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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-rrx 24901 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 24902 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 24933). 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 24903 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 24904* 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 24905 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 24906* 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 24907* 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 24908 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 24909* The distance over generalized Euclidean spaces. Compare with df-rrn 36689. (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 24910 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 24911* 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 24912 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 24913 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 24914 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 24915* 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 24916* 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 24917* 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 24918* 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 24919* 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 24920* 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 24921* The value of the Euclidean metric. Compare with rrnmval 36691. (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 24922* The value of the Euclidean metric. Compare with rrnval 36690. (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 24923* Lemma for rrxmet 24924. (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 24924* 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 24925* 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 24926 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 24927* 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 24928 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 24929* 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 24930 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 24931 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 24932 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 24933 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 24934* 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 24935* 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 24936* 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 24937 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 24938* 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 24939 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 24940* Lemma for minvec 24952. 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 24941* Lemma for minvec 24952. 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 24942* Lemma for minvec 24952. 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 24943* Lemma for minvec 24952. 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 24944* Lemma for minvec 24952. 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 24945* Lemma for minvec 24952. 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 24946* Lemma for minvec 24952. 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 24947* Lemma for minvec 24952. 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 24948* Lemma for minvec 24952. 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 24949* Lemma for minvec 24952. Discharge the assumptions in minveclem4 24948. (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 24950* Lemma for minvec 24952. 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 24951* Lemma for minvec 24952. 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 24952* 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 24953* Lemma for pjth 24955. (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 24954 Lemma for pjth 24955. (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 24955 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 24956 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 24957 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 24958 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 24959 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 24960* 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 24961* 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 24962* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(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 24963* 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 24964* Lemma for pmltpc 24966. (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 24965* Lemma for pmltpc 24966. (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 24966* 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 24967* Lemma for ivth 24970. 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))
 
Theoremivthlem2 24968* Lemma for ivth 24970. Show that the supremum of 𝑆 cannot be less than π‘ˆ. If it was, continuity of 𝐹 implies that there are points just above the supremum that are also less than π‘ˆ, a contradiction. (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}    &   πΆ = sup(𝑆, ℝ, < )    β‡’   (πœ‘ β†’ Β¬ (πΉβ€˜πΆ) < π‘ˆ)
 
Theoremivthlem3 24969* Lemma for ivth 24970, the intermediate value theorem. Show that (πΉβ€˜πΆ) cannot be greater than π‘ˆ, 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.)
(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}    &   πΆ = sup(𝑆, ℝ, < )    β‡’   (πœ‘ β†’ (𝐢 ∈ (𝐴(,)𝐡) ∧ (πΉβ€˜πΆ) = π‘ˆ))
 
Theoremivth 24970* 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.)
(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΅)))    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ (𝐴(,)𝐡)(πΉβ€˜π‘) = π‘ˆ)
 
Theoremivth2 24971* The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-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΄)))    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ (𝐴(,)𝐡)(πΉβ€˜π‘) = π‘ˆ)
 
Theoremivthle 24972* The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-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΅)))    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ (𝐴[,]𝐡)(πΉβ€˜π‘) = π‘ˆ)
 
Theoremivthle2 24973* The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-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΄)))    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ (𝐴[,]𝐡)(πΉβ€˜π‘) = π‘ˆ)
 
Theoremivthicc 24974* 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝑀 ∈ (𝐴[,]𝐡))    &   (πœ‘ β†’ 𝑁 ∈ (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)    β‡’   (πœ‘ β†’ ((πΉβ€˜π‘€)[,](πΉβ€˜π‘)) βŠ† ran 𝐹)
 
Theoremevthicc 24975* Specialization of the Extreme Value Theorem to a closed interval of ℝ. (Contributed by Mario Carneiro, 12-Aug-2014.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cn→ℝ))    β‡’   (πœ‘ β†’ (βˆƒπ‘₯ ∈ (𝐴[,]𝐡)βˆ€π‘¦ ∈ (𝐴[,]𝐡)(πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯) ∧ βˆƒπ‘§ ∈ (𝐴[,]𝐡)βˆ€π‘€ ∈ (𝐴[,]𝐡)(πΉβ€˜π‘§) ≀ (πΉβ€˜π‘€)))
 
Theoremevthicc2 24976* Combine ivthicc 24974 with evthicc 24975 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cn→ℝ))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ ran 𝐹 = (π‘₯[,]𝑦))
 
Theoremcniccbdd 24977* A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐹 ∈ ((𝐴[,]𝐡)–cnβ†’β„‚)) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ (𝐴[,]𝐡)(absβ€˜(πΉβ€˜π‘¦)) ≀ π‘₯)
 
13.2  Integrals
 
13.2.1  Lebesgue measure
 
Syntaxcovol 24978 Extend class notation with the outer Lebesgue measure.
class vol*
 
Syntaxcvol 24979 Extend class notation with the Lebesgue measure.
class vol
 
Definitiondf-ovol 24980* Define the outer Lebesgue measure for subsets of the reals. Here 𝑓 is a function from the positive integers to pairs βŸ¨π‘Ž, π‘βŸ© with π‘Ž ≀ 𝑏, and the outer volume of the set π‘₯ is the infimum over all such functions such that the union of the open intervals (π‘Ž, 𝑏) covers π‘₯ of the sum of 𝑏 βˆ’ π‘Ž. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
vol* = (π‘₯ ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(π‘₯ βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
 
Definitiondf-vol 24981* 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* β†Ύ {π‘₯ ∣ βˆ€π‘¦ ∈ (β—‘vol* β€œ ℝ)(vol*β€˜π‘¦) = ((vol*β€˜(𝑦 ∩ π‘₯)) + (vol*β€˜(𝑦 βˆ– π‘₯)))})
 
Theoremovolfcl 24982 Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑁 ∈ β„•) β†’ ((1st β€˜(πΉβ€˜π‘)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘)) ≀ (2nd β€˜(πΉβ€˜π‘))))
 
Theoremovolfioo 24983* Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴 βŠ† ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝐹) ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘› ∈ β„• ((1st β€˜(πΉβ€˜π‘›)) < 𝑧 ∧ 𝑧 < (2nd β€˜(πΉβ€˜π‘›)))))
 
Theoremovolficc 24984* Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴 βŠ† ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (𝐴 βŠ† βˆͺ ran ([,] ∘ 𝐹) ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘› ∈ β„• ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
 
Theoremovolficcss 24985 Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)
(𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ βˆͺ ran ([,] ∘ 𝐹) βŠ† ℝ)
 
Theoremovolfsval 24986 The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ βˆ’ ) ∘ 𝐹)    β‡’   ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑁 ∈ β„•) β†’ (πΊβ€˜π‘) = ((2nd β€˜(πΉβ€˜π‘)) βˆ’ (1st β€˜(πΉβ€˜π‘))))
 
Theoremovolfsf 24987 Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ βˆ’ ) ∘ 𝐹)    β‡’   (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ 𝐺:β„•βŸΆ(0[,)+∞))
 
Theoremovolsf 24988 Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ βˆ’ ) ∘ 𝐹)    &   π‘† = seq1( + , 𝐺)    β‡’   (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ 𝑆:β„•βŸΆ(0[,)+∞))
 
Theoremovolval 24989* The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
𝑀 = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}    β‡’   (𝐴 βŠ† ℝ β†’ (vol*β€˜π΄) = inf(𝑀, ℝ*, < ))
 
Theoremelovolmlem 24990 Lemma for elovolm 24991 and related theorems. (Contributed by BJ, 23-Jul-2022.)
(𝐹 ∈ ((𝐴 ∩ (ℝ Γ— ℝ)) ↑m β„•) ↔ 𝐹:β„•βŸΆ(𝐴 ∩ (ℝ Γ— ℝ)))
 
Theoremelovolm 24991* Elementhood in the set 𝑀 of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}    β‡’   (𝐡 ∈ 𝑀 ↔ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝐡 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )))
 
Theoremelovolmr 24992* Sufficient condition for elementhood in the set 𝑀. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}    &   π‘† = seq1( + , ((abs ∘ βˆ’ ) ∘ 𝐹))    β‡’   ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝐴 βŠ† βˆͺ ran ((,) ∘ 𝐹)) β†’ sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
 
Theoremovolmge0 24993* The set 𝑀 is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}    β‡’   (𝐡 ∈ 𝑀 β†’ 0 ≀ 𝐡)
 
Theoremovolcl 24994 The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 βŠ† ℝ β†’ (vol*β€˜π΄) ∈ ℝ*)
 
Theoremovollb 24995 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 ∘ βˆ’ ) ∘ 𝐹))    β‡’   ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝐴 βŠ† βˆͺ ran ((,) ∘ 𝐹)) β†’ (vol*β€˜π΄) ≀ sup(ran 𝑆, ℝ*, < ))
 
Theoremovolgelb 24996* 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 ∘ βˆ’ ) ∘ 𝑔))    β‡’   ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ ∧ 𝐡 ∈ ℝ+) β†’ βˆƒπ‘” ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ*, < ) ≀ ((vol*β€˜π΄) + 𝐡)))
 
Theoremovolge0 24997 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 βŠ† ℝ β†’ 0 ≀ (vol*β€˜π΄))
 
Theoremovolf 24998 The domain and codomain of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
vol*:𝒫 β„βŸΆ(0[,]+∞)
 
Theoremovollecl 24999 If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 βŠ† ℝ ∧ 𝐡 ∈ ℝ ∧ (vol*β€˜π΄) ≀ 𝐡) β†’ (vol*β€˜π΄) ∈ ℝ)
 
Theoremovolsslem 25000* Lemma for ovolss 25001. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
𝑀 = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}    &   π‘ = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐡 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}    β‡’   ((𝐴 βŠ† 𝐡 ∧ 𝐡 βŠ† ℝ) β†’ (vol*β€˜π΄) ≀ (vol*β€˜π΅))
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