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Theorem iscau 25324
Description: Express the property "𝐹 is a Cauchy sequence of metric 𝐷". Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition 𝐹 ⊆ (ℂ × 𝑋) allows to use objects more general than sequences when convenient; see the comment in df-lm 23253. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
iscau (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝐹 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝐹𝑘)(ball‘𝐷)𝑥))))
Distinct variable groups:   𝑥,𝑘,𝐷   𝑘,𝐹,𝑥   𝑘,𝑋,𝑥

Proof of Theorem iscau
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 caufval 25323 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (Cau‘𝐷) = {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)})
21eleq2d 2825 . 2 (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)}))
3 reseq1 5994 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ↾ (ℤ𝑘)) = (𝐹 ↾ (ℤ𝑘)))
4 eqidd 2736 . . . . . 6 (𝑓 = 𝐹 → (ℤ𝑘) = (ℤ𝑘))
5 fveq1 6906 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
65oveq1d 7446 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑘)(ball‘𝐷)𝑥) = ((𝐹𝑘)(ball‘𝐷)𝑥))
73, 4, 6feq123d 6726 . . . . 5 (𝑓 = 𝐹 → ((𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥) ↔ (𝐹 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝐹𝑘)(ball‘𝐷)𝑥)))
87rexbidv 3177 . . . 4 (𝑓 = 𝐹 → (∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥) ↔ ∃𝑘 ∈ ℤ (𝐹 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝐹𝑘)(ball‘𝐷)𝑥)))
98ralbidv 3176 . . 3 (𝑓 = 𝐹 → (∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥) ↔ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝐹 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝐹𝑘)(ball‘𝐷)𝑥)))
109elrab 3695 . 2 (𝐹 ∈ {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)} ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝐹 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝐹𝑘)(ball‘𝐷)𝑥)))
112, 10bitrdi 287 1 (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝐹 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝐹𝑘)(ball‘𝐷)𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  cres 5691  wf 6559  cfv 6563  (class class class)co 7431  pm cpm 8866  cc 11151  cz 12611  cuz 12876  +crp 13032  ∞Metcxmet 21367  ballcbl 21369  Cauccau 25301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-xr 11297  df-xmet 21375  df-cau 25304
This theorem is referenced by:  iscau2  25325  caufpm  25330  lmcau  25361
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