P. 253 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iscau 25201*	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 23142. (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 25202*	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 25203*	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 25204*	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 25205*	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 25206	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 25207	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 25208	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 25209*	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 25210	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 25211	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 25212	A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)
⊢ 𝐷 ∈ (CMet‘𝑋)    ⇒   ⊢ 𝐷 ∈ (Met‘𝑋)
Theorem	cmetcaulem 25213*	Lemma for cmetcau 25214. (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 25214	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 25215*	Lemma for iscmet3 25218. (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 25216*	Lemma for iscmet3 25218. (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 25217*	Lemma for iscmet3 25218. (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 25218*	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 25219	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 25220	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 25221	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 25222	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 25223	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 25224*	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 25225*	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 25226*	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 25227*	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 25228	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 25229	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 25230*	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 10323. 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 25231*	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 25232	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 25233*	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 25234*	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 25235*	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 25236*	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 25237*	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 25238	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 25239	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 25240	A complete subspace of a metric space is closed in the parent space. Formerly part of proof for cmetss 25241. (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 25241	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 25242*	If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 25225, metss2 24425, 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 25243*	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 25244	A compact metric space is complete. One half of heibor 37860. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
Theorem	cfilucfil3 25245	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 25246	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 25247	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 25248	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 25249*	Lemma for bcth 25254. 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 25250*	Lemma for bcth 25254. 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 25251*	Lemma for bcth 25254. 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 25252*	Lemma for bcth 25254. 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 25253*	Lemma for bcth 25254. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 13888, 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 25250) 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 25252). 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 25251) 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 25254*	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 25253 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 25255*	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 25256*	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 25257	Extend class notation with the class of complete metric spaces.
class CMetSp
Syntax	cbn 25258	Extend class notation with the class of Banach spaces.
class Ban
Syntax	chl 25259	Extend class notation with the class of subcomplex Hilbert spaces.
class ℂHil
Definition	df-cms 25260*	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 25261	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 25262	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 25263	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 25264	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 25265	A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)
Theorem	bnnlm 25266	A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
Theorem	bnngp 25267	A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp)
Theorem	bnlmod 25268	A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ LMod)
Theorem	bncms 25269	A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
Theorem	iscms 25270	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 25271	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 25272	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))
Theorem	cmsms 25273	A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝐺 ∈ CMetSp → 𝐺 ∈ MetSp)
Theorem	cmspropd 25274	Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))    ⇒   ⊢ (𝜑 → (𝐾 ∈ CMetSp ↔ 𝐿 ∈ CMetSp))
Theorem	cmssmscld 25275	The restriction of a metric space is closed if it is complete. (Contributed by AV, 9-Oct-2022.)
⊢ 𝐾 = (𝑀 ↾s 𝐴)    &   ⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐽 = (TopOpen‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ⊆ 𝑋 ∧ 𝐾 ∈ CMetSp) → 𝐴 ∈ (Clsd‘𝐽))
Theorem	cmsss 25276	The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐾 = (𝑀 ↾s 𝐴)    &   ⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐽 = (TopOpen‘𝑀)    ⇒   ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐾 ∈ CMetSp ↔ 𝐴 ∈ (Clsd‘𝐽)))
Theorem	lssbn 25277	A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    ⇒   ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ Ban ↔ 𝑈 ∈ (Clsd‘𝐽)))
Theorem	cmetcusp1 25278	If the uniform set of a complete metric space is the uniform structure generated by its metric, then it is a complete uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
⊢ 𝑋 = (Base‘𝐹)    &   ⊢ 𝐷 = ((dist‘𝐹) ↾ (𝑋 × 𝑋))    &   ⊢ 𝑈 = (UnifSt‘𝐹)    ⇒   ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ CMetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐹 ∈ CUnifSp)
Theorem	cmetcusp 25279	The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (CMet‘𝑋)) → (toUnifSp‘(metUnif‘𝐷)) ∈ CUnifSp)
Theorem	cncms 25280	The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ℂfld ∈ CMetSp
Theorem	cnflduss 25281	The uniform structure of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ 𝑈 = (UnifSt‘ℂfld)    ⇒   ⊢ 𝑈 = (metUnif‘(abs ∘ − ))
Theorem	cnfldcusp 25282	The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
⊢ ℂfld ∈ CUnifSp
Theorem	resscdrg 25283	The real numbers are a subset of any complete subfield in the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐹 = (ℂfld ↾s 𝐾)    ⇒   ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ℝ ⊆ 𝐾)
Theorem	cncdrg 25284	The only complete subfields of the complex numbers are ℝ and ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐹 = (ℂfld ↾s 𝐾)    ⇒   ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → 𝐾 ∈ {ℝ, ℂ})
Theorem	srabn 25285	The subring algebra over a complete normed ring is a Banach space iff the subring is a closed division ring. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    ⇒   ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ Ban ↔ (𝑆 ∈ (Clsd‘𝐽) ∧ (𝑊 ↾s 𝑆) ∈ DivRing)))
Theorem	rlmbn 25286	The ring module over a complete normed division ring is a Banach space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → (ringLMod‘𝑅) ∈ Ban)
Theorem	ishl 25287	The predicate "is a subcomplex Hilbert space". A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
Theorem	hlbn 25288	Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
Theorem	hlcph 25289	Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
Theorem	hlphl 25290	Every subcomplex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil)
Theorem	hlcms 25291	Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
Theorem	hlprlem 25292	Lemma for hlpr 25294. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂHil → (𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐾) ∈ DivRing ∧ (ℂfld ↾s 𝐾) ∈ CMetSp))
Theorem	hlress 25293	The scalar field of a subcomplex Hilbert space contains ℝ. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂHil → ℝ ⊆ 𝐾)
Theorem	hlpr 25294	The scalar field of a subcomplex Hilbert space is either ℝ or ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂHil → 𝐾 ∈ {ℝ, ℂ})
Theorem	ishl2 25295	A Hilbert space is a complete subcomplex pre-Hilbert space over ℝ or ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}))
Theorem	cphssphl 25296	A Banach subspace of a subcomplex pre-Hilbert space is a subcomplex Hilbert space. (Contributed by NM, 11-Apr-2008.) (Revised by AV, 25-Sep-2022.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ Ban) → 𝑋 ∈ ℂHil)
Theorem	cmslssbn 25297	A complete linear subspace of a normed vector space is a Banach space. We furthermore have to assume that the field of scalars is complete since this is a requirement in the current definition of Banach spaces df-bn 25261. (Contributed by AV, 8-Oct-2022.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ Ban)
Theorem	cmscsscms 25298	A closed subspace of a complete metric space which is also a subcomplex pre-Hilbert space is a complete metric space. Remark: the assumption that the Banach space must be a (subcomplex) pre-Hilbert space is required because the definition of ClSubSp is based on an inner product. If ClSubSp was generalized to arbitrary topological spaces (or at least topological modules), this assumption could be omitted. (Contributed by AV, 8-Oct-2022.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (ClSubSp‘𝑊)    ⇒   ⊢ (((𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil) ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ CMetSp)
Theorem	bncssbn 25299	A closed subspace of a Banach space which is also a subcomplex pre-Hilbert space is a Banach space. Remark: the assumption that the Banach space must be a (subcomplex) pre-Hilbert space is required because the definition of ClSubSp is based on an inner product. If ClSubSp was generalized for arbitrary topological spaces, this assuption could be omitted. (Contributed by AV, 8-Oct-2022.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (ClSubSp‘𝑊)    ⇒   ⊢ (((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ Ban)
Theorem	cssbn 25300	A complete subspace of a normed vector space with a complete scalar field is a Banach space. Remark: In contrast to ClSubSp, a complete subspace is defined by "a linear subspace in which all Cauchy sequences converge to a point in the subspace". This is closer to the original, but deprecated definition Cℋ (df-ch 31196) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 25297. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈))    ⇒   ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ Ban)