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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | blpnf 23001 | The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑃(ball‘𝐷)+∞) = 𝑋) | ||
Theorem | bldisj 23002 | Two balls are disjoint if the center-to-center distance is more than the sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.) |
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ* ∧ (𝑅 +𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) = ∅) | ||
Theorem | blgt0 23003 | A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅) | ||
Theorem | bl2in 23004 | Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑅)) = ∅) | ||
Theorem | xblss2ps 23005 | One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 23008 for extended metrics, we have to assume the balls are a finite distance apart, or else 𝑃 will not even be in the infinity ball around 𝑄. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑋)) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝜑 → 𝑄 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ (𝜑 → 𝑆 ∈ ℝ*) & ⊢ (𝜑 → (𝑃𝐷𝑄) ∈ ℝ) & ⊢ (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅)) ⇒ ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆)) | ||
Theorem | xblss2 23006 | One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 23008 for extended metrics, we have to assume the balls are a finite distance apart, or else 𝑃 will not even be in the infinity ball around 𝑄. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝜑 → 𝑄 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ (𝜑 → 𝑆 ∈ ℝ*) & ⊢ (𝜑 → (𝑃𝐷𝑄) ∈ ℝ) & ⊢ (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅)) ⇒ ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆)) | ||
Theorem | blss2ps 23007 | One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ (𝑃𝐷𝑄) ≤ (𝑆 − 𝑅))) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆)) | ||
Theorem | blss2 23008 | One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.) |
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ (𝑃𝐷𝑄) ≤ (𝑆 − 𝑅))) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆)) | ||
Theorem | blhalf 23009 | A ball of radius 𝑅 / 2 is contained in a ball of radius 𝑅 centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.) |
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑍 ∈ (𝑌(ball‘𝑀)(𝑅 / 2)))) → (𝑌(ball‘𝑀)(𝑅 / 2)) ⊆ (𝑍(ball‘𝑀)𝑅)) | ||
Theorem | blfps 23010 | Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) | ||
Theorem | blf 23011 | Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) | ||
Theorem | blrnps 23012* | Membership in the range of the ball function. Note that ran (ball‘𝐷) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟))) | ||
Theorem | blrn 23013* | Membership in the range of the ball function. Note that ran (ball‘𝐷) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟))) | ||
Theorem | xblcntrps 23014 | A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅)) | ||
Theorem | xblcntr 23015 | A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅)) | ||
Theorem | blcntrps 23016 | A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅)) | ||
Theorem | blcntr 23017 | A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅)) | ||
Theorem | xbln0 23018 | A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → ((𝑃(ball‘𝐷)𝑅) ≠ ∅ ↔ 0 < 𝑅)) | ||
Theorem | bln0 23019 | A ball is not empty. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ≠ ∅) | ||
Theorem | blelrnps 23020 | A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷)) | ||
Theorem | blelrn 23021 | A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷)) | ||
Theorem | blssm 23022 | A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋) | ||
Theorem | unirnblps 23023 | The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋) | ||
Theorem | unirnbl 23024 | The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋) | ||
Theorem | blin 23025 | The intersection of two balls with the same center is the smaller of them. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ*)) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑃(ball‘𝐷)𝑆)) = (𝑃(ball‘𝐷)if(𝑅 ≤ 𝑆, 𝑅, 𝑆))) | ||
Theorem | ssblps 23026 | The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ*) ∧ 𝑅 ≤ 𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆)) | ||
Theorem | ssbl 23027 | The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) |
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ*) ∧ 𝑅 ≤ 𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆)) | ||
Theorem | blssps 23028* | Any point 𝑃 in a ball 𝐵 can be centered in another ball that is a subset of 𝐵. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵) | ||
Theorem | blss 23029* | Any point 𝑃 in a ball 𝐵 can be centered in another ball that is a subset of 𝐵. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵) | ||
Theorem | blssexps 23030* | Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝐴)) | ||
Theorem | blssex 23031* | Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝐴)) | ||
Theorem | ssblex 23032* | A nested ball exists whose radius is less than any desired amount. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆))) | ||
Theorem | blin2 23033* | Given any two balls and a point in their intersection, there is a ball contained in the intersection with the given center point. (Contributed by Mario Carneiro, 12-Nov-2013.) |
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶)) | ||
Theorem | blbas 23034 | The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.) |
⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases) | ||
Theorem | blres 23035 | A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.) |
⊢ 𝐶 = (𝐷 ↾ (𝑌 × 𝑌)) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋 ∩ 𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑅) = ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌)) | ||
Theorem | xmeterval 23036 | Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ ∼ = (◡𝐷 “ ℝ) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ))) | ||
Theorem | xmeter 23037 | The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ ∼ = (◡𝐷 “ ℝ) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∼ Er 𝑋) | ||
Theorem | xmetec 23038 | The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 8335, infinity balls are either identical or disjoint, quite unlike the usual situation with Euclidean balls which admit many kinds of overlap. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ ∼ = (◡𝐷 “ ℝ) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → [𝑃] ∼ = (𝑃(ball‘𝐷)+∞)) | ||
Theorem | blssec 23039 | A ball centered at 𝑃 is contained in the set of points finitely separated from 𝑃. This is just an application of ssbl 23027 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ ∼ = (◡𝐷 “ ℝ) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) ⊆ [𝑃] ∼ ) | ||
Theorem | blpnfctr 23040 | The infinity ball in an extended metric acts like an ultrametric ball in that every point in the ball is also its center. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ (𝑃(ball‘𝐷)+∞)) → (𝑃(ball‘𝐷)+∞) = (𝐴(ball‘𝐷)+∞)) | ||
Theorem | xmetresbl 23041 | An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 23038, this shows that any extended metric space can be "factored" into the disjoint union of proper metric spaces, with points in the same region measured by that region's metric, and points in different regions being distance +∞ from each other. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ 𝐵 = (𝑃(ball‘𝐷)𝑅) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵)) | ||
Theorem | mopnval 23042 | An open set is a subset of a metric space which includes a ball around each of its points. Definition 1.3-2 of [Kreyszig] p. 18. The object (MetOpen‘𝐷) is the family of all open sets in the metric space determined by the metric 𝐷. By mopntop 23044, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 23045. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷))) | ||
Theorem | mopntopon 23043 | The set of open sets of a metric space 𝑋 is a topology on 𝑋. Remark in [Kreyszig] p. 19. This theorem connects the two concepts and makes available the theorems for topologies for use with metric spaces. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | ||
Theorem | mopntop 23044 | The set of open sets of a metric space is a topology. (Contributed by NM, 28-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) | ||
Theorem | mopnuni 23045 | The union of all open sets in a metric space is its underlying set. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) | ||
Theorem | elmopn 23046* | The defining property of an open set of a metric space. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))) | ||
Theorem | mopnfss 23047 | The family of open sets of a metric space is a collection of subsets of the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ⊆ 𝒫 𝑋) | ||
Theorem | mopnm 23048 | The base set of a metric space is open. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ 𝐽) | ||
Theorem | elmopn2 23049* | A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴))) | ||
Theorem | mopnss 23050 | An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.) |
⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) | ||
Theorem | isxms 23051 | Express the predicate "〈𝑋, 𝐷〉 is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ 𝐽 = (TopOpen‘𝐾) & ⊢ 𝑋 = (Base‘𝐾) & ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) | ||
Theorem | isxms2 23052 | Express the predicate "〈𝑋, 𝐷〉 is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ 𝐽 = (TopOpen‘𝐾) & ⊢ 𝑋 = (Base‘𝐾) & ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷))) | ||
Theorem | isms 23053 | Express the predicate "〈𝑋, 𝐷〉 is a metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) |
⊢ 𝐽 = (TopOpen‘𝐾) & ⊢ 𝑋 = (Base‘𝐾) & ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋))) | ||
Theorem | isms2 23054 | Express the predicate "〈𝑋, 𝐷〉 is a metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) |
⊢ 𝐽 = (TopOpen‘𝐾) & ⊢ 𝑋 = (Base‘𝐾) & ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝐾 ∈ MetSp ↔ (𝐷 ∈ (Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷))) | ||
Theorem | xmstopn 23055 | The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ 𝐽 = (TopOpen‘𝐾) & ⊢ 𝑋 = (Base‘𝐾) & ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷)) | ||
Theorem | mstopn 23056 | The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ 𝐽 = (TopOpen‘𝐾) & ⊢ 𝑋 = (Base‘𝐾) & ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝐾 ∈ MetSp → 𝐽 = (MetOpen‘𝐷)) | ||
Theorem | xmstps 23057 | An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp) | ||
Theorem | msxms 23058 | A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) | ||
Theorem | mstps 23059 | A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝑀 ∈ MetSp → 𝑀 ∈ TopSp) | ||
Theorem | xmsxmet 23060 | The distance function, suitably truncated, is an extended metric on 𝑋. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝑀 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋)) | ||
Theorem | msmet 23061 | The distance function, suitably truncated, is a metric on 𝑋. (Contributed by Mario Carneiro, 12-Nov-2013.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝑀 ∈ MetSp → 𝐷 ∈ (Met‘𝑋)) | ||
Theorem | msf 23062 | The distance function of a metric space is a function into the real numbers. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝑀 ∈ MetSp → 𝐷:(𝑋 × 𝑋)⟶ℝ) | ||
Theorem | xmsxmet2 23063 | The distance function, suitably truncated, is an extended metric on 𝑋. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ (𝑀 ∈ ∞MetSp → (𝐷 ↾ (𝑋 × 𝑋)) ∈ (∞Met‘𝑋)) | ||
Theorem | msmet2 23064 | The distance function, suitably truncated, is a metric on 𝑋. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ (𝑀 ∈ MetSp → (𝐷 ↾ (𝑋 × 𝑋)) ∈ (Met‘𝑋)) | ||
Theorem | mscl 23065 | Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐷 = (dist‘𝑀) ⇒ ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ) | ||
Theorem | xmscl 23066 | 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 23067 | 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 23068 | 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 23069 | 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 23070 | 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 23071 | 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 23072 | 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 23073 | 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 23074 | 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 23075 | 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 23076 | 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 23077 | 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 23078 | 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 23079 | 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 23080 | 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 23081 | 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 23082 | 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 23083 | 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 23084 | 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 23085 | 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 23086 | Lemma for tmsbas 23087, tmsds 23088, and tmstopn 23089. (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 23087 | 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 23088 | 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 23089 | 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 23090 | 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 23091 | 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 23092 | 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 23093 | 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 23094 | 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 23095* |
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 23109) - 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 23096* | 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 23097* | 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 23098* | 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 23099 | 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 23100 | 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) |
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