Home | Metamath
Proof Explorer Theorem List (p. 231 of 449) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28623) |
Hilbert Space Explorer
(28624-30146) |
Users' Mathboxes
(30147-44804) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | xmscl 23001 | Closure of the distance function of an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | ||
Theorem | xmsge0 23002 | The distance function in an extended metric space is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | ||
Theorem | xmseq0 23003 | The distance between two points in an extended metric space is zero iff the two points are identical. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)) | ||
Theorem | xmssym 23004 | The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴)) | ||
Theorem | xmstri2 23005 | Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ ∞MetSp ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) | ||
Theorem | mstri2 23006 | Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ MetSp ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵))) | ||
Theorem | xmstri 23007 | Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ ∞MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵))) | ||
Theorem | mstri 23008 | Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐶𝐷𝐵))) | ||
Theorem | xmstri3 23009 | Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ ∞MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐵𝐷𝐶))) | ||
Theorem | mstri3 23010 | Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶))) | ||
Theorem | msrtri 23011 | Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵)) | ||
Theorem | xmspropd 23012 | Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵))) & ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) ⇒ ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp)) | ||
Theorem | mspropd 23013 | Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵))) & ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) ⇒ ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp)) | ||
Theorem | setsmsbas 23014 | The base set 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 → 𝑋 = (Base‘𝐾)) | ||
Theorem | setsmsds 23015 | The distance function 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 → (dist‘𝑀) = (dist‘𝐾)) | ||
Theorem | setsmstset 23016 | 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 23017 | 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 23018 | 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 23019 | 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 23020 | 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 23021 | Lemma for tmsbas 23022, tmsds 23023, and tmstopn 23024. (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 | tmsbas 23022 | 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 23023 | 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 23024 | 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 23025 | 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 23026 | 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 23027 | 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 23028 | 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 23029 | 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 23030* |
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 23044) - 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 23031* | 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 23032* | 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 23033* | 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 23034 | 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 23035 | 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 23036 | 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 23037 | 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 23038 | 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 23039 | 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 23040* | 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 23041 | 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 23042* | 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 23043* | 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 23044* | 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 23045* | 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 23046 | 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 23047* | 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 23048* | 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 23049* | 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 23050* | Lemma for metss2 23051. (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 23051* | 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 23052* | 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 23053* | 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 23054* | 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 23055* | 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 23056* | 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 23057* | 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 23058* | 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 23059 | 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 23060 | 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 23061 | 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 23062* | 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 23063 | 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 23064 | 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 23065 | 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 23066 | Lemma for prdsms 23070. 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 23067 | Lemma for prdsms 23070. 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 23068* | Lemma for prdsxms 23069. 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 23069 | 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 23070 | 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 23071 | The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ ∞MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ ∞MetSp) | ||
Theorem | pwsms 23072 | The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ MetSp) | ||
Theorem | xpsxms 23073 | 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 23074 | 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 23075 | 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 23076 | 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 23077 | 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 23078 | 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 23079* | 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 23080* | 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 23081* | 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 23080 (and Munkres' metcn 23082) for compatibility with df-lm 21767. 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 23082* | 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 23083* | Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 23080. (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 23084* | Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 23081. (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 23085* | Epsilon-delta property of a metric space function continuous at 𝑃. A variation of metcnpi2 23084 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 23086* | 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 23087* | 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 23088* | 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 23089* | 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 23090* | 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 23091* | 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 23092* | 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 23093* | 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 23094* | 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 23095* | 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 23096* | 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 23097* | 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 23098* | 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 23797. (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 23099 | 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 23100* | 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 23797. (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)))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |