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Theorem List for Metamath Proof Explorer - 22701-22800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtchphl 22701 Augmentation of a pre-Hilbert space with a norm does not affect whether it is still a pre-Hilbert space because all the original components are the same. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)       (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil)
 
Theoremtchnmfval 22702* The norm of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑁 = (norm‘𝐺)    &   𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       (𝑊 ∈ Grp → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
 
Theoremtchnmval 22703 The norm of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑁 = (norm‘𝐺)    &   𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ Grp ∧ 𝑋𝑉) → (𝑁𝑋) = (√‘(𝑋 , 𝑋)))
 
Theoremcphtchnm 22704 The norm of a norm-augmented complex pre-Hilbert space is the same as the original norm on it. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑁 = (norm‘𝑊)       (𝑊 ∈ ℂPreHil → 𝑁 = (norm‘𝐺))
 
Theoremtchds 22705 The distance of a pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝐺 = (toℂHil‘𝑊)    &   𝑁 = (norm‘𝐺)    &    = (-g𝑊)       (𝑊 ∈ Grp → (𝑁 ) = (dist‘𝐺))
 
Theoremtchclm 22706 Lemma for tchcph 22711. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))       (𝜑𝑊 ∈ ℂMod)
 
Theoremtchcphlem3 22707 Lemma for tchcph 22711: real closure of an inner product of a vector with itself. (Contributed by Mario Carneiro, 10-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)       ((𝜑𝑋𝑉) → (𝑋 , 𝑋) ∈ ℝ)
 
Theoremipcau2 22708* The Cauchy-Schwarz inequality for a complex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)    &   ((𝜑 ∧ (𝑥𝐾𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝑉) → 0 ≤ (𝑥 , 𝑥))    &   𝐾 = (Base‘𝐹)    &   𝑁 = (norm‘𝐺)    &   𝐶 = ((𝑌 , 𝑋) / (𝑌 , 𝑌))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((𝑁𝑋) · (𝑁𝑌)))
 
Theoremtchcphlem1 22709* Lemma for tchcph 22711: the triangle inequality. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)    &   ((𝜑 ∧ (𝑥𝐾𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝑉) → 0 ≤ (𝑥 , 𝑥))    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (√‘((𝑋 𝑌) , (𝑋 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))
 
Theoremtchcphlem2 22710* Lemma for tchcph 22711: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)    &   ((𝜑 ∧ (𝑥𝐾𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝑉) → 0 ≤ (𝑥 , 𝑥))    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝑉)       (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌))))
 
Theoremtchcph 22711* The standard definition of a norm turns any pre-Hilbert space over a quadratically closed subfield of into a complex pre-Hilbert space (which allows access to a norm, metric, and topology). (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)    &   ((𝜑 ∧ (𝑥𝐾𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝑉) → 0 ≤ (𝑥 , 𝑥))       (𝜑𝐺 ∈ ℂPreHil)
 
Theoremipcau 22712 The Cauchy-Schwarz inequality for a complex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝑋𝑉𝑌𝑉) → (abs‘(𝑋 , 𝑌)) ≤ ((𝑁𝑋) · (𝑁𝑌)))
 
Theoremnmparlem 22713 Lemma for nmpar 22714. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝑁 = (norm‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)       (𝜑 → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 𝐵))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))
 
Theoremnmpar 22714 A complex pre-Hilbert space satisfies the parallelogram law. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 𝐵))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))
 
Theoremipcnlem2 22715 The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐷 = (dist‘𝑊)    &   𝑁 = (norm‘𝑊)    &   𝑇 = ((𝑅 / 2) / ((𝑁𝐴) + 1))    &   𝑈 = ((𝑅 / 2) / ((𝑁𝐵) + 𝑇))    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝐴𝐷𝑋) < 𝑈)    &   (𝜑 → (𝐵𝐷𝑌) < 𝑇)       (𝜑 → (abs‘((𝐴 , 𝐵) − (𝑋 , 𝑌))) < 𝑅)
 
Theoremipcnlem1 22716* The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐷 = (dist‘𝑊)    &   𝑁 = (norm‘𝑊)    &   𝑇 = ((𝑅 / 2) / ((𝑁𝐴) + 1))    &   𝑈 = ((𝑅 / 2) / ((𝑁𝐵) + 𝑇))    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 ∈ ℝ+)       (𝜑 → ∃𝑟 ∈ ℝ+𝑥𝑉𝑦𝑉 (((𝐴𝐷𝑥) < 𝑟 ∧ (𝐵𝐷𝑦) < 𝑟) → (abs‘((𝐴 , 𝐵) − (𝑥 , 𝑦))) < 𝑅))
 
Theoremipcn 22717 The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
, = (·if𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐾 = (TopOpen‘ℂfld)       (𝑊 ∈ ℂPreHil → , ∈ ((𝐽 ×t 𝐽) Cn 𝐾))
 
Theoremcnmpt1ip 22718* Continuity of inner product; analogue of cnmpt12f 21182 which cannot be used directly because ·𝑖 is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐽 = (TopOpen‘𝑊)    &   𝐶 = (TopOpen‘ℂfld)    &    , = (·𝑖𝑊)    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))       (𝜑 → (𝑥𝑋 ↦ (𝐴 , 𝐵)) ∈ (𝐾 Cn 𝐶))
 
Theoremcnmpt2ip 22719* Continuity of inner product; analogue of cnmpt22f 21191 which cannot be used directly because ·𝑖 is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐽 = (TopOpen‘𝑊)    &   𝐶 = (TopOpen‘ℂfld)    &    , = (·𝑖𝑊)    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑𝐿 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 , 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐶))
 
Theoremcsscld 22720 A "closed subspace" in a complex pre-Hilbert space is actually closed in the topology induced by the norm, thus justifying the terminology "closed subspace". (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝑆𝐶) → 𝑆 ∈ (Clsd‘𝐽))
 
Theoremclsocv 22721 The orthogonal complement of the closure of a subset is the same as the orthogonal complement of the subset itself. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑂 = (ocv‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝑆𝑉) → (𝑂‘((cls‘𝐽)‘𝑆)) = (𝑂𝑆))
 
12.5.4  Convergence and completeness
 
Syntaxccfil 22722 Extend class notation with the set of Cauchy filters.
class CauFil
 
Syntaxcca 22723 Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space.
class Cau
 
Syntaxcms 22724 Extend class notation with class of complete metric spaces.
class CMet
 
Definitiondf-cfil 22725* Define the set of Cauchy filters on a metric space. A Cauchy filter is a filter on the set such that for every 0 < 𝑥 there is an element of the filter whose metric diameter is less than 𝑥. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil = (𝑑 ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
 
Definitiondf-cau 22726* Define a function on metric spaces whose value is the set of Cauchy sequences of the space. (Contributed by NM, 8-Sep-2006.)
Cau = (𝑑 ran ∞Met ↦ {𝑓 ∈ (dom dom 𝑑pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ (𝑓 ↾ (ℤ𝑗)):(ℤ𝑗)⟶((𝑓𝑗)(ball‘𝑑)𝑥)})
 
Definitiondf-cmet 22727* Define the class of complete metrics. (Contributed by Mario Carneiro, 1-May-2014.)
CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
 
Theoremlmmbr 22728* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 ⊆ (ℂ × 𝑋) allows us to use objects more general than sequences when convenient; see the comment in df-lm 20746. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥))))
 
Theoremlmmbr2 22729* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 ⊆ (ℂ × 𝑋) allows us to use objects more general than sequences when convenient; see the comment in df-lm 20746. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷𝑃) < 𝑥))))
 
Theoremlmmbr3 22730* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷𝑃) < 𝑥))))
 
Theoremlmmcvg 22731* Convergence property of a converging sequence. (Contributed by NM, 1-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝑅 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐴𝑋 ∧ (𝐴𝐷𝑃) < 𝑅))
 
Theoremlmmbrf 22732* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. This version of lmmbr2 22729 presupposes that 𝐹 is a function. (Contributed by NM, 20-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   (𝜑𝐹:𝑍𝑋)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐴𝐷𝑃) < 𝑥)))
 
Theoremlmnn 22733* A condition that implies convergence. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝐹:ℕ⟶𝑋)    &   ((𝜑𝑘 ∈ ℕ) → ((𝐹𝑘)𝐷𝑃) < (1 / 𝑘))       (𝜑𝐹(⇝𝑡𝐽)𝑃)
 
Theoremcfilfval 22734* The set of Cauchy filters on a metric space. (Contributed by Mario Carneiro, 13-Oct-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
 
Theoremiscfil 22735* The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
 
Theoremiscfil2 22736* The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐹𝑧𝑦𝑤𝑦 (𝑧𝐷𝑤) < 𝑥)))
 
Theoremcfilfil 22737 A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋))
 
Theoremcfili 22738* Property of a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝐹𝑦𝑥𝑧𝑥 (𝑦𝐷𝑧) < 𝑅)
 
Theoremcfil3i 22739* A Cauchy filter contains balls of any pre-chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝐹)
 
Theoremcfilss 22740 A filter finer than a Cauchy filter is Cauchy. (Contributed by Mario Carneiro, 13-Oct-2015.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → 𝐺 ∈ (CauFil‘𝐷))
 
Theoremfgcfil 22741* The Cauchy filter condition for a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+𝑦𝐵𝑧𝑦𝑤𝑦 (𝑧𝐷𝑤) < 𝑥))
 
Theoremfmcfil 22742* The Cauchy filter condition for a filter map. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (((𝑋 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+𝑦𝐵𝑧𝑦𝑤𝑦 ((𝐹𝑧)𝐷(𝐹𝑤)) < 𝑥))
 
Theoremiscfil3 22743* A filter is Cauchy iff it contains a ball of any chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ+𝑥𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)))
 
Theoremcfilfcls 22744 Similar to ultrafilters (uffclsflim 21548), the cluster points and limit points of a Cauchy filter coincide. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)    &   𝑋 = dom dom 𝐷       (𝐹 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))
 
Theoremcaufval 22745* The set of Cauchy sequences on a metric space. (Contributed by NM, 8-Sep-2006.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (Cau‘𝐷) = {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)})
 
Theoremiscau 22746* 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 us to use objects more general than sequences when convenient; see the comment in df-lm 20746. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝐹 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝐹𝑘)(ball‘𝐷)𝑥))))
 
Theoremiscau2 22747* Express the property "𝐹 is a Cauchy sequence of metric 𝐷," using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))))
 
Theoremiscau3 22748* Express the Cauchy sequence property in the more conventional three-quantifier form. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥))))
 
Theoremiscau4 22749* Express the property "𝐹 is a Cauchy sequence of metric 𝐷," using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑗𝑍) → (𝐹𝑗) = 𝐵)       (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥))))
 
Theoremiscauf 22750* Express the property "𝐹 is a Cauchy sequence of metric 𝐷 " presupposing 𝐹 is a function. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑗𝑍) → (𝐹𝑗) = 𝐵)    &   (𝜑𝐹:𝑍𝑋)       (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵𝐷𝐴) < 𝑥))
 
Theoremcaun0 22751 A metric with a Cauchy sequence cannot be empty. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝑋 ≠ ∅)
 
Theoremcaufpm 22752 Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋pm ℂ))
 
Theoremcaucfil 22753 A Cauchy sequence predicate can be expressed in terms of the Cauchy filter predicate for a suitably chosen filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑍 = (ℤ𝑀)    &   𝐿 = ((𝑋 FilMap 𝐹)‘(ℤ𝑍))       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐿 ∈ (CauFil‘𝐷)))
 
Theoremiscmet 22754* The property "𝐷 is a complete metric." meaning all Cauchy filters converge to a point in the space. (Contributed by Mario Carneiro, 1-May-2014.) (Revised by Mario Carneiro, 13-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
 
Theoremcmetcvg 22755 The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅)
 
Theoremcmetmet 22756 A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 29-Jan-2014.)
(𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
 
Theoremcmetmeti 22757 A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)
𝐷 ∈ (CMet‘𝑋)       𝐷 ∈ (Met‘𝑋)
 
Theoremcmetcaulem 22758* Lemma for cmetcau 22759. (Contributed by Mario Carneiro, 14-Oct-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝐹 ∈ (Cau‘𝐷))    &   𝐺 = (𝑥 ∈ ℕ ↦ if(𝑥 ∈ dom 𝐹, (𝐹𝑥), 𝑃))       (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
 
Theoremcmetcau 22759 The convergence of a Cauchy sequence in a complete metric space. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡𝐽))
 
Theoremiscmet3lem3 22760* Lemma for iscmet3 22763. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝑅 ∈ ℝ+) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((1 / 2)↑𝑘) < 𝑅)
 
Theoremiscmet3lem1 22761* Lemma for iscmet3 22763. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆𝑘)∀𝑣 ∈ (𝑆𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))    &   (𝜑 → ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝐹𝑘) ∈ (𝑆𝑛))       (𝜑𝐹 ∈ (Cau‘𝐷))
 
Theoremiscmet3lem2 22762* Lemma for iscmet3 22763. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆𝑘)∀𝑣 ∈ (𝑆𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))    &   (𝜑 → ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝐹𝑘) ∈ (𝑆𝑛))    &   (𝜑𝐺 ∈ (Fil‘𝑋))    &   (𝜑𝑆:ℤ⟶𝐺)    &   (𝜑𝐹 ∈ dom (⇝𝑡𝐽))       (𝜑 → (𝐽 fLim 𝐺) ≠ ∅)
 
Theoremiscmet3 22763* The property "𝐷 is a complete metric" expressed in terms of functions on (or any other upper integer set). Thus, we only have to look at functions on , and not all possible Cauchy filters, to determine completeness. (The proof uses countable choice.) (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 5-May-2014.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))       (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))))
 
Theoremiscmet2 22764 A metric 𝐷 is complete iff all Cauchy sequences converge to a point in the space. The proof uses countable choice. Part of Definition 1.4-3 of [Kreyszig] p. 28. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡𝐽)))
 
Theoremcfilresi 22765 A Cauchy filter on a metric subspace extends to a Cauchy filter in the larger space. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝐹) ∈ (CauFil‘𝐷))
 
Theoremcfilres 22766 Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))))
 
Theoremcaussi 22767 Cauchy sequence on a metric subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
(𝐷 ∈ (∞Met‘𝑋) → (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ⊆ (Cau‘𝐷))
 
Theoremcauss 22768 Cauchy sequence on a metric subspace. (Contributed by NM, 29-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))))
 
Theoremequivcfil 22769* If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), all the 𝐷-Cauchy filters are also 𝐶-Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)
(𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
 
Theoremequivcau 22770* If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), all the 𝐷-Cauchy sequences are also 𝐶-Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)
(𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))
 
Theoremlmle 22771* If the distance from each member of a converging sequence to a given point is less than or equal to a given amount, so is the convergence value. (Contributed by NM, 23-Dec-2007.) (Proof shortened by Mario Carneiro, 1-May-2014.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝑄𝑋)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑘𝑍) → (𝑄𝐷(𝐹𝑘)) ≤ 𝑅)       (𝜑 → (𝑄𝐷𝑃) ≤ 𝑅)
 
Theoremlmclim 22772 Relate a limit on the metric space of complex numbers to our complex number limit notation. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝑍 ⊆ dom 𝐹) → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹𝑃)))
 
Theoremlmclimf 22773 Relate a limit on the metric space of complex numbers to our complex number limit notation. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡𝐽)𝑃𝐹𝑃))
 
Theoremmetelcls 22774* A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 9016. The statement can be generalized to first-countable spaces, not just metrizable spaces. (Contributed by NM, 8-Nov-2007.) (Proof shortened by Mario Carneiro, 1-May-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆𝑋)       (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))
 
Theoremmetcld 22775* A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of [Kreyszig] p. 30. (Contributed by NM, 11-Nov-2007.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ∀𝑥𝑓((𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑥) → 𝑥𝑆)))
 
Theoremmetcld2 22776 A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of [Kreyszig] p. 30. (Contributed by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((⇝𝑡𝐽) “ (𝑆𝑚 ℕ)) ⊆ 𝑆))
 
Theoremcaubl 22777* Sufficient condition to ensure a sequence of nested balls is Cauchy. (Contributed by Mario Carneiro, 18-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
(𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))    &   (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟)       (𝜑 → (1st𝐹) ∈ (Cau‘𝐷))
 
Theoremcaublcls 22778* The convergent point of a sequence of nested balls is in the closures of any of the balls (i.e. it is in the intersection of the closures). Indeed, it is the only point in the intersection because a metric space is Hausdorff, but we don't prove this here. (Contributed by Mario Carneiro, 21-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
(𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))    &   𝐽 = (MetOpen‘𝐷)       ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹𝐴))))
 
Theoremmetcnp4 22779* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. Theorem 14-4.3 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 4-May-2014.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   (𝜑𝐶 ∈ (∞Met‘𝑋))    &   (𝜑𝐷 ∈ (∞Met‘𝑌))    &   (𝜑𝑃𝑋)       (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋𝑓(⇝𝑡𝐽)𝑃) → (𝐹𝑓)(⇝𝑡𝐾)(𝐹𝑃)))))
 
Theoremmetcn4 22780* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous. Theorem 10.3 of [Munkres] p. 128. (Contributed by NM, 13-Jun-2007.) (Revised by Mario Carneiro, 4-May-2014.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   (𝜑𝐶 ∈ (∞Met‘𝑋))    &   (𝜑𝐷 ∈ (∞Met‘𝑌))    &   (𝜑𝐹:𝑋𝑌)       (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ∀𝑥(𝑓(⇝𝑡𝐽)𝑥 → (𝐹𝑓)(⇝𝑡𝐾)(𝐹𝑥)))))
 
Theoremiscmet3i 22781* Properties that determine a complete metric space. (Contributed by NM, 15-Apr-2007.) (Revised by Mario Carneiro, 5-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐷 ∈ (Met‘𝑋)    &   ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝑋) → 𝑓 ∈ dom (⇝𝑡𝐽))       𝐷 ∈ (CMet‘𝑋)
 
Theoremlmcau 22782 Every convergent sequence in a metric space is a Cauchy sequence. Theorem 1.4-5 of [Kreyszig] p. 28. (Contributed by NM, 29-Jan-2008.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → dom (⇝𝑡𝐽) ⊆ (Cau‘𝐷))
 
Theoremflimcfil 22783 Every convergent filter in a metric space is a Cauchy filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝐹)) → 𝐹 ∈ (CauFil‘𝐷))
 
Theoremcmetss 22784 A subspace of a complete metric space is complete iff it is closed in the parent space. Theorem 1.4-7 of [Kreyszig] p. 30. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (CMet‘𝑋) → ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ 𝑌 ∈ (Clsd‘𝐽)))
 
Theoremequivcmet 22785* If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 22770, metss2 22029, this theorem does not have a one-directional form - it is possible for a metric 𝐶 that is strongly finer than the complete metric 𝐷 to be incomplete and vice versa. Consider 𝐷 = the metric on induced by the usual homeomorphism from (0, 1) against the usual metric 𝐶 on and against the discrete metric 𝐸 on . Then both 𝐶 and 𝐸 are complete but 𝐷 is not, and 𝐶 is strongly finer than 𝐷, which is strongly finer than 𝐸. (Contributed by Mario Carneiro, 15-Sep-2015.)
(𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝑆 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐷𝑦) ≤ (𝑆 · (𝑥𝐶𝑦)))       (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋)))
 
Theoremrelcmpcmet 22786* If 𝐷 is a metric space such that all the balls of some fixed size are relatively compact, then 𝐷 is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑𝑥𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)       (𝜑𝐷 ∈ (CMet‘𝑋))
 
Theoremcmpcmet 22787 A compact metric space is complete. One half of heibor 32680. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)       (𝜑𝐷 ∈ (CMet‘𝑋))
 
Theoremcfilucfil3 22788 Given a metric 𝐷 and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFilu‘(metUnif‘𝐷))) ↔ 𝐶 ∈ (CauFil‘𝐷)))
 
Theoremcfilucfil4 22789 Given a metric 𝐷 and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ (Fil‘𝑋)) → (𝐶 ∈ (CauFilu‘(metUnif‘𝐷)) ↔ 𝐶 ∈ (CauFil‘𝐷)))
 
Theoremcncmet 22790 The set of complex numbers is a complete metric space under the absolute value metric. (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝐷 = (abs ∘ − )       𝐷 ∈ (CMet‘ℂ)
 
Theoremrecmet 22791 The real numbers are a complete metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (CMet‘ℝ)
 
12.5.5  Baire's Category Theorem
 
Theorembcthlem1 22792* Lemma for bcth 22797. Substitutions for the function 𝐹. (Contributed by Mario Carneiro, 9-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})       ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+))) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))
 
Theorembcthlem2 22793* Lemma for bcth 22797. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})    &   (𝜑𝑀:ℕ⟶(Clsd‘𝐽))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐶𝑋)    &   (𝜑𝑔:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)    &   (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))       (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔𝑛)))
 
Theorembcthlem3 22794* Lemma for bcth 22797. The limit point of the centers in the sequence is in the intersection of every ball in the sequence. (Contributed by Mario Carneiro, 7-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})    &   (𝜑𝑀:ℕ⟶(Clsd‘𝐽))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐶𝑋)    &   (𝜑𝑔:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)    &   (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))       ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝐴 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔𝐴)))
 
Theorembcthlem4 22795* Lemma for bcth 22797. Given any open ball (𝐶(ball‘𝐷)𝑅) as starting point (and in particular, a ball in int( ran 𝑀)), the limit point 𝑥 of the centers of the induced sequence of balls 𝑔 is outside ran 𝑀. Note that a set 𝐴 has empty interior iff every nonempty open set 𝑈 contains points outside 𝐴, i.e. (𝑈𝐴) ≠ ∅. (Contributed by Mario Carneiro, 7-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})    &   (𝜑𝑀:ℕ⟶(Clsd‘𝐽))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐶𝑋)    &   (𝜑𝑔:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)    &   (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))       (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)
 
Theorembcthlem5 22796* Lemma for bcth 22797. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 12513, which in the form used here accepts a "selection" function 𝐹 from each element of 𝐾 to a nonempty subset of 𝐾, and the result function 𝑔 maps 𝑔(𝑛 + 1) to an element of 𝐹(𝑛, 𝑔(𝑛)). The trick here is thus in the choice of 𝐹 and 𝐾: we let 𝐾 be the set of all tagged nonempty open sets (tagged here meaning that we have a point and an open set, in an ordered pair), and 𝐹(𝑘, ⟨𝑥, 𝑧⟩) gives the set of all balls of size less than 1 / 𝑘, tagged by their centers, whose closures fit within the given open set 𝑧 and miss 𝑀(𝑘).

Since 𝑀(𝑘) is closed, 𝑧𝑀(𝑘) is open and also nonempty, since 𝑧 is nonempty and 𝑀(𝑘) has empty interior. Then there is some ball contained in it, and hence our function 𝐹 is valid (it never maps to the empty set). Now starting at a point in the interior of ran 𝑀, DC gives us the function 𝑔 all whose elements are constrained by 𝐹 acting on the previous value. (This is all proven in this lemma.) Now 𝑔 is a sequence of tagged open balls, forming an inclusion chain (see bcthlem2 22793) and whose sizes tend to zero, since they are bounded above by 1 / 𝑘. Thus, the centers of these balls form a Cauchy sequence, and converge to a point 𝑥 (see bcthlem4 22795). Since the inclusion chain also ensures the closure of each ball is in the previous ball, the point 𝑥 must be in all these balls (see bcthlem3 22794) and hence misses each 𝑀(𝑘), contradicting the fact that 𝑥 is in the interior of ran 𝑀 (which was the starting point). (Contributed by Mario Carneiro, 6-Jan-2014.)

𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})    &   (𝜑𝑀:ℕ⟶(Clsd‘𝐽))    &   (𝜑 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)       (𝜑 → ((int‘𝐽)‘ ran 𝑀) = ∅)
 
Theorembcth 22797* 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 22796 for an overview of the proof. (Contributed by NM, 28-Oct-2007.) (Proof shortened by Mario Carneiro, 6-Jan-2014.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘ ran 𝑀) ≠ ∅) → ∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅)
 
Theorembcth2 22798* Baire's Category Theorem, version 2: If countably many closed sets cover 𝑋, then one of them has an interior. (Contributed by Mario Carneiro, 10-Jan-2014.)
𝐽 = (MetOpen‘𝐷)       (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑋 ≠ ∅) ∧ (𝑀:ℕ⟶(Clsd‘𝐽) ∧ ran 𝑀 = 𝑋)) → ∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅)
 
Theorembcth3 22799* Baire's Category Theorem, version 3: The intersection of countably many dense open sets is dense. (Contributed by Mario Carneiro, 10-Jan-2014.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋) → ((cls‘𝐽)‘ ran 𝑀) = 𝑋)
 
12.5.6  Banach spaces and complex Hilbert spaces
 
Syntaxccms 22800 Extend class notation with the class of all complete normed groups.
class CMetSp
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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