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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | blcom 23001 | Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) |
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝑃 ∈ (𝐴(ball‘𝐷)𝑅))) | ||
Theorem | xblpnfps 23002 | The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝐴 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) ∈ ℝ))) | ||
Theorem | xblpnf 23003 | The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝐴 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) ∈ ℝ))) | ||
Theorem | blpnf 23004 | 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 23005 | 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 23006 | 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 23007 | 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 23008 | 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 23011 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 23009 | 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 23011 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 23010 | 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 23011 | 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 23012 | 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 23013 | 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 23014 | 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 23015* | 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 23016* | 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 23017 | 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 23018 | 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 23019 | 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 23020 | 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 23021 | 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 23022 | 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 23023 | 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 23024 | 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 23025 | 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 23026 | 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 23027 | 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 23028 | 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 23029 | 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 23030 | 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 23031* | 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 23032* | 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 23033* | 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 23034* | 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 23035* | 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 23036* | 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 23037 | 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 23038 | 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 23039 | 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 23040 | The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ ∼ = (◡𝐷 “ ℝ) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∼ Er 𝑋) | ||
Theorem | xmetec 23041 | The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 8324, 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 23042 | A ball centered at 𝑃 is contained in the set of points finitely separated from 𝑃. This is just an application of ssbl 23030 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 23043 | 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 23044 | An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 23041, 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 23045 | 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 23047, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 23048. (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 23046 | 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 23047 | 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 23048 | 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 23049* | 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 23050 | 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 23051 | 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 23052* | 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 23053 | 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 23054 | 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 23055 | 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 23056 | 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 23057 | 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 23058 | 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 23059 | 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 23060 | An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp) | ||
Theorem | msxms 23061 | A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) | ||
Theorem | mstps 23062 | A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝑀 ∈ MetSp → 𝑀 ∈ TopSp) | ||
Theorem | xmsxmet 23063 | 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 23064 | 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 23065 | 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 23066 | 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 23067 | 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 23068 | 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 23069 | 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 23070 | 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 23071 | 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 23072 | 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 23073 | 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 23074 | 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 23075 | 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 23076 | 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 23077 | 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 23078 | 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 23079 | 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 23080 | 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 23081 | 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 23082 | 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 23083 | 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 23084 | 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 23085 | 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 23086 | 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 23087 | 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 23088 | 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 23089 | Lemma for tmsbas 23090, tmsds 23091, and tmstopn 23092. (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 23090 | 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 23091 | 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 23092 | 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 23093 | 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 23094 | 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 23095 | 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 23096 | 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 23097 | 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 23098* |
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 23112) - 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 23099* | 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 23100* | 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) |
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