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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtmsds 22001 The metric of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐾 = (toMetSp‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾))
 
Theoremtmstopn 22002 The topology of a constructed metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐾 = (toMetSp‘𝐷)    &   𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (TopOpen‘𝐾))
 
Theoremtmsxms 22003 The constructed metric space is an extended metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐾 = (toMetSp‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐾 ∈ ∞MetSp)
 
Theoremtmsms 22004 The constructed metric space is a metric space given a metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐾 = (toMetSp‘𝐷)       (𝐷 ∈ (Met‘𝑋) → 𝐾 ∈ MetSp)
 
Theoremimasf1obl 22005 The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑈)    &   (𝜑𝐸 ∈ (∞Met‘𝑉))    &   (𝜑𝑃𝑉)    &   (𝜑𝑆 ∈ ℝ*)       (𝜑 → ((𝐹𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆)))
 
Theoremimasf1oxms 22006 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅 ∈ ∞MetSp)       (𝜑𝑈 ∈ ∞MetSp)
 
Theoremimasf1oms 22007 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅 ∈ MetSp)       (𝜑𝑈 ∈ MetSp)
 
Theoremprdsbl 22008* A ball in the product metric for finite index set is the Cartesian product of balls in all coordinates. For infinite index set this is no longer true; instead the correct statement is that a *closed ball* is the product of closed balls in each coordinate (where closed ball means a set of the form in blcld 22022) - for a counterexample the point 𝑝 in ℝ↑ℕ whose 𝑛-th coordinate is 1 − 1 / 𝑛 is in X𝑛 ∈ ℕball(0, 1) but is not in the 1-ball of the product (since 𝑑(0, 𝑝) = 1).

The last assumption, 0 < 𝐴, is needed only in the case 𝐼 = ∅, when the right side evaluates to {∅} and the left evaluates to if 𝐴 ≤ 0 and {∅} if 0 < 𝐴. (Contributed by Mario Carneiro, 28-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))    &   (𝜑𝑃𝐵)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → 0 < 𝐴)       (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥𝐼 ((𝑃𝑥)(ball‘𝐸)𝐴))
 
Theoremmopni 22009* An open set of a metric space includes a ball around each of its points. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝐽𝑃𝐴) → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃𝑥𝑥𝐴))
 
Theoremmopni2 22010* An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝐽𝑃𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)
 
Theoremmopni3 22011* An open set of a metric space includes an arbitrarily small ball around each of its points. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝐽𝑃𝐴) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴))
 
Theoremblssopn 22012 The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ⊆ 𝐽)
 
Theoremunimopn 22013 The union of a collection of open sets of a metric space is open. Theorem T2 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝐽) → 𝐴𝐽)
 
Theoremmopnin 22014 The intersection of two open sets of a metric space is open. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
 
Theoremmopn0 22015 The empty set is an open set of a metric space. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → ∅ ∈ 𝐽)
 
Theoremrnblopn 22016 A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷)) → 𝐵𝐽)
 
Theoremblopn 22017 A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝐽)
 
Theoremneibl 22018* The neighborhoods around a point 𝑃 of a metric space are those subsets containing a ball around 𝑃. Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁𝑋 ∧ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝑁)))
 
Theoremblnei 22019 A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃}))
 
Theoremlpbl 22020* Every ball around a limit point 𝑃 of a subset 𝑆 includes a member of 𝑆 (even if 𝑃𝑆). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝐽 = (MetOpen‘𝐷)       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝑆 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))
 
Theoremblsscls2 22021* A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝑆 = {𝑧𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑇 ∈ ℝ*𝑅 < 𝑇)) → 𝑆 ⊆ (𝑃(ball‘𝐷)𝑇))
 
Theoremblcld 22022* A "closed ball" in a metric space is actually closed. (Contributed by Mario Carneiro, 31-Dec-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
𝐽 = (MetOpen‘𝐷)    &   𝑆 = {𝑧𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑆 ∈ (Clsd‘𝐽))
 
Theoremblcls 22023* The closure of an open ball in a metric space is contained in the corresponding closed ball. (Equality need not hold; for example, with the discrete metric, the closed ball of radius 1 is the whole space, but the open ball of radius 1 is just a point, whose closure is also a point.) (Contributed by Mario Carneiro, 31-Dec-2013.)
𝐽 = (MetOpen‘𝐷)    &   𝑆 = {𝑧𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ 𝑆)
 
Theoremblsscls 22024 If two concentric balls have different radii, the closure of the smaller one is contained in the larger one. (Contributed by Mario Carneiro, 5-Jan-2014.)
𝐽 = (MetOpen‘𝐷)       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑆 ∈ ℝ*𝑅 < 𝑆)) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ (𝑃(ball‘𝐷)𝑆))
 
Theoremmetss 22025* Two ways of saying that metric 𝐷 generates a finer topology than metric 𝐶. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽𝐾 ↔ ∀𝑥𝑋𝑟 ∈ ℝ+𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
 
Theoremmetequiv 22026* Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeff Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 = 𝐾 ↔ ∀𝑥𝑋 (∀𝑟 ∈ ℝ+𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∀𝑎 ∈ ℝ+𝑏 ∈ ℝ+ (𝑥(ball‘𝐶)𝑏) ⊆ (𝑥(ball‘𝐷)𝑎))))
 
Theoremmetequiv2 22027* If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑥𝑋𝑟 ∈ ℝ+𝑠 ∈ ℝ+ (𝑠𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → 𝐽 = 𝐾))
 
Theoremmetss2lem 22028* Lemma for metss2 22029. (Contributed by Mario Carneiro, 14-Sep-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   (𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       ((𝜑 ∧ (𝑥𝑋𝑆 ∈ ℝ+)) → (𝑥(ball‘𝐷)(𝑆 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑆))
 
Theoremmetss2 22029* If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), then 𝐷 generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   (𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       (𝜑𝐽𝐾)
 
Theoremcomet 22030* The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐹:(0[,]+∞)⟶ℝ*)    &   ((𝜑𝑥 ∈ (0[,]+∞)) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0))    &   ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))       (𝜑 → (𝐹𝐷) ∈ (∞Met‘𝑋))
 
Theoremstdbdmetval 22031* Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))       ((𝑅𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
 
Theoremstdbdxmet 22032* The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
 
Theoremstdbdmet 22033* The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (Met‘𝑋))
 
Theoremstdbdbl 22034* The standard bounded metric corresponding to 𝐶 generates the same balls as 𝐶 for radii less than 𝑅. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → (𝑃(ball‘𝐷)𝑆) = (𝑃(ball‘𝐶)𝑆))
 
Theoremstdbdmopn 22035* The standard bounded metric corresponding to 𝐶 generates the same topology as 𝐶. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    &   𝐽 = (MetOpen‘𝐶)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐽 = (MetOpen‘𝐷))
 
Theoremmopnex 22036* The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → ∃𝑑 ∈ (Met‘𝑋)𝐽 = (MetOpen‘𝑑))
 
Theoremmethaus 22037 The topology generated by a metric space is Hausdorff. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus)
 
Theoremmet1stc 22038 The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ 1st𝜔)
 
Theoremmet2ndci 22039 A separable metric space (a metric space with a countable dense subset) is second-countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ 2nd𝜔)
 
Theoremmet2ndc 22040* A metric space is second-countable iff it is separable (has a countable dense subset). (Contributed by Mario Carneiro, 13-Apr-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → (𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
 
Theoremmetrest 22041 Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
𝐷 = (𝐶 ↾ (𝑌 × 𝑌))    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) = 𝐾)
 
Theoremressxms 22042 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ ∞MetSp)
 
Theoremressms 22043 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐾 ∈ MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ MetSp)
 
Theoremprdsmslem1 22044 Lemma for prdsms 22048. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   𝐷 = (dist‘𝑌)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅:𝐼⟶MetSp)       (𝜑𝐷 ∈ (Met‘𝐵))
 
Theoremprdsxmslem1 22045 Lemma for prdsms 22048. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   𝐷 = (dist‘𝑌)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅:𝐼⟶∞MetSp)       (𝜑𝐷 ∈ (∞Met‘𝐵))
 
Theoremprdsxmslem2 22046* Lemma for prdsxms 22047. The topology generated by the supremum metric is the same as the product topology, when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   𝐷 = (dist‘𝑌)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅:𝐼⟶∞MetSp)    &   𝐽 = (TopOpen‘𝑌)    &   𝑉 = (Base‘(𝑅𝑘))    &   𝐸 = ((dist‘(𝑅𝑘)) ↾ (𝑉 × 𝑉))    &   𝐾 = (TopOpen‘(𝑅𝑘))    &   𝐶 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))}       (𝜑𝐽 = (MetOpen‘𝐷))
 
Theoremprdsxms 22047 The indexed product structure is an extended metric space when the index set is finite. (Although the extended metric is still valid when the index set is infinite, it no longer agrees with the product topology, which is not metrizable in any case.) (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑆Xs𝑅)       ((𝑆𝑊𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑌 ∈ ∞MetSp)
 
Theoremprdsms 22048 The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑆Xs𝑅)       ((𝑆𝑊𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑌 ∈ MetSp)
 
Theorempwsxms 22049 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ ∞MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ ∞MetSp)
 
Theorempwsms 22050 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ MetSp)
 
Theoremxpsxms 22051 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)       ((𝑅 ∈ ∞MetSp ∧ 𝑆 ∈ ∞MetSp) → 𝑇 ∈ ∞MetSp)
 
Theoremxpsms 22052 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)       ((𝑅 ∈ MetSp ∧ 𝑆 ∈ MetSp) → 𝑇 ∈ MetSp)
 
Theoremtmsxps 22053 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))       (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
 
Theoremtmsxpsmopn 22054 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   𝐽 = (MetOpen‘𝑀)    &   𝐾 = (MetOpen‘𝑁)    &   𝐿 = (MetOpen‘𝑃)       (𝜑𝐿 = (𝐽 ×t 𝐾))
 
Theoremtmsxpsval 22055 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)       (𝜑 → (⟨𝐴, 𝐵𝑃𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
 
Theoremtmsxpsval2 22056 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)       (𝜑 → (⟨𝐴, 𝐵𝑃𝐶, 𝐷⟩) = if((𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐵𝑁𝐷), (𝐴𝑀𝐶)))
 
12.4.5  Continuity in metric spaces
 
Theoremmetcnp3 22057* Two ways to express that 𝐹 is continuous at 𝑃 for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
 
Theoremmetcnp 22058* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹𝑃)𝐷(𝐹𝑤)) < 𝑦))))
 
Theoremmetcnp2 22059* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. The distance arguments are swapped compared to metcnp 22058 (and Munkres' metcn 22060) for compatibility with df-lm 20746. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑤𝐶𝑃) < 𝑧 → ((𝐹𝑤)𝐷(𝐹𝑃)) < 𝑦))))
 
Theoremmetcn 22060* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon" 𝑦 there is a positive "delta" 𝑧 such that a distance less than delta in 𝐶 maps to a distance less than epsilon in 𝐷. (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝐹𝑥)𝐷(𝐹𝑤)) < 𝑦))))
 
Theoremmetcnpi 22061* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 22058. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+𝑦𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹𝑃)𝐷(𝐹𝑦)) < 𝐴))
 
Theoremmetcnpi2 22062* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 22059. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+𝑦𝑋 ((𝑦𝐶𝑃) < 𝑥 → ((𝐹𝑦)𝐷(𝐹𝑃)) < 𝐴))
 
Theoremmetcnpi3 22063* Epsilon-delta property of a metric space function continuous at 𝑃. A variation of metcnpi2 22062 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+𝑦𝑋 ((𝑦𝐶𝑃) ≤ 𝑥 → ((𝐹𝑦)𝐷(𝐹𝑃)) ≤ 𝐴))
 
Theoremtxmetcnp 22064* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐿 = (MetOpen‘𝐸)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))))
 
Theoremtxmetcn 22065* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐿 = (MetOpen‘𝐸)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → (𝐹 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑥𝑋𝑦𝑌𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧))))
 
12.4.6  The uniform structure generated by a metric
 
Theoremmetuval 22066* Value of the uniform structure generated by metric 𝐷. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))))
 
Theoremmetustel 22067* Define a filter base 𝐹 generated by a metric 𝐷. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
 
Theoremmetustss 22068* Range of the elements of the filter base generated by the metric 𝐷. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
 
Theoremmetustrel 22069* Elements of the filter base generated by the metric 𝐷 are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → Rel 𝐴)
 
Theoremmetustto 22070* Any two elements of the filter base generated by the metric 𝐷 can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵𝐵𝐴))
 
Theoremmetustid 22071* The identity diagonal is included in all elements of the filter base generated by the metric 𝐷. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)
 
Theoremmetustsym 22072* Elements of the filter base generated by the metric 𝐷 are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 = 𝐴)
 
Theoremmetustexhalf 22073* For any element 𝐴 of the filter base generated by the metric 𝐷, the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) → ∃𝑣𝐹 (𝑣𝑣) ⊆ 𝐴)
 
Theoremmetustfbas 22074* The filter base generated by a metric 𝐷. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
 
Theoremmetust 22075* The uniform structure generated by a metric 𝐷. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
 
Theoremcfilucfil 22076* Given a metric 𝐷 and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 22735. (Contributed by Thierry Arnoux, 29-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
 
Theoremmetuust 22077 The uniform structure generated by metric 𝐷 is a uniform structure. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
 
Theoremcfilucfil2 22078* Given a metric 𝐷 and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 22735. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐶 ∈ (CauFilu‘(metUnif‘𝐷)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
 
Theoremblval2 22079 The ball around a point 𝑃, alternative definition. (Contributed by Thierry Arnoux, 7-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) = ((𝐷 “ (0[,)𝑅)) “ {𝑃}))
 
Theoremelbl4 22080 Membership in a ball, alternative definition. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝐴𝑋𝐵𝑋)) → (𝐵 ∈ (𝐴(ball‘𝐷)𝑅) ↔ 𝐵(𝐷 “ (0[,)𝑅))𝐴))
 
Theoremmetuel 22081* Elementhood in the uniform structure generated by a metric 𝐷 (Contributed by Thierry Arnoux, 8-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ (metUnif‘𝐷) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉)))
 
Theoremmetuel2 22082* Elementhood in the uniform structure generated by a metric 𝐷 (Contributed by Thierry Arnoux, 24-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝑈 = (metUnif‘𝐷)       ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉𝑈 ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦))))
 
Theoremmetustbl 22083* The "section" image of an entourage at a point 𝑃 always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ∈ (metUnif‘𝐷) ∧ 𝑃𝑋) → ∃𝑎 ∈ ran (ball‘𝐷)(𝑃𝑎𝑎 ⊆ (𝑉 “ {𝑃})))
 
Theorempsmetutop 22084 The topology induced by a uniform structure generated by a metric 𝐷 is generated by that metric's open balls. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))
 
Theoremxmetutop 22085 The topology induced by a uniform structure generated by an extended metric 𝐷 is that metric's open sets. (Contributed by Thierry Arnoux, 11-Mar-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷))
 
Theoremxmsusp 22086 If the uniform set of a metric space is the uniform structure generated by its metric, then it is a uniform space. (Contributed by Thierry Arnoux, 14-Dec-2017.)
𝑋 = (Base‘𝐹)    &   𝐷 = ((dist‘𝐹) ↾ (𝑋 × 𝑋))    &   𝑈 = (UnifSt‘𝐹)       ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐹 ∈ UnifSp)
 
Theoremrestmetu 22087 The uniform structure generated by the restriction of a metric is its trace. (Contributed by Thierry Arnoux, 18-Dec-2017.)
((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))))
 
Theoremmetucn 22088* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 22060. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝑈 = (metUnif‘𝐶)    &   𝑉 = (metUnif‘𝐷)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝑌 ≠ ∅)    &   (𝜑𝐶 ∈ (PsMet‘𝑋))    &   (𝜑𝐷 ∈ (PsMet‘𝑌))       (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))))
 
12.4.7  Examples of metric spaces
 
Theoremdscmet 22089* The discrete metric on any set 𝑋. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if(𝑥 = 𝑦, 0, 1))       (𝑋𝑉𝐷 ∈ (Met‘𝑋))
 
Theoremdscopn 22090* The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if(𝑥 = 𝑦, 0, 1))       (𝑋𝑉 → (MetOpen‘𝐷) = 𝒫 𝑋)
 
Theoremnrmmetd 22091* Show that a group norm generates a metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐹:𝑋⟶ℝ)    &   ((𝜑𝑥𝑋) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))       (𝜑 → (𝐹 ) ∈ (Met‘𝑋))
 
Theoremabvmet 22092 An absolute value 𝐹 generates a metric defined by 𝑑(𝑥, 𝑦) = 𝐹(𝑥𝑦), analogously to cnmet 22294. (In fact, the ring structure is not needed at all; the group properties abveq0 18556 and abvtri 18560, abvneg 18564 are sufficient.) (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝐴 = (AbsVal‘𝑅)    &    = (-g𝑅)       (𝐹𝐴 → (𝐹 ) ∈ (Met‘𝑋))
 
12.4.8  Normed algebraic structures
 
Syntaxcnm 22093 Norm of a normed ring.
class norm
 
Syntaxcngp 22094 The class of all normed groups.
class NrmGrp
 
Syntaxctng 22095 Make a normed group from a norm and a group.
class toNrmGrp
 
Syntaxcnrg 22096 Normed ring.
class NrmRing
 
Syntaxcnlm 22097 Normed module.
class NrmMod
 
Syntaxcnvc 22098 Normed vector space.
class NrmVec
 
Definitiondf-nm 22099* Define the norm on a group or ring (when it makes sense) in terms of the distance to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))))
 
Definitiondf-ngp 22100 Define a normed group, which is a group with a right-translation-invariant metric. This is not a standard notion, but is helpful as the most general context in which a metric-like norm makes sense. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔)}
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