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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cfiluweak 22901 | A Cauchy filter base is also a Cauchy filter base on any coarser uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.) |
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu‘𝑈)) | ||
Theorem | neipcfilu 22902 | In an uniform space, a neighboring filter is a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.) |
⊢ 𝑋 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑈 = (UnifSt‘𝑊) ⇒ ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu‘𝑈)) | ||
Syntax | ccusp 22903 | Extend class notation with the class of all complete uniform spaces. |
class CUnifSp | ||
Definition | df-cusp 22904* | Define the class of all complete uniform spaces. Definition 3 of [BourbakiTop1] p. II.15. (Contributed by Thierry Arnoux, 1-Dec-2017.) |
⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)} | ||
Theorem | iscusp 22905* | The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.) |
⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) | ||
Theorem | cuspusp 22906 | A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.) |
⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp) | ||
Theorem | cuspcvg 22907 | In a complete uniform space, any Cauchy filter 𝐶 has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) ⇒ ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅) | ||
Theorem | iscusp2 22908* | The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑈 = (UnifSt‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) ⇒ ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅))) | ||
Theorem | cnextucn 22909* | Extension by continuity. Proposition 11 of [BourbakiTop1] p. II.20. Given a topology 𝐽 on 𝑋, a subset 𝐴 dense in 𝑋, this states a condition for 𝐹 from 𝐴 to a space 𝑌 Hausdorff and complete to be extensible by continuity. (Contributed by Thierry Arnoux, 4-Dec-2017.) |
⊢ 𝑋 = (Base‘𝑉) & ⊢ 𝑌 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑉) & ⊢ 𝐾 = (TopOpen‘𝑊) & ⊢ 𝑈 = (UnifSt‘𝑊) & ⊢ (𝜑 → 𝑉 ∈ TopSp) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → 𝑊 ∈ CUnifSp) & ⊢ (𝜑 → 𝐾 ∈ Haus) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑌) & ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈)) ⇒ ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾)) | ||
Theorem | ucnextcn 22910 | Extension by continuity. Theorem 2 of [BourbakiTop1] p. II.20. Given an uniform space on a set 𝑋, a subset 𝐴 dense in 𝑋, and a function 𝐹 uniformly continuous from 𝐴 to 𝑌, that function can be extended by continuity to the whole 𝑋, and its extension is uniformly continuous. (Contributed by Thierry Arnoux, 25-Jan-2018.) |
⊢ 𝑋 = (Base‘𝑉) & ⊢ 𝑌 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑉) & ⊢ 𝐾 = (TopOpen‘𝑊) & ⊢ 𝑆 = (UnifSt‘𝑉) & ⊢ 𝑇 = (UnifSt‘(𝑉 ↾s 𝐴)) & ⊢ 𝑈 = (UnifSt‘𝑊) & ⊢ (𝜑 → 𝑉 ∈ TopSp) & ⊢ (𝜑 → 𝑉 ∈ UnifSp) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → 𝑊 ∈ CUnifSp) & ⊢ (𝜑 → 𝐾 ∈ Haus) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → 𝐹 ∈ (𝑇 Cnu𝑈)) & ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋) ⇒ ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾)) | ||
Theorem | ispsmet 22911* | Express the predicate "𝐷 is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) | ||
Theorem | psmetdmdm 22912 | Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷) | ||
Theorem | psmetf 22913 | The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | ||
Theorem | psmetcl 22914 | Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | ||
Theorem | psmet0 22915 | The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) | ||
Theorem | psmettri2 22916 | Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) | ||
Theorem | psmetsym 22917 | The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴)) | ||
Theorem | psmettri 22918 | Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵))) | ||
Theorem | psmetge0 22919 | The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | ||
Theorem | psmetxrge0 22920 | The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞)) | ||
Theorem | psmetres2 22921 | Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅)) | ||
Theorem | psmetlecl 22922 | Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ ℝ ∧ (𝐴𝐷𝐵) ≤ 𝐶)) → (𝐴𝐷𝐵) ∈ ℝ) | ||
Theorem | distspace 22923 | A set 𝑋 together with a (distance) function 𝐷 which is a pseudometric is a distance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set 𝑋 equipped with a distance 𝐷, which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.) (Revised by AV, 5-Jul-2022.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ (𝐴𝐷𝐴) = 0) ∧ (0 ≤ (𝐴𝐷𝐵) ∧ (𝐴𝐷𝐵) = (𝐵𝐷𝐴)))) | ||
Syntax | cxms 22924 | Extend class notation with the class of extended metric spaces. |
class ∞MetSp | ||
Syntax | cms 22925 | Extend class notation with the class of metric spaces. |
class MetSp | ||
Syntax | ctms 22926 | Extend class notation with the function mapping a metric to the metric space it defines. |
class toMetSp | ||
Definition | df-xms 22927 | Define the (proper) class of extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} | ||
Definition | df-ms 22928 | Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.) |
⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))} | ||
Definition | df-tms 22929 | Define the function mapping a metric to the metric space which it defines. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({〈(Base‘ndx), dom dom 𝑑〉, 〈(dist‘ndx), 𝑑〉} sSet 〈(TopSet‘ndx), (MetOpen‘𝑑)〉)) | ||
Theorem | ismet 22930* | Express the predicate "𝐷 is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))) | ||
Theorem | isxmet 22931* | Express the predicate "𝐷 is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) | ||
Theorem | ismeti 22932* | Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ 𝑋 ∈ V & ⊢ 𝐷:(𝑋 × 𝑋)⟶ℝ & ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)) & ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))) ⇒ ⊢ 𝐷 ∈ (Met‘𝑋) | ||
Theorem | isxmetd 22933* | Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 9-Apr-2024.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) ⇒ ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | ||
Theorem | isxmet2d 22934* | It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample: 𝐷(𝑥, 𝑦) = if(𝑥 = 𝑦, 0, -∞) satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ (𝑥𝐷𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) ≤ 0 ↔ 𝑥 = 𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))) ⇒ ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | ||
Theorem | metflem 22935* | Lemma for metf 22937 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))) | ||
Theorem | xmetf 22936 | Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | ||
Theorem | metf 22937 | Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) |
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | ||
Theorem | xmetcl 22938 | Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | ||
Theorem | metcl 22939 | Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ) | ||
Theorem | ismet2 22940 | An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷:(𝑋 × 𝑋)⟶ℝ)) | ||
Theorem | metxmet 22941 | A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | ||
Theorem | xmetdmdm 22942 | Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷) | ||
Theorem | metdmdm 22943 | Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 = dom dom 𝐷) | ||
Theorem | xmetunirn 22944 | Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) | ||
Theorem | xmeteq0 22945 | The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)) | ||
Theorem | meteq0 22946 | The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4. (Contributed by NM, 30-Aug-2006.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)) | ||
Theorem | xmettri2 22947 | Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) | ||
Theorem | mettri2 22948 | Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵))) | ||
Theorem | xmet0 22949 | The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) | ||
Theorem | met0 22950 | The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by NM, 30-Aug-2006.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) | ||
Theorem | xmetge0 22951 | The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | ||
Theorem | metge0 22952 | The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | ||
Theorem | xmetlecl 22953 | Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ ℝ ∧ (𝐴𝐷𝐵) ≤ 𝐶)) → (𝐴𝐷𝐵) ∈ ℝ) | ||
Theorem | xmetsym 22954 | The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴)) | ||
Theorem | xmetpsmet 22955 | An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | ||
Theorem | xmettpos 22956 | The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → tpos 𝐷 = 𝐷) | ||
Theorem | metsym 22957 | The distance function of a metric space is symmetric. Definition 14-1.1(c) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴)) | ||
Theorem | xmettri 22958 | Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵))) | ||
Theorem | mettri 22959 | Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐶𝐷𝐵))) | ||
Theorem | xmettri3 22960 | Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐵𝐷𝐶))) | ||
Theorem | mettri3 22961 | Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶))) | ||
Theorem | xmetrtri 22962 | One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵)) | ||
Theorem | xmetrtri2 22963 | The reverse triangle inequality for the distance function of an extended metric. In order to express the "extended absolute value function", we use the distance function xrsdsval 20135 defined on the extended real structure. (Contributed by Mario Carneiro, 4-Sep-2015.) |
⊢ 𝐾 = (dist‘ℝ*𝑠) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐶)𝐾(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵)) | ||
Theorem | metrtri 22964 | Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵)) | ||
Theorem | xmetgt0 22965 | The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≠ 𝐵 ↔ 0 < (𝐴𝐷𝐵))) | ||
Theorem | metgt0 22966 | The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of [Gleason] p. 223 and its converse. (Contributed by NM, 27-Aug-2006.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≠ 𝐵 ↔ 0 < (𝐴𝐷𝐵))) | ||
Theorem | metn0 22967 | A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅)) | ||
Theorem | xmetres2 22968 | Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅)) | ||
Theorem | metreslem 22969 | Lemma for metres 22972. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅)))) | ||
Theorem | metres2 22970 | Lemma for metres 22972. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘𝑅)) | ||
Theorem | xmetres 22971 | A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘(𝑋 ∩ 𝑅))) | ||
Theorem | metres 22972 | A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘(𝑋 ∩ 𝑅))) | ||
Theorem | 0met 22973 | The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ ∅ ∈ (Met‘∅) | ||
Theorem | prdsdsf 22974* | The product metric is a function into the nonnegative extended reals. In general this means that it is not a metric but rather an *extended* metric (even when all the factors are metrics), but it will be a metric when restricted to regions where it does not take infinite values. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑉 = (Base‘𝑅) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) ⇒ ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶(0[,]+∞)) | ||
Theorem | prdsxmetlem 22975* | The product metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑉 = (Base‘𝑅) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) | ||
Theorem | prdsxmet 22976* | The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 22975. (Contributed by Mario Carneiro, 26-Sep-2015.) |
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑉 = (Base‘𝑅) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) | ||
Theorem | prdsmet 22977* | The product metric is a metric when the index set is finite. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑉 = (Base‘𝑅) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵)) | ||
Theorem | ressprdsds 22978* | Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.) |
⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅))) & ⊢ (𝜑 → 𝐻 = (𝑇Xs(𝑥 ∈ 𝐼 ↦ (𝑅 ↾s 𝐴)))) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ 𝐸 = (dist‘𝐻) & ⊢ (𝜑 → 𝑆 ∈ 𝑈) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐸 = (𝐷 ↾ (𝐵 × 𝐵))) | ||
Theorem | resspwsds 22979 | Restriction of a power metric. (Contributed by Mario Carneiro, 16-Sep-2015.) |
⊢ (𝜑 → 𝑌 = (𝑅 ↑s 𝐼)) & ⊢ (𝜑 → 𝐻 = ((𝑅 ↾s 𝐴) ↑s 𝐼)) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ 𝐸 = (dist‘𝐻) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐸 = (𝐷 ↾ (𝐵 × 𝐵))) | ||
Theorem | imasdsf1olem 22980* | Lemma for imasdsf1o 22981. (Contributed by Mario Carneiro, 21-Aug-2015.) (Proof shortened by AV, 6-Oct-2020.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ 𝑊 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) & ⊢ 𝑆 = {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} & ⊢ 𝑇 = ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) ⇒ ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌)) | ||
Theorem | imasdsf1o 22981 | The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌)) | ||
Theorem | imasf1oxmet 22982 | The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) | ||
Theorem | imasf1omet 22983 | The image of a metric is a metric. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ (Met‘𝑉)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵)) | ||
Theorem | xpsdsfn 22984 | Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝑃 = (dist‘𝑇) ⇒ ⊢ (𝜑 → 𝑃 Fn ((𝑋 × 𝑌) × (𝑋 × 𝑌))) | ||
Theorem | xpsdsfn2 22985 | Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝑃 = (dist‘𝑇) ⇒ ⊢ (𝜑 → 𝑃 Fn ((Base‘𝑇) × (Base‘𝑇))) | ||
Theorem | xpsxmetlem 22986* | Lemma for xpsxmet 22987. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝑃 = (dist‘𝑇) & ⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋)) & ⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌)) & ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌)) ⇒ ⊢ (𝜑 → (dist‘((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉})) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}))) | ||
Theorem | xpsxmet 22987 | A product metric of extended metrics is an extended metric. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝑃 = (dist‘𝑇) & ⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋)) & ⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌)) & ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌)) ⇒ ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌))) | ||
Theorem | xpsdsval 22988 | Value of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝑃 = (dist‘𝑇) & ⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋)) & ⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌)) & ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉𝑃〈𝐶, 𝐷〉) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < )) | ||
Theorem | xpsmet 22989 | The direct product of two metric spaces. Definition 14-1.5 of [Gleason] p. 225. (Contributed by NM, 20-Jun-2007.) (Revised by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝑃 = (dist‘𝑇) & ⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋)) & ⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌)) & ⊢ (𝜑 → 𝑀 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑁 ∈ (Met‘𝑌)) ⇒ ⊢ (𝜑 → 𝑃 ∈ (Met‘(𝑋 × 𝑌))) | ||
Theorem | blfvalps 22990* | The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})) | ||
Theorem | blfval 22991* | The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Proof shortened by Thierry Arnoux, 11-Feb-2018.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})) | ||
Theorem | blvalps 22992* | The ball around a point 𝑃 is the set of all points whose distance from 𝑃 is less than the ball's radius 𝑅. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅}) | ||
Theorem | blval 22993* | The ball around a point 𝑃 is the set of all points whose distance from 𝑃 is less than the ball's radius 𝑅. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅}) | ||
Theorem | elblps 22994 | Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅))) | ||
Theorem | elbl 22995 | Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅))) | ||
Theorem | elbl2ps 22996 | Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅)) | ||
Theorem | elbl2 22997 | Membership in a ball. (Contributed by NM, 9-Mar-2007.) |
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅)) | ||
Theorem | elbl3ps 22998 | Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.) |
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝐷𝑃) < 𝑅)) | ||
Theorem | elbl3 22999 | Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.) |
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝐷𝑃) < 𝑅)) | ||
Theorem | blcomps 23000 | Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝑃 ∈ (𝐴(ball‘𝐷)𝑅))) |
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