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Type | Label | Description |
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Statement | ||
Definition | df-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.) |
β’ β^ = (π β V β¦ (toβPreHilβ(βfld freeLMod π))) | ||
Definition | df-ehl 24902 | Define a function generating the real Euclidean spaces of finite dimension. The case π = 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.) |
β’ πΌhil = (π β β0 β¦ (β^β(1...π))) | ||
Theorem | rrxval 24903 | Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
β’ π» = (β^βπΌ) β β’ (πΌ β π β π» = (toβPreHilβ(βfld freeLMod πΌ))) | ||
Theorem | rrxbase 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.) |
β’ π» = (β^βπΌ) & β’ π΅ = (Baseβπ») β β’ (πΌ β π β π΅ = {π β (β βm πΌ) β£ π finSupp 0}) | ||
Theorem | rrxprds 24905 | Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
β’ π» = (β^βπΌ) & β’ π΅ = (Baseβπ») β β’ (πΌ β π β π» = (toβPreHilβ((βfldXs(πΌ Γ {((subringAlg ββfld)ββ)})) βΎs π΅))) | ||
Theorem | rrxip 24906* | The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
β’ π» = (β^βπΌ) & β’ π΅ = (Baseβπ») β β’ (πΌ β π β (π β (β βm πΌ), π β (β βm πΌ) β¦ (βfld Ξ£g (π₯ β πΌ β¦ ((πβπ₯) Β· (πβπ₯))))) = (Β·πβπ»)) | ||
Theorem | rrxnm 24907* | The norm of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
β’ π» = (β^βπΌ) & β’ π΅ = (Baseβπ») β β’ (πΌ β π β (π β π΅ β¦ (ββ(βfld Ξ£g (π₯ β πΌ β¦ ((πβπ₯)β2))))) = (normβπ»)) | ||
Theorem | rrxcph 24908 | Generalized Euclidean real spaces are subcomplex pre-Hilbert spaces. (Contributed by Thierry Arnoux, 23-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.) |
β’ π» = (β^βπΌ) & β’ π΅ = (Baseβπ») β β’ (πΌ β π β π» β βPreHil) | ||
Theorem | rrxds 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.) |
β’ π» = (β^βπΌ) & β’ π΅ = (Baseβπ») β β’ (πΌ β π β (π β π΅, π β π΅ β¦ (ββ(βfld Ξ£g (π₯ β πΌ β¦ (((πβπ₯) β (πβπ₯))β2))))) = (distβπ»)) | ||
Theorem | rrxvsca 24910 | The scalar product over generalized Euclidean spaces is the componentwise real number multiplication. (Contributed by Thierry Arnoux, 18-Jan-2023.) |
β’ π» = (β^βπΌ) & β’ π΅ = (Baseβπ») & β’ β = ( Β·π βπ») & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ (π β π΄ β β) & β’ (π β π β (Baseβπ»)) β β’ (π β ((π΄ β π)βπ½) = (π΄ Β· (πβπ½))) | ||
Theorem | rrxplusgvscavalb 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.) |
β’ π» = (β^βπΌ) & β’ π΅ = (Baseβπ») & β’ β = ( Β·π βπ») & β’ (π β πΌ β π) & β’ (π β π΄ β β) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ β = (+gβπ») & β’ (π β πΆ β β) β β’ (π β (π = ((π΄ β π) β (πΆ β π)) β βπ β πΌ (πβπ) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ))))) | ||
Theorem | rrxsca 24912 | The field of real numbers is the scalar field of the generalized real Euclidean space. (Contributed by AV, 15-Jan-2023.) |
β’ π» = (β^βπΌ) β β’ (πΌ β π β (Scalarβπ») = βfld) | ||
Theorem | rrx0 24913 | The zero ("origin") in a generalized real Euclidean space. (Contributed by AV, 11-Feb-2023.) |
β’ π» = (β^βπΌ) & β’ 0 = (πΌ Γ {0}) β β’ (πΌ β π β (0gβπ») = 0 ) | ||
Theorem | rrx0el 24914 | The zero ("origin") in a generalized real Euclidean space is an element of its base set. (Contributed by AV, 11-Feb-2023.) |
β’ 0 = (πΌ Γ {0}) & β’ π = (β βm πΌ) β β’ (πΌ β π β 0 β π) | ||
Theorem | csbren 24915* | Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.) |
β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β π΅ β β) & β’ ((π β§ π β π΄) β πΆ β β) β β’ (π β (Ξ£π β π΄ (π΅ Β· πΆ)β2) β€ (Ξ£π β π΄ (π΅β2) Β· Ξ£π β π΄ (πΆβ2))) | ||
Theorem | trirn 24916* | Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.) |
β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β π΅ β β) & β’ ((π β§ π β π΄) β πΆ β β) β β’ (π β (ββΞ£π β π΄ ((π΅ + πΆ)β2)) β€ ((ββΞ£π β π΄ (π΅β2)) + (ββΞ£π β π΄ (πΆβ2)))) | ||
Theorem | rrxf 24917* | Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
β’ π = {β β (β βm πΌ) β£ β finSupp 0} & β’ (π β πΉ β π) β β’ (π β πΉ:πΌβΆβ) | ||
Theorem | rrxfsupp 24918* | Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
β’ π = {β β (β βm πΌ) β£ β finSupp 0} & β’ (π β πΉ β π) β β’ (π β (πΉ supp 0) β Fin) | ||
Theorem | rrxsuppss 24919* | Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
β’ π = {β β (β βm πΌ) β£ β finSupp 0} & β’ (π β πΉ β π) β β’ (π β (πΉ supp 0) β πΌ) | ||
Theorem | rrxmvallem 24920* | Support of the function used for building the distance . (Contributed by Thierry Arnoux, 30-Jun-2019.) |
β’ π = {β β (β βm πΌ) β£ β finSupp 0} β β’ ((πΌ β π β§ πΉ β π β§ πΊ β π) β ((π β πΌ β¦ (((πΉβπ) β (πΊβπ))β2)) supp 0) β ((πΉ supp 0) βͺ (πΊ supp 0))) | ||
Theorem | rrxmval 24921* | The value of the Euclidean metric. Compare with rrnmval 36691. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
β’ π = {β β (β βm πΌ) β£ β finSupp 0} & β’ π· = (distβ(β^βπΌ)) β β’ ((πΌ β π β§ πΉ β π β§ πΊ β π) β (πΉπ·πΊ) = (ββΞ£π β ((πΉ supp 0) βͺ (πΊ supp 0))(((πΉβπ) β (πΊβπ))β2))) | ||
Theorem | rrxmfval 24922* | The value of the Euclidean metric. Compare with rrnval 36690. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
β’ π = {β β (β βm πΌ) β£ β finSupp 0} & β’ π· = (distβ(β^βπΌ)) β β’ (πΌ β π β π· = (π β π, π β π β¦ (ββΞ£π β ((π supp 0) βͺ (π supp 0))(((πβπ) β (πβπ))β2)))) | ||
Theorem | rrxmetlem 24923* | Lemma for rrxmet 24924. (Contributed by Thierry Arnoux, 5-Jul-2019.) |
β’ π = {β β (β βm πΌ) β£ β finSupp 0} & β’ π· = (distβ(β^βπΌ)) & β’ (π β πΌ β π) & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β π΄ β πΌ) & β’ (π β π΄ β Fin) & β’ (π β ((πΉ supp 0) βͺ (πΊ supp 0)) β π΄) β β’ (π β Ξ£π β ((πΉ supp 0) βͺ (πΊ supp 0))(((πΉβπ) β (πΊβπ))β2) = Ξ£π β π΄ (((πΉβπ) β (πΊβπ))β2)) | ||
Theorem | rrxmet 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.) |
β’ π = {β β (β βm πΌ) β£ β finSupp 0} & β’ π· = (distβ(β^βπΌ)) β β’ (πΌ β π β π· β (Metβπ)) | ||
Theorem | rrxdstprj1 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.) |
β’ π = {β β (β βm πΌ) β£ β finSupp 0} & β’ π· = (distβ(β^βπΌ)) & β’ π = ((abs β β ) βΎ (β Γ β)) β β’ (((πΌ β π β§ π΄ β πΌ) β§ (πΉ β π β§ πΊ β π)) β ((πΉβπ΄)π(πΊβπ΄)) β€ (πΉπ·πΊ)) | ||
Theorem | rrxbasefi 24926 | The base of the generalized real Euclidean space, when the dimension of the space is finite. This justifies the use of (β βm π) for the development of the Lebesgue measure theory for n-dimensional real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
β’ (π β π β Fin) & β’ π» = (β^βπ) & β’ π΅ = (Baseβπ») β β’ (π β π΅ = (β βm π)) | ||
Theorem | rrxdsfi 24927* | The distance over generalized Euclidean spaces. Finite dimensional case. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
β’ π» = (β^βπΌ) & β’ π΅ = (β βm πΌ) β β’ (πΌ β Fin β (distβπ») = (π β π΅, π β π΅ β¦ (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)))) | ||
Theorem | rrxmetfi 24928 | Euclidean space is a metric space. Finite dimensional version. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
β’ π· = (distβ(β^βπΌ)) β β’ (πΌ β Fin β π· β (Metβ(β βm πΌ))) | ||
Theorem | rrxdsfival 24929* | The value of the Euclidean distance function in a generalized real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.) |
β’ π = (β βm πΌ) & β’ π· = (distβ(β^βπΌ)) β β’ ((πΌ β Fin β§ πΉ β π β§ πΊ β π) β (πΉπ·πΊ) = (ββΞ£π β πΌ (((πΉβπ) β (πΊβπ))β2))) | ||
Theorem | ehlval 24930 | Value of the Euclidean space of dimension π. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
β’ πΈ = (πΌhilβπ) β β’ (π β β0 β πΈ = (β^β(1...π))) | ||
Theorem | ehlbase 24931 | The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
β’ πΈ = (πΌhilβπ) β β’ (π β β0 β (β βm (1...π)) = (BaseβπΈ)) | ||
Theorem | ehl0base 24932 | The base of the Euclidean space of dimension 0 consists only of one element, the empty set. (Contributed by AV, 12-Feb-2023.) |
β’ πΈ = (πΌhilβ0) β β’ (BaseβπΈ) = {β } | ||
Theorem | ehl0 24933 | The Euclidean space of dimension 0 consists of the neutral element only. (Contributed by AV, 12-Feb-2023.) |
β’ πΈ = (πΌhilβ0) & β’ 0 = (0gβπΈ) β β’ (BaseβπΈ) = { 0 } | ||
Theorem | ehleudis 24934* | The Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.) |
β’ πΌ = (1...π) & β’ πΈ = (πΌhilβπ) & β’ π = (β βm πΌ) & β’ π· = (distβπΈ) β β’ (π β β0 β π· = (π β π, π β π β¦ (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)))) | ||
Theorem | ehleudisval 24935* | The value of the Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.) |
β’ πΌ = (1...π) & β’ πΈ = (πΌhilβπ) & β’ π = (β βm πΌ) & β’ π· = (distβπΈ) β β’ ((π β β0 β§ πΉ β π β§ πΊ β π) β (πΉπ·πΊ) = (ββΞ£π β πΌ (((πΉβπ) β (πΊβπ))β2))) | ||
Theorem | ehl1eudis 24936* | The Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.) |
β’ πΈ = (πΌhilβ1) & β’ π = (β βm {1}) & β’ π· = (distβπΈ) β β’ π· = (π β π, π β π β¦ (absβ((πβ1) β (πβ1)))) | ||
Theorem | ehl1eudisval 24937 | The value of the Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.) |
β’ πΈ = (πΌhilβ1) & β’ π = (β βm {1}) & β’ π· = (distβπΈ) β β’ ((πΉ β π β§ πΊ β π) β (πΉπ·πΊ) = (absβ((πΉβ1) β (πΊβ1)))) | ||
Theorem | ehl2eudis 24938* | The Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.) |
β’ πΈ = (πΌhilβ2) & β’ π = (β βm {1, 2}) & β’ π· = (distβπΈ) β β’ π· = (π β π, π β π β¦ (ββ((((πβ1) β (πβ1))β2) + (((πβ2) β (πβ2))β2)))) | ||
Theorem | ehl2eudisval 24939 | The value of the Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.) |
β’ πΈ = (πΌhilβ2) & β’ π = (β βm {1, 2}) & β’ π· = (distβπΈ) β β’ ((πΉ β π β§ πΊ β π) β (πΉπ·πΊ) = (ββ((((πΉβ1) β (πΊβ1))β2) + (((πΉβ2) β (πΊβ2))β2)))) | ||
Theorem | minveclem1 24940* | Lemma for minvec 24952. The set of all distances from points of π to π΄ are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) β β’ (π β (π β β β§ π β β β§ βπ€ β π 0 β€ π€)) | ||
Theorem | minveclem4c 24941* | Lemma for minvec 24952. The infimum of the distances to π΄ is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) β β’ (π β π β β) | ||
Theorem | minveclem2 24942* | Lemma for minvec 24952. Any two points πΎ and πΏ in π are close to each other if they are close to the infimum of distance to π΄. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) & β’ π· = ((distβπ) βΎ (π Γ π)) & β’ (π β π΅ β β) & β’ (π β 0 β€ π΅) & β’ (π β πΎ β π) & β’ (π β πΏ β π) & β’ (π β ((π΄π·πΎ)β2) β€ ((πβ2) + π΅)) & β’ (π β ((π΄π·πΏ)β2) β€ ((πβ2) + π΅)) β β’ (π β ((πΎπ·πΏ)β2) β€ (4 Β· π΅)) | ||
Theorem | minveclem3a 24943* | Lemma for minvec 24952. π· is a complete metric when restricted to π. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) & β’ π· = ((distβπ) βΎ (π Γ π)) β β’ (π β (π· βΎ (π Γ π)) β (CMetβπ)) | ||
Theorem | minveclem3b 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.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) & β’ π· = ((distβπ) βΎ (π Γ π)) & β’ πΉ = ran (π β β+ β¦ {π¦ β π β£ ((π΄π·π¦)β2) β€ ((πβ2) + π)}) β β’ (π β πΉ β (fBasβπ)) | ||
Theorem | minveclem3 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.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) & β’ π· = ((distβπ) βΎ (π Γ π)) & β’ πΉ = ran (π β β+ β¦ {π¦ β π β£ ((π΄π·π¦)β2) β€ ((πβ2) + π)}) β β’ (π β (πfilGenπΉ) β (CauFilβ(π· βΎ (π Γ π)))) | ||
Theorem | minveclem4a 24946* | Lemma for minvec 24952. πΉ converges to a point π in π. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) & β’ π· = ((distβπ) βΎ (π Γ π)) & β’ πΉ = ran (π β β+ β¦ {π¦ β π β£ ((π΄π·π¦)β2) β€ ((πβ2) + π)}) & β’ π = βͺ (π½ fLim (πfilGenπΉ)) β β’ (π β π β ((π½ fLim (πfilGenπΉ)) β© π)) | ||
Theorem | minveclem4b 24947* | Lemma for minvec 24952. The convergent point of the Cauchy sequence πΉ is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) & β’ π· = ((distβπ) βΎ (π Γ π)) & β’ πΉ = ran (π β β+ β¦ {π¦ β π β£ ((π΄π·π¦)β2) β€ ((πβ2) + π)}) & β’ π = βͺ (π½ fLim (πfilGenπΉ)) β β’ (π β π β π) | ||
Theorem | minveclem4 24948* | Lemma for minvec 24952. The convergent point of the Cauchy sequence πΉ attains the minimum distance, and so is closer to π΄ than any other point in π. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) & β’ π· = ((distβπ) βΎ (π Γ π)) & β’ πΉ = ran (π β β+ β¦ {π¦ β π β£ ((π΄π·π¦)β2) β€ ((πβ2) + π)}) & β’ π = βͺ (π½ fLim (πfilGenπΉ)) & β’ π = (((((π΄π·π) + π) / 2)β2) β (πβ2)) β β’ (π β βπ₯ β π βπ¦ β π (πβ(π΄ β π₯)) β€ (πβ(π΄ β π¦))) | ||
Theorem | minveclem5 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.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) & β’ π· = ((distβπ) βΎ (π Γ π)) β β’ (π β βπ₯ β π βπ¦ β π (πβ(π΄ β π₯)) β€ (πβ(π΄ β π¦))) | ||
Theorem | minveclem6 24950* | Lemma for minvec 24952. Any minimal point is less than π away from π΄. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) & β’ π· = ((distβπ) βΎ (π Γ π)) β β’ ((π β§ π₯ β π) β (((π΄π·π₯)β2) β€ ((πβ2) + 0) β βπ¦ β π (πβ(π΄ β π₯)) β€ (πβ(π΄ β π¦)))) | ||
Theorem | minveclem7 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.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) & β’ π = inf(π , β, < ) & β’ π· = ((distβπ) βΎ (π Γ π)) β β’ (π β β!π₯ β π βπ¦ β π (πβ(π΄ β π₯)) β€ (πβ(π΄ β π¦))) | ||
Theorem | minvec 24952* | Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace π that minimizes the distance to an arbitrary vector π΄ 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.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (normβπ) & β’ (π β π β βPreHil) & β’ (π β π β (LSubSpβπ)) & β’ (π β (π βΎs π) β CMetSp) & β’ (π β π΄ β π) β β’ (π β β!π₯ β π βπ¦ β π (πβ(π΄ β π₯)) β€ (πβ(π΄ β π¦))) | ||
Theorem | pjthlem1 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.) |
β’ π = (Baseβπ) & β’ π = (normβπ) & β’ + = (+gβπ) & β’ β = (-gβπ) & β’ , = (Β·πβπ) & β’ πΏ = (LSubSpβπ) & β’ (π β π β βHil) & β’ (π β π β πΏ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β βπ₯ β π (πβπ΄) β€ (πβ(π΄ β π₯))) & β’ π = ((π΄ , π΅) / ((π΅ , π΅) + 1)) β β’ (π β (π΄ , π΅) = 0) | ||
Theorem | pjthlem2 24954 | Lemma for pjth 24955. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) |
β’ π = (Baseβπ) & β’ π = (normβπ) & β’ + = (+gβπ) & β’ β = (-gβπ) & β’ , = (Β·πβπ) & β’ πΏ = (LSubSpβπ) & β’ (π β π β βHil) & β’ (π β π β πΏ) & β’ (π β π΄ β π) & β’ π½ = (TopOpenβπ) & β’ β = (LSSumβπ) & β’ π = (ocvβπ) & β’ (π β π β (Clsdβπ½)) β β’ (π β π΄ β (π β (πβπ))) | ||
Theorem | pjth 24955 | Projection Theorem: Any Hilbert space vector π΄ can be decomposed uniquely into a member π₯ of a closed subspace π» and a member π¦ 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.) |
β’ π = (Baseβπ) & β’ β = (LSSumβπ) & β’ π = (ocvβπ) & β’ π½ = (TopOpenβπ) & β’ πΏ = (LSubSpβπ) β β’ ((π β βHil β§ π β πΏ β§ π β (Clsdβπ½)) β (π β (πβπ)) = π) | ||
Theorem | pjth2 24956 | Projection Theorem with abbreviations: A topologically closed subspace is a projection subspace. (Contributed by Mario Carneiro, 17-Oct-2015.) |
β’ π½ = (TopOpenβπ) & β’ πΏ = (LSubSpβπ) & β’ πΎ = (projβπ) β β’ ((π β βHil β§ π β πΏ β§ π β (Clsdβπ½)) β π β dom πΎ) | ||
Theorem | cldcss 24957 | Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.) |
β’ π = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ πΏ = (LSubSpβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β βHil β (π β πΆ β (π β πΏ β§ π β (Clsdβπ½)))) | ||
Theorem | cldcss2 24958 | Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.) |
β’ π = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ πΏ = (LSubSpβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β βHil β πΆ = (πΏ β© (Clsdβπ½))) | ||
Theorem | hlhil 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.) |
β’ (π β βHil β π β Hil) | ||
Theorem | addcncf 24960* | The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β (π₯ β π β¦ π΄) β (πβcnββ)) & β’ (π β (π₯ β π β¦ π΅) β (πβcnββ)) β β’ (π β (π₯ β π β¦ (π΄ + π΅)) β (πβcnββ)) | ||
Theorem | subcncf 24961* | The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β (π₯ β π β¦ π΄) β (πβcnββ)) & β’ (π β (π₯ β π β¦ π΅) β (πβcnββ)) β β’ (π β (π₯ β π β¦ (π΄ β π΅)) β (πβcnββ)) | ||
Theorem | mulcncf 24962* | The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ (π β (π₯ β π β¦ π΄) β (πβcnββ)) & β’ (π β (π₯ β π β¦ π΅) β (πβcnββ)) β β’ (π β (π₯ β π β¦ (π΄ Β· π΅)) β (πβcnββ)) | ||
Theorem | divcncf 24963* | The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β (π₯ β π β¦ π΄) β (πβcnββ)) & β’ (π β (π₯ β π β¦ π΅) β (πβcnβ(β β {0}))) β β’ (π β (π₯ β π β¦ (π΄ / π΅)) β (πβcnββ)) | ||
Theorem | pmltpclem1 24964* | Lemma for pmltpc 24966. (Contributed by Mario Carneiro, 1-Jul-2014.) |
β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π΄ < π΅) & β’ (π β π΅ < πΆ) & β’ (π β (((πΉβπ΄) < (πΉβπ΅) β§ (πΉβπΆ) < (πΉβπ΅)) β¨ ((πΉβπ΅) < (πΉβπ΄) β§ (πΉβπ΅) < (πΉβπΆ)))) β β’ (π β βπ β π βπ β π βπ β π (π < π β§ π < π β§ (((πΉβπ) < (πΉβπ) β§ (πΉβπ) < (πΉβπ)) β¨ ((πΉβπ) < (πΉβπ) β§ (πΉβπ) < (πΉβπ))))) | ||
Theorem | pmltpclem2 24965* | Lemma for pmltpc 24966. (Contributed by Mario Carneiro, 1-Jul-2014.) |
β’ (π β πΉ β (β βpm β)) & β’ (π β π΄ β dom πΉ) & β’ (π β π β π΄) & β’ (π β π β π΄) & β’ (π β π β π΄) & β’ (π β π β π΄) & β’ (π β π β€ π) & β’ (π β π β€ π) & β’ (π β Β¬ (πΉβπ) β€ (πΉβπ)) & β’ (π β Β¬ (πΉβπ) β€ (πΉβπ)) β β’ (π β βπ β π΄ βπ β π΄ βπ β π΄ (π < π β§ π < π β§ (((πΉβπ) < (πΉβπ) β§ (πΉβπ) < (πΉβπ)) β¨ ((πΉβπ) < (πΉβπ) β§ (πΉβπ) < (πΉβπ))))) | ||
Theorem | pmltpc 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.) |
β’ ((πΉ β (β βpm β) β§ π΄ β dom πΉ) β (βπ₯ β π΄ βπ¦ β π΄ (π₯ β€ π¦ β (πΉβπ₯) β€ (πΉβπ¦)) β¨ βπ₯ β π΄ βπ¦ β π΄ (π₯ β€ π¦ β (πΉβπ¦) β€ (πΉβπ₯)) β¨ βπ β π΄ βπ β π΄ βπ β π΄ (π < π β§ π < π β§ (((πΉβπ) < (πΉβπ) β§ (πΉβπ) < (πΉβπ)) β¨ ((πΉβπ) < (πΉβπ) β§ (πΉβπ) < (πΉβπ)))))) | ||
Theorem | ivthlem1 24967* | Lemma for ivth 24970. The set π of all π₯ values with (πΉβπ₯) less than π is lower bounded by π΄ and upper bounded by π΅. (Contributed by Mario Carneiro, 17-Jun-2014.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π β β) & β’ (π β π΄ < π΅) & β’ (π β (π΄[,]π΅) β π·) & β’ (π β πΉ β (π·βcnββ)) & β’ ((π β§ π₯ β (π΄[,]π΅)) β (πΉβπ₯) β β) & β’ (π β ((πΉβπ΄) < π β§ π < (πΉβπ΅))) & β’ π = {π₯ β (π΄[,]π΅) β£ (πΉβπ₯) β€ π} β β’ (π β (π΄ β π β§ βπ§ β π π§ β€ π΅)) | ||
Theorem | ivthlem2 24968* | Lemma for ivth 24970. Show that the supremum of π cannot be less than π. If it was, continuity of πΉ implies that there are points just above the supremum that are also less than π, a contradiction. (Contributed by Mario Carneiro, 17-Jun-2014.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π β β) & β’ (π β π΄ < π΅) & β’ (π β (π΄[,]π΅) β π·) & β’ (π β πΉ β (π·βcnββ)) & β’ ((π β§ π₯ β (π΄[,]π΅)) β (πΉβπ₯) β β) & β’ (π β ((πΉβπ΄) < π β§ π < (πΉβπ΅))) & β’ π = {π₯ β (π΄[,]π΅) β£ (πΉβπ₯) β€ π} & β’ πΆ = sup(π, β, < ) β β’ (π β Β¬ (πΉβπΆ) < π) | ||
Theorem | ivthlem3 24969* | Lemma for ivth 24970, the intermediate value theorem. Show that (πΉβπΆ) cannot be greater than π, 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.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π β β) & β’ (π β π΄ < π΅) & β’ (π β (π΄[,]π΅) β π·) & β’ (π β πΉ β (π·βcnββ)) & β’ ((π β§ π₯ β (π΄[,]π΅)) β (πΉβπ₯) β β) & β’ (π β ((πΉβπ΄) < π β§ π < (πΉβπ΅))) & β’ π = {π₯ β (π΄[,]π΅) β£ (πΉβπ₯) β€ π} & β’ πΆ = sup(π, β, < ) β β’ (π β (πΆ β (π΄(,)π΅) β§ (πΉβπΆ) = π)) | ||
Theorem | ivth 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.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π β β) & β’ (π β π΄ < π΅) & β’ (π β (π΄[,]π΅) β π·) & β’ (π β πΉ β (π·βcnββ)) & β’ ((π β§ π₯ β (π΄[,]π΅)) β (πΉβπ₯) β β) & β’ (π β ((πΉβπ΄) < π β§ π < (πΉβπ΅))) β β’ (π β βπ β (π΄(,)π΅)(πΉβπ) = π) | ||
Theorem | ivth2 24971* | The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π β β) & β’ (π β π΄ < π΅) & β’ (π β (π΄[,]π΅) β π·) & β’ (π β πΉ β (π·βcnββ)) & β’ ((π β§ π₯ β (π΄[,]π΅)) β (πΉβπ₯) β β) & β’ (π β ((πΉβπ΅) < π β§ π < (πΉβπ΄))) β β’ (π β βπ β (π΄(,)π΅)(πΉβπ) = π) | ||
Theorem | ivthle 24972* | The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π β β) & β’ (π β π΄ < π΅) & β’ (π β (π΄[,]π΅) β π·) & β’ (π β πΉ β (π·βcnββ)) & β’ ((π β§ π₯ β (π΄[,]π΅)) β (πΉβπ₯) β β) & β’ (π β ((πΉβπ΄) β€ π β§ π β€ (πΉβπ΅))) β β’ (π β βπ β (π΄[,]π΅)(πΉβπ) = π) | ||
Theorem | ivthle2 24973* | The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π β β) & β’ (π β π΄ < π΅) & β’ (π β (π΄[,]π΅) β π·) & β’ (π β πΉ β (π·βcnββ)) & β’ ((π β§ π₯ β (π΄[,]π΅)) β (πΉβπ₯) β β) & β’ (π β ((πΉβπ΅) β€ π β§ π β€ (πΉβπ΄))) β β’ (π β βπ β (π΄[,]π΅)(πΉβπ) = π) | ||
Theorem | ivthicc 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.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π β (π΄[,]π΅)) & β’ (π β π β (π΄[,]π΅)) & β’ (π β (π΄[,]π΅) β π·) & β’ (π β πΉ β (π·βcnββ)) & β’ ((π β§ π₯ β (π΄[,]π΅)) β (πΉβπ₯) β β) β β’ (π β ((πΉβπ)[,](πΉβπ)) β ran πΉ) | ||
Theorem | evthicc 24975* | Specialization of the Extreme Value Theorem to a closed interval of β. (Contributed by Mario Carneiro, 12-Aug-2014.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π΄ β€ π΅) & β’ (π β πΉ β ((π΄[,]π΅)βcnββ)) β β’ (π β (βπ₯ β (π΄[,]π΅)βπ¦ β (π΄[,]π΅)(πΉβπ¦) β€ (πΉβπ₯) β§ βπ§ β (π΄[,]π΅)βπ€ β (π΄[,]π΅)(πΉβπ§) β€ (πΉβπ€))) | ||
Theorem | evthicc2 24976* | Combine ivthicc 24974 with evthicc 24975 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π΄ β€ π΅) & β’ (π β πΉ β ((π΄[,]π΅)βcnββ)) β β’ (π β βπ₯ β β βπ¦ β β ran πΉ = (π₯[,]π¦)) | ||
Theorem | cniccbdd 24977* | A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.) |
β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β βπ₯ β β βπ¦ β (π΄[,]π΅)(absβ(πΉβπ¦)) β€ π₯) | ||
Syntax | covol 24978 | Extend class notation with the outer Lebesgue measure. |
class vol* | ||
Syntax | cvol 24979 | Extend class notation with the Lebesgue measure. |
class vol | ||
Definition | df-ovol 24980* | Define the outer Lebesgue measure for subsets of the reals. Here π is a function from the positive integers to pairs β¨π, πβ© with π β€ π, and the outer volume of the set π₯ is the infimum over all such functions such that the union of the open intervals (π, π) covers π₯ of the sum of π β π. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.) |
β’ vol* = (π₯ β π« β β¦ inf({π¦ β β* β£ βπ β (( β€ β© (β Γ β)) βm β)(π₯ β βͺ ran ((,) β π) β§ π¦ = sup(ran seq1( + , ((abs β β ) β π)), β*, < ))}, β*, < )) | ||
Definition | df-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 π΄ β dom vol. (Contributed by Mario Carneiro, 17-Mar-2014.) |
β’ vol = (vol* βΎ {π₯ β£ βπ¦ β (β‘vol* β β)(vol*βπ¦) = ((vol*β(π¦ β© π₯)) + (vol*β(π¦ β π₯)))}) | ||
Theorem | ovolfcl 24982 | Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ ((πΉ:ββΆ( β€ β© (β Γ β)) β§ π β β) β ((1st β(πΉβπ)) β β β§ (2nd β(πΉβπ)) β β β§ (1st β(πΉβπ)) β€ (2nd β(πΉβπ)))) | ||
Theorem | ovolfioo 24983* | Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ ((π΄ β β β§ πΉ:ββΆ( β€ β© (β Γ β))) β (π΄ β βͺ ran ((,) β πΉ) β βπ§ β π΄ βπ β β ((1st β(πΉβπ)) < π§ β§ π§ < (2nd β(πΉβπ))))) | ||
Theorem | ovolficc 24984* | Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ ((π΄ β β β§ πΉ:ββΆ( β€ β© (β Γ β))) β (π΄ β βͺ ran ([,] β πΉ) β βπ§ β π΄ βπ β β ((1st β(πΉβπ)) β€ π§ β§ π§ β€ (2nd β(πΉβπ))))) | ||
Theorem | ovolficcss 24985 | Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.) |
β’ (πΉ:ββΆ( β€ β© (β Γ β)) β βͺ ran ([,] β πΉ) β β) | ||
Theorem | ovolfsval 24986 | The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ πΊ = ((abs β β ) β πΉ) β β’ ((πΉ:ββΆ( β€ β© (β Γ β)) β§ π β β) β (πΊβπ) = ((2nd β(πΉβπ)) β (1st β(πΉβπ)))) | ||
Theorem | ovolfsf 24987 | Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ πΊ = ((abs β β ) β πΉ) β β’ (πΉ:ββΆ( β€ β© (β Γ β)) β πΊ:ββΆ(0[,)+β)) | ||
Theorem | ovolsf 24988 | Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ πΊ = ((abs β β ) β πΉ) & β’ π = seq1( + , πΊ) β β’ (πΉ:ββΆ( β€ β© (β Γ β)) β π:ββΆ(0[,)+β)) | ||
Theorem | ovolval 24989* | The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.) |
β’ π = {π¦ β β* β£ βπ β (( β€ β© (β Γ β)) βm β)(π΄ β βͺ ran ((,) β π) β§ π¦ = sup(ran seq1( + , ((abs β β ) β π)), β*, < ))} β β’ (π΄ β β β (vol*βπ΄) = inf(π, β*, < )) | ||
Theorem | elovolmlem 24990 | Lemma for elovolm 24991 and related theorems. (Contributed by BJ, 23-Jul-2022.) |
β’ (πΉ β ((π΄ β© (β Γ β)) βm β) β πΉ:ββΆ(π΄ β© (β Γ β))) | ||
Theorem | elovolm 24991* | Elementhood in the set π of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ π = {π¦ β β* β£ βπ β (( β€ β© (β Γ β)) βm β)(π΄ β βͺ ran ((,) β π) β§ π¦ = sup(ran seq1( + , ((abs β β ) β π)), β*, < ))} β β’ (π΅ β π β βπ β (( β€ β© (β Γ β)) βm β)(π΄ β βͺ ran ((,) β π) β§ π΅ = sup(ran seq1( + , ((abs β β ) β π)), β*, < ))) | ||
Theorem | elovolmr 24992* | Sufficient condition for elementhood in the set π. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ π = {π¦ β β* β£ βπ β (( β€ β© (β Γ β)) βm β)(π΄ β βͺ ran ((,) β π) β§ π¦ = sup(ran seq1( + , ((abs β β ) β π)), β*, < ))} & β’ π = seq1( + , ((abs β β ) β πΉ)) β β’ ((πΉ:ββΆ( β€ β© (β Γ β)) β§ π΄ β βͺ ran ((,) β πΉ)) β sup(ran π, β*, < ) β π) | ||
Theorem | ovolmge0 24993* | The set π is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ π = {π¦ β β* β£ βπ β (( β€ β© (β Γ β)) βm β)(π΄ β βͺ ran ((,) β π) β§ π¦ = sup(ran seq1( + , ((abs β β ) β π)), β*, < ))} β β’ (π΅ β π β 0 β€ π΅) | ||
Theorem | ovolcl 24994 | The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ (π΄ β β β (vol*βπ΄) β β*) | ||
Theorem | ovollb 24995 | The outer volume is a lower bound on the sum of all interval coverings of π΄. (Contributed by Mario Carneiro, 15-Jun-2014.) |
β’ π = seq1( + , ((abs β β ) β πΉ)) β β’ ((πΉ:ββΆ( β€ β© (β Γ β)) β§ π΄ β βͺ ran ((,) β πΉ)) β (vol*βπ΄) β€ sup(ran π, β*, < )) | ||
Theorem | ovolgelb 24996* | The outer volume is the greatest lower bound on the sum of all interval coverings of π΄. (Contributed by Mario Carneiro, 15-Jun-2014.) |
β’ π = seq1( + , ((abs β β ) β π)) β β’ ((π΄ β β β§ (vol*βπ΄) β β β§ π΅ β β+) β βπ β (( β€ β© (β Γ β)) βm β)(π΄ β βͺ ran ((,) β π) β§ sup(ran π, β*, < ) β€ ((vol*βπ΄) + π΅))) | ||
Theorem | ovolge0 24997 | The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.) |
β’ (π΄ β β β 0 β€ (vol*βπ΄)) | ||
Theorem | ovolf 24998 | The domain and codomain of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.) |
β’ vol*:π« ββΆ(0[,]+β) | ||
Theorem | ovollecl 24999 | If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.) |
β’ ((π΄ β β β§ π΅ β β β§ (vol*βπ΄) β€ π΅) β (vol*βπ΄) β β) | ||
Theorem | ovolsslem 25000* | Lemma for ovolss 25001. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.) |
β’ π = {π¦ β β* β£ βπ β (( β€ β© (β Γ β)) βm β)(π΄ β βͺ ran ((,) β π) β§ π¦ = sup(ran seq1( + , ((abs β β ) β π)), β*, < ))} & β’ π = {π¦ β β* β£ βπ β (( β€ β© (β Γ β)) βm β)(π΅ β βͺ ran ((,) β π) β§ π¦ = sup(ran seq1( + , ((abs β β ) β π)), β*, < ))} β β’ ((π΄ β π΅ β§ π΅ β β) β (vol*βπ΄) β€ (vol*βπ΅)) |
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