝ+))
53, 4bi2anan9 912 . . . . . . . . . 10 ((𝑥 = 𝑦𝑟 = 𝑚) → ((𝑥𝑋𝑟 ∈ ℝ+) ↔ (𝑦𝑋𝑚 ∈ ℝ+)))
6 simpr 475 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑟 = 𝑚) → 𝑟 = 𝑚)
76breq1d 4587 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑟 = 𝑚) → (𝑟 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑘)))
8 oveq12 6536 . . . . . . . . . . . . 13 ((𝑥 = 𝑦𝑟 = 𝑚) → (𝑥(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)𝑚))
98fveq2d 6092 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑟 = 𝑚) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) = ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)))
109sseq1d 3594 . . . . . . . . . . 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 742 . . . . . . . . . 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 742 . . . . . . . . 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 4648 . . . . . . . 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 6535 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
1514breq2d 4589 . . . . . . . . . . 11 (𝑘 = 𝑛 → (𝑚 < (1 / 𝑘) ↔ 𝑚 < (1 / 𝑛)))
16 fveq2 6088 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑀𝑘) = (𝑀𝑛))
1716difeq2d 3689 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛)))
1817sseq2d 3595 . . . . . . . . . . 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 742 . . . . . . . . . 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 735 . . . . . . . . 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 4642 . . . . . . . 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 2655 . . . . . . 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 6088 . . . . . . . . . . . 12 (𝑧 = 𝑔 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝑔))
2423difeq1d 3688 . . . . . . . . . . 11 (𝑧 = 𝑔 → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛)) = (((ball‘𝐷)‘𝑔) ∖ (𝑀𝑛)))
2524sseq2d 3595 . . . . . . . . . 10 (𝑧 = 𝑔 → (((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑛)) ↔ ((cls‘𝐽)‘(𝑦(ball‘𝐷)𝑚)) ⊆ (((ball‘𝐷)‘𝑔) ∖ (𝑀𝑛))))
2625anbi2d 735 . . . . . . . . 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 735 . . . . . . . 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 4642 . . . . . . 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, 28cbvmpt2v 6611 . . . . . 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 787 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅) → 𝑀:ℕ⟶(Clsd‘𝐽))
31 simpr 475 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅) → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
3216fveq2d 6092 . . . . . . . . 9 (𝑘 = 𝑛 → ((int‘𝐽)‘(𝑀𝑘)) = ((int‘𝐽)‘(𝑀𝑛)))
3332eqeq1d 2611 . . . . . . . 8 (𝑘 = 𝑛 → (((int‘𝐽)‘(𝑀𝑘)) = ∅ ↔ ((int‘𝐽)‘(𝑀𝑛)) = ∅))
3433cbvralv 3146 . . . . . . 7 (∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅ ↔ ∀𝑛 ∈ ℕ ((int‘𝐽)‘(𝑀𝑛)) = ∅)
3531, 34sylib 206 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅) → ∀𝑛 ∈ ℕ ((int‘𝐽)‘(𝑀𝑛)) = ∅)
361, 2, 29, 30, 35bcthlem5 22850 . . . . 5 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) ∧ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅) → ((int‘𝐽)‘ ran 𝑀) = ∅)
3736ex 448 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅ → ((int‘𝐽)‘ ran 𝑀) = ∅))
3837necon3ad 2794 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽)) → (((int‘𝐽)‘ ran 𝑀) ≠ ∅ → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅))
39383impia 1252 . 2 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘ ran 𝑀) ≠ ∅) → ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
40 df-ne 2781 . . . 4 (((int‘𝐽)‘(𝑀𝑘)) ≠ ∅ ↔ ¬ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
4140rexbii 3022 . . 3 (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅ ↔ ∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
42 rexnal 2977 . . 3 (∃𝑘 ∈ ℕ ¬ ((int‘𝐽)‘(𝑀𝑘)) = ∅ ↔ ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
4341, 42bitri 262 . 2 (∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅ ↔ ¬ ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
4439, 43sylibr 222 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 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  cdif 3536  wss 3539  c0 3873   cuni 4366   class class class wbr 4577  {copab 4636   × cxp 5026  ran crn 5029  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  1c1 9793   < clt 9930   / cdiv 10533  cn 10867  +crp 11664  ballcbl 19500  MetOpencmopn 19503  Clsdccld 20572  intcnt 20573  clsccl 20574  CMetcms 22778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-dc 9128  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-sup 8208  df-inf 8209  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-ico 12008  df-rest 15852  df-topgen 15873  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-fbas 19510  df-fg 19511  df-top 20463  df-bases 20464  df-topon 20465  df-cld 20575  df-ntr 20576  df-cls 20577  df-nei 20654  df-lm 20785  df-fil 21402  df-fm 21494  df-flim 21495  df-flf 21496  df-cfil 22779  df-cau 22780  df-cmet 22781
This theorem is referenced by:  bcth2  22852  bcth3  22853
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