HomeHome Metamath Proof Explorer
Theorem List (p. 254 of 482)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-30715)
  Hilbert Space Explorer  Hilbert Space Explorer
(30716-32238)
  Users' Mathboxes  Users' Mathboxes
(32239-48161)
 

Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-ehl 25301 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 25332). 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 25302 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 25303* 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 25304 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 25305* 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 25306* 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 25307 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 25308* The distance over generalized Euclidean spaces. Compare with df-rrn 37234. (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 25309 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 25310* 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 25311 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 25312 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 25313 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 25314* 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 25315* 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 25316* 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 25317* 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 25318* 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 25319* 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 25320* The value of the Euclidean metric. Compare with rrnmval 37236. (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 25321* The value of the Euclidean metric. Compare with rrnval 37235. (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 25322* Lemma for rrxmet 25323. (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 25323* 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 25324* 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 25325 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 25326* 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 25327 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 25328* 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 25329 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 25330 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 25331 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 25332 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 25333* 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 25334* 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 25335* 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 25336 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 25337* 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 25338 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 25339* Lemma for minvec 25351. 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 25340* Lemma for minvec 25351. 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 25341* Lemma for minvec 25351. 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 25342* Lemma for minvec 25351. 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 25343* Lemma for minvec 25351. 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 25344* Lemma for minvec 25351. 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 25345* Lemma for minvec 25351. 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 25346* Lemma for minvec 25351. 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 25347* Lemma for minvec 25351. 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 25348* Lemma for minvec 25351. Discharge the assumptions in minveclem4 25347. (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 25349* Lemma for minvec 25351. 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 25350* Lemma for minvec 25351. 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 25351* 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 25352* Lemma for pjth 25354. (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 25353 Lemma for pjth 25354. (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 25354 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 25355 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 25356 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 25357 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 25358 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 25359* 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 25360* 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 25361* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.) Avoid ax-mulf 11210. (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 25362* Obsolete version of mulcncf 25361 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 25363* 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 25364* Lemma for pmltpc 25366. (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 25365* Lemma for pmltpc 25366. (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 25366* 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 25367* Lemma for ivth 25370. 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 25368* Lemma for ivth 25370. 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 25369* Lemma for ivth 25370, 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 25370* 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 25371* 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 25372* 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 25373* 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 25374* 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 25375* 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 25376* Combine ivthicc 25374 with evthicc 25375 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 25377* 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 25378 Extend class notation with the outer Lebesgue measure.
class vol*
 
Syntaxcvol 25379 Extend class notation with the Lebesgue measure.
class vol
 
Definitiondf-ovol 25380* 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 25381* 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 25382 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 25383* 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 25384* 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 25385 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 25386 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 25387 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 25388 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 25389* 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 25390 Lemma for elovolm 25391 and related theorems. (Contributed by BJ, 23-Jul-2022.)
(𝐹 ∈ ((𝐴 ∩ (ℝ Γ— ℝ)) ↑m β„•) ↔ 𝐹:β„•βŸΆ(𝐴 ∩ (ℝ Γ— ℝ)))
 
Theoremelovolm 25391* 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 25392* 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 25393* 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 25394 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 25395 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 25396* 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 25397 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 βŠ† ℝ β†’ 0 ≀ (vol*β€˜π΄))
 
Theoremovolf 25398 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 25399 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 25400* Lemma for ovolss 25401. (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΅))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 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 47901-48000 481 48001-48100 482 48101-48161
  Copyright terms: Public domain < Previous  Next >