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
Syntax	chl 25301	Extend class notation with the class of subcomplex Hilbert spaces.
class ℂHil
Definition	df-cms 25302*	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 25303	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 25304	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 25305	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 25306	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 25307	A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)
Theorem	bnnlm 25308	A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
Theorem	bnngp 25309	A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp)
Theorem	bnlmod 25310	A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ LMod)
Theorem	bncms 25311	A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
Theorem	iscms 25312	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 25313	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 25314	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 25315	A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝐺 ∈ CMetSp → 𝐺 ∈ MetSp)
Theorem	cmspropd 25316	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 25317	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 25318	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 25319	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 25320	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 25321	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 25322	The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ℂfld ∈ CMetSp
Theorem	cnflduss 25323	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 25324	The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
⊢ ℂfld ∈ CUnifSp
Theorem	resscdrg 25325	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 25326	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 25327	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 25328	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 25329	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 25330	Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
Theorem	hlcph 25331	Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
Theorem	hlphl 25332	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 25333	Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
Theorem	hlprlem 25334	Lemma for hlpr 25336. (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 25335	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 25336	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 25337	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 25338	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 25339	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 25303. (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 25340	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 25341	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 25342	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 31292) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 25339. (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)
Theorem	csschl 25343	A complete subspace of a complex pre-Hilbert space is a complex Hilbert space. Remarks: (a) 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 31292) of closed subspaces of a Hilbert space. (b) This theorem does not hold for arbitrary subcomplex (pre-)Hilbert spaces, because the scalar field as restriction of the field of the complex numbers need not be closed. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈))    &   ⊢ (Scalar‘𝑊) = ℂfld    ⇒   ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (𝑋 ∈ ℂHil ∧ (Scalar‘𝑋) = ℂfld))
Theorem	cmslsschl 25344	A complete linear subspace of a subcomplex Hilbert space is a subcomplex Hilbert space. (Contributed by AV, 8-Oct-2022.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂHil ∧ 𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂHil)
Theorem	chlcsschl 25345	A closed subspace of a subcomplex Hilbert space is a subcomplex Hilbert space. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 8-Oct-2022.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (ClSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂHil)
12.5.7.1  The complete ordered field of the real numbers
Theorem	retopn 25346	The topology of the real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld)
Theorem	recms 25347	The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.)
⊢ ℝfld ∈ CMetSp
Theorem	reust 25348	The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.)
⊢ (UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ)))
Theorem	recusp 25349	The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
⊢ ℝfld ∈ CUnifSp
12.5.8  Euclidean spaces
Syntax	crrx 25350	Extend class notation with generalized real Euclidean spaces.
class ℝ^
Syntax	cehl 25351	Extend class notation with real Euclidean spaces.
class 𝔼hil
Definition	df-rrx 25352	Define the function associating with a set the free real vector space on that set, equipped with the natural inner product and norm. 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 ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
Definition	df-ehl 25353	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 (see ehl0 25384). 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)))
Theorem	rrxval 25354	Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐻 = (ℝ^‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
Theorem	rrxbase 25355*	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)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0})
Theorem	rrxprds 25356	Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐻 = (ℝ^‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵)))
Theorem	rrxip 25357*	The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐻 = (ℝ^‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝐻))
Theorem	rrxnm 25358*	The norm of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐻 = (ℝ^‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵 ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))) = (norm‘𝐻))
Theorem	rrxcph 25359	Generalized Euclidean real spaces are subcomplex pre-Hilbert spaces. (Contributed by Thierry Arnoux, 23-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.)
⊢ 𝐻 = (ℝ^‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐻 ∈ ℂPreHil)
Theorem	rrxds 25360*	The distance over generalized Euclidean spaces. Compare with df-rrn 38147. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)
⊢ 𝐻 = (ℝ^‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘𝐻))
Theorem	rrxvsca 25361	The scalar product over generalized Euclidean spaces is the componentwise real number multiplication. (Contributed by Thierry Arnoux, 18-Jan-2023.)
⊢ 𝐻 = (ℝ^‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐻)    &   ⊢ ∙ = ( ·𝑠 ‘𝐻)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))    ⇒   ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽)))
Theorem	rrxplusgvscavalb 25362*	The result of the addition combined with scalar multiplication in a generalized Euclidean space is defined by its coordinate-wise operations. (Contributed by AV, 21-Jan-2023.)
⊢ 𝐻 = (ℝ^‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐻)    &   ⊢ ∙ = ( ·𝑠 ‘𝐻)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ ✚ = (+g‘𝐻)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))))
Theorem	rrxsca 25363	The field of real numbers is the scalar field of the generalized real Euclidean space. (Contributed by AV, 15-Jan-2023.)
⊢ 𝐻 = (ℝ^‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → (Scalar‘𝐻) = ℝfld)
Theorem	rrx0 25364	The zero ("origin") in a generalized real Euclidean space. (Contributed by AV, 11-Feb-2023.)
⊢ 𝐻 = (ℝ^‘𝐼)    &   ⊢ 0 = (𝐼 × {0})    ⇒   ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = 0 )
Theorem	rrx0el 25365	The zero ("origin") in a generalized real Euclidean space is an element of its base set. (Contributed by AV, 11-Feb-2023.)
⊢ 0 = (𝐼 × {0})    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 0 ∈ 𝑃)
Theorem	csbren 25366*	Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶)↑2) ≤ (Σ𝑘 ∈ 𝐴 (𝐵↑2) · Σ𝑘 ∈ 𝐴 (𝐶↑2)))
Theorem	trirn 25367*	Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (√‘Σ𝑘 ∈ 𝐴 ((𝐵 + 𝐶)↑2)) ≤ ((√‘Σ𝑘 ∈ 𝐴 (𝐵↑2)) + (√‘Σ𝑘 ∈ 𝐴 (𝐶↑2))))
Theorem	rrxf 25368*	Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝐹:𝐼⟶ℝ)
Theorem	rrxfsupp 25369*	Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin)
Theorem	rrxsuppss 25370*	Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 supp 0) ⊆ 𝐼)
Theorem	rrxmvallem 25371*	Support of the function used for building the distance . (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝑘 ∈ 𝐼 ↦ (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)))
Theorem	rrxmval 25372*	The value of the Euclidean metric. Compare with rrnmval 38149. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
Theorem	rrxmfval 25373*	The value of the Euclidean metric. Compare with rrnval 38148. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
Theorem	rrxmetlem 25374*	Lemma for rrxmet 25375. (Contributed by Thierry Arnoux, 5-Jul-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐴)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = Σ𝑘 ∈ 𝐴 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
Theorem	rrxmet 25375*	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 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))
Theorem	rrxdstprj1 25376*	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 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    &   ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))
Theorem	rrxbasefi 25377	The base of the generalized real Euclidean space, when the dimension of the space is finite. This justifies the use of (ℝ ↑m 𝑋) for the development of the Lebesgue measure theory for n-dimensional real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝐻 = (ℝ^‘𝑋)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ (𝜑 → 𝐵 = (ℝ ↑m 𝑋))
Theorem	rrxdsfi 25378*	The distance over generalized Euclidean spaces. Finite dimensional case. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐻 = (ℝ^‘𝐼)    &   ⊢ 𝐵 = (ℝ ↑m 𝐼)    ⇒   ⊢ (𝐼 ∈ Fin → (dist‘𝐻) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
Theorem	rrxmetfi 25379	Euclidean space is a metric space. Finite dimensional version. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼)))
Theorem	rrxdsfival 25380*	The value of the Euclidean distance function in a generalized real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
⊢ 𝑋 = (ℝ ↑m 𝐼)    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
Theorem	ehlval 25381	Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐸 = (𝔼hil‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁)))
Theorem	ehlbase 25382	The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐸 = (𝔼hil‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑m (1...𝑁)) = (Base‘𝐸))
Theorem	ehl0base 25383	The base of the Euclidean space of dimension 0 consists only of one element, the empty set. (Contributed by AV, 12-Feb-2023.)
⊢ 𝐸 = (𝔼hil‘0)    ⇒   ⊢ (Base‘𝐸) = {∅}
Theorem	ehl0 25384	The Euclidean space of dimension 0 consists of the neutral element only. (Contributed by AV, 12-Feb-2023.)
⊢ 𝐸 = (𝔼hil‘0)    &   ⊢ 0 = (0g‘𝐸)    ⇒   ⊢ (Base‘𝐸) = { 0 }
Theorem	ehleudis 25385*	The Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
⊢ 𝐼 = (1...𝑁)    &   ⊢ 𝐸 = (𝔼hil‘𝑁)    &   ⊢ 𝑋 = (ℝ ↑m 𝐼)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
Theorem	ehleudisval 25386*	The value of the Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
⊢ 𝐼 = (1...𝑁)    &   ⊢ 𝐸 = (𝔼hil‘𝑁)    &   ⊢ 𝑋 = (ℝ ↑m 𝐼)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
Theorem	ehl1eudis 25387*	The Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐸 = (𝔼hil‘1)    &   ⊢ 𝑋 = (ℝ ↑m {1})    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1))))
Theorem	ehl1eudisval 25388	The value of the Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐸 = (𝔼hil‘1)    &   ⊢ 𝑋 = (ℝ ↑m {1})    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (abs‘((𝐹‘1) − (𝐺‘1))))
Theorem	ehl2eudis 25389*	The Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐸 = (𝔼hil‘2)    &   ⊢ 𝑋 = (ℝ ↑m {1, 2})    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))
Theorem	ehl2eudisval 25390	The value of the Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐸 = (𝔼hil‘2)    &   ⊢ 𝑋 = (ℝ ↑m {1, 2})    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2))))
12.5.9  Minimizing Vector Theorem
Theorem	minveclem1 25391*	Lemma for minvec 25403. 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    ⇒   ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
Theorem	minveclem4c 25392*	Lemma for minvec 25403. 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    ⇒   ⊢ (𝜑 → 𝑆 ∈ ℝ)
Theorem	minveclem2 25393*	Lemma for minvec 25403. 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑌)    &   ⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵))    &   ⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵))    ⇒   ⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))
Theorem	minveclem3a 25394*	Lemma for minvec 25403. 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
Theorem	minveclem3b 25395*	Lemma for minvec 25403. 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    ⇒   ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))
Theorem	minveclem3 25396*	Lemma for minvec 25403. 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    ⇒   ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))
Theorem	minveclem4a 25397*	Lemma for minvec 25403. 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹))    ⇒   ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
Theorem	minveclem4b 25398*	Lemma for minvec 25403. 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹))    ⇒   ⊢ (𝜑 → 𝑃 ∈ 𝑋)
Theorem	minveclem4 25399*	Lemma for minvec 25403. 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹))    &   ⊢ 𝑇 = (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
Theorem	minveclem5 25400*	Lemma for minvec 25403. Discharge the assumptions in minveclem4 25399. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑋 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))