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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreust 23901 The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.)
(UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ)))
 
Theoremrecusp 23902 The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
fld ∈ CUnifSp
 
12.5.8  Euclidean spaces
 
Syntaxcrrx 23903 Extend class notation with generalized real Euclidean spaces.
class ℝ^
 
Syntaxcehl 23904 Extend class notation with real Euclidean spaces.
class 𝔼hil
 
Definitiondf-rrx 23905 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 ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
 
Definitiondf-ehl 23906 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 23937). Members of this family of spaces are Hilbert spaces, as shown in - ehlhl . (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
 
Theoremrrxval 23907 Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)       (𝐼𝑉𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
 
Theoremrrxbase 23908* The base of the generalized real Euclidean space is the set of functions with finite support. (Contributed by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉𝐵 = {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0})
 
Theoremrrxprds 23909 Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵)))
 
Theoremrrxip 23910* The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉 → (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))) = (·𝑖𝐻))
 
Theoremrrxnm 23911* The norm of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉 → (𝑓𝐵 ↦ (√‘(ℝfld Σg (𝑥𝐼 ↦ ((𝑓𝑥)↑2))))) = (norm‘𝐻))
 
Theoremrrxcph 23912 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^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉𝐻 ∈ ℂPreHil)
 
Theoremrrxds 23913* The distance over generalized Euclidean spaces. Compare with df-rrn 34974. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉 → (𝑓𝐵, 𝑔𝐵 ↦ (√‘(ℝfld Σg (𝑥𝐼 ↦ (((𝑓𝑥) − (𝑔𝑥))↑2))))) = (dist‘𝐻))
 
Theoremrrxvsca 23914 The scalar product over generalized Euclidean spaces is the componentwise real number multiplication. (Contributed by Thierry Arnoux, 18-Jan-2023.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)    &    = ( ·𝑠𝐻)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝐼)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑋 ∈ (Base‘𝐻))       (𝜑 → ((𝐴 𝑋)‘𝐽) = (𝐴 · (𝑋𝐽)))
 
Theoremrrxplusgvscavalb 23915* 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^‘𝐼)    &   𝐵 = (Base‘𝐻)    &    = ( ·𝑠𝐻)    &   (𝜑𝐼𝑉)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    = (+g𝐻)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝑍 = ((𝐴 𝑋) (𝐶 𝑌)) ↔ ∀𝑖𝐼 (𝑍𝑖) = ((𝐴 · (𝑋𝑖)) + (𝐶 · (𝑌𝑖)))))
 
Theoremrrxsca 23916 The field of real numbers is the scalar field of the generalized real Euclidean space. (Contributed by AV, 15-Jan-2023.)
𝐻 = (ℝ^‘𝐼)       (𝐼𝑉 → (Scalar‘𝐻) = ℝfld)
 
Theoremrrx0 23917 The zero ("origin") in a generalized real Euclidean space. (Contributed by AV, 11-Feb-2023.)
𝐻 = (ℝ^‘𝐼)    &    0 = (𝐼 × {0})       (𝐼𝑉 → (0g𝐻) = 0 )
 
Theoremrrx0el 23918 The zero ("origin") in a generalized real Euclidean space is an element of its base set. (Contributed by AV, 11-Feb-2023.)
0 = (𝐼 × {0})    &   𝑃 = (ℝ ↑m 𝐼)       (𝐼𝑉0𝑃)
 
Theoremcsbren 23919* Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)       (𝜑 → (Σ𝑘𝐴 (𝐵 · 𝐶)↑2) ≤ (Σ𝑘𝐴 (𝐵↑2) · Σ𝑘𝐴 (𝐶↑2)))
 
Theoremtrirn 23920* Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)       (𝜑 → (√‘Σ𝑘𝐴 ((𝐵 + 𝐶)↑2)) ≤ ((√‘Σ𝑘𝐴 (𝐵↑2)) + (√‘Σ𝑘𝐴 (𝐶↑2))))
 
Theoremrrxf 23921* Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑𝐹:𝐼⟶ℝ)
 
Theoremrrxfsupp 23922* Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑 → (𝐹 supp 0) ∈ Fin)
 
Theoremrrxsuppss 23923* Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑 → (𝐹 supp 0) ⊆ 𝐼)
 
Theoremrrxmvallem 23924* Support of the function used for building the distance . (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}       ((𝐼𝑉𝐹𝑋𝐺𝑋) → ((𝑘𝐼 ↦ (((𝐹𝑘) − (𝐺𝑘))↑2)) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)))
 
Theoremrrxmval 23925* The value of the Euclidean metric. Compare with rrnmval 34976. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))       ((𝐼𝑉𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹𝑘) − (𝐺𝑘))↑2)))
 
Theoremrrxmfval 23926* The value of the Euclidean metric. Compare with rrnval 34975. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))       (𝐼𝑉𝐷 = (𝑓𝑋, 𝑔𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓𝑘) − (𝑔𝑘))↑2))))
 
Theoremrrxmetlem 23927* Lemma for rrxmet 23928. (Contributed by Thierry Arnoux, 5-Jul-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))    &   (𝜑𝐼𝑉)    &   (𝜑𝐹𝑋)    &   (𝜑𝐺𝑋)    &   (𝜑𝐴𝐼)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐴)       (𝜑 → Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹𝑘) − (𝐺𝑘))↑2) = Σ𝑘𝐴 (((𝐹𝑘) − (𝐺𝑘))↑2))
 
Theoremrrxmet 23928* Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.) (Revised by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))       (𝐼𝑉𝐷 ∈ (Met‘𝑋))
 
Theoremrrxdstprj1 23929* The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))    &   𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))       (((𝐼𝑉𝐴𝐼) ∧ (𝐹𝑋𝐺𝑋)) → ((𝐹𝐴)𝑀(𝐺𝐴)) ≤ (𝐹𝐷𝐺))
 
Theoremrrxbasefi 23930 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.)
(𝜑𝑋 ∈ Fin)    &   𝐻 = (ℝ^‘𝑋)    &   𝐵 = (Base‘𝐻)       (𝜑𝐵 = (ℝ ↑m 𝑋))
 
Theoremrrxdsfi 23931* The distance over generalized Euclidean spaces. Finite dimensional case. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (ℝ ↑m 𝐼)       (𝐼 ∈ Fin → (dist‘𝐻) = (𝑓𝐵, 𝑔𝐵 ↦ (√‘Σ𝑘𝐼 (((𝑓𝑘) − (𝑔𝑘))↑2))))
 
Theoremrrxmetfi 23932 Euclidean space is a metric space. Finite dimensional version. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐷 = (dist‘(ℝ^‘𝐼))       (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼)))
 
Theoremrrxdsfival 23933* 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)    &   𝐷 = (dist‘(ℝ^‘𝐼))       ((𝐼 ∈ Fin ∧ 𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘𝐼 (((𝐹𝑘) − (𝐺𝑘))↑2)))
 
Theoremehlval 23934 Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐸 = (𝔼hil𝑁)       (𝑁 ∈ ℕ0𝐸 = (ℝ^‘(1...𝑁)))
 
Theoremehlbase 23935 The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐸 = (𝔼hil𝑁)       (𝑁 ∈ ℕ0 → (ℝ ↑m (1...𝑁)) = (Base‘𝐸))
 
Theoremehl0base 23936 The base of the Euclidean space of dimension 0 consists only of one element, the empty set. (Contributed by AV, 12-Feb-2023.)
𝐸 = (𝔼hil‘0)       (Base‘𝐸) = {∅}
 
Theoremehl0 23937 The Euclidean space of dimension 0 consists of the neutral element only. (Contributed by AV, 12-Feb-2023.)
𝐸 = (𝔼hil‘0)    &    0 = (0g𝐸)       (Base‘𝐸) = { 0 }
 
Theoremehleudis 23938* The Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
𝐼 = (1...𝑁)    &   𝐸 = (𝔼hil𝑁)    &   𝑋 = (ℝ ↑m 𝐼)    &   𝐷 = (dist‘𝐸)       (𝑁 ∈ ℕ0𝐷 = (𝑓𝑋, 𝑔𝑋 ↦ (√‘Σ𝑘𝐼 (((𝑓𝑘) − (𝑔𝑘))↑2))))
 
Theoremehleudisval 23939* The value of the Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
𝐼 = (1...𝑁)    &   𝐸 = (𝔼hil𝑁)    &   𝑋 = (ℝ ↑m 𝐼)    &   𝐷 = (dist‘𝐸)       ((𝑁 ∈ ℕ0𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘𝐼 (((𝐹𝑘) − (𝐺𝑘))↑2)))
 
Theoremehl1eudis 23940* The Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.)
𝐸 = (𝔼hil‘1)    &   𝑋 = (ℝ ↑m {1})    &   𝐷 = (dist‘𝐸)       𝐷 = (𝑓𝑋, 𝑔𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1))))
 
Theoremehl1eudisval 23941 The value of the Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.)
𝐸 = (𝔼hil‘1)    &   𝑋 = (ℝ ↑m {1})    &   𝐷 = (dist‘𝐸)       ((𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (abs‘((𝐹‘1) − (𝐺‘1))))
 
Theoremehl2eudis 23942* The Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.)
𝐸 = (𝔼hil‘2)    &   𝑋 = (ℝ ↑m {1, 2})    &   𝐷 = (dist‘𝐸)       𝐷 = (𝑓𝑋, 𝑔𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))
 
Theoremehl2eudisval 23943 The value of the Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.)
𝐸 = (𝔼hil‘2)    &   𝑋 = (ℝ ↑m {1, 2})    &   𝐷 = (dist‘𝐸)       ((𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2))))
 
12.5.9  Minimizing Vector Theorem
 
Theoremminveclem1 23944* Lemma for minvec 23956. The set of all distances from points of 𝑌 to 𝐴 are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))       (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤𝑅 0 ≤ 𝑤))
 
Theoremminveclem4c 23945* Lemma for minvec 23956. The infimum of the distances to 𝐴 is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )       (𝜑𝑆 ∈ ℝ)
 
Theoremminveclem2 23946* Lemma for minvec 23956. Any two points 𝐾 and 𝐿 in 𝑌 are close to each other if they are close to the infimum of distance to 𝐴. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑𝐾𝑌)    &   (𝜑𝐿𝑌)    &   (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵))    &   (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵))       (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))
 
Theoremminveclem3a 23947* Lemma for minvec 23956. 𝐷 is a complete metric when restricted to 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
 
Theoremminveclem3b 23948* Lemma for minvec 23956. The set of vectors within a fixed distance of the infimum forms a filter base. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})       (𝜑𝐹 ∈ (fBas‘𝑌))
 
Theoremminveclem3 23949* Lemma for minvec 23956. The filter formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})       (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))
 
Theoremminveclem4a 23950* Lemma for minvec 23956. 𝐹 converges to a point 𝑃 in 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   𝑃 = (𝐽 fLim (𝑋filGen𝐹))       (𝜑𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
 
Theoremminveclem4b 23951* Lemma for minvec 23956. The convergent point of the Cauchy sequence 𝐹 is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   𝑃 = (𝐽 fLim (𝑋filGen𝐹))       (𝜑𝑃𝑋)
 
Theoremminveclem4 23952* Lemma for minvec 23956. The convergent point of the Cauchy sequence 𝐹 attains the minimum distance, and so is closer to 𝐴 than any other point in 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   𝑃 = (𝐽 fLim (𝑋filGen𝐹))    &   𝑇 = (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))       (𝜑 → ∃𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))
 
Theoremminveclem5 23953* Lemma for minvec 23956. Discharge the assumptions in minveclem4 23952. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       (𝜑 → ∃𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))
 
Theoremminveclem6 23954* Lemma for minvec 23956. Any minimal point is less than 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       ((𝜑𝑥𝑌) → (((𝐴𝐷𝑥)↑2) ≤ ((𝑆↑2) + 0) ↔ ∀𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦))))
 
Theoremminveclem7 23955* Lemma for minvec 23956. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       (𝜑 → ∃!𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))
 
Theoremminvec 23956* Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace 𝑊 that minimizes the distance to an arbitrary vector 𝐴 in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Proof shortened by AV, 3-Oct-2020.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)       (𝜑 → ∃!𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))
 
12.5.10  Projection Theorem
 
Theorempjthlem1 23957* Lemma for pjth 23959. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 17-Oct-2015.) (Proof shortened by AV, 10-Jul-2022.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &    , = (·𝑖𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ ℂHil)    &   (𝜑𝑈𝐿)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑈)    &   (𝜑 → ∀𝑥𝑈 (𝑁𝐴) ≤ (𝑁‘(𝐴 𝑥)))    &   𝑇 = ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))       (𝜑 → (𝐴 , 𝐵) = 0)
 
Theorempjthlem2 23958 Lemma for pjth 23959. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &    , = (·𝑖𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ ℂHil)    &   (𝜑𝑈𝐿)    &   (𝜑𝐴𝑉)    &   𝐽 = (TopOpen‘𝑊)    &    = (LSSum‘𝑊)    &   𝑂 = (ocv‘𝑊)    &   (𝜑𝑈 ∈ (Clsd‘𝐽))       (𝜑𝐴 ∈ (𝑈 (𝑂𝑈)))
 
Theorempjth 23959 Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.)
𝑉 = (Base‘𝑊)    &    = (LSSum‘𝑊)    &   𝑂 = (ocv‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ ℂHil ∧ 𝑈𝐿𝑈 ∈ (Clsd‘𝐽)) → (𝑈 (𝑂𝑈)) = 𝑉)
 
Theorempjth2 23960 Projection Theorem with abbreviations: A topologically closed subspace is a projection subspace. (Contributed by Mario Carneiro, 17-Oct-2015.)
𝐽 = (TopOpen‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   𝐾 = (proj‘𝑊)       ((𝑊 ∈ ℂHil ∧ 𝑈𝐿𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ dom 𝐾)
 
Theoremcldcss 23961 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑊 ∈ ℂHil → (𝑈𝐶 ↔ (𝑈𝐿𝑈 ∈ (Clsd‘𝐽))))
 
Theoremcldcss2 23962 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑊 ∈ ℂHil → 𝐶 = (𝐿 ∩ (Clsd‘𝐽)))
 
Theoremhlhil 23963 Corollary of the Projection Theorem: A subcomplex Hilbert space is a Hilbert space (in the algebraic sense, meaning that all algebraically closed subspaces have a projection decomposition). (Contributed by Mario Carneiro, 17-Oct-2015.)
(𝑊 ∈ ℂHil → 𝑊 ∈ Hil)
 
PART 13  BASIC REAL AND COMPLEX ANALYSIS
 
13.1  Continuity
 
Theoremmulcncf 23964* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋cn→ℂ))
 
Theoremdivcncf 23965* The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→(ℂ ∖ {0})))       (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐵)) ∈ (𝑋cn→ℂ))
 
13.1.1  Intermediate value theorem
 
Theorempmltpclem1 23966* Lemma for pmltpc 23968. (Contributed by Mario Carneiro, 1-Jul-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵 < 𝐶)    &   (𝜑 → (((𝐹𝐴) < (𝐹𝐵) ∧ (𝐹𝐶) < (𝐹𝐵)) ∨ ((𝐹𝐵) < (𝐹𝐴) ∧ (𝐹𝐵) < (𝐹𝐶))))       (𝜑 → ∃𝑎𝑆𝑏𝑆𝑐𝑆 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐)))))
 
Theorempmltpclem2 23967* Lemma for pmltpc 23968. (Contributed by Mario Carneiro, 1-Jul-2014.)
(𝜑𝐹 ∈ (ℝ ↑pm ℝ))    &   (𝜑𝐴 ⊆ dom 𝐹)    &   (𝜑𝑈𝐴)    &   (𝜑𝑉𝐴)    &   (𝜑𝑊𝐴)    &   (𝜑𝑋𝐴)    &   (𝜑𝑈𝑉)    &   (𝜑𝑊𝑋)    &   (𝜑 → ¬ (𝐹𝑈) ≤ (𝐹𝑉))    &   (𝜑 → ¬ (𝐹𝑋) ≤ (𝐹𝑊))       (𝜑 → ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐)))))
 
Theorempmltpc 23968* Any function on the reals is either increasing, decreasing, or has a triple of points in a vee formation. (This theorem was created on demand by Mario Carneiro for the 6PCM conference in Bialystok, 1-Jul-2014.) (Contributed by Mario Carneiro, 1-Jul-2014.)
((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)) ∨ ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
 
Theoremivthlem1 23969* Lemma for ivth 23972. The set 𝑆 of all 𝑥 values with (𝐹𝑥) less than 𝑈 is lower bounded by 𝐴 and upper bounded by 𝐵. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}       (𝜑 → (𝐴𝑆 ∧ ∀𝑧𝑆 𝑧𝐵))
 
Theoremivthlem2 23970* Lemma for ivth 23972. Show that the supremum of 𝑆 cannot be less than 𝑈. If it was, continuity of 𝐹 implies that there are points just above the supremum that are also less than 𝑈, a contradiction. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}    &   𝐶 = sup(𝑆, ℝ, < )       (𝜑 → ¬ (𝐹𝐶) < 𝑈)
 
Theoremivthlem3 23971* Lemma for ivth 23972, the intermediate value theorem. Show that (𝐹𝐶) cannot be greater than 𝑈, and so establish the existence of a root of the function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 17-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}    &   𝐶 = sup(𝑆, ℝ, < )       (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ∧ (𝐹𝐶) = 𝑈))
 
Theoremivth 23972* The intermediate value theorem, increasing case. This is Metamath 100 proof #79. (Contributed by Paul Chapman, 22-Jan-2008.) (Proof shortened by Mario Carneiro, 30-Apr-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))       (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivth2 23973* The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐵) < 𝑈𝑈 < (𝐹𝐴)))       (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivthle 23974* The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) ≤ 𝑈𝑈 ≤ (𝐹𝐵)))       (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivthle2 23975* The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐵) ≤ 𝑈𝑈 ≤ (𝐹𝐴)))       (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivthicc 23976* The interval between any two points of a continuous real function is contained in the range of the function. Equivalently, the range of a continuous real function is convex. (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ (𝐴[,]𝐵))    &   (𝜑𝑁 ∈ (𝐴[,]𝐵))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)       (𝜑 → ((𝐹𝑀)[,](𝐹𝑁)) ⊆ ran 𝐹)
 
Theoremevthicc 23977* Specialization of the Extreme Value Theorem to a closed interval of . (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))       (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹𝑧) ≤ (𝐹𝑤)))
 
Theoremevthicc2 23978* Combine ivthicc 23976 with evthicc 23977 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))       (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran 𝐹 = (𝑥[,]𝑦))
 
Theoremcniccbdd 23979* A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹𝑦)) ≤ 𝑥)
 
13.2  Integrals
 
13.2.1  Lebesgue measure
 
Syntaxcovol 23980 Extend class notation with the outer Lebesgue measure.
class vol*
 
Syntaxcvol 23981 Extend class notation with the Lebesgue measure.
class vol
 
Definitiondf-ovol 23982* Define the outer Lebesgue measure for subsets of the reals. Here 𝑓 is a function from the positive integers to pairs 𝑎, 𝑏 with 𝑎𝑏, and the outer volume of the set 𝑥 is the infimum over all such functions such that the union of the open intervals (𝑎, 𝑏) covers 𝑥 of the sum of 𝑏𝑎. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
 
Definitiondf-vol 23983* Define the Lebesgue measure, which is just the outer measure with a peculiar domain of definition. The property of being Lebesgue-measurable can be expressed as 𝐴 ∈ dom vol. (Contributed by Mario Carneiro, 17-Mar-2014.)
vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})
 
Theoremovolfcl 23984 Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
 
Theoremovolfioo 23985* Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
 
Theoremovolficc 23986* Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
 
Theoremovolficcss 23987 Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)
(𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
 
Theoremovolfsval 23988 The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
 
Theoremovolfsf 23989 Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))
 
Theoremovolsf 23990 Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)    &   𝑆 = seq1( + , 𝐺)       (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
 
Theoremovolval 23991* The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
 
Theoremelovolmlem 23992 Lemma for elovolm 23993 and related theorems. (Contributed by BJ, 23-Jul-2022.)
(𝐹 ∈ ((𝐴 ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶(𝐴 ∩ (ℝ × ℝ)))
 
Theoremelovolm 23993* Elementhood in the set 𝑀 of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝐵𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))
 
Theoremelovolmr 23994* Sufficient condition for elementhood in the set 𝑀. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐹)) → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
 
Theoremovolmge0 23995* The set 𝑀 is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝐵𝑀 → 0 ≤ 𝐵)
 
Theoremovolcl 23996 The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
 
Theoremovollb 23997 The outer volume is a lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
 
Theoremovolgelb 23998* The outer volume is the greatest lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑔))       ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
 
Theoremovolge0 23999 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
 
Theoremovolf 24000 The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
vol*:𝒫 ℝ⟶(0[,]+∞)
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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 44701-44740
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