P. 254 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnmpt2ip 25301*	Continuity of inner product; analogue of cnmpt22f 23704 which cannot be used directly because ·𝑖 is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐶 = (TopOpen‘ℂfld)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 , 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐶))
Theorem	csscld 25302	A "closed subspace" in a subcomplex 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 = (ClSubSp‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ∈ 𝐶) → 𝑆 ∈ (Clsd‘𝐽))
Theorem	clsocv 25303	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 ∧ 𝑆 ⊆ 𝑉) → (𝑂‘((cls‘𝐽)‘𝑆)) = (𝑂‘𝑆))
Theorem	cphsscph 25304	A subspace of a subcomplex pre-Hilbert space is a subcomplex pre-Hilbert space. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 25-Sep-2022.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂPreHil)
12.5.5  Convergence and completeness
Syntax	ccfil 25305	Extend class notation with the class of Cauchy filters.
class CauFil
Syntax	ccau 25306	Extend class notation with the class of Cauchy sequences.
class Cau
Syntax	ccmet 25307	Extend class notation with the class of complete metrics.
class CMet
Definition	df-cfil 25308*	Define the set of Cauchy filters on a given extended 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+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
Definition	df-cau 25309*	Define the set of Cauchy sequences on a given extended metric space. (Contributed by NM, 8-Sep-2006.)
⊢ Cau = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (dom dom 𝑑 ↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶((𝑓‘𝑗)(ball‘𝑑)𝑥)})
Definition	df-cmet 25310*	Define the set of complete metrics on a given set. (Contributed by Mario Carneiro, 1-May-2014.)
⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
Theorem	lmmbr 25311*	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 to use objects more general than sequences when convenient; see the comment in df-lm 23258. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    ⇒   ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ran ℤ≥(𝐹 ↾ 𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥))))
Theorem	lmmbr2 25312*	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 to use objects more general than sequences when convenient; see the comment in df-lm 23258. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    ⇒   ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑃) < 𝑥))))
Theorem	lmmbr3 25313*	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)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑃) < 𝑥))))
Theorem	lmmcvg 25314*	Convergence property of a converging sequence. (Contributed by NM, 1-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝑃) < 𝑅))
Theorem	lmmbrf 25315*	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 25312 presupposes that 𝐹 is a function. (Contributed by NM, 20-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    ⇒   ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐴𝐷𝑃) < 𝑥)))
Theorem	lmnn 25316*	A condition that implies convergence. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹:ℕ⟶𝑋)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷𝑃) < (1 / 𝑘))    ⇒   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)
Theorem	cfilfval 25317*	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+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
Theorem	iscfil 25318*	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+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
Theorem	iscfil2 25319*	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+ ∃𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)))
Theorem	cfilfil 25320	A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋))
Theorem	cfili 25321*	Property of a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦𝐷𝑧) < 𝑅)
Theorem	cfil3i 25322*	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+) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝐹)
Theorem	cfilss 25323	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) ∧ 𝐹 ⊆ 𝐺)) → 𝐺 ∈ (CauFil‘𝐷))
Theorem	fgcfil 25324*	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+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
Theorem	fmcfil 25325*	The Cauchy filter condition for a filter map. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (((𝑋 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))
Theorem	iscfil3 25326*	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+ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)))
Theorem	cfilfcls 25327	Similar to ultrafilters (uffclsflim 24060), 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))
Theorem	caufval 25328*	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 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)})
Theorem	iscau 25329*	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 23258. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝐹‘𝑘)(ball‘𝐷)𝑥))))
Theorem	iscau2 25330*	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 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))))
Theorem	iscau3 25331*	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 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))))
Theorem	iscau4 25332*	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 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥))))
Theorem	iscauf 25333*	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 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = 𝐵)    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵𝐷𝐴) < 𝑥))
Theorem	caun0 25334	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 ≠ ∅)
Theorem	caufpm 25335	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 ∈ (𝑋 ↑pm ℂ))
Theorem	caucfil 25336	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 = (ℤ≥‘𝑀)    &   ⊢ 𝐿 = ((𝑋 FilMap 𝐹)‘(ℤ≥ “ 𝑍))    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐿 ∈ (CauFil‘𝐷)))
Theorem	iscmet 25337*	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) ≠ ∅))
Theorem	cmetcvg 25338	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) ≠ ∅)
Theorem	cmetmet 25339	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))
Theorem	cmetmeti 25340	A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)
⊢ 𝐷 ∈ (CMet‘𝑋)    ⇒   ⊢ 𝐷 ∈ (Met‘𝑋)
Theorem	cmetcaulem 25341*	Lemma for cmetcau 25342. (Contributed by Mario Carneiro, 14-Oct-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))    &   ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ if(𝑥 ∈ dom 𝐹, (𝐹‘𝑥), 𝑃))    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽))
Theorem	cmetcau 25342	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 (⇝𝑡‘𝐽))
Theorem	iscmet3lem3 25343*	Lemma for iscmet3 25346. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑅 ∈ ℝ+) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((1 / 2)↑𝑘) < 𝑅)
Theorem	iscmet3lem1 25344*	Lemma for iscmet3 25346. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    &   ⊢ (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))    &   ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))
Theorem	iscmet3lem2 25345*	Lemma for iscmet3 25346. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    &   ⊢ (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))    &   ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛))    &   ⊢ (𝜑 → 𝐺 ∈ (Fil‘𝑋))    &   ⊢ (𝜑 → 𝑆:ℤ⟶𝐺)    &   ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽))    ⇒   ⊢ (𝜑 → (𝐽 fLim 𝐺) ≠ ∅)
Theorem	iscmet3 25346*	The property "𝐷 is a complete metric" expressed in terms of functions on ℕ (or any other upper integer set). Thus, we only have to look at functions on ℕ, 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 = (ℤ≥‘𝑀)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    ⇒   ⊢ (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))))
Theorem	iscmet2 25347	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 (⇝𝑡‘𝐽)))
Theorem	cfilresi 25348	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))
Theorem	cfilres 25349	Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ↾t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))))
Theorem	caussi 25350	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))
Theorem	causs 25351	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)))))
Theorem	equivcfil 25352*	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.)
⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    ⇒   ⊢ (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
Theorem	equivcau 25353*	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.)
⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    ⇒   ⊢ (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))
Theorem	lmle 25354*	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 = (ℤ≥‘𝑀)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑄𝐷(𝐹‘𝑘)) ≤ 𝑅)    ⇒   ⊢ (𝜑 → (𝑄𝐷𝑃) ≤ 𝑅)
Theorem	nglmle 25355*	If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value. (Contributed by NM, 25-Dec-2007.) (Revised by AV, 16-Oct-2021.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝑁 = (norm‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ NrmGrp)    &   ⊢ (𝜑 → 𝐹:ℕ⟶𝑋)    &   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) ≤ 𝑅)    ⇒   ⊢ (𝜑 → (𝑁‘𝑃) ≤ 𝑅)
Theorem	lmclim 25356	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 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑍 ⊆ dom 𝐹) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹 ⇝ 𝑃)))
Theorem	lmclimf 25357	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 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃))
Theorem	metelcls 25358*	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 10504. 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)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)))
Theorem	metcld 25359*	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) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ∀𝑥∀𝑓((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆)))
Theorem	metcld2 25360	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) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) ⊆ 𝑆))
Theorem	caubl 25361*	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.)
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 × ℝ+))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))    &   ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ (2nd ‘(𝐹‘𝑛)) < 𝑟)    ⇒   ⊢ (𝜑 → (1st ∘ 𝐹) ∈ (Cau‘𝐷))
Theorem	caublcls 25362*	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.)
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 × ℝ+))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹‘𝐴))))
Theorem	metcnp4 25363*	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)    &   ⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ 𝑓(⇝𝑡‘𝐽)𝑃) → (𝐹 ∘ 𝑓)(⇝𝑡‘𝐾)(𝐹‘𝑃)))))
Theorem	metcn4 25364*	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)    &   ⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌))    &   ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ∀𝑥(𝑓(⇝𝑡‘𝐽)𝑥 → (𝐹 ∘ 𝑓)(⇝𝑡‘𝐾)(𝐹‘𝑥)))))
Theorem	iscmet3i 25365*	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 (⇝𝑡‘𝐽))    ⇒   ⊢ 𝐷 ∈ (CMet‘𝑋)
Theorem	lmcau 25366	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 (⇝𝑡‘𝐽) ⊆ (Cau‘𝐷))
Theorem	flimcfil 25367	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))
Theorem	metsscmetcld 25368	A complete subspace of a metric space is closed in the parent space. Formerly part of proof for cmetss 25369. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 9-Oct-2022.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌 ∈ (Clsd‘𝐽))
Theorem	cmetss 25369	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.) (Proof shortened by AV, 9-Oct-2022.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (CMet‘𝑋) → ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ 𝑌 ∈ (Clsd‘𝐽)))
Theorem	equivcmet 25370*	If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 25353, metss2 24546, 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 ℝ induced by the usual homeomorphism from (0, 1) against the usual metric 𝐶 on ℝ and against the discrete metric 𝐸 on ℝ. 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.)
⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑆 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ (𝑆 · (𝑥𝐶𝑦)))    ⇒   ⊢ (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋)))
Theorem	relcmpcmet 25371*	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)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
Theorem	cmpcmet 25372	A compact metric space is complete. One half of heibor 37781. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
Theorem	cfilucfil3 25373	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)))
Theorem	cfilucfil4 25374	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)))
Theorem	cncmet 25375	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)
Theorem	recmet 25376	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)
12.5.6  Baire's Category Theorem
Theorem	bcthlem1 25377*	Lemma for bcth 25382. Substitutions for the function 𝐹. (Contributed by Mario Carneiro, 9-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ 𝐹 = (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))))})    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+))) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))))
Theorem	bcthlem2 25378*	Lemma for bcth 25382. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ 𝐹 = (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))))})    &   ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+))    &   ⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)    &   ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))    ⇒   ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
Theorem	bcthlem3 25379*	Lemma for bcth 25382. 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)    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ 𝐹 = (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))))})    &   ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+))    &   ⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)    &   ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))    ⇒   ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝐴 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘𝐴)))
Theorem	bcthlem4 25380*	Lemma for bcth 25382. 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. (𝑈 ∖ 𝐴) ≠ ∅. (Contributed by Mario Carneiro, 7-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ 𝐹 = (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))))})    &   ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+))    &   ⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)    &   ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))    ⇒   ⊢ (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅)
Theorem	bcthlem5 25381*	Lemma for bcth 25382. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 14035, 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, 𝑧 ∖ 𝑀(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 25378) 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 25380). 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 25379) 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)    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ 𝐹 = (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))))})    &   ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))    &   ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)    ⇒   ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) = ∅)
Theorem	bcth 25382*	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 25381 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)‘(𝑀‘𝑘)) ≠ ∅)
Theorem	bcth2 25383*	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)‘(𝑀‘𝑘)) ≠ ∅)
Theorem	bcth3 25384*	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)‘(𝑀‘𝑘)) = 𝑋) → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋)
12.5.7  Banach spaces and subcomplex Hilbert spaces
Syntax	ccms 25385	Extend class notation with the class of complete metric spaces.
class CMetSp
Syntax	cbn 25386	Extend class notation with the class of Banach spaces.
class Ban
Syntax	chl 25387	Extend class notation with the class of subcomplex Hilbert spaces.
class ℂHil
Definition	df-cms 25388*	Define the class of complete metric spaces. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
Definition	df-bn 25389	Define the class of all Banach spaces. A Banach space is a normed vector space such that both the vector space and the scalar field are complete under their respective norm-induced metrics. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
Definition	df-hl 25390	Define the class of all subcomplex Hilbert spaces. A subcomplex Hilbert space is a Banach space which is also an inner product space over a subfield of the field of complex numbers closed under square roots of nonnegative reals. (Contributed by Steve Rodriguez, 28-Apr-2007.)
⊢ ℂHil = (Ban ∩ ℂPreHil)
Theorem	isbn 25391	A Banach space is a normed vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
Theorem	bnsca 25392	The scalar field of a Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ Ban → 𝐹 ∈ CMetSp)
Theorem	bnnvc 25393	A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)
Theorem	bnnlm 25394	A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
Theorem	bnngp 25395	A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp)
Theorem	bnlmod 25396	A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ LMod)
Theorem	bncms 25397	A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
Theorem	iscms 25398	A complete metric space is a metric space with a complete metric. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
Theorem	cmscmet 25399	The induced metric on a complete normed group is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))
Theorem	bncmet 25400	The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑀 ∈ Ban → 𝐷 ∈ (CMet‘𝑋))