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Theorem List for Intuitionistic Logic Explorer - 12401-12500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhmeocn 12401 A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremhmeocnvcn 12402 The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
 
Theoremhmeocnv 12403 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾Homeo𝐽))
 
Theoremhmeof1o2 12404 A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋1-1-onto𝑌)
 
Theoremhmeof1o 12405 A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto𝑌)
 
Theoremhmeoima 12406 The image of an open set by a homeomorphism is an open set. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐾)
 
Theoremhmeoopn 12407 Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐹𝐴) ∈ 𝐾))
 
Theoremhmeocld 12408 Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐾)))
 
Theoremhmeontr 12409 Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))
 
Theoremhmeoimaf1o 12410* The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐺 = (𝑥𝐽 ↦ (𝐹𝑥))       (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽1-1-onto𝐾)
 
Theoremhmeores 12411 The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))))
 
Theoremhmeoco 12412 The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺𝐹) ∈ (𝐽Homeo𝐿))
 
Theoremidhmeo 12413 The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽Homeo𝐽))
 
Theoremhmeocnvb 12414 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
(Rel 𝐹 → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
 
Theoremtxhmeo 12415* Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑋 = 𝐽    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐽Homeo𝐿))    &   (𝜑𝐺 ∈ (𝐾Homeo𝑀))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀)))
 
Theoremtxswaphmeolem 12416* Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
 
Theoremtxswaphmeo 12417* There is a homeomorphism from 𝑋 × 𝑌 to 𝑌 × 𝑋. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)))
 
7.2  Metric spaces
 
7.2.1  Pseudometric spaces
 
Theorempsmetrel 12418 The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
Rel PsMet
 
Theoremispsmet 12419* 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 12420 Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
 
Theorempsmetf 12421 The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
 
Theorempsmetcl 12422 Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)
 
Theorempsmet0 12423 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 12424 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 12425 The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
 
Theorempsmettri 12426 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 12427 The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → 0 ≤ (𝐴𝐷𝐵))
 
Theorempsmetxrge0 12428 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 12429 Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))
 
Theorempsmetlecl 12430 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 12431 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) ∧ 𝐴𝑋𝐵𝑋) → ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ (𝐴𝐷𝐴) = 0) ∧ (0 ≤ (𝐴𝐷𝐵) ∧ (𝐴𝐷𝐵) = (𝐵𝐷𝐴))))
 
7.2.2  Basic metric space properties
 
Syntaxcxms 12432 Extend class notation with the class of extended metric spaces.
class ∞MetSp
 
Syntaxcms 12433 Extend class notation with the class of metric spaces.
class MetSp
 
Syntaxctms 12434 Extend class notation with the function mapping a metric to the metric space it defines.
class toMetSp
 
Definitiondf-xms 12435 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))))}
 
Definitiondf-ms 12436 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))}
 
Definitiondf-tms 12437 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)⟩))
 
Theoremmetrel 12438 The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
Rel Met
 
Theoremxmetrel 12439 The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
Rel ∞Met
 
Theoremismet 12440* 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 12441* 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 12442* 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 12443* 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 12444* 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 12445* Lemma for metf 12447 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 12446 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
 
Theoremmetf 12447 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)
(𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
 
Theoremxmetcl 12448 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 12449 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 12450 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 12451 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
 
Theoremxmetdmdm 12452 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷)
 
Theoremmetdmdm 12453 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝐷 ∈ (Met‘𝑋) → 𝑋 = dom dom 𝐷)
 
Theoremxmetunirn 12454 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 12455 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 12456 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 12457 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 12458 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 12459 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 12460 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 12461 The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → 0 ≤ (𝐴𝐷𝐵))
 
Theoremmetge0 12462 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 12463 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 12464 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
 
Theoremxmetpsmet 12465 An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))
 
Theoremxmettpos 12466 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐷 ∈ (∞Met‘𝑋) → tpos 𝐷 = 𝐷)
 
Theoremmetsym 12467 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 12468 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 12469 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 12470 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 12471 Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)
((𝐷 ∈ (Met‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶)))
 
Theoremxmetrtri 12472 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))
 
Theoremmetrtri 12473 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 21-Apr-2023.)
((𝐷 ∈ (Met‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))
 
Theoremmetn0 12474 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 12475 Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅))
 
Theoremmetreslem 12476 Lemma for metres 12479. (Contributed by Mario Carneiro, 24-Aug-2015.)
(dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))
 
Theoremmetres2 12477 Lemma for metres 12479. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
((𝐷 ∈ (Met‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘𝑅))
 
Theoremxmetres 12478 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 12479 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 12480 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
∅ ∈ (Met‘∅)
 
7.2.3  Metric space balls
 
Theoremblfvalps 12481* 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 12482* 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}))
 
Theoremblex 12483 A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.)
(𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) ∈ V)
 
Theoremblvalps 12484* 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})
 
Theoremblval 12485* 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
 
Theoremelblps 12486 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
 
Theoremelbl 12487 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
 
Theoremelbl2ps 12488 Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))
 
Theoremelbl2 12489 Membership in a ball. (Contributed by NM, 9-Mar-2007.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))
 
Theoremelbl3ps 12490 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
(((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝐷𝑃) < 𝑅))
 
Theoremelbl3 12491 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝐷𝑃) < 𝑅))
 
Theoremblcomps 12492 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*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝑃 ∈ (𝐴(ball‘𝐷)𝑅)))
 
Theoremblcom 12493 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝑃 ∈ (𝐴(ball‘𝐷)𝑅)))
 
Theoremxblpnfps 12494 The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋) → (𝐴 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝐴𝑋 ∧ (𝑃𝐷𝐴) ∈ ℝ)))
 
Theoremxblpnf 12495 The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → (𝐴 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝐴𝑋 ∧ (𝑃𝐷𝐴) ∈ ℝ)))
 
Theoremblpnf 12496 The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐷 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (𝑃(ball‘𝐷)+∞) = 𝑋)
 
Theorembldisj 12497 Two balls are disjoint if the center-to-center distance is more than the sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑄𝑋) ∧ (𝑅 ∈ ℝ*𝑆 ∈ ℝ* ∧ (𝑅 +𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) = ∅)
 
Theoremblgt0 12498 A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)
 
Theorembl2in 12499 Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
(((𝐷 ∈ (Met‘𝑋) ∧ 𝑃𝑋𝑄𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑅)) = ∅)
 
Theoremxblss2ps 12500 One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 12503 for extended metrics, we have to assume the balls are a finite distance apart, or else 𝑃 will not even be in the infinity ball around 𝑄. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(𝜑𝐷 ∈ (PsMet‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝑄𝑋)    &   (𝜑𝑅 ∈ ℝ*)    &   (𝜑𝑆 ∈ ℝ*)    &   (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)    &   (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))       (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
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