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Theorem List for Metamath Proof Explorer - 23101-23200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcaublcls 23101* 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 23102* 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 23103* 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 23104* 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 23105 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 23106 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 23107 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 23108* If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 23092, metss2 22311, 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 23109* 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 23110 A compact metric space is complete. One half of heibor 33600. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)       (𝜑𝐷 ∈ (CMet‘𝑋))

Theoremcfilucfil3 23111 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 23112 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 23113 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 23114 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

Theorembcthlem1 23115* Lemma for bcth 23120. 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 23116* Lemma for bcth 23120. 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 23117* Lemma for bcth 23120. 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 23118* Lemma for bcth 23120. 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 23119* Lemma for bcth 23120. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 12778, 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 23116) 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 23118). 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 23117) 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 23120* 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 23119 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 23121* 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 23122* 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)

12.5.7  Banach spaces and subcomplex Hilbert spaces

Syntaxccms 23123 Extend class notation with the class of all complete normed groups.
class CMetSp

Syntaxcbn 23124 Extend class notation with the class of all Banach spaces.
class Ban

Syntaxchl 23125 Extend class notation with the class of all subcomplex Hilbert spaces.
class ℂHil

Definitiondf-cms 23126* Define the class of all complete metric spaces. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}

Definitiondf-bn 23127 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}

Definitiondf-hl 23128 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 quadratically closed subfield of the field of complex numbers. (Contributed by Steve Rodriguez, 28-Apr-2007.)
ℂHil = (Ban ∩ ℂPreHil)

Theoremisbn 23129 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))

Theorembnsca 23130 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)

Theorembnnvc 23131 A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ Ban → 𝑊 ∈ NrmVec)

Theorembnnlm 23132 A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ Ban → 𝑊 ∈ NrmMod)

Theorembnngp 23133 A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ Ban → 𝑊 ∈ NrmGrp)

Theorembnlmod 23134 A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ Ban → 𝑊 ∈ LMod)

Theorembncms 23135 A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ Ban → 𝑊 ∈ CMetSp)

Theoremiscms 23136 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)))

Theoremcmscmet 23137 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))

Theorembncmet 23138 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))

Theoremcmsms 23139 A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝐺 ∈ CMetSp → 𝐺 ∈ MetSp)

Theoremcmspropd 23140 Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ CMetSp ↔ 𝐿 ∈ CMetSp))

Theoremcmsss 23141 The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐾 = (𝑀s 𝐴)    &   𝑋 = (Base‘𝑀)    &   𝐽 = (TopOpen‘𝑀)       ((𝑀 ∈ CMetSp ∧ 𝐴𝑋) → (𝐾 ∈ CMetSp ↔ 𝐴 ∈ (Clsd‘𝐽)))

Theoremlssbn 23142 A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ Ban ∧ 𝑈𝑆) → (𝑋 ∈ Ban ↔ 𝑈 ∈ (Clsd‘𝐽)))

Theoremcmetcusp1 23143 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)

Theoremcmetcusp 23144 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)

Theoremcncms 23145 The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
fld ∈ CMetSp

Theoremcnflduss 23146 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 ∘ − ))

Theoremcnfldcusp 23147 The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
fld ∈ CUnifSp

Theoremresscdrg 23148 The real numbers are a subset of any complete subfield in the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (ℂflds 𝐾)       ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ℝ ⊆ 𝐾)

Theoremcncdrg 23149 The only complete subfields of the complex numbers are and . (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (ℂflds 𝐾)       ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → 𝐾 ∈ {ℝ, ℂ})

Theoremsrabn 23150 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) ∧ (𝑊s 𝑆) ∈ DivRing)))

Theoremrlmbn 23151 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)

Theoremishl 23152 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))

Theoremhlbn 23153 Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
(𝑊 ∈ ℂHil → 𝑊 ∈ Ban)

Theoremhlcph 23154 Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)

Theoremhlphl 23155 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)

Theoremhlcms 23156 Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
(𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)

Theoremhlprlem 23157 Lemma for hlpr 23159. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂHil → (𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝐾) ∈ DivRing ∧ (ℂflds 𝐾) ∈ CMetSp))

Theoremhlress 23158 The scalar field of a subcomplex Hilbert space contains . (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂHil → ℝ ⊆ 𝐾)

Theoremhlpr 23159 The scalar field of a subcomplex Hilbert space is either or . (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂHil → 𝐾 ∈ {ℝ, ℂ})

Theoremishl2 23160 A Hilbert space is a complete subcomplex pre-Hilbert space over or . (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}))

12.5.7.1  The complete ordered field of the real numbers

Theoremretopn 23161 The topology of the real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
(topGen‘ran (,)) = (TopOpen‘ℝfld)

Theoremrecms 23162 The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.)
fld ∈ CMetSp

Theoremreust 23163 The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.)
(UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ)))

Theoremrecusp 23164 The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
fld ∈ CUnifSp

12.5.8  Euclidean spaces

Syntaxcrrx 23165 Extend class notation with generalized real Euclidean spaces.
class ℝ^

Syntaxcehl 23166 Extend class notation with real Euclidean spaces.
class 𝔼hil

Definitiondf-rrx 23167 Define the function associating with a set the free real vector space on that set, equipped with the natural inner product. This is the direct sum of copies of the field of real numbers indexed by that set. We call it here a "generalized real Euclidean space", but note that it need not be complete (for instance if the given set is infinite countable). (Contributed by Thierry Arnoux, 16-Jun-2019.)
ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))

Definitiondf-ehl 23168 Define a function generating the real Euclidean spaces of finite dimension. The case 𝑛 = 0 corresponds to a space of dimension 0, that is, limited to a neutral element. Members of this family of spaces are Hilbert spaces, as shown in - ehlhl . (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))

Theoremrrxval 23169 Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)       (𝐼𝑉𝐻 = (toℂHil‘(ℝfld freeLMod 𝐼)))

Theoremrrxbase 23170* The base of the generalized real Euclidean space is the set of functions with finite support. (Contributed by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉𝐵 = {𝑓 ∈ (ℝ ↑𝑚 𝐼) ∣ 𝑓 finSupp 0})

Theoremrrxprds 23171 Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉𝐻 = (toℂHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵)))

Theoremrrxip 23172* The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉 → (𝑓 ∈ (ℝ ↑𝑚 𝐼), 𝑔 ∈ (ℝ ↑𝑚 𝐼) ↦ (ℝfld Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))) = (·𝑖𝐻))

Theoremrrxnm 23173* The norm of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉 → (𝑓𝐵 ↦ (√‘(ℝfld Σg (𝑥𝐼 ↦ ((𝑓𝑥)↑2))))) = (norm‘𝐻))

Theoremrrxcph 23174 Generalized Euclidean real spaces are pre-Hilbert spaces. (Contributed by Thierry Arnoux, 23-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉𝐻 ∈ ℂPreHil)

Theoremrrxds 23175* The distance over generalized Euclidean spaces. Compare with df-rrn 33605. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉 → (𝑓𝐵, 𝑔𝐵 ↦ (√‘(ℝfld Σg (𝑥𝐼 ↦ (((𝑓𝑥) − (𝑔𝑥))↑2))))) = (dist‘𝐻))

Theoremcsbren 23176* Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)       (𝜑 → (Σ𝑘𝐴 (𝐵 · 𝐶)↑2) ≤ (Σ𝑘𝐴 (𝐵↑2) · Σ𝑘𝐴 (𝐶↑2)))

Theoremtrirn 23177* Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)       (𝜑 → (√‘Σ𝑘𝐴 ((𝐵 + 𝐶)↑2)) ≤ ((√‘Σ𝑘𝐴 (𝐵↑2)) + (√‘Σ𝑘𝐴 (𝐶↑2))))

Theoremrrxf 23178* Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑𝐹:𝐼⟶ℝ)

Theoremrrxfsupp 23179* Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑 → (𝐹 supp 0) ∈ Fin)

Theoremrrxsuppss 23180* Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑 → (𝐹 supp 0) ⊆ 𝐼)

Theoremrrxmvallem 23181* Support of the function used for building the distance . (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}       ((𝐼𝑉𝐹𝑋𝐺𝑋) → ((𝑘𝐼 ↦ (((𝐹𝑘) − (𝐺𝑘))↑2)) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)))

Theoremrrxmval 23182* The value of the Euclidean metric. Compare with rrnmval 33607. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))       ((𝐼𝑉𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹𝑘) − (𝐺𝑘))↑2)))

Theoremrrxmfval 23183* The value of the Euclidean metric. Compare with rrnval 33606. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))       (𝐼𝑉𝐷 = (𝑓𝑋, 𝑔𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓𝑘) − (𝑔𝑘))↑2))))

Theoremrrxmetlem 23184* Lemma for rrxmet 23185. (Contributed by Thierry Arnoux, 5-Jul-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))    &   (𝜑𝐼𝑉)    &   (𝜑𝐹𝑋)    &   (𝜑𝐺𝑋)    &   (𝜑𝐴𝐼)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐴)       (𝜑 → Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹𝑘) − (𝐺𝑘))↑2) = Σ𝑘𝐴 (((𝐹𝑘) − (𝐺𝑘))↑2))

Theoremrrxmet 23185* Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.) (Revised by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))       (𝐼𝑉𝐷 ∈ (Met‘𝑋))

Theoremrrxdstprj1 23186* The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))    &   𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))       (((𝐼𝑉𝐴𝐼) ∧ (𝐹𝑋𝐺𝑋)) → ((𝐹𝐴)𝑀(𝐺𝐴)) ≤ (𝐹𝐷𝐺))

Theoremehlval 23187 Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐸 = (𝔼hil𝑁)       (𝑁 ∈ ℕ0𝐸 = (ℝ^‘(1...𝑁)))

Theoremehlbase 23188 The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐸 = (𝔼hil𝑁)       (𝑁 ∈ ℕ0 → (ℝ ↑𝑚 (1...𝑁)) = (Base‘𝐸))

12.5.9  Minimizing Vector Theorem

Theoremminveclem1 23189* Lemma for minvec 23201. The set of all distances from points of 𝑌 to 𝐴 are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))       (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤𝑅 0 ≤ 𝑤))

Theoremminveclem4c 23190* Lemma for minvec 23201. The infimum of the distances to 𝐴 is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )       (𝜑𝑆 ∈ ℝ)

Theoremminveclem2 23191* Lemma for minvec 23201. Any two points 𝐾 and 𝐿 in 𝑌 are close to each other if they are close to the infimum of distance to 𝐴. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑𝐾𝑌)    &   (𝜑𝐿𝑌)    &   (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵))    &   (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵))       (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))

Theoremminveclem3a 23192* Lemma for minvec 23201. 𝐷 is a complete metric when restricted to 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))

Theoremminveclem3b 23193* Lemma for minvec 23201. The set of vectors within a fixed distance of the infimum forms a filter base. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})       (𝜑𝐹 ∈ (fBas‘𝑌))

Theoremminveclem3 23194* Lemma for minvec 23201. The filter formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})       (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))

Theoremminveclem4a 23195* Lemma for minvec 23201. 𝐹 converges to a point 𝑃 in 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   𝑃 = (𝐽 fLim (𝑋filGen𝐹))       (𝜑𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))

Theoremminveclem4b 23196* Lemma for minvec 23201. The convergent point of the Cauchy sequence 𝐹 is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   𝑃 = (𝐽 fLim (𝑋filGen𝐹))       (𝜑𝑃𝑋)

Theoremminveclem4 23197* Lemma for minvec 23201. The convergent point of the Cauchy sequence 𝐹 attains the minimum distance, and so is closer to 𝐴 than any other point in 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   𝑃 = (𝐽 fLim (𝑋filGen𝐹))    &   𝑇 = (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))       (𝜑 → ∃𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))

Theoremminveclem5 23198* Lemma for minvec 23201. Discharge the assumptions in minveclem4 23197. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       (𝜑 → ∃𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))

Theoremminveclem6 23199* Lemma for minvec 23201. Any minimal point is less than 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       ((𝜑𝑥𝑌) → (((𝐴𝐷𝑥)↑2) ≤ ((𝑆↑2) + 0) ↔ ∀𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦))))

Theoremminveclem7 23200* Lemma for minvec 23201. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       (𝜑 → ∃!𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))

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