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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcncdrg 23201 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 23202 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 23203 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 23204 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 23205 Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
(𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
 
Theoremhlcph 23206 Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
 
Theoremhlphl 23207 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 23208 Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
(𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
 
Theoremhlprlem 23209 Lemma for hlpr 23211. (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 23210 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 23211 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 23212 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 23213 The topology of the real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
(topGen‘ran (,)) = (TopOpen‘ℝfld)
 
Theoremrecms 23214 The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.)
fld ∈ CMetSp
 
Theoremreust 23215 The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.)
(UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ)))
 
Theoremrecusp 23216 The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
fld ∈ CUnifSp
 
12.5.8  Euclidean spaces
 
Syntaxcrrx 23217 Extend class notation with generalized real Euclidean spaces.
class ℝ^
 
Syntaxcehl 23218 Extend class notation with real Euclidean spaces.
class 𝔼hil
 
Definitiondf-rrx 23219 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 23220 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 23221 Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)       (𝐼𝑉𝐻 = (toℂHil‘(ℝfld freeLMod 𝐼)))
 
Theoremrrxbase 23222* 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 23223 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 23224* 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 23225* 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 23226 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 23227* The distance over generalized Euclidean spaces. Compare with df-rrn 33755. (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 23228* 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 23229* 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 23230* Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑𝐹:𝐼⟶ℝ)
 
Theoremrrxfsupp 23231* 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 23232* Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑 → (𝐹 supp 0) ⊆ 𝐼)
 
Theoremrrxmvallem 23233* 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 23234* The value of the Euclidean metric. Compare with rrnmval 33757. (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 23235* The value of the Euclidean metric. Compare with rrnval 33756. (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 23236* Lemma for rrxmet 23237. (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 23237* 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 23238* 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 23239 Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐸 = (𝔼hil𝑁)       (𝑁 ∈ ℕ0𝐸 = (ℝ^‘(1...𝑁)))
 
Theoremehlbase 23240 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 23241* Lemma for minvec 23253. 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 23242* Lemma for minvec 23253. 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 23243* Lemma for minvec 23253. 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 23244* Lemma for minvec 23253. 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 23245* Lemma for minvec 23253. 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 23246* Lemma for minvec 23253. 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 23247* Lemma for minvec 23253. 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 23248* Lemma for minvec 23253. 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 23249* Lemma for minvec 23253. 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 23250* Lemma for minvec 23253. Discharge the assumptions in minveclem4 23249. (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 23251* Lemma for minvec 23253. 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 23252* Lemma for minvec 23253. 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)))
 
Theoremminvec 23253* Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace 𝑊 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 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)       (𝜑 → ∃!𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))
 
12.5.10  Projection Theorem
 
Theorempjthlem1 23254* Lemma for pjth 23256. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 17-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &    , = (·𝑖𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ ℂHil)    &   (𝜑𝑈𝐿)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑈)    &   (𝜑 → ∀𝑥𝑈 (𝑁𝐴) ≤ (𝑁‘(𝐴 𝑥)))    &   𝑇 = ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))       (𝜑 → (𝐴 , 𝐵) = 0)
 
Theorempjthlem2 23255 Lemma for pjth 23256. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &    , = (·𝑖𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ ℂHil)    &   (𝜑𝑈𝐿)    &   (𝜑𝐴𝑉)    &   𝐽 = (TopOpen‘𝑊)    &    = (LSSum‘𝑊)    &   𝑂 = (ocv‘𝑊)    &   (𝜑𝑈 ∈ (Clsd‘𝐽))       (𝜑𝐴 ∈ (𝑈 (𝑂𝑈)))
 
Theorempjth 23256 Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely into a member 𝑥 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 = (Base‘𝑊)    &    = (LSSum‘𝑊)    &   𝑂 = (ocv‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ ℂHil ∧ 𝑈𝐿𝑈 ∈ (Clsd‘𝐽)) → (𝑈 (𝑂𝑈)) = 𝑉)
 
Theorempjth2 23257 Projection Theorem with abbreviations: A topologically closed subspace is a projection subspace. (Contributed by Mario Carneiro, 17-Oct-2015.)
𝐽 = (TopOpen‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   𝐾 = (proj‘𝑊)       ((𝑊 ∈ ℂHil ∧ 𝑈𝐿𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ dom 𝐾)
 
Theoremcldcss 23258 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊 ∈ ℂHil → (𝑈𝐶 ↔ (𝑈𝐿𝑈 ∈ (Clsd‘𝐽))))
 
Theoremcldcss2 23259 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊 ∈ ℂHil → 𝐶 = (𝐿 ∩ (Clsd‘𝐽)))
 
Theoremhlhil 23260 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.)
(𝑊 ∈ ℂHil → 𝑊 ∈ Hil)
 
PART 13  BASIC REAL AND COMPLEX ANALYSIS
 
13.1  Continuity
 
Theoremmulcncf 23261* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋cn→ℂ))
 
Theoremdivcncf 23262* The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→(ℂ ∖ {0})))       (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐵)) ∈ (𝑋cn→ℂ))
 
13.1.1  Intermediate value theorem
 
Theorempmltpclem1 23263* Lemma for pmltpc 23265. (Contributed by Mario Carneiro, 1-Jul-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵 < 𝐶)    &   (𝜑 → (((𝐹𝐴) < (𝐹𝐵) ∧ (𝐹𝐶) < (𝐹𝐵)) ∨ ((𝐹𝐵) < (𝐹𝐴) ∧ (𝐹𝐵) < (𝐹𝐶))))       (𝜑 → ∃𝑎𝑆𝑏𝑆𝑐𝑆 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐)))))
 
Theorempmltpclem2 23264* Lemma for pmltpc 23265. (Contributed by Mario Carneiro, 1-Jul-2014.)
(𝜑𝐹 ∈ (ℝ ↑pm ℝ))    &   (𝜑𝐴 ⊆ dom 𝐹)    &   (𝜑𝑈𝐴)    &   (𝜑𝑉𝐴)    &   (𝜑𝑊𝐴)    &   (𝜑𝑋𝐴)    &   (𝜑𝑈𝑉)    &   (𝜑𝑊𝑋)    &   (𝜑 → ¬ (𝐹𝑈) ≤ (𝐹𝑉))    &   (𝜑 → ¬ (𝐹𝑋) ≤ (𝐹𝑊))       (𝜑 → ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐)))))
 
Theorempmltpc 23265* 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 ⊆ dom 𝐹) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)) ∨ ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
 
Theoremivthlem1 23266* Lemma for ivth 23269. The set 𝑆 of all 𝑥 values with (𝐹𝑥) less than 𝑈 is lower bounded by 𝐴 and upper bounded by 𝐵. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}       (𝜑 → (𝐴𝑆 ∧ ∀𝑧𝑆 𝑧𝐵))
 
Theoremivthlem2 23267* Lemma for ivth 23269. Show that the supremum of 𝑆 cannot be less than 𝑈. If it was, continuity of 𝐹 implies that there are points just above the supremum that are also less than 𝑈, a contradiction. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}    &   𝐶 = sup(𝑆, ℝ, < )       (𝜑 → ¬ (𝐹𝐶) < 𝑈)
 
Theoremivthlem3 23268* Lemma for ivth 23269, the intermediate value theorem. Show that (𝐹𝐶) cannot be greater than 𝑈, and so establish the existence of a root of the function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 17-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}    &   𝐶 = sup(𝑆, ℝ, < )       (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ∧ (𝐹𝐶) = 𝑈))
 
Theoremivth 23269* The intermediate value theorem, increasing case. This is Metamath 100 proof #79. (Contributed by Paul Chapman, 22-Jan-2008.) (Proof shortened by Mario Carneiro, 30-Apr-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))       (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivth2 23270* The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐵) < 𝑈𝑈 < (𝐹𝐴)))       (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivthle 23271* The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) ≤ 𝑈𝑈 ≤ (𝐹𝐵)))       (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivthle2 23272* The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐵) ≤ 𝑈𝑈 ≤ (𝐹𝐴)))       (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivthicc 23273* The interval between any two points of a continuous real function is contained in the range of the function. Equivalently, the range of a continuous real function is convex. (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ (𝐴[,]𝐵))    &   (𝜑𝑁 ∈ (𝐴[,]𝐵))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)       (𝜑 → ((𝐹𝑀)[,](𝐹𝑁)) ⊆ ran 𝐹)
 
Theoremevthicc 23274* Specialization of the Extreme Value Theorem to a closed interval of . (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))       (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹𝑧) ≤ (𝐹𝑤)))
 
Theoremevthicc2 23275* Combine ivthicc 23273 with evthicc 23274 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))       (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran 𝐹 = (𝑥[,]𝑦))
 
Theoremcniccbdd 23276* A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹𝑦)) ≤ 𝑥)
 
13.2  Integrals
 
13.2.1  Lebesgue measure
 
Syntaxcovol 23277 Extend class notation with the outer Lebesgue measure.
class vol*
 
Syntaxcvol 23278 Extend class notation with the Lebesgue measure.
class vol
 
Definitiondf-ovol 23279* Define the outer Lebesgue measure for subsets of the reals. Here 𝑓 is a function from the positive integers to pairs 𝑎, 𝑏 with 𝑎𝑏, and the outer volume of the set 𝑥 is the infimum over all such functions such that the union of the open intervals (𝑎, 𝑏) covers 𝑥 of the sum of 𝑏𝑎. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
 
Definitiondf-vol 23280* Define the Lebesgue measure, which is just the outer measure with a peculiar domain of definition. The property of being Lebesgue-measurable can be expressed as 𝐴 ∈ dom vol. (Contributed by Mario Carneiro, 17-Mar-2014.)
vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})
 
Theoremovolfcl 23281 Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
 
Theoremovolfioo 23282* Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
 
Theoremovolficc 23283* Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
 
Theoremovolficcss 23284 Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)
(𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
 
Theoremovolfsval 23285 The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
 
Theoremovolfsf 23286 Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))
 
Theoremovolsf 23287 Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)    &   𝑆 = seq1( + , 𝐺)       (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
 
Theoremovolval 23288* The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
 
Theoremelovolm 23289* Elementhood in the set 𝑀 of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝐵𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))
 
Theoremelovolmr 23290* Sufficient condition for elementhood in the set 𝑀. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐹)) → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
 
Theoremovolmge0 23291* The set 𝑀 is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝐵𝑀 → 0 ≤ 𝐵)
 
Theoremovolcl 23292 The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
 
Theoremovollb 23293 The outer volume is a lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
 
Theoremovolgelb 23294* The outer volume is the greatest lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑔))       ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
 
Theoremovolge0 23295 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
 
Theoremovolf 23296 The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
vol*:𝒫 ℝ⟶(0[,]+∞)
 
Theoremovollecl 23297 If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ (vol*‘𝐴) ≤ 𝐵) → (vol*‘𝐴) ∈ ℝ)
 
Theoremovolsslem 23298* Lemma for ovolss 23299. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    &   𝑁 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
 
Theoremovolss 23299 The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
 
Theoremovolsscl 23300 If a set is contained in another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐴) ∈ ℝ)
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