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Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmetreslem 22901 Lemma for metres 22904. (Contributed by Mario Carneiro, 24-Aug-2015.)
(dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))
 
Theoremmetres2 22902 Lemma for metres 22904. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
((𝐷 ∈ (Met‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘𝑅))
 
Theoremxmetres 22903 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 22904 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 22905 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
∅ ∈ (Met‘∅)
 
Theoremprdsdsf 22906* 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 22907* 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 22908* The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 22907. (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 22909* 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 22910* 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 22911 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 22912* Lemma for imasdsf1o 22913. (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 22913 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 22914 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 22915 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 22916 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 22917 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 22918* Lemma for xpsxmet 22919. (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⟩, ⟨1o, 𝑆⟩})) ∈ (∞Met‘ran (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})))
 
Theoremxpsxmet 22919 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 22920 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 22921 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 22922* 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 22923* 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 22924* 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 22925* 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 22926 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 22927 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 22928 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 22929 Membership in a ball. (Contributed by NM, 9-Mar-2007.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))
 
Theoremelbl3ps 22930 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 22931 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 22932 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 22933 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 22934 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 22935 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 22936 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 22937 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 22938 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 22939 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 22940 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 22943 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))
 
Theoremxblss2 22941 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 22943 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.)
(𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝑄𝑋)    &   (𝜑𝑅 ∈ ℝ*)    &   (𝜑𝑆 ∈ ℝ*)    &   (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)    &   (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))       (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
 
Theoremblss2ps 22942 One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑄𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ (𝑃𝐷𝑄) ≤ (𝑆𝑅))) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
 
Theoremblss2 22943 One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑄𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ (𝑃𝐷𝑄) ≤ (𝑆𝑅))) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
 
Theoremblhalf 22944 A ball of radius 𝑅 / 2 is contained in a ball of radius 𝑅 centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.)
(((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑍 ∈ (𝑌(ball‘𝑀)(𝑅 / 2)))) → (𝑌(ball‘𝑀)(𝑅 / 2)) ⊆ (𝑍(ball‘𝑀)𝑅))
 
Theoremblfps 22945 Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
 
Theoremblf 22946 Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
 
Theoremblrnps 22947* Membership in the range of the ball function. Note that ran (ball‘𝐷) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥𝑋𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))
 
Theoremblrn 22948* Membership in the range of the ball function. Note that ran (ball‘𝐷) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥𝑋𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))
 
Theoremxblcntrps 22949 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
 
Theoremxblcntr 22950 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
 
Theoremblcntrps 22951 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
 
Theoremblcntr 22952 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
 
Theoremxbln0 22953 A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → ((𝑃(ball‘𝐷)𝑅) ≠ ∅ ↔ 0 < 𝑅))
 
Theorembln0 22954 A ball is not empty. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ≠ ∅)
 
Theoremblelrnps 22955 A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
 
Theoremblelrn 22956 A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
 
Theoremblssm 22957 A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋)
 
Theoremunirnblps 22958 The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) = 𝑋)
 
Theoremunirnbl 22959 The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
(𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) = 𝑋)
 
Theoremblin 22960 The intersection of two balls with the same center is the smaller of them. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑆 ∈ ℝ*)) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑃(ball‘𝐷)𝑆)) = (𝑃(ball‘𝐷)if(𝑅𝑆, 𝑅, 𝑆)))
 
Theoremssblps 22961 The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑆 ∈ ℝ*) ∧ 𝑅𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))
 
Theoremssbl 22962 The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑆 ∈ ℝ*) ∧ 𝑅𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))
 
Theoremblssps 22963* Any point 𝑃 in a ball 𝐵 can be centered in another ball that is a subset of 𝐵. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)
 
Theoremblss 22964* Any point 𝑃 in a ball 𝐵 can be centered in another ball that is a subset of 𝐵. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)
 
Theoremblssexps 22965* Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋) → (∃𝑥 ∈ ran (ball‘𝐷)(𝑃𝑥𝑥𝐴) ↔ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝐴))
 
Theoremblssex 22966* Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → (∃𝑥 ∈ ran (ball‘𝐷)(𝑃𝑥𝑥𝐴) ↔ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝐴))
 
Theoremssblex 22967* A nested ball exists whose radius is less than any desired amount. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ+𝑆 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)))
 
Theoremblin2 22968* Given any two balls and a point in their intersection, there is a ball contained in the intersection with the given center point. (Contributed by Mario Carneiro, 12-Nov-2013.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶))
 
Theoremblbas 22969 The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)
(𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases)
 
Theoremblres 22970 A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
𝐶 = (𝐷 ↾ (𝑌 × 𝑌))       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑅) = ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌))
 
Theoremxmeterval 22971 Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       (𝐷 ∈ (∞Met‘𝑋) → (𝐴 𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ)))
 
Theoremxmeter 22972 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       (𝐷 ∈ (∞Met‘𝑋) → Er 𝑋)
 
Theoremxmetec 22973 The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 8331, infinity balls are either identical or disjoint, quite unlike the usual situation with Euclidean balls which admit many kinds of overlap. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → [𝑃] = (𝑃(ball‘𝐷)+∞))
 
Theoremblssec 22974 A ball centered at 𝑃 is contained in the set of points finitely separated from 𝑃. This is just an application of ssbl 22962 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) ⊆ [𝑃] )
 
Theoremblpnfctr 22975 The infinity ball in an extended metric acts like an ultrametric ball in that every point in the ball is also its center. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝐴 ∈ (𝑃(ball‘𝐷)+∞)) → (𝑃(ball‘𝐷)+∞) = (𝐴(ball‘𝐷)+∞))
 
Theoremxmetresbl 22976 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 22973, this shows that any extended metric space can be "factored" into the disjoint union of proper metric spaces, with points in the same region measured by that region's metric, and points in different regions being distance +∞ from each other. (Contributed by Mario Carneiro, 23-Aug-2015.)
𝐵 = (𝑃(ball‘𝐷)𝑅)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))
 
12.4.4  Open sets of a metric space
 
Theoremmopnval 22977 An open set is a subset of a metric space which includes a ball around each of its points. Definition 1.3-2 of [Kreyszig] p. 18. The object (MetOpen‘𝐷) is the family of all open sets in the metric space determined by the metric 𝐷. By mopntop 22979, the open sets of a metric space form a topology 𝐽, whose base set is 𝐽 by mopnuni 22980. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷)))
 
Theoremmopntopon 22978 The set of open sets of a metric space 𝑋 is a topology on 𝑋. Remark in [Kreyszig] p. 19. This theorem connects the two concepts and makes available the theorems for topologies for use with metric spaces. (Contributed by Mario Carneiro, 24-Aug-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
 
Theoremmopntop 22979 The set of open sets of a metric space is a topology. (Contributed by NM, 28-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
 
Theoremmopnuni 22980 The union of all open sets in a metric space is its underlying set. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
 
Theoremelmopn 22981* The defining property of an open set of a metric space. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → (𝐴𝐽 ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑦 ∈ ran (ball‘𝐷)(𝑥𝑦𝑦𝐴))))
 
Theoremmopnfss 22982 The family of open sets of a metric space is a collection of subsets of the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ⊆ 𝒫 𝑋)
 
Theoremmopnm 22983 The base set of a metric space is open. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝑋𝐽)
 
Theoremelmopn2 22984* A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → (𝐴𝐽 ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴)))
 
Theoremmopnss 22985 An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
 
Theoremisxms 22986 Express the predicate "𝑋, 𝐷 is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘𝐾)    &   𝑋 = (Base‘𝐾)    &   𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))       (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
 
Theoremisxms2 22987 Express the predicate "𝑋, 𝐷 is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘𝐾)    &   𝑋 = (Base‘𝐾)    &   𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))       (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))
 
Theoremisms 22988 Express the predicate "𝑋, 𝐷 is a metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
𝐽 = (TopOpen‘𝐾)    &   𝑋 = (Base‘𝐾)    &   𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))       (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))
 
Theoremisms2 22989 Express the predicate "𝑋, 𝐷 is a metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
𝐽 = (TopOpen‘𝐾)    &   𝑋 = (Base‘𝐾)    &   𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))       (𝐾 ∈ MetSp ↔ (𝐷 ∈ (Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))
 
Theoremxmstopn 22990 The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐽 = (TopOpen‘𝐾)    &   𝑋 = (Base‘𝐾)    &   𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))       (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
 
Theoremmstopn 22991 The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐽 = (TopOpen‘𝐾)    &   𝑋 = (Base‘𝐾)    &   𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))       (𝐾 ∈ MetSp → 𝐽 = (MetOpen‘𝐷))
 
Theoremxmstps 22992 An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
 
Theoremmsxms 22993 A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
 
Theoremmstps 22994 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ MetSp → 𝑀 ∈ TopSp)
 
Theoremxmsxmet 22995 The distance function, suitably truncated, is an extended metric on 𝑋. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑋 = (Base‘𝑀)    &   𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))       (𝑀 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋))
 
Theoremmsmet 22996 The distance function, suitably truncated, is a metric on 𝑋. (Contributed by Mario Carneiro, 12-Nov-2013.)
𝑋 = (Base‘𝑀)    &   𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))       (𝑀 ∈ MetSp → 𝐷 ∈ (Met‘𝑋))
 
Theoremmsf 22997 The distance function of a metric space is a function into the real numbers. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝑋 = (Base‘𝑀)    &   𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))       (𝑀 ∈ MetSp → 𝐷:(𝑋 × 𝑋)⟶ℝ)
 
Theoremxmsxmet2 22998 The distance function, suitably truncated, is an extended metric on 𝑋. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝑀)    &   𝐷 = (dist‘𝑀)       (𝑀 ∈ ∞MetSp → (𝐷 ↾ (𝑋 × 𝑋)) ∈ (∞Met‘𝑋))
 
Theoremmsmet2 22999 The distance function, suitably truncated, is a metric on 𝑋. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝑀)    &   𝐷 = (dist‘𝑀)       (𝑀 ∈ MetSp → (𝐷 ↾ (𝑋 × 𝑋)) ∈ (Met‘𝑋))
 
Theoremmscl 23000 Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝑀)    &   𝐷 = (dist‘𝑀)       ((𝑀 ∈ MetSp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ)
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