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Theorem List for Metamath Proof Explorer - 23101-23200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremimasf1oms 23101 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 23102* 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 23116) - 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 23103* 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 23104* 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 23105* 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 23106 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 23107 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 23108 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 23109 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 23110 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 23111 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 23112* 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 23113 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 23114* 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 23115* 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 23116* 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 23117* 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 23118 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 23119* 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 23120* 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 23121* 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 23122* Lemma for metss2 23123. (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 23123* 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 23124* 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 23125* 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 23126* 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 23127* 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 23128* 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 23129* 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 23130* 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 23131 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 23132 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 23133 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 23134* 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 23135 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 23136 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ ∞MetSp)

Theoremressms 23137 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐾 ∈ MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ MetSp)

Theoremprdsmslem1 23138 Lemma for prdsms 23142. 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 23139 Lemma for prdsms 23142. 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 23140* Lemma for prdsxms 23141. 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 23141 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 23142 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 23143 A power of an extended metric space is an extended metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ ∞MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ ∞MetSp)

Theorempwsms 23144 A power of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ MetSp)

Theoremxpsxms 23145 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 23146 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 23147 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 23148 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 23149 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 23150 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 23151* 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 23152* 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 23153* 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 23152 (and Munkres' metcn 23154) for compatibility with df-lm 21838. 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 23154* 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 23155* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 23152. (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 23156* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 23153. (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 23157* Epsilon-delta property of a metric space function continuous at 𝑃. A variation of metcnpi2 23156 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 23158* 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 23159* 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 23160* 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 23161* 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 23162* 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 23163* 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 23164* 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 23165* 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.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)

Theoremmetustsym 23166* 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 23167* 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 23168* The filter base generated by a metric 𝐷. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))

Theoremmetust 23169* 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 23170* 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 23873. (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 23171 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 23172* 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 23873. (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 23173 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 23174 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 23175* 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 23176* 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 23177* 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 23178 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 23179 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 23180 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 23181 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 23182* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 23154. (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 23183* 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 23184* 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 23185* Show that a group norm generates a metric. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised 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 23186 An absolute value 𝐹 generates a metric defined by 𝑑(𝑥, 𝑦) = 𝐹(𝑥𝑦), analogously to cnmet 23381. (In fact, the ring structure is not needed at all; the group properties abveq0 19594 and abvtri 19598, abvneg 19602 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

In the following, the norm of a normed algebraic structure (group, left module, vector space) is defined by the (given) distance function (the norm 𝑁 of an element is its distance 𝐷 from the zero element, see nmval 23200: (𝑁𝐴) = (𝐴𝐷 0 )). By this definition, the norm function 𝑁 is actually a norm (satisfying the properties of being a function into the reals, subadditivity/triangle inequality (𝑛‘(𝑥 + 𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦)), absolute homogeneity ( n(sx) = |s| n(x) ) [Remark: for group norms, some authors (e.g. Juris Steprans in "A characterization of free abelian groups") demand that n(sx) = |s| n(x) for all s in ZZ, whereas other authors (e.g. N. H. Bingham and A. J. Ostaszewski in "Normed versus topological groups: Dichotomy and duality") only require inversion symmetry, i.e. n(-x) = n(x). The definition df-ngp 23194 of a group norm follows the second aproach, see nminv 23231.] and positive definiteness/point-separating ( n(x) = 0 <-> x = 0 ) if the structure is a metric space with a right-translation-invariant metric (see nmf 23225, nmtri 23236, nmvs 23286 and nmeq0 23228). An alternate definition of a normed group (i.e. a group equipped with a norm) without using the properties of a metric space is given by theorem tngngp3 23266. For a structure being a group, the (arbitrary) distance function can be restricted to the elements of the group without affecting the norm, see nmfval2 23201.

Usually, however, the norm of a normed structure is given, and the corresponding metric ("induced metric") is achieved by defining a distance function based on the norm (the distance 𝐷 between two elements is the norm 𝑁 of their difference, see ngpds 23214: (𝐴𝐷𝐵) = (𝑁‘(𝐴 𝐵))). The operation toNrmGrp does exactly this, i.e. it adds a distance function (and a topology) to a structure (which should be at least a group) corresponding to a given norm in the just shown way: (dist‘𝑇) = (𝑁 ), see also tngds 23258. By this, the enhanced structure becomes a normed structure if the induced metric is in fact a metric (see tngngp2 23262) resp. if the norm is in fact a norm (see tngngpd 23263). If the norm is derived from a given metric, as done with df-nm 23193, the induced metric is the original metric restricted to the base set: (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)), see nrmtngdist 23267, and the norm remains the same: (norm‘𝑇) = (norm‘𝐺), see nrmtngnrm 23268.

Syntaxcnm 23187 Norm of a normed ring.
class norm

Syntaxcngp 23188 The class of all normed groups.
class NrmGrp

Syntaxctng 23189 Make a normed group from a norm and a group.
class toNrmGrp

Syntaxcnrg 23190 Normed ring.
class NrmRing

Syntaxcnlm 23191 Normed module.
class NrmMod

Syntaxcnvc 23192 Normed vector space.
class NrmVec

Definitiondf-nm 23193* 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 23194 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)}

Definitiondf-tng 23195* Define a function that fills in the topology and metric components of a structure given a group and a norm on it. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))

Definitiondf-nrg 23196 A normed ring is a ring with an induced topology and metric such that the metric is translation-invariant and the norm (distance from 0) is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}

Definitiondf-nlm 23197* A normed (left) module is a module which is also a normed group over a normed ring, such that the norm distributes over scalar multiplication. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmMod = {𝑤 ∈ (NrmGrp ∩ LMod) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ NrmRing ∧ ∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))}

Definitiondf-nvc 23198 A normed vector space is a normed module which is also a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec = (NrmMod ∩ LVec)

Theoremnmfval 23199* The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝐷 = (dist‘𝑊)       𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))

Theoremnmval 23200 The value of the norm function. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝐷 = (dist‘𝑊)       (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))

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