P. 246 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24501-24600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	neibl 24501*	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) ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝑁)))
Theorem	blnei 24502	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) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃}))
Theorem	lpbl 24503*	Every ball around a limit point 𝑃 of a subset 𝑆 includes a member of 𝑆 (even if 𝑃 ∉ 𝑆). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝑆 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))
Theorem	blsscls2 24504*	A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝑆 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑇 ∈ ℝ* ∧ 𝑅 < 𝑇)) → 𝑆 ⊆ (𝑃(ball‘𝐷)𝑇))
Theorem	blcld 24505*	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 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑆 ∈ (Clsd‘𝐽))
Theorem	blcls 24506*	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 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ 𝑆)
Theorem	blsscls 24507	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) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ* ∧ 𝑅 < 𝑆)) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ (𝑃(ball‘𝐷)𝑆))
Theorem	metss 24508*	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)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
Theorem	metequiv 24509*	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 ↔ ∀𝑥 ∈ 𝑋 (∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∀𝑎 ∈ ℝ+ ∃𝑏 ∈ ℝ+ (𝑥(ball‘𝐶)𝑏) ⊆ (𝑥(ball‘𝐷)𝑎))))
Theorem	metequiv2 24510*	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)) → (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → 𝐽 = 𝐾))
Theorem	metss2lem 24511*	Lemma for metss2 24512. (Contributed by Mario Carneiro, 14-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    ⇒   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑆 ∈ ℝ+)) → (𝑥(ball‘𝐷)(𝑆 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑆))
Theorem	metss2 24512*	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)    &   ⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    ⇒   ⊢ (𝜑 → 𝐽 ⊆ 𝐾)
Theorem	comet 24513*	The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐹:(0[,]+∞)⟶ℝ*)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]+∞)) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹‘𝑥) +𝑒 (𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐷) ∈ (∞Met‘𝑋))
Theorem	stdbdmetval 24514*	Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
Theorem	stdbdxmet 24515*	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 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
Theorem	stdbdmet 24516*	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 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (Met‘𝑋))
Theorem	stdbdbl 24517*	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 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    ⇒   ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → (𝑃(ball‘𝐷)𝑆) = (𝑃(ball‘𝐶)𝑆))
Theorem	stdbdmopn 24518*	The standard bounded metric corresponding to 𝐶 generates the same topology as 𝐶. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    &   ⊢ 𝐽 = (MetOpen‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐽 = (MetOpen‘𝐷))
Theorem	mopnex 24519*	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))
Theorem	methaus 24520	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)
Theorem	met1stc 24521	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)
Theorem	met2ndci 24522	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) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ 2ndω)
Theorem	met2ndc 24523*	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)))
Theorem	metrest 24524	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) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = 𝐾)
Theorem	ressxms 24525	The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ ∞MetSp)
Theorem	ressms 24526	The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ MetSp)
Theorem	prdsmslem1 24527	Lemma for prdsms 24531. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅:𝐼⟶MetSp)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵))
Theorem	prdsxmslem1 24528	Lemma for prdsms 24531. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
Theorem	prdsxmslem2 24529*	Lemma for prdsxms 24530. 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)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp)    &   ⊢ 𝐽 = (TopOpen‘𝑌)    &   ⊢ 𝑉 = (Base‘(𝑅‘𝑘))    &   ⊢ 𝐸 = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉))    &   ⊢ 𝐾 = (TopOpen‘(𝑅‘𝑘))    &   ⊢ 𝐶 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))}    ⇒   ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))
Theorem	prdsxms 24530	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)    ⇒   ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑌 ∈ ∞MetSp)
Theorem	prdsms 24531	The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    ⇒   ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑌 ∈ MetSp)
Theorem	pwsxms 24532	A power of an extended metric space is an extended metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    ⇒   ⊢ ((𝑅 ∈ ∞MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ ∞MetSp)
Theorem	pwsms 24533	A power of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    ⇒   ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ MetSp)
Theorem	xpsxms 24534	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)
Theorem	xpsms 24535	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)
Theorem	tmsxps 24536	Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    ⇒   ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
Theorem	tmsxpsmopn 24537	Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    &   ⊢ 𝐽 = (MetOpen‘𝑀)    &   ⊢ 𝐾 = (MetOpen‘𝑁)    &   ⊢ 𝐿 = (MetOpen‘𝑃)    ⇒   ⊢ (𝜑 → 𝐿 = (𝐽 ×t 𝐾))
Theorem	tmsxpsval 24538	Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
Theorem	tmsxpsval2 24539	Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩) = if((𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐵𝑁𝐷), (𝐴𝑀𝐶)))
12.4.5  Continuity in metric spaces
Theorem	metcnp3 24540*	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) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
Theorem	metcnp 24541*	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) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))))
Theorem	metcnp2 24542*	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 24541 (and Munkres' metcn 24543) for compatibility with df-lm 23224. 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) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑤𝐶𝑃) < 𝑧 → ((𝐹‘𝑤)𝐷(𝐹‘𝑃)) < 𝑦))))
Theorem	metcn 24543*	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:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝐹‘𝑥)𝐷(𝐹‘𝑤)) < 𝑦))))
Theorem	metcnpi 24544*	Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 24541. (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+ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝐴))
Theorem	metcnpi2 24545*	Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 24542. (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+ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) < 𝑥 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴))
Theorem	metcnpi3 24546*	Epsilon-delta property of a metric space function continuous at 𝑃. A variation of metcnpi2 24545 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+ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) ≤ 𝑥 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴))
Theorem	txmetcnp 24547*	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)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))))
Theorem	txmetcn 24548*	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 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧))))
12.4.6  The uniform structure generated by a metric
Theorem	metuval 24549*	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)))))
Theorem	metustel 24550*	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) → (𝐵 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎))))
Theorem	metustss 24551*	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) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
Theorem	metustrel 24552*	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) ∧ 𝐴 ∈ 𝐹) → Rel 𝐴)
Theorem	metustto 24553*	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) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
Theorem	metustid 24554*	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) ∧ 𝐴 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)
Theorem	metustsym 24555*	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) ∧ 𝐴 ∈ 𝐹) → ◡𝐴 = 𝐴)
Theorem	metustexhalf 24556*	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)) ∧ 𝐴 ∈ 𝐹) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 𝐴)
Theorem	metustfbas 24557*	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)))
Theorem	metust 24558*	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))
Theorem	cfilucfil 24559*	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 25284. (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+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
Theorem	metuust 24560	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))
Theorem	cfilucfil2 24561*	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 25284. (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+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
Theorem	blval2 24562	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) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) = ((◡𝐷 “ (0[,)𝑅)) “ {𝑃}))
Theorem	elbl4 24563	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+) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵 ∈ (𝐴(ball‘𝐷)𝑅) ↔ 𝐵(◡𝐷 “ (0[,)𝑅))𝐴))
Theorem	metuel 24564*	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)))𝑤 ⊆ 𝑉)))
Theorem	metuel2 24565*	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)) → (𝑉 ∈ 𝑈 ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦))))
Theorem	metustbl 24566*	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) ∧ 𝑃 ∈ 𝑋) → ∃𝑎 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑎 ∧ 𝑎 ⊆ (𝑉 “ {𝑃})))
Theorem	psmetutop 24567	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)))
Theorem	xmetutop 24568	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))
Theorem	xmsusp 24569	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)
Theorem	restmetu 24570	The uniform structure generated by the restriction of a metric is its trace. (Contributed by Thierry Arnoux, 18-Dec-2017.)
⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))))
Theorem	metucn 24571*	Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 24543. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
⊢ 𝑈 = (metUnif‘𝐶)    &   ⊢ 𝑉 = (metUnif‘𝐷)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝑌 ≠ ∅)    &   ⊢ (𝜑 → 𝐶 ∈ (PsMet‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑌))    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑑 ∈ ℝ+ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑))))
12.4.7  Examples of metric spaces
Theorem	dscmet 24572*	The discrete metric on any set 𝑋. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if(𝑥 = 𝑦, 0, 1))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))
Theorem	dscopn 24573*	The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if(𝑥 = 𝑦, 0, 1))    ⇒   ⊢ (𝑋 ∈ 𝑉 → (MetOpen‘𝐷) = 𝒫 𝑋)
Theorem	nrmmetd 24574*	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)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 − 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → (𝐹 ∘ − ) ∈ (Met‘𝑋))
Theorem	abvmet 24575	An absolute value 𝐹 generates a metric defined by 𝑑(𝑥, 𝑦) = 𝐹(𝑥 − 𝑦), analogously to cnmet 24779. (In fact, the ring structure is not needed at all; the group properties abveq0 20797 and abvtri 20801, abvneg 20805 are sufficient.) (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ − = (-g‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐴 → (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 24589: (𝑁‘𝐴) = (𝐴𝐷 0 )). By this definition, the norm function 𝑁 is actually a norm (satisfying the properties: being a function into the reals; subadditivity/triangle inequality (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)); 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 ∈ ℤ, 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 24583 of a group norm follows the second approach, see nminv 24621.] and positive definiteness/point-separation ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0)) if the structure is a metric space with a right-translation-invariant metric (see nmf 24615, nmtri 24626, nmvs 24684 and nmeq0 24618). An alternate definition of a normed group (i.e., a group equipped with a norm) not using the properties of a metric space is given by Theorem tngngp3 24664. The norm can be expressed as the distance to zero (nmfval 24588), so in a structure with a zero (a "pointed set", for instance a monoid or a group), the norm can be expressed as the distance restricted to the elements of the base set to zero (nmfval0 24590). Usually, however, the norm of a normed structure is given, and the corresponding metric ("induced metric") is defined as the distance function based on the norm (the distance 𝐷 between two elements is the norm 𝑁 of their difference, see ngpds 24604: (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 for the difference of two elements to make sense) corresponding to a given norm as explained above: (dist‘𝑇) = (𝑁 ∘ − ), see also tngds 24655. By this, the enhanced structure becomes a normed structure if the induced metric is in fact a metric (see tngngp2 24660) or a norm (see tngngpd 24661). If the norm is derived from a given metric, as done with df-nm 24582, the induced metric is the original metric restricted to the base set: (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)), see nrmtngdist 24665, and the norm remains the same: (norm‘𝑇) = (norm‘𝐺), see nrmtngnrm 24666.
Syntax	cnm 24576	Norm of a normed ring.
class norm
Syntax	cngp 24577	The class of all normed groups.
class NrmGrp
Syntax	ctng 24578	Make a normed group from a norm and a group.
class toNrmGrp
Syntax	cnrg 24579	Normed ring.
class NrmRing
Syntax	cnlm 24580	Normed module.
class NrmMod
Syntax	cnvc 24581	Normed vector space.
class NrmVec
Definition	df-nm 24582*	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)(0g‘𝑤))))
Definition	df-ngp 24583	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) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)}
Definition	df-tng 24584*	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 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
Definition	df-nrg 24585	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)}
Definition	df-nlm 24586*	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( ·𝑠 ‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))}
Definition	df-nvc 24587	A normed vector space is a normed module which is also a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ NrmVec = (NrmMod ∩ LVec)
Theorem	nmfval 24588*	The value of the norm function as the distance to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑁 = (norm‘𝑊)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    ⇒   ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
Theorem	nmval 24589	The value of the norm as the distance to zero. 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 = (0g‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    ⇒   ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 ))
Theorem	nmfval0 24590*	The value of the norm function on a structure containing a zero as the distance restricted to the elements of the base set to zero. Examples of structures containing a "zero" are groups (see nmfval2 24591 proved from this theorem and grpidcl 18960) or more generally monoids (see mndidcl 18742), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 24591. (Revised by BJ, 27-Aug-2024.)
⊢ 𝑁 = (norm‘𝑊)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    &   ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))    ⇒   ⊢ ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
Theorem	nmfval2 24591*	The value of the norm function on a group as the distance restricted to the elements of the base set to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑁 = (norm‘𝑊)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    &   ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
Theorem	nmval2 24592	The value of the norm on a group as the distance restricted to the elements of the base set to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑁 = (norm‘𝑊)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    &   ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))    ⇒   ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐸 0 ))
Theorem	nmf2 24593	The norm on a metric group is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑁 = (norm‘𝑊)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    &   ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))    ⇒   ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ)
Theorem	nmpropd 24594	Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿))    &   ⊢ (𝜑 → (dist‘𝐾) = (dist‘𝐿))    ⇒   ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Theorem	nmpropd2 24595*	Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐾 ∈ Grp)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    ⇒   ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Theorem	isngp 24596	The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑁 = (norm‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    ⇒   ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷))
Theorem	isngp2 24597	The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑁 = (norm‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) = 𝐸))
Theorem	isngp3 24598*	The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝑁 = (norm‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐷𝑦) = (𝑁‘(𝑥 − 𝑦))))
Theorem	ngpgrp 24599	A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
Theorem	ngpms 24600	A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)