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Theorem List for Metamath Proof Explorer - 25101-25200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcmetcusp1 25101 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 = (Baseβ€˜πΉ)    &   π· = ((distβ€˜πΉ) β†Ύ (𝑋 Γ— 𝑋))    &   π‘ˆ = (UnifStβ€˜πΉ)    β‡’   ((𝑋 β‰  βˆ… ∧ 𝐹 ∈ CMetSp ∧ π‘ˆ = (metUnifβ€˜π·)) β†’ 𝐹 ∈ CUnifSp)
 
Theoremcmetcusp 25102 The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
((𝑋 β‰  βˆ… ∧ 𝐷 ∈ (CMetβ€˜π‘‹)) β†’ (toUnifSpβ€˜(metUnifβ€˜π·)) ∈ CUnifSp)
 
Theoremcncms 25103 The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
β„‚fld ∈ CMetSp
 
Theoremcnflduss 25104 The uniform structure of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
π‘ˆ = (UnifStβ€˜β„‚fld)    β‡’   π‘ˆ = (metUnifβ€˜(abs ∘ βˆ’ ))
 
Theoremcnfldcusp 25105 The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
β„‚fld ∈ CUnifSp
 
Theoremresscdrg 25106 The real numbers are a subset of any complete subfield in the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (β„‚fld β†Ύs 𝐾)    β‡’   ((𝐾 ∈ (SubRingβ€˜β„‚fld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) β†’ ℝ βŠ† 𝐾)
 
Theoremcncdrg 25107 The only complete subfields of the complex numbers are ℝ and β„‚. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (β„‚fld β†Ύs 𝐾)    β‡’   ((𝐾 ∈ (SubRingβ€˜β„‚fld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) β†’ 𝐾 ∈ {ℝ, β„‚})
 
Theoremsrabn 25108 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 β€˜π‘Š)β€˜π‘†)    &   π½ = (TopOpenβ€˜π‘Š)    β‡’   ((π‘Š ∈ NrmRing ∧ π‘Š ∈ CMetSp ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) β†’ (𝐴 ∈ Ban ↔ (𝑆 ∈ (Clsdβ€˜π½) ∧ (π‘Š β†Ύs 𝑆) ∈ DivRing)))
 
Theoremrlmbn 25109 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) β†’ (ringLModβ€˜π‘…) ∈ Ban)
 
Theoremishl 25110 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.)
(π‘Š ∈ β„‚Hil ↔ (π‘Š ∈ Ban ∧ π‘Š ∈ β„‚PreHil))
 
Theoremhlbn 25111 Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
(π‘Š ∈ β„‚Hil β†’ π‘Š ∈ Ban)
 
Theoremhlcph 25112 Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(π‘Š ∈ β„‚Hil β†’ π‘Š ∈ β„‚PreHil)
 
Theoremhlphl 25113 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.)
(π‘Š ∈ β„‚Hil β†’ π‘Š ∈ PreHil)
 
Theoremhlcms 25114 Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
(π‘Š ∈ β„‚Hil β†’ π‘Š ∈ CMetSp)
 
Theoremhlprlem 25115 Lemma for hlpr 25117. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ β„‚Hil β†’ (𝐾 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝐾) ∈ DivRing ∧ (β„‚fld β†Ύs 𝐾) ∈ CMetSp))
 
Theoremhlress 25116 The scalar field of a subcomplex Hilbert space contains ℝ. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ β„‚Hil β†’ ℝ βŠ† 𝐾)
 
Theoremhlpr 25117 The scalar field of a subcomplex Hilbert space is either ℝ or β„‚. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ β„‚Hil β†’ 𝐾 ∈ {ℝ, β„‚})
 
Theoremishl2 25118 A Hilbert space is a complete subcomplex pre-Hilbert space over ℝ or β„‚. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ β„‚Hil ↔ (π‘Š ∈ CMetSp ∧ π‘Š ∈ β„‚PreHil ∧ 𝐾 ∈ {ℝ, β„‚}))
 
Theoremcphssphl 25119 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 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ π‘ˆ ∈ 𝑆 ∧ 𝑋 ∈ Ban) β†’ 𝑋 ∈ β„‚Hil)
 
Theoremcmslssbn 25120 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 25084. (Contributed by AV, 8-Oct-2022.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (((π‘Š ∈ NrmVec ∧ (Scalarβ€˜π‘Š) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ π‘ˆ ∈ 𝑆)) β†’ 𝑋 ∈ Ban)
 
Theoremcmscsscms 25121 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 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (ClSubSpβ€˜π‘Š)    β‡’   (((π‘Š ∈ CMetSp ∧ π‘Š ∈ β„‚PreHil) ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ CMetSp)
 
Theorembncssbn 25122 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 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (ClSubSpβ€˜π‘Š)    β‡’   (((π‘Š ∈ Ban ∧ π‘Š ∈ β„‚PreHil) ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ Ban)
 
Theoremcssbn 25123 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 Cβ„‹ (df-ch 30741) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 25120. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π· = ((distβ€˜π‘Š) β†Ύ (π‘ˆ Γ— π‘ˆ))    β‡’   (((π‘Š ∈ NrmVec ∧ (Scalarβ€˜π‘Š) ∈ CMetSp ∧ π‘ˆ ∈ 𝑆) ∧ (Cauβ€˜π·) βŠ† dom (β‡π‘‘β€˜(MetOpenβ€˜π·))) β†’ 𝑋 ∈ Ban)
 
Theoremcsschl 25124 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 Cβ„‹ (df-ch 30741) 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 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π· = ((distβ€˜π‘Š) β†Ύ (π‘ˆ Γ— π‘ˆ))    &   (Scalarβ€˜π‘Š) = β„‚fld    β‡’   ((π‘Š ∈ β„‚PreHil ∧ π‘ˆ ∈ 𝑆 ∧ (Cauβ€˜π·) βŠ† dom (β‡π‘‘β€˜(MetOpenβ€˜π·))) β†’ (𝑋 ∈ β„‚Hil ∧ (Scalarβ€˜π‘‹) = β„‚fld))
 
Theoremcmslsschl 25125 A complete linear subspace of a subcomplex Hilbert space is a subcomplex Hilbert space. (Contributed by AV, 8-Oct-2022.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚Hil ∧ 𝑋 ∈ CMetSp ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ β„‚Hil)
 
Theoremchlcsschl 25126 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 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (ClSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚Hil ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ β„‚Hil)
 
12.5.7.1  The complete ordered field of the real numbers
 
Theoremretopn 25127 The topology of the real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
(topGenβ€˜ran (,)) = (TopOpenβ€˜β„fld)
 
Theoremrecms 25128 The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.)
ℝfld ∈ CMetSp
 
Theoremreust 25129 The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.)
(UnifStβ€˜β„fld) = (metUnifβ€˜((distβ€˜β„fld) β†Ύ (ℝ Γ— ℝ)))
 
Theoremrecusp 25130 The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
ℝfld ∈ CUnifSp
 
12.5.8  Euclidean spaces
 
Syntaxcrrx 25131 Extend class notation with generalized real Euclidean spaces.
class ℝ^
 
Syntaxcehl 25132 Extend class notation with real Euclidean spaces.
class 𝔼hil
 
Definitiondf-rrx 25133 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 ↦ (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝑖)))
 
Definitiondf-ehl 25134 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 25165). Members of this family of spaces are Hilbert spaces, as shown in - ehlhl . (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝔼hil = (𝑛 ∈ β„•0 ↦ (ℝ^β€˜(1...𝑛)))
 
Theoremrrxval 25135 Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^β€˜πΌ)    β‡’   (𝐼 ∈ 𝑉 β†’ 𝐻 = (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝐼)))
 
Theoremrrxbase 25136* 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^β€˜πΌ)    &   π΅ = (Baseβ€˜π»)    β‡’   (𝐼 ∈ 𝑉 β†’ 𝐡 = {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0})
 
Theoremrrxprds 25137 Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^β€˜πΌ)    &   π΅ = (Baseβ€˜π»)    β‡’   (𝐼 ∈ 𝑉 β†’ 𝐻 = (toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)))
 
Theoremrrxip 25138* The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^β€˜πΌ)    &   π΅ = (Baseβ€˜π»)    β‡’   (𝐼 ∈ 𝑉 β†’ (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯))))) = (Β·π‘–β€˜π»))
 
Theoremrrxnm 25139* The norm of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^β€˜πΌ)    &   π΅ = (Baseβ€˜π»)    β‡’   (𝐼 ∈ 𝑉 β†’ (𝑓 ∈ 𝐡 ↦ (βˆšβ€˜(ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)↑2))))) = (normβ€˜π»))
 
Theoremrrxcph 25140 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^β€˜πΌ)    &   π΅ = (Baseβ€˜π»)    β‡’   (𝐼 ∈ 𝑉 β†’ 𝐻 ∈ β„‚PreHil)
 
Theoremrrxds 25141* The distance over generalized Euclidean spaces. Compare with df-rrn 36997. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)
𝐻 = (ℝ^β€˜πΌ)    &   π΅ = (Baseβ€˜π»)    β‡’   (𝐼 ∈ 𝑉 β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (βˆšβ€˜(ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ (((π‘“β€˜π‘₯) βˆ’ (π‘”β€˜π‘₯))↑2))))) = (distβ€˜π»))
 
Theoremrrxvsca 25142 The scalar product over generalized Euclidean spaces is the componentwise real number multiplication. (Contributed by Thierry Arnoux, 18-Jan-2023.)
𝐻 = (ℝ^β€˜πΌ)    &   π΅ = (Baseβ€˜π»)    &    βˆ™ = ( ·𝑠 β€˜π»)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐽 ∈ 𝐼)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜π»))    β‡’   (πœ‘ β†’ ((𝐴 βˆ™ 𝑋)β€˜π½) = (𝐴 Β· (π‘‹β€˜π½)))
 
Theoremrrxplusgvscavalb 25143* 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^β€˜πΌ)    &   π΅ = (Baseβ€˜π»)    &    βˆ™ = ( ·𝑠 β€˜π»)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &    ✚ = (+gβ€˜π»)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝑍 = ((𝐴 βˆ™ 𝑋) ✚ (𝐢 βˆ™ π‘Œ)) ↔ βˆ€π‘– ∈ 𝐼 (π‘β€˜π‘–) = ((𝐴 Β· (π‘‹β€˜π‘–)) + (𝐢 Β· (π‘Œβ€˜π‘–)))))
 
Theoremrrxsca 25144 The field of real numbers is the scalar field of the generalized real Euclidean space. (Contributed by AV, 15-Jan-2023.)
𝐻 = (ℝ^β€˜πΌ)    β‡’   (𝐼 ∈ 𝑉 β†’ (Scalarβ€˜π») = ℝfld)
 
Theoremrrx0 25145 The zero ("origin") in a generalized real Euclidean space. (Contributed by AV, 11-Feb-2023.)
𝐻 = (ℝ^β€˜πΌ)    &    0 = (𝐼 Γ— {0})    β‡’   (𝐼 ∈ 𝑉 β†’ (0gβ€˜π») = 0 )
 
Theoremrrx0el 25146 The zero ("origin") in a generalized real Euclidean space is an element of its base set. (Contributed by AV, 11-Feb-2023.)
0 = (𝐼 Γ— {0})    &   π‘ƒ = (ℝ ↑m 𝐼)    β‡’   (𝐼 ∈ 𝑉 β†’ 0 ∈ 𝑃)
 
Theoremcsbren 25147* Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ (Ξ£π‘˜ ∈ 𝐴 (𝐡 Β· 𝐢)↑2) ≀ (Ξ£π‘˜ ∈ 𝐴 (𝐡↑2) Β· Ξ£π‘˜ ∈ 𝐴 (𝐢↑2)))
 
Theoremtrirn 25148* Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ (βˆšβ€˜Ξ£π‘˜ ∈ 𝐴 ((𝐡 + 𝐢)↑2)) ≀ ((βˆšβ€˜Ξ£π‘˜ ∈ 𝐴 (𝐡↑2)) + (βˆšβ€˜Ξ£π‘˜ ∈ 𝐴 (𝐢↑2))))
 
Theoremrrxf 25149* Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}    &   (πœ‘ β†’ 𝐹 ∈ 𝑋)    β‡’   (πœ‘ β†’ 𝐹:πΌβŸΆβ„)
 
Theoremrrxfsupp 25150* Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}    &   (πœ‘ β†’ 𝐹 ∈ 𝑋)    β‡’   (πœ‘ β†’ (𝐹 supp 0) ∈ Fin)
 
Theoremrrxsuppss 25151* Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}    &   (πœ‘ β†’ 𝐹 ∈ 𝑋)    β‡’   (πœ‘ β†’ (𝐹 supp 0) βŠ† 𝐼)
 
Theoremrrxmvallem 25152* Support of the function used for building the distance . (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}    β‡’   ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) β†’ ((π‘˜ ∈ 𝐼 ↦ (((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜))↑2)) supp 0) βŠ† ((𝐹 supp 0) βˆͺ (𝐺 supp 0)))
 
Theoremrrxmval 25153* The value of the Euclidean metric. Compare with rrnmval 36999. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}    &   π· = (distβ€˜(ℝ^β€˜πΌ))    β‡’   ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) β†’ (𝐹𝐷𝐺) = (βˆšβ€˜Ξ£π‘˜ ∈ ((𝐹 supp 0) βˆͺ (𝐺 supp 0))(((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜))↑2)))
 
Theoremrrxmfval 25154* The value of the Euclidean metric. Compare with rrnval 36998. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}    &   π· = (distβ€˜(ℝ^β€˜πΌ))    β‡’   (𝐼 ∈ 𝑉 β†’ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (βˆšβ€˜Ξ£π‘˜ ∈ ((𝑓 supp 0) βˆͺ (𝑔 supp 0))(((π‘“β€˜π‘˜) βˆ’ (π‘”β€˜π‘˜))↑2))))
 
Theoremrrxmetlem 25155* Lemma for rrxmet 25156. (Contributed by Thierry Arnoux, 5-Jul-2019.)
𝑋 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}    &   π· = (distβ€˜(ℝ^β€˜πΌ))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹 ∈ 𝑋)    &   (πœ‘ β†’ 𝐺 ∈ 𝑋)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐼)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ ((𝐹 supp 0) βˆͺ (𝐺 supp 0)) βŠ† 𝐴)    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ ((𝐹 supp 0) βˆͺ (𝐺 supp 0))(((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜))↑2) = Ξ£π‘˜ ∈ 𝐴 (((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜))↑2))
 
Theoremrrxmet 25156* 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 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}    &   π· = (distβ€˜(ℝ^β€˜πΌ))    β‡’   (𝐼 ∈ 𝑉 β†’ 𝐷 ∈ (Metβ€˜π‘‹))
 
Theoremrrxdstprj1 25157* 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 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}    &   π· = (distβ€˜(ℝ^β€˜πΌ))    &   π‘€ = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))    β‡’   (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) β†’ ((πΉβ€˜π΄)𝑀(πΊβ€˜π΄)) ≀ (𝐹𝐷𝐺))
 
Theoremrrxbasefi 25158 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.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   π» = (ℝ^β€˜π‘‹)    &   π΅ = (Baseβ€˜π»)    β‡’   (πœ‘ β†’ 𝐡 = (ℝ ↑m 𝑋))
 
Theoremrrxdsfi 25159* The distance over generalized Euclidean spaces. Finite dimensional case. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐻 = (ℝ^β€˜πΌ)    &   π΅ = (ℝ ↑m 𝐼)    β‡’   (𝐼 ∈ Fin β†’ (distβ€˜π») = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (βˆšβ€˜Ξ£π‘˜ ∈ 𝐼 (((π‘“β€˜π‘˜) βˆ’ (π‘”β€˜π‘˜))↑2))))
 
Theoremrrxmetfi 25160 Euclidean space is a metric space. Finite dimensional version. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐷 = (distβ€˜(ℝ^β€˜πΌ))    β‡’   (𝐼 ∈ Fin β†’ 𝐷 ∈ (Metβ€˜(ℝ ↑m 𝐼)))
 
Theoremrrxdsfival 25161* 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)    &   π· = (distβ€˜(ℝ^β€˜πΌ))    β‡’   ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) β†’ (𝐹𝐷𝐺) = (βˆšβ€˜Ξ£π‘˜ ∈ 𝐼 (((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜))↑2)))
 
Theoremehlval 25162 Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐸 = (𝔼hilβ€˜π‘)    β‡’   (𝑁 ∈ β„•0 β†’ 𝐸 = (ℝ^β€˜(1...𝑁)))
 
Theoremehlbase 25163 The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐸 = (𝔼hilβ€˜π‘)    β‡’   (𝑁 ∈ β„•0 β†’ (ℝ ↑m (1...𝑁)) = (Baseβ€˜πΈ))
 
Theoremehl0base 25164 The base of the Euclidean space of dimension 0 consists only of one element, the empty set. (Contributed by AV, 12-Feb-2023.)
𝐸 = (𝔼hilβ€˜0)    β‡’   (Baseβ€˜πΈ) = {βˆ…}
 
Theoremehl0 25165 The Euclidean space of dimension 0 consists of the neutral element only. (Contributed by AV, 12-Feb-2023.)
𝐸 = (𝔼hilβ€˜0)    &    0 = (0gβ€˜πΈ)    β‡’   (Baseβ€˜πΈ) = { 0 }
 
Theoremehleudis 25166* The Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
𝐼 = (1...𝑁)    &   πΈ = (𝔼hilβ€˜π‘)    &   π‘‹ = (ℝ ↑m 𝐼)    &   π· = (distβ€˜πΈ)    β‡’   (𝑁 ∈ β„•0 β†’ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (βˆšβ€˜Ξ£π‘˜ ∈ 𝐼 (((π‘“β€˜π‘˜) βˆ’ (π‘”β€˜π‘˜))↑2))))
 
Theoremehleudisval 25167* The value of the Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
𝐼 = (1...𝑁)    &   πΈ = (𝔼hilβ€˜π‘)    &   π‘‹ = (ℝ ↑m 𝐼)    &   π· = (distβ€˜πΈ)    β‡’   ((𝑁 ∈ β„•0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) β†’ (𝐹𝐷𝐺) = (βˆšβ€˜Ξ£π‘˜ ∈ 𝐼 (((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜))↑2)))
 
Theoremehl1eudis 25168* The Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.)
𝐸 = (𝔼hilβ€˜1)    &   π‘‹ = (ℝ ↑m {1})    &   π· = (distβ€˜πΈ)    β‡’   π· = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (absβ€˜((π‘“β€˜1) βˆ’ (π‘”β€˜1))))
 
Theoremehl1eudisval 25169 The value of the Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.)
𝐸 = (𝔼hilβ€˜1)    &   π‘‹ = (ℝ ↑m {1})    &   π· = (distβ€˜πΈ)    β‡’   ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) β†’ (𝐹𝐷𝐺) = (absβ€˜((πΉβ€˜1) βˆ’ (πΊβ€˜1))))
 
Theoremehl2eudis 25170* The Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.)
𝐸 = (𝔼hilβ€˜2)    &   π‘‹ = (ℝ ↑m {1, 2})    &   π· = (distβ€˜πΈ)    β‡’   π· = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (βˆšβ€˜((((π‘“β€˜1) βˆ’ (π‘”β€˜1))↑2) + (((π‘“β€˜2) βˆ’ (π‘”β€˜2))↑2))))
 
Theoremehl2eudisval 25171 The value of the Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.)
𝐸 = (𝔼hilβ€˜2)    &   π‘‹ = (ℝ ↑m {1, 2})    &   π· = (distβ€˜πΈ)    β‡’   ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) β†’ (𝐹𝐷𝐺) = (βˆšβ€˜((((πΉβ€˜1) βˆ’ (πΊβ€˜1))↑2) + (((πΉβ€˜2) βˆ’ (πΊβ€˜2))↑2))))
 
12.5.9  Minimizing Vector Theorem
 
Theoremminveclem1 25172* Lemma for minvec 25184. The set of all distances from points of π‘Œ to 𝐴 are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    β‡’   (πœ‘ β†’ (𝑅 βŠ† ℝ ∧ 𝑅 β‰  βˆ… ∧ βˆ€π‘€ ∈ 𝑅 0 ≀ 𝑀))
 
Theoremminveclem4c 25173* Lemma for minvec 25184. 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 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    β‡’   (πœ‘ β†’ 𝑆 ∈ ℝ)
 
Theoremminveclem2 25174* Lemma for minvec 25184. Any two points 𝐾 and 𝐿 in π‘Œ 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 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    &   π· = ((distβ€˜π‘ˆ) β†Ύ (𝑋 Γ— 𝑋))    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 0 ≀ 𝐡)    &   (πœ‘ β†’ 𝐾 ∈ π‘Œ)    &   (πœ‘ β†’ 𝐿 ∈ π‘Œ)    &   (πœ‘ β†’ ((𝐴𝐷𝐾)↑2) ≀ ((𝑆↑2) + 𝐡))    &   (πœ‘ β†’ ((𝐴𝐷𝐿)↑2) ≀ ((𝑆↑2) + 𝐡))    β‡’   (πœ‘ β†’ ((𝐾𝐷𝐿)↑2) ≀ (4 Β· 𝐡))
 
Theoremminveclem3a 25175* Lemma for minvec 25184. 𝐷 is a complete metric when restricted to π‘Œ. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    &   π· = ((distβ€˜π‘ˆ) β†Ύ (𝑋 Γ— 𝑋))    β‡’   (πœ‘ β†’ (𝐷 β†Ύ (π‘Œ Γ— π‘Œ)) ∈ (CMetβ€˜π‘Œ))
 
Theoremminveclem3b 25176* Lemma for minvec 25184. 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 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    &   π· = ((distβ€˜π‘ˆ) β†Ύ (𝑋 Γ— 𝑋))    &   πΉ = ran (π‘Ÿ ∈ ℝ+ ↦ {𝑦 ∈ π‘Œ ∣ ((𝐴𝐷𝑦)↑2) ≀ ((𝑆↑2) + π‘Ÿ)})    β‡’   (πœ‘ β†’ 𝐹 ∈ (fBasβ€˜π‘Œ))
 
Theoremminveclem3 25177* Lemma for minvec 25184. 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 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    &   π· = ((distβ€˜π‘ˆ) β†Ύ (𝑋 Γ— 𝑋))    &   πΉ = ran (π‘Ÿ ∈ ℝ+ ↦ {𝑦 ∈ π‘Œ ∣ ((𝐴𝐷𝑦)↑2) ≀ ((𝑆↑2) + π‘Ÿ)})    β‡’   (πœ‘ β†’ (π‘ŒfilGen𝐹) ∈ (CauFilβ€˜(𝐷 β†Ύ (π‘Œ Γ— π‘Œ))))
 
Theoremminveclem4a 25178* Lemma for minvec 25184. 𝐹 converges to a point 𝑃 in π‘Œ. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    &   π· = ((distβ€˜π‘ˆ) β†Ύ (𝑋 Γ— 𝑋))    &   πΉ = ran (π‘Ÿ ∈ ℝ+ ↦ {𝑦 ∈ π‘Œ ∣ ((𝐴𝐷𝑦)↑2) ≀ ((𝑆↑2) + π‘Ÿ)})    &   π‘ƒ = βˆͺ (𝐽 fLim (𝑋filGen𝐹))    β‡’   (πœ‘ β†’ 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ π‘Œ))
 
Theoremminveclem4b 25179* Lemma for minvec 25184. 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 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    &   π· = ((distβ€˜π‘ˆ) β†Ύ (𝑋 Γ— 𝑋))    &   πΉ = ran (π‘Ÿ ∈ ℝ+ ↦ {𝑦 ∈ π‘Œ ∣ ((𝐴𝐷𝑦)↑2) ≀ ((𝑆↑2) + π‘Ÿ)})    &   π‘ƒ = βˆͺ (𝐽 fLim (𝑋filGen𝐹))    β‡’   (πœ‘ β†’ 𝑃 ∈ 𝑋)
 
Theoremminveclem4 25180* Lemma for minvec 25184. The convergent point of the Cauchy sequence 𝐹 attains the minimum distance, and so is closer to 𝐴 than any other point in π‘Œ. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    &   π· = ((distβ€˜π‘ˆ) β†Ύ (𝑋 Γ— 𝑋))    &   πΉ = ran (π‘Ÿ ∈ ℝ+ ↦ {𝑦 ∈ π‘Œ ∣ ((𝐴𝐷𝑦)↑2) ≀ ((𝑆↑2) + π‘Ÿ)})    &   π‘ƒ = βˆͺ (𝐽 fLim (𝑋filGen𝐹))    &   π‘‡ = (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) βˆ’ (𝑆↑2))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ π‘Œ βˆ€π‘¦ ∈ π‘Œ (π‘β€˜(𝐴 βˆ’ π‘₯)) ≀ (π‘β€˜(𝐴 βˆ’ 𝑦)))
 
Theoremminveclem5 25181* Lemma for minvec 25184. Discharge the assumptions in minveclem4 25180. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    &   π· = ((distβ€˜π‘ˆ) β†Ύ (𝑋 Γ— 𝑋))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ π‘Œ βˆ€π‘¦ ∈ π‘Œ (π‘β€˜(𝐴 βˆ’ π‘₯)) ≀ (π‘β€˜(𝐴 βˆ’ 𝑦)))
 
Theoremminveclem6 25182* Lemma for minvec 25184. 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 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    &   π· = ((distβ€˜π‘ˆ) β†Ύ (𝑋 Γ— 𝑋))    β‡’   ((πœ‘ ∧ π‘₯ ∈ π‘Œ) β†’ (((𝐴𝐷π‘₯)↑2) ≀ ((𝑆↑2) + 0) ↔ βˆ€π‘¦ ∈ π‘Œ (π‘β€˜(𝐴 βˆ’ π‘₯)) ≀ (π‘β€˜(𝐴 βˆ’ 𝑦))))
 
Theoremminveclem7 25183* Lemma for minvec 25184. 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 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   π½ = (TopOpenβ€˜π‘ˆ)    &   π‘… = ran (𝑦 ∈ π‘Œ ↦ (π‘β€˜(𝐴 βˆ’ 𝑦)))    &   π‘† = inf(𝑅, ℝ, < )    &   π· = ((distβ€˜π‘ˆ) β†Ύ (𝑋 Γ— 𝑋))    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ π‘Œ βˆ€π‘¦ ∈ π‘Œ (π‘β€˜(𝐴 βˆ’ π‘₯)) ≀ (π‘β€˜(𝐴 βˆ’ 𝑦)))
 
Theoremminvec 25184* Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace π‘Š that minimizes the distance to an arbitrary vector 𝐴 in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Proof shortened by AV, 3-Oct-2020.)
𝑋 = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (normβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘ˆ ∈ β„‚PreHil)    &   (πœ‘ β†’ π‘Œ ∈ (LSubSpβ€˜π‘ˆ))    &   (πœ‘ β†’ (π‘ˆ β†Ύs π‘Œ) ∈ CMetSp)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ π‘Œ βˆ€π‘¦ ∈ π‘Œ (π‘β€˜(𝐴 βˆ’ π‘₯)) ≀ (π‘β€˜(𝐴 βˆ’ 𝑦)))
 
12.5.10  Projection Theorem
 
Theorempjthlem1 25185* Lemma for pjth 25187. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 17-Oct-2015.) (Proof shortened by AV, 10-Jul-2022.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   πΏ = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ β„‚Hil)    &   (πœ‘ β†’ π‘ˆ ∈ 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ π‘ˆ (π‘β€˜π΄) ≀ (π‘β€˜(𝐴 βˆ’ π‘₯)))    &   π‘‡ = ((𝐴 , 𝐡) / ((𝐡 , 𝐡) + 1))    β‡’   (πœ‘ β†’ (𝐴 , 𝐡) = 0)
 
Theorempjthlem2 25186 Lemma for pjth 25187. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   πΏ = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ β„‚Hil)    &   (πœ‘ β†’ π‘ˆ ∈ 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   π½ = (TopOpenβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘‚ = (ocvβ€˜π‘Š)    &   (πœ‘ β†’ π‘ˆ ∈ (Clsdβ€˜π½))    β‡’   (πœ‘ β†’ 𝐴 ∈ (π‘ˆ βŠ• (π‘‚β€˜π‘ˆ)))
 
Theorempjth 25187 Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely into a member π‘₯ of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.)
𝑉 = (Baseβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘‚ = (ocvβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   πΏ = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚Hil ∧ π‘ˆ ∈ 𝐿 ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (π‘ˆ βŠ• (π‘‚β€˜π‘ˆ)) = 𝑉)
 
Theorempjth2 25188 Projection Theorem with abbreviations: A topologically closed subspace is a projection subspace. (Contributed by Mario Carneiro, 17-Oct-2015.)
𝐽 = (TopOpenβ€˜π‘Š)    &   πΏ = (LSubSpβ€˜π‘Š)    &   πΎ = (projβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚Hil ∧ π‘ˆ ∈ 𝐿 ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ π‘ˆ ∈ dom 𝐾)
 
Theoremcldcss 25189 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   πΏ = (LSubSpβ€˜π‘Š)    &   πΆ = (ClSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ β„‚Hil β†’ (π‘ˆ ∈ 𝐢 ↔ (π‘ˆ ∈ 𝐿 ∧ π‘ˆ ∈ (Clsdβ€˜π½))))
 
Theoremcldcss2 25190 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   πΏ = (LSubSpβ€˜π‘Š)    &   πΆ = (ClSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ β„‚Hil β†’ 𝐢 = (𝐿 ∩ (Clsdβ€˜π½)))
 
Theoremhlhil 25191 Corollary of the Projection Theorem: A subcomplex Hilbert space is a Hilbert space (in the algebraic sense, meaning that all algebraically closed subspaces have a projection decomposition). (Contributed by Mario Carneiro, 17-Oct-2015.)
(π‘Š ∈ β„‚Hil β†’ π‘Š ∈ Hil)
 
PART 13  BASIC REAL AND COMPLEX ANALYSIS
 
13.1  Continuity
 
Theoremaddcncf 25192* The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 + 𝐡)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremsubcncf 25193* The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 βˆ’ 𝐡)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremmulcncf 25194* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
(πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡)) ∈ (𝑋–cnβ†’β„‚))
 
TheoremmulcncfOLD 25195* Obsolete version of mulcncf 25194 as of 9-Apr-2025. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremdivcncf 25196* The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cnβ†’(β„‚ βˆ– {0})))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 / 𝐡)) ∈ (𝑋–cnβ†’β„‚))
 
13.1.1  Intermediate value theorem
 
Theorempmltpclem1 25197* Lemma for pmltpc 25199. (Contributed by Mario Carneiro, 1-Jul-2014.)
(πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ 𝑆)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐡 < 𝐢)    &   (πœ‘ β†’ (((πΉβ€˜π΄) < (πΉβ€˜π΅) ∧ (πΉβ€˜πΆ) < (πΉβ€˜π΅)) ∨ ((πΉβ€˜π΅) < (πΉβ€˜π΄) ∧ (πΉβ€˜π΅) < (πΉβ€˜πΆ))))    β‡’   (πœ‘ β†’ βˆƒπ‘Ž ∈ 𝑆 βˆƒπ‘ ∈ 𝑆 βˆƒπ‘ ∈ 𝑆 (π‘Ž < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((πΉβ€˜π‘Ž) < (πΉβ€˜π‘) ∧ (πΉβ€˜π‘) < (πΉβ€˜π‘)) ∨ ((πΉβ€˜π‘) < (πΉβ€˜π‘Ž) ∧ (πΉβ€˜π‘) < (πΉβ€˜π‘)))))
 
Theorempmltpclem2 25198* Lemma for pmltpc 25199. (Contributed by Mario Carneiro, 1-Jul-2014.)
(πœ‘ β†’ 𝐹 ∈ (ℝ ↑pm ℝ))    &   (πœ‘ β†’ 𝐴 βŠ† dom 𝐹)    &   (πœ‘ β†’ π‘ˆ ∈ 𝐴)    &   (πœ‘ β†’ 𝑉 ∈ 𝐴)    &   (πœ‘ β†’ π‘Š ∈ 𝐴)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ π‘ˆ ≀ 𝑉)    &   (πœ‘ β†’ π‘Š ≀ 𝑋)    &   (πœ‘ β†’ Β¬ (πΉβ€˜π‘ˆ) ≀ (πΉβ€˜π‘‰))    &   (πœ‘ β†’ Β¬ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘Š))    β‡’   (πœ‘ β†’ βˆƒπ‘Ž ∈ 𝐴 βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ ∈ 𝐴 (π‘Ž < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((πΉβ€˜π‘Ž) < (πΉβ€˜π‘) ∧ (πΉβ€˜π‘) < (πΉβ€˜π‘)) ∨ ((πΉβ€˜π‘) < (πΉβ€˜π‘Ž) ∧ (πΉβ€˜π‘) < (πΉβ€˜π‘)))))
 
Theorempmltpc 25199* Any function on the reals is either increasing, decreasing, or has a triple of points in a vee formation. (This theorem was created on demand by Mario Carneiro for the 6PCM conference in Bialystok, 1-Jul-2014.) (Contributed by Mario Carneiro, 1-Jul-2014.)
((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 βŠ† dom 𝐹) β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)) ∨ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯)) ∨ βˆƒπ‘Ž ∈ 𝐴 βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ ∈ 𝐴 (π‘Ž < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((πΉβ€˜π‘Ž) < (πΉβ€˜π‘) ∧ (πΉβ€˜π‘) < (πΉβ€˜π‘)) ∨ ((πΉβ€˜π‘) < (πΉβ€˜π‘Ž) ∧ (πΉβ€˜π‘) < (πΉβ€˜π‘))))))
 
Theoremivthlem1 25200* Lemma for ivth 25203. The set 𝑆 of all π‘₯ values with (πΉβ€˜π‘₯) less than π‘ˆ is lower bounded by 𝐴 and upper bounded by 𝐡. (Contributed by Mario Carneiro, 17-Jun-2014.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   π‘† = {π‘₯ ∈ (𝐴[,]𝐡) ∣ (πΉβ€˜π‘₯) ≀ π‘ˆ}    β‡’   (πœ‘ β†’ (𝐴 ∈ 𝑆 ∧ βˆ€π‘§ ∈ 𝑆 𝑧 ≀ 𝐡))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-47939
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