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Theorem List for Metamath Proof Explorer - 21901-22000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-cfilu 21901* Define the set of Cauchy filter bases on a uniform space. A Cauchy filter base is a filter base on the set such that for every entourage 𝑣, there is an element 𝑎 of the filter "small enough in 𝑣 " i.e. such that every pair {𝑥, 𝑦} of points in 𝑎 is related by 𝑣". Definition 2 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 16-Nov-2017.)
CauFilu = (𝑢 ran UnifOn ↦ {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})

Theoremiscfilu 21902* The predicate "𝐹 is a Cauchy filter base on uniform space 𝑈." (Contributed by Thierry Arnoux, 18-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))

Theoremcfilufbas 21903 A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → 𝐹 ∈ (fBas‘𝑋))

Theoremcfiluexsm 21904* For a Cauchy filter base and any entourage 𝑉, there is an element of the filter small in 𝑉. (Contributed by Thierry Arnoux, 19-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈) ∧ 𝑉𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉)

Theoremfmucndlem 21905* Lemma for fmucnd 21906. (Contributed by Thierry Arnoux, 19-Nov-2017.)
((𝐹 Fn 𝑋𝐴𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝐴 × 𝐴)) = ((𝐹𝐴) × (𝐹𝐴)))

Theoremfmucnd 21906* 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.)
(𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝐹 ∈ (𝑈 Cnu𝑉))    &   (𝜑𝐶 ∈ (CauFilu𝑈))    &   𝐷 = ran (𝑎𝐶 ↦ (𝐹𝑎))       (𝜑𝐷 ∈ (CauFilu𝑉))

Theoremcfilufg 21907 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 ∈ (CauFilu𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu𝑈))

Theoremtrcfilu 21908 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 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹t 𝐴) ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))))

Theoremcfiluweak 21909 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) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu𝑈))

Theoremneipcfilu 21910 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 ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈))

12.3.6  Complete uniform spaces

Syntaxccusp 21911 Extend class notation with the class of all complete uniform spaces.
class CUnifSp

Definitiondf-cusp 21912* 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) ≠ ∅)}

Theoremiscusp 21913* 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) ≠ ∅)))

Theoremcuspusp 21914 A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
(𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)

Theoremcuspcvg 21915 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) ≠ ∅)

Theoremiscusp2 21916* 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 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))

Theoremcnextucn 21917* 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)    &   (𝜑𝑉 ∈ TopSp)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝑊 ∈ CUnifSp)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐴𝑋)    &   (𝜑𝐹:𝐴𝑌)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)    &   ((𝜑𝑥𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))       (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Theoremucnextcn 21918 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‘(𝑉s 𝐴))    &   𝑈 = (UnifSt‘𝑊)    &   (𝜑𝑉 ∈ TopSp)    &   (𝜑𝑉 ∈ UnifSp)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝑊 ∈ CUnifSp)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐴𝑋)    &   (𝜑𝐹 ∈ (𝑇 Cnu𝑈))    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)       (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

12.4  Metric spaces

12.4.1  Pseudometric spaces

Theoremispsmet 21919* Express the predicate "𝐷 is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))

Theorempsmetdmdm 21920 Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)

Theorempsmetf 21921 The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)

Theorempsmetcl 21922 Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)

Theorempsmet0 21923 The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)

Theorempsmettri2 21924 Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))

Theorempsmetsym 21925 The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))

Theorempsmettri 21926 Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵)))

Theorempsmetge0 21927 The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → 0 ≤ (𝐴𝐷𝐵))

Theorempsmetxrge0 21928 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[,]+∞))

Theorempsmetres2 21929 Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))

Theorempsmetlecl 21930 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶 ∈ ℝ ∧ (𝐴𝐷𝐵) ≤ 𝐶)) → (𝐴𝐷𝐵) ∈ ℝ)

Theoremdistspace 21931 A structure 𝐺 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 non-negative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.)
𝑋 = (Base‘𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ (𝐴𝐷𝐴) = 0) ∧ (0 ≤ (𝐴𝐷𝐵) ∧ (𝐴𝐷𝐵) = (𝐵𝐷𝐴))))

12.4.2  Basic metric space properties

Syntaxcxme 21932 Extend class notation with the class of all extended metric spaces.
class ∞MetSp

Syntaxcmt 21933 Extend class notation with the class of all metric spaces.
class MetSp

Syntaxctmt 21934 Extend class notation with the function mapping a metric to a metric space.
class toMetSp

Definitiondf-xms 21935 Define the (proper) class of all extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}

Definitiondf-ms 21936 Define the (proper) class of all metric spaces. (Contributed by NM, 27-Aug-2006.)
MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}

Definitiondf-tms 21937 Define the function mapping a metric to a metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp = (𝑑 ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))

Theoremismet 21938* Express the predicate "𝐷 is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
(𝑋𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))

Theoremisxmet 21939* Express the predicate "𝐷 is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝑋𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))

Theoremismeti 21940* Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
𝑋 ∈ V    &   𝐷:(𝑋 × 𝑋)⟶ℝ    &   ((𝑥𝑋𝑦𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))    &   ((𝑥𝑋𝑦𝑋𝑧𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))       𝐷 ∈ (Met‘𝑋)

Theoremisxmetd 21941* Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝜑𝑋 ∈ V)    &   (𝜑𝐷:(𝑋 × 𝑋)⟶ℝ*)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))       (𝜑𝐷 ∈ (∞Met‘𝑋))

Theoremisxmet2d 21942* 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.)
(𝜑𝑋 ∈ V)    &   (𝜑𝐷:(𝑋 × 𝑋)⟶ℝ*)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 0 ≤ (𝑥𝐷𝑦))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥𝐷𝑦) ≤ 0 ↔ 𝑥 = 𝑦))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))       (𝜑𝐷 ∈ (∞Met‘𝑋))

Theoremmetflem 21943* Lemma for metf 21945 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
(𝐷 ∈ (Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))

Theoremxmetf 21944 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)

Theoremmetf 21945 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)
(𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ)

Theoremxmetcl 21946 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) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)

Theoremmetcl 21947 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) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ)

Theoremismet2 21948 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))

Theoremmetxmet 21949 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))

Theoremxmetdmdm 21950 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷)

Theoremmetdmdm 21951 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝐷 ∈ (Met‘𝑋) → 𝑋 = dom dom 𝐷)

Theoremxmetunirn 21952 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))

Theoremxmeteq0 21953 The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))

Theoremmeteq0 21954 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) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))

Theoremxmettri2 21955 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))

Theoremmettri2 21956 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) ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵)))

Theoremxmet0 21957 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) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)

Theoremmet0 21958 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) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)

Theoremxmetge0 21959 The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → 0 ≤ (𝐴𝐷𝐵))

Theoremmetge0 21960 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) ∧ 𝐴𝑋𝐵𝑋) → 0 ≤ (𝐴𝐷𝐵))

Theoremxmetlecl 21961 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶 ∈ ℝ ∧ (𝐴𝐷𝐵) ≤ 𝐶)) → (𝐴𝐷𝐵) ∈ ℝ)

Theoremxmetsym 21962 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))

Theoremxmetpsmet 21963 An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))

Theoremxmettpos 21964 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐷 ∈ (∞Met‘𝑋) → tpos 𝐷 = 𝐷)

Theoremmetsym 21965 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) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))

Theoremxmettri 21966 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) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵)))

Theoremmettri 21967 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) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐶𝐷𝐵)))

Theoremxmettri3 21968 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐵𝐷𝐶)))

Theoremmettri3 21969 Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)
((𝐷 ∈ (Met‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶)))

Theoremxmetrtri 21970 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) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵))

Theoremxmetrtri2 21971 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 19609 defined on the extended real structure. (Contributed by Mario Carneiro, 4-Sep-2015.)
𝐾 = (dist‘ℝ*𝑠)       ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐶)𝐾(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵))

Theoremmetrtri 21972 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.)
((𝐷 ∈ (Met‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))

Theoremxmetgt0 21973 The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵 ↔ 0 < (𝐴𝐷𝐵)))

Theoremmetgt0 21974 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) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵 ↔ 0 < (𝐴𝐷𝐵)))

Theoremmetn0 21975 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 ≠ ∅))

Theoremxmetres2 21976 Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅))

Theoremmetreslem 21977 Lemma for metres 21980. (Contributed by Mario Carneiro, 24-Aug-2015.)
(dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))

Theoremmetres2 21978 Lemma for metres 21980. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
((𝐷 ∈ (Met‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘𝑅))

Theoremxmetres 21979 A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘(𝑋𝑅)))

Theoremmetres 21980 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‘(𝑋𝑅)))

Theorem0met 21981 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
∅ ∈ (Met‘∅)

Theoremprdsdsf 21982* 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(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   𝑉 = (Base‘𝑅)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑌)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼𝑋)    &   ((𝜑𝑥𝐼) → 𝑅𝑍)    &   ((𝜑𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))       (𝜑𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))

Theoremprdsxmetlem 21983* The product metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   𝑉 = (Base‘𝑅)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑌)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼𝑋)    &   ((𝜑𝑥𝐼) → 𝑅𝑍)    &   ((𝜑𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))       (𝜑𝐷 ∈ (∞Met‘𝐵))

Theoremprdsxmet 21984* The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 21983. (Contributed by Mario Carneiro, 26-Sep-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   𝑉 = (Base‘𝑅)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑌)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼𝑋)    &   ((𝜑𝑥𝐼) → 𝑅𝑍)    &   ((𝜑𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))       (𝜑𝐷 ∈ (∞Met‘𝐵))

Theoremprdsmet 21985* The product metric is a metric when the index set is finite. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   𝑉 = (Base‘𝑅)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑌)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   ((𝜑𝑥𝐼) → 𝑅𝑍)    &   ((𝜑𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))       (𝜑𝐷 ∈ (Met‘𝐵))

Theoremressprdsds 21986* Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
(𝜑𝑌 = (𝑆Xs(𝑥𝐼𝑅)))    &   (𝜑𝐻 = (𝑇Xs(𝑥𝐼 ↦ (𝑅s 𝐴))))    &   𝐵 = (Base‘𝐻)    &   𝐷 = (dist‘𝑌)    &   𝐸 = (dist‘𝐻)    &   (𝜑𝑆𝑈)    &   (𝜑𝑇𝑉)    &   (𝜑𝐼𝑊)    &   ((𝜑𝑥𝐼) → 𝑅𝑋)    &   ((𝜑𝑥𝐼) → 𝐴𝑍)       (𝜑𝐸 = (𝐷 ↾ (𝐵 × 𝐵)))

Theoremresspwsds 21987 Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
(𝜑𝑌 = (𝑅s 𝐼))    &   (𝜑𝐻 = ((𝑅s 𝐴) ↑s 𝐼))    &   𝐵 = (Base‘𝐻)    &   𝐷 = (dist‘𝑌)    &   𝐸 = (dist‘𝐻)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝐴𝑋)       (𝜑𝐸 = (𝐷 ↾ (𝐵 × 𝐵)))

Theoremimasdsf1olem 21988* Lemma for imasdsf1o 21989. (Contributed by Mario Carneiro, 21-Aug-2015.) (Proof shortened by AV, 6-Oct-2020.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑈)    &   (𝜑𝐸 ∈ (∞Met‘𝑉))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   𝑊 = (ℝ*𝑠s (ℝ* ∖ {-∞}))    &   𝑆 = { ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = (𝐹𝑋) ∧ (𝐹‘(2nd ‘(𝑛))) = (𝐹𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))}    &   𝑇 = 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔)))       (𝜑 → ((𝐹𝑋)𝐷(𝐹𝑌)) = (𝑋𝐸𝑌))

Theoremimasdsf1o 21989 The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑈)    &   (𝜑𝐸 ∈ (∞Met‘𝑉))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → ((𝐹𝑋)𝐷(𝐹𝑌)) = (𝑋𝐸𝑌))

Theoremimasf1oxmet 21990 The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑈)    &   (𝜑𝐸 ∈ (∞Met‘𝑉))       (𝜑𝐷 ∈ (∞Met‘𝐵))

Theoremimasf1omet 21991 The image of a metric is a metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑈)    &   (𝜑𝐸 ∈ (Met‘𝑉))       (𝜑𝐷 ∈ (Met‘𝐵))

Theoremxpsdsfn 21992 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)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝑃 = (dist‘𝑇)       (𝜑𝑃 Fn ((𝑋 × 𝑌) × (𝑋 × 𝑌)))

Theoremxpsdsfn2 21993 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)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝑃 = (dist‘𝑇)       (𝜑𝑃 Fn ((Base‘𝑇) × (Base‘𝑇)))

Theoremxpsxmetlem 21994* Lemma for xpsxmet 21995. (Contributed by Mario Carneiro, 21-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝑃 = (dist‘𝑇)    &   𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))    &   𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))       (𝜑 → (dist‘((Scalar‘𝑅)Xs({𝑅} +𝑐 {𝑆}))) ∈ (∞Met‘ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))))

Theoremxpsxmet 21995 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)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝑃 = (dist‘𝑇)    &   𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))    &   𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))       (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))

Theoremxpsdsval 21996 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)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝑃 = (dist‘𝑇)    &   𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))    &   𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)       (𝜑 → (⟨𝐴, 𝐵𝑃𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))

Theoremxpsmet 21997 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)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝑃 = (dist‘𝑇)    &   𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))    &   𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))    &   (𝜑𝑀 ∈ (Met‘𝑋))    &   (𝜑𝑁 ∈ (Met‘𝑌))       (𝜑𝑃 ∈ (Met‘(𝑋 × 𝑌)))

12.4.3  Metric space balls

Theoremblfvalps 21998* 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) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))

Theoremblfval 21999* 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) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))

Theoremblvalps 22000* 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})

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