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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hlbn 25201 | Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) |
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) | ||
Theorem | hlcph 25202 | Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) | ||
Theorem | hlphl 25203 | Every subcomplex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) | ||
Theorem | hlcms 25204 | Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.) |
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp) | ||
Theorem | hlprlem 25205 | Lemma for hlpr 25207. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂHil → (𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐾) ∈ DivRing ∧ (ℂfld ↾s 𝐾) ∈ CMetSp)) | ||
Theorem | hlress 25206 | The scalar field of a subcomplex Hilbert space contains ℝ. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂHil → ℝ ⊆ 𝐾) | ||
Theorem | hlpr 25207 | The scalar field of a subcomplex Hilbert space is either ℝ or ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂHil → 𝐾 ∈ {ℝ, ℂ}) | ||
Theorem | ishl2 25208 | A Hilbert space is a complete subcomplex pre-Hilbert space over ℝ or ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ})) | ||
Theorem | cphssphl 25209 | A Banach subspace of a subcomplex pre-Hilbert space is a subcomplex Hilbert space. (Contributed by NM, 11-Apr-2008.) (Revised by AV, 25-Sep-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ Ban) → 𝑋 ∈ ℂHil) | ||
Theorem | cmslssbn 25210 | A complete linear subspace of a normed vector space is a Banach space. We furthermore have to assume that the field of scalars is complete since this is a requirement in the current definition of Banach spaces df-bn 25174. (Contributed by AV, 8-Oct-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ Ban) | ||
Theorem | cmscsscms 25211 | A closed subspace of a complete metric space which is also a subcomplex pre-Hilbert space is a complete metric space. Remark: the assumption that the Banach space must be a (subcomplex) pre-Hilbert space is required because the definition of ClSubSp is based on an inner product. If ClSubSp was generalized to arbitrary topological spaces (or at least topological modules), this assumption could be omitted. (Contributed by AV, 8-Oct-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (ClSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil) ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ CMetSp) | ||
Theorem | bncssbn 25212 | A closed subspace of a Banach space which is also a subcomplex pre-Hilbert space is a Banach space. Remark: the assumption that the Banach space must be a (subcomplex) pre-Hilbert space is required because the definition of ClSubSp is based on an inner product. If ClSubSp was generalized for arbitrary topological spaces, this assuption could be omitted. (Contributed by AV, 8-Oct-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (ClSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ Ban) | ||
Theorem | cssbn 25213 | A complete subspace of a normed vector space with a complete scalar field is a Banach space. Remark: In contrast to ClSubSp, a complete subspace is defined by "a linear subspace in which all Cauchy sequences converge to a point in the subspace". This is closer to the original, but deprecated definition Cℋ (df-ch 30898) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 25210. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈)) ⇒ ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ Ban) | ||
Theorem | csschl 25214 | A complete subspace of a complex pre-Hilbert space is a complex Hilbert space. Remarks: (a) In contrast to ClSubSp, a complete subspace is defined by "a linear subspace in which all Cauchy sequences converge to a point in the subspace". This is closer to the original, but deprecated definition Cℋ (df-ch 30898) of closed subspaces of a Hilbert space. (b) This theorem does not hold for arbitrary subcomplex (pre-)Hilbert spaces, because the scalar field as restriction of the field of the complex numbers need not be closed. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈)) & ⊢ (Scalar‘𝑊) = ℂfld ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (𝑋 ∈ ℂHil ∧ (Scalar‘𝑋) = ℂfld)) | ||
Theorem | cmslsschl 25215 | A complete linear subspace of a subcomplex Hilbert space is a subcomplex Hilbert space. (Contributed by AV, 8-Oct-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂHil ∧ 𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂHil) | ||
Theorem | chlcsschl 25216 | A closed subspace of a subcomplex Hilbert space is a subcomplex Hilbert space. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 8-Oct-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (ClSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂHil) | ||
Theorem | retopn 25217 | The topology of the real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | ||
Theorem | recms 25218 | The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ ℝfld ∈ CMetSp | ||
Theorem | reust 25219 | The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.) |
⊢ (UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) | ||
Theorem | recusp 25220 | The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
⊢ ℝfld ∈ CUnifSp | ||
Syntax | crrx 25221 | Extend class notation with generalized real Euclidean spaces. |
class ℝ^ | ||
Syntax | cehl 25222 | Extend class notation with real Euclidean spaces. |
class 𝔼hil | ||
Definition | df-rrx 25223 | 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))) | ||
Definition | df-ehl 25224 | 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 25255). 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))) | ||
Theorem | rrxval 25225 | Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) | ||
Theorem | rrxbase 25226* | 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) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0}) | ||
Theorem | rrxprds 25227 | Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵))) | ||
Theorem | rrxip 25228* | The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝐻)) | ||
Theorem | rrxnm 25229* | The norm of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵 ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))) = (norm‘𝐻)) | ||
Theorem | rrxcph 25230 | 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) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐻 ∈ ℂPreHil) | ||
Theorem | rrxds 25231* | The distance over generalized Euclidean spaces. Compare with df-rrn 37150. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘𝐻)) | ||
Theorem | rrxvsca 25232 | 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) & ⊢ ∙ = ( ·𝑠 ‘𝐻) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻)) ⇒ ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) | ||
Theorem | rrxplusgvscavalb 25233* | 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) & ⊢ ∙ = ( ·𝑠 ‘𝐻) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ ✚ = (+g‘𝐻) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) | ||
Theorem | rrxsca 25234 | The field of real numbers is the scalar field of the generalized real Euclidean space. (Contributed by AV, 15-Jan-2023.) |
⊢ 𝐻 = (ℝ^‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → (Scalar‘𝐻) = ℝfld) | ||
Theorem | rrx0 25235 | The zero ("origin") in a generalized real Euclidean space. (Contributed by AV, 11-Feb-2023.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 0 = (𝐼 × {0}) ⇒ ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = 0 ) | ||
Theorem | rrx0el 25236 | 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) ⇒ ⊢ (𝐼 ∈ 𝑉 → 0 ∈ 𝑃) | ||
Theorem | csbren 25237* | Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶)↑2) ≤ (Σ𝑘 ∈ 𝐴 (𝐵↑2) · Σ𝑘 ∈ 𝐴 (𝐶↑2))) | ||
Theorem | trirn 25238* | Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (√‘Σ𝑘 ∈ 𝐴 ((𝐵 + 𝐶)↑2)) ≤ ((√‘Σ𝑘 ∈ 𝐴 (𝐵↑2)) + (√‘Σ𝑘 ∈ 𝐴 (𝐶↑2)))) | ||
Theorem | rrxf 25239* | Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐹 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐹:𝐼⟶ℝ) | ||
Theorem | rrxfsupp 25240* | Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐹 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin) | ||
Theorem | rrxsuppss 25241* | Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐹 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹 supp 0) ⊆ 𝐼) | ||
Theorem | rrxmvallem 25242* | Support of the function used for building the distance . (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝑘 ∈ 𝐼 ↦ (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0))) | ||
Theorem | rrxmval 25243* | The value of the Euclidean metric. Compare with rrnmval 37152. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))) | ||
Theorem | rrxmfval 25244* | The value of the Euclidean metric. Compare with rrnval 37151. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) | ||
Theorem | rrxmetlem 25245* | Lemma for rrxmet 25246. (Contributed by Thierry Arnoux, 5-Jul-2019.) |
⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ (𝜑 → 𝐺 ∈ 𝑋) & ⊢ (𝜑 → 𝐴 ⊆ 𝐼) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐴) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = Σ𝑘 ∈ 𝐴 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) | ||
Theorem | rrxmet 25246* | 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 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋)) | ||
Theorem | rrxdstprj1 25247* | 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 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) & ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺)) | ||
Theorem | rrxbasefi 25248 | 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.) |
⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ 𝐻 = (ℝ^‘𝑋) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝜑 → 𝐵 = (ℝ ↑m 𝑋)) | ||
Theorem | rrxdsfi 25249* | 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) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) | ||
Theorem | rrxmetfi 25250 | 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))) | ||
Theorem | rrxdsfival 25251* | 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 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))) | ||
Theorem | ehlval 25252 | Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝐸 = (𝔼hil‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁))) | ||
Theorem | ehlbase 25253 | The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝐸 = (𝔼hil‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑m (1...𝑁)) = (Base‘𝐸)) | ||
Theorem | ehl0base 25254 | 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) = {∅} | ||
Theorem | ehl0 25255 | The Euclidean space of dimension 0 consists of the neutral element only. (Contributed by AV, 12-Feb-2023.) |
⊢ 𝐸 = (𝔼hil‘0) & ⊢ 0 = (0g‘𝐸) ⇒ ⊢ (Base‘𝐸) = { 0 } | ||
Theorem | ehleudis 25256* | The Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.) |
⊢ 𝐼 = (1...𝑁) & ⊢ 𝐸 = (𝔼hil‘𝑁) & ⊢ 𝑋 = (ℝ ↑m 𝐼) & ⊢ 𝐷 = (dist‘𝐸) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) | ||
Theorem | ehleudisval 25257* | 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 = (𝔼hil‘𝑁) & ⊢ 𝑋 = (ℝ ↑m 𝐼) & ⊢ 𝐷 = (dist‘𝐸) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))) | ||
Theorem | ehl1eudis 25258* | 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 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1)))) | ||
Theorem | ehl1eudisval 25259 | 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) ⇒ ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (abs‘((𝐹‘1) − (𝐺‘1)))) | ||
Theorem | ehl2eudis 25260* | 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 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2)))) | ||
Theorem | ehl2eudisval 25261 | 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) ⇒ ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2)))) | ||
Theorem | minveclem1 25262* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) ⇒ ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) | ||
Theorem | minveclem4c 25263* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) ⇒ ⊢ (𝜑 → 𝑆 ∈ ℝ) | ||
Theorem | minveclem2 25264* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) & ⊢ (𝜑 → 𝐾 ∈ 𝑌) & ⊢ (𝜑 → 𝐿 ∈ 𝑌) & ⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵)) & ⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵)) ⇒ ⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵)) | ||
Theorem | minveclem3a 25265* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) | ||
Theorem | minveclem3b 25266* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) & ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ⇒ ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌)) | ||
Theorem | minveclem3 25267* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) & ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ⇒ ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) | ||
Theorem | minveclem4a 25268* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) & ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) & ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹)) ⇒ ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌)) | ||
Theorem | minveclem4b 25269* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) & ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) & ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹)) ⇒ ⊢ (𝜑 → 𝑃 ∈ 𝑋) | ||
Theorem | minveclem4 25270* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) & ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) & ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹)) & ⊢ 𝑇 = (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦))) | ||
Theorem | minveclem5 25271* | Lemma for minvec 25274. Discharge the assumptions in minveclem4 25270. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦))) | ||
Theorem | minveclem6 25272* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (((𝐴𝐷𝑥)↑2) ≤ ((𝑆↑2) + 0) ↔ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) | ||
Theorem | minveclem7 25273* | Lemma for minvec 25274. 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐽 = (TopOpen‘𝑈) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦))) | ||
Theorem | minvec 25274* | 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) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (norm‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) & ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) & ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦))) | ||
Theorem | pjthlem1 25275* | Lemma for pjth 25277. (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) & ⊢ + = (+g‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐿 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ ℂHil) & ⊢ (𝜑 → 𝑈 ∈ 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − 𝑥))) & ⊢ 𝑇 = ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ⇒ ⊢ (𝜑 → (𝐴 , 𝐵) = 0) | ||
Theorem | pjthlem2 25276 | Lemma for pjth 25277. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐿 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ ℂHil) & ⊢ (𝜑 → 𝑈 ∈ 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑂 = (ocv‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐽)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝑈 ⊕ (𝑂‘𝑈))) | ||
Theorem | pjth 25277 | 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 ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑈 ⊕ (𝑂‘𝑈)) = 𝑉) | ||
Theorem | pjth2 25278 | 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 ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ dom 𝐾) | ||
Theorem | cldcss 25279 | 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 → (𝑈 ∈ 𝐶 ↔ (𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)))) | ||
Theorem | cldcss2 25280 | 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))) | ||
Theorem | hlhil 25281 | 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) | ||
Theorem | addcncf 25282* | The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝑋–cn→ℂ)) | ||
Theorem | subcncf 25283* | The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 − 𝐵)) ∈ (𝑋–cn→ℂ)) | ||
Theorem | mulcncf 25284* | The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.) Avoid ax-mulf 11185. (Revised by GG, 16-Mar-2025.) |
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋–cn→ℂ)) | ||
Theorem | mulcncfOLD 25285* | Obsolete version of mulcncf 25284 as of 9-Apr-2025. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋–cn→ℂ)) | ||
Theorem | divcncf 25286* | The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→(ℂ ∖ {0}))) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)) ∈ (𝑋–cn→ℂ)) | ||
Theorem | pmltpclem1 25287* | Lemma for pmltpc 25289. (Contributed by Mario Carneiro, 1-Jul-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐵 < 𝐶) & ⊢ (𝜑 → (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝐶) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝐶)))) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))) | ||
Theorem | pmltpclem2 25288* | Lemma for pmltpc 25289. (Contributed by Mario Carneiro, 1-Jul-2014.) |
⊢ (𝜑 → 𝐹 ∈ (ℝ ↑pm ℝ)) & ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) & ⊢ (𝜑 → 𝑉 ∈ 𝐴) & ⊢ (𝜑 → 𝑊 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑈 ≤ 𝑉) & ⊢ (𝜑 → 𝑊 ≤ 𝑋) & ⊢ (𝜑 → ¬ (𝐹‘𝑈) ≤ (𝐹‘𝑉)) & ⊢ (𝜑 → ¬ (𝐹‘𝑋) ≤ (𝐹‘𝑊)) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))) | ||
Theorem | pmltpc 25289* | 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) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)) ∨ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))) | ||
Theorem | ivthlem1 25290* | Lemma for ivth 25293. The set 𝑆 of all 𝑥 values with (𝐹‘𝑥) less than 𝑈 is lower bounded by 𝐴 and upper bounded by 𝐵. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵)) | ||
Theorem | ivthlem2 25291* | Lemma for ivth 25293. 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} & ⊢ 𝐶 = sup(𝑆, ℝ, < ) ⇒ ⊢ (𝜑 → ¬ (𝐹‘𝐶) < 𝑈) | ||
Theorem | ivthlem3 25292* | Lemma for ivth 25293, the intermediate value theorem. Show that (𝐹‘𝐶) 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} & ⊢ 𝐶 = sup(𝑆, ℝ, < ) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ∧ (𝐹‘𝐶) = 𝑈)) | ||
Theorem | ivth 25293* | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈) | ||
Theorem | ivth2 25294* | The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈) | ||
Theorem | ivthle 25295* | The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) ≤ 𝑈 ∧ 𝑈 ≤ (𝐹‘𝐵))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈) | ||
Theorem | ivthle2 25296* | The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐵) ≤ 𝑈 ∧ 𝑈 ≤ (𝐹‘𝐴))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈) | ||
Theorem | ivthicc 25297* | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑁 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐹‘𝑀)[,](𝐹‘𝑁)) ⊆ ran 𝐹) | ||
Theorem | evthicc 25298* | Specialization of the Extreme Value Theorem to a closed interval of ℝ. (Contributed by Mario Carneiro, 12-Aug-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤))) | ||
Theorem | evthicc2 25299* | Combine ivthicc 25297 with evthicc 25298 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran 𝐹 = (𝑥[,]𝑦)) | ||
Theorem | cniccbdd 25300* | 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‘(𝐹‘𝑦)) ≤ 𝑥) |
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