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Type | Label | Description |
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Statement | ||
Theorem | fmucnd 22901* | The image of a Cauchy filter base by an uniformly continuous function is a Cauchy filter base. Deduction form. Proposition 3 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 18-Nov-2017.) |
⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋)) & ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌)) & ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉)) & ⊢ (𝜑 → 𝐶 ∈ (CauFilu‘𝑈)) & ⊢ 𝐷 = ran (𝑎 ∈ 𝐶 ↦ (𝐹 “ 𝑎)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (CauFilu‘𝑉)) | ||
Theorem | cfilufg 22902 | The filter generated by a Cauchy filter base is still a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.) |
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu‘𝑈)) | ||
Theorem | trcfilu 22903 | Condition for the trace of a Cauchy filter base to be a Cauchy filter base for the restricted uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.) |
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) | ||
Theorem | cfiluweak 22904 | 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 22905 | 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 22906 | Extend class notation with the class of all complete uniform spaces. |
class CUnifSp | ||
Definition | df-cusp 22907* | 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 22908* | 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 22909 | A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.) |
⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp) | ||
Theorem | cuspcvg 22910 | 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 22911* | 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 22912* | 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 22913 | 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 22914* | 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 22915 | Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷) | ||
Theorem | psmetf 22916 | 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 22917 | 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 22918 | 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 22919 | 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 22920 | 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 22921 | 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 22922 | 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 22923 | 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 22924 | Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅)) | ||
Theorem | psmetlecl 22925 | 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 22926 | 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 22927 | Extend class notation with the class of extended metric spaces. |
class ∞MetSp | ||
Syntax | cms 22928 | Extend class notation with the class of metric spaces. |
class MetSp | ||
Syntax | ctms 22929 | Extend class notation with the function mapping a metric to the metric space it defines. |
class toMetSp | ||
Definition | df-xms 22930 | 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 22931 | 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 22932 | 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 22933* | 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 22934* | 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 22935* | 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 22936* | Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (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 22937* | 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 22938* | Lemma for metf 22940 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 22939 | 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 22940 | Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) |
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | ||
Theorem | xmetcl 22941 | 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 22942 | 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 22943 | 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 22944 | A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | ||
Theorem | xmetdmdm 22945 | Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷) | ||
Theorem | metdmdm 22946 | Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 = dom dom 𝐷) | ||
Theorem | xmetunirn 22947 | 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 22948 | 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 22949 | 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 22950 | 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 22951 | 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 22952 | 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 22953 | 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 22954 | 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 22955 | 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 22956 | 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 22957 | 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 22958 | An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | ||
Theorem | xmettpos 22959 | 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 22960 | 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 22961 | 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 22962 | 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 22963 | 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 22964 | 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 22965 | 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 22966 | 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 20589 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 22967 | 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 22968 | 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 22969 | 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 22970 | 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 22971 | Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅)) | ||
Theorem | metreslem 22972 | Lemma for metres 22975. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅)))) | ||
Theorem | metres2 22973 | Lemma for metres 22975. (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 22974 | 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 22975 | 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 22976 | The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ ∅ ∈ (Met‘∅) | ||
Theorem | prdsdsf 22977* | 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 22978* | 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 22979* | The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 22978. (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 22980* | 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 22981* | 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 22982 | 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 22983* | Lemma for imasdsf1o 22984. (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 22984 | 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 22985 | 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 22986 | 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 22987 | 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 22988 | 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 22989* | Lemma for xpsxmet 22990. (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 22990 | 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 22991 | 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 22992 | 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 22993* | 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 22994* | 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 22995* | 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 22996* | 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 22997 | 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 22998 | 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 22999 | 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 23000 | 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)) |
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