HomeTheorem List (p. 246 of 491)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | setsmsbasOLD 24501 | Obsolete version of setsmsbas 24500 as of 12-Nov-2024. The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) & ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) ⇒ ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) | ||
| Theorem | setsmsds 24502 | The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 11-Nov-2024.) |
| ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) & ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) ⇒ ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) | ||
| Theorem | setsmsdsOLD 24503 | Obsolete version of setsmsds 24502 as of 11-Nov-2024. The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) & ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) ⇒ ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) | ||
| Theorem | setsmstset 24504 | The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) & ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) ⇒ ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) | ||
| Theorem | setsmstopn 24505 | The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) & ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) ⇒ ⊢ (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾)) | ||
| Theorem | setsxms 24506 | The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) & ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ (∞Met‘𝑋))) | ||
| Theorem | setsms 24507 | The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) & ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋))) | ||
| Theorem | tmsval 24508 | For any metric there is an associated metric space. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩} & ⊢ 𝐾 = (toMetSp‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) | ||
| Theorem | tmslem 24509 | Lemma for tmsbas 24511, tmsds 24512, and tmstopn 24513. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩} & ⊢ 𝐾 = (toMetSp‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾))) | ||
| Theorem | tmslemOLD 24510 | Obsolete version of tmslem 24509 as of 28-Oct-2024. Lemma for tmsbas 24511, tmsds 24512, and tmstopn 24513. (Contributed by Mario Carneiro, 2-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩} & ⊢ 𝐾 = (toMetSp‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾))) | ||
| Theorem | tmsbas 24511 | The base set of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐾 = (toMetSp‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾)) | ||
| Theorem | tmsds 24512 | The metric of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐾 = (toMetSp‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾)) | ||
| Theorem | tmstopn 24513 | 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)) | ||
| Theorem | tmsxms 24514 | The constructed metric space is an extended metric space. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐾 = (toMetSp‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 ∈ ∞MetSp) | ||
| Theorem | tmsms 24515 | 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) | ||
| Theorem | imasf1obl 24516 | The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) & ⊢ (𝜑 → 𝑃 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ ℝ*) ⇒ ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆))) | ||
| Theorem | imasf1oxms 24517 | The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ ∞MetSp) ⇒ ⊢ (𝜑 → 𝑈 ∈ ∞MetSp) | ||
| Theorem | imasf1oms 24518 | The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ MetSp) ⇒ ⊢ (𝜑 → 𝑈 ∈ MetSp) | ||
| Theorem | prdsbl 24519* |
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 24533) - 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(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑉 = (Base‘𝑅) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 0 < 𝐴) ⇒ ⊢ (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)) | ||
| Theorem | mopni 24520* | 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) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) | ||
| Theorem | mopni2 24521* | 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) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴) | ||
| Theorem | mopni3 24522* | 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) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)) | ||
| Theorem | blssopn 24523 | 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) | ||
| Theorem | unimopn 24524 | 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) ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽) | ||
| Theorem | mopnin 24525 | 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) ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) | ||
| Theorem | mopn0 24526 | 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) | ||
| Theorem | rnblopn 24527 | 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)) → 𝐵 ∈ 𝐽) | ||
| Theorem | blopn 24528 | 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) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝐽) | ||
| Theorem | neibl 24529* | 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 24530 | 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 24531* | 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 24532* | 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 24533* | 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 24534* | 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 24535 | 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 24536* | 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 24537* | 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 24538* | 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 24539* | Lemma for metss2 24540. (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 24540* | 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 24541* | 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 24542* | 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 24543* | 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 24544* | 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 24545* | 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 24546* | 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 24547* | 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 24548 | 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 24549 | 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 24550 | 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 24551* | 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 24552 | 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 24553 | 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 24554 | 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 24555 | Lemma for prdsms 24559. 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 24556 | Lemma for prdsms 24559. 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 24557* | Lemma for prdsxms 24558. 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 24558 | 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 24559 | 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 24560 | 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 24561 | 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 24562 | 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 24563 | 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 24564 | 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 24565 | 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 24566 | 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 24567 | 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))) | ||
| Theorem | metcnp3 24568* | 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 24569* | 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 24570* | 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 24569 (and Munkres' metcn 24571) for compatibility with df-lm 23252. 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 24571* | 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 24572* | Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 24569. (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 24573* | Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 24570. (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 24574* | Epsilon-delta property of a metric space function continuous at 𝑃. A variation of metcnpi2 24573 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 24575* | 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 24576* | 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)))) | ||
| Theorem | metuval 24577* | 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 24578* | 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 24579* | 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 24580* | 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 24581* | 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 24582* | 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 24583* | 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 24584* | 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 24585* | 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 24586* | 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 24587* | 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 25312. (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 24588 | 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 24589* | 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 25312. (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 24590 | 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 24591 | 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 24592* | 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 24593* | 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 24594* | 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 24595 | 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 24596 | 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 24597 | 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 24598 | 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 24599* | Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 24571. (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)))) | ||
| Theorem | dscmet 24600* | 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)) | ||
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