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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremblval 22001* The ball around a point 𝑃 is the set of all points whose distance from 𝑃 is less than the ball's radius 𝑅. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})

Theoremelblps 22002 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))

Theoremelbl 22003 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))

Theoremelbl2ps 22004 Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))

Theoremelbl2 22005 Membership in a ball. (Contributed by NM, 9-Mar-2007.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))

Theoremelbl3ps 22006 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
(((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝐷𝑃) < 𝑅))

Theoremelbl3 22007 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝐷𝑃) < 𝑅))

Theoremblcomps 22008 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝑃 ∈ (𝐴(ball‘𝐷)𝑅)))

Theoremblcom 22009 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋𝐴𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝑃 ∈ (𝐴(ball‘𝐷)𝑅)))

Theoremxblpnfps 22010 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) ∧ 𝑃𝑋) → (𝐴 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝐴𝑋 ∧ (𝑃𝐷𝐴) ∈ ℝ)))

Theoremxblpnf 22011 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) ∧ 𝑃𝑋) → (𝐴 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝐴𝑋 ∧ (𝑃𝐷𝐴) ∈ ℝ)))

Theoremblpnf 22012 The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐷 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (𝑃(ball‘𝐷)+∞) = 𝑋)

Theorembldisj 22013 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) ∧ 𝑃𝑋𝑄𝑋) ∧ (𝑅 ∈ ℝ*𝑆 ∈ ℝ* ∧ (𝑅 +𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) = ∅)

Theoremblgt0 22014 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)

Theorembl2in 22015 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) ∧ 𝑃𝑋𝑄𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑅)) = ∅)

Theoremxblss2ps 22016 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 22019 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.)
(𝜑𝐷 ∈ (PsMet‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝑄𝑋)    &   (𝜑𝑅 ∈ ℝ*)    &   (𝜑𝑆 ∈ ℝ*)    &   (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)    &   (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))       (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))

Theoremxblss2 22017 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 22019 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.)
(𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝑄𝑋)    &   (𝜑𝑅 ∈ ℝ*)    &   (𝜑𝑆 ∈ ℝ*)    &   (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)    &   (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))       (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))

Theoremblss2ps 22018 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) ∧ 𝑃𝑋𝑄𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ (𝑃𝐷𝑄) ≤ (𝑆𝑅))) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))

Theoremblss2 22019 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) ∧ 𝑃𝑋𝑄𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ (𝑃𝐷𝑄) ≤ (𝑆𝑅))) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))

Theoremblhalf 22020 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) ∧ 𝑌𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑍 ∈ (𝑌(ball‘𝑀)(𝑅 / 2)))) → (𝑌(ball‘𝑀)(𝑅 / 2)) ⊆ (𝑍(ball‘𝑀)𝑅))

Theoremblfps 22021 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)

Theoremblf 22022 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)

Theoremblrnps 22023* 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) ↔ ∃𝑥𝑋𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))

Theoremblrn 22024* 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) ↔ ∃𝑥𝑋𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))

Theoremxblcntrps 22025 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) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))

Theoremxblcntr 22026 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))

Theoremblcntrps 22027 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) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))

Theoremblcntr 22028 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))

Theoremxbln0 22029 A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → ((𝑃(ball‘𝐷)𝑅) ≠ ∅ ↔ 0 < 𝑅))

Theorembln0 22030 A ball is not empty. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ≠ ∅)

Theoremblelrnps 22031 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))

Theoremblelrn 22032 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))

Theoremblssm 22033 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋)

Theoremunirnblps 22034 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)

Theoremunirnbl 22035 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)

Theoremblin 22036 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) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑆 ∈ ℝ*)) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑃(ball‘𝐷)𝑆)) = (𝑃(ball‘𝐷)if(𝑅𝑆, 𝑅, 𝑆)))

Theoremssblps 22037 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) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑆 ∈ ℝ*) ∧ 𝑅𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))

Theoremssbl 22038 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) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑆 ∈ ℝ*) ∧ 𝑅𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))

Theoremblssps 22039* 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) ∧ 𝑃𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)

Theoremblss 22040* 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) ∧ 𝑃𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)

Theoremblssexps 22041* 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) ∧ 𝑃𝑋) → (∃𝑥 ∈ ran (ball‘𝐷)(𝑃𝑥𝑥𝐴) ↔ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝐴))

Theoremblssex 22042* 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) ∧ 𝑃𝑋) → (∃𝑥 ∈ ran (ball‘𝐷)(𝑃𝑥𝑥𝐴) ↔ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝐴))

Theoremssblex 22043* 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) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ+𝑆 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)))

Theoremblin2 22044* 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 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶))

Theoremblbas 22045 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)

Theoremblres 22046 A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
𝐶 = (𝐷 ↾ (𝑌 × 𝑌))       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑅) = ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌))

Theoremxmeterval 22047 Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       (𝐷 ∈ (∞Met‘𝑋) → (𝐴 𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ)))

Theoremxmeter 22048 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       (𝐷 ∈ (∞Met‘𝑋) → Er 𝑋)

Theoremxmetec 22049 The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 7681, 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) ∧ 𝑃𝑋) → [𝑃] = (𝑃(ball‘𝐷)+∞))

Theoremblssec 22050 A ball centered at 𝑃 is contained in the set of points finitely separated from 𝑃. This is just an application of ssbl 22038 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) ⊆ [𝑃] )

Theoremblpnfctr 22051 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) ∧ 𝑃𝑋𝐴 ∈ (𝑃(ball‘𝐷)+∞)) → (𝑃(ball‘𝐷)+∞) = (𝐴(ball‘𝐷)+∞))

Theoremxmetresbl 22052 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 22049, 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))

12.4.4  Open sets of a metric space

Theoremmopnval 22053 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 22055, the open sets of a metric space form a topology 𝐽, whose base set is 𝐽 by mopnuni 22056. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷)))

Theoremmopntopon 22054 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))

Theoremmopntop 22055 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)

Theoremmopnuni 22056 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)

Theoremelmopn 22057* 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) → (𝐴𝐽 ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑦 ∈ ran (ball‘𝐷)(𝑥𝑦𝑦𝐴))))

Theoremmopnfss 22058 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)

Theoremmopnm 22059 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) → 𝑋𝐽)

Theoremelmopn2 22060* 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) → (𝐴𝐽 ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴)))

Theoremmopnss 22061 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) ∧ 𝐴𝐽) → 𝐴𝑋)

Theoremisxms 22062 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)))

Theoremisxms2 22063 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)))

Theoremisms 22064 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)))

Theoremisms2 22065 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)))

Theoremxmstopn 22066 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))

Theoremmstopn 22067 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))

Theoremxmstps 22068 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

Theoremmsxms 22069 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Theoremmstps 22070 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ MetSp → 𝑀 ∈ TopSp)

Theoremxmsxmet 22071 The distance function, suitably truncated, is a metric on 𝑋. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑋 = (Base‘𝑀)    &   𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))       (𝑀 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋))

Theoremmsmet 22072 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))

Theoremmsf 22073 Mapping of the distance function of a metric space. (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)

Theoremxmsxmet2 22074 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))

Theoremmsmet2 22075 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))

Theoremmscl 22076 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 ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ)

Theoremxmscl 22077 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 ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)

Theoremxmsge0 22078 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 ∧ 𝐴𝑋𝐵𝑋) → 0 ≤ (𝐴𝐷𝐵))

Theoremxmseq0 22079 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 ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))

Theoremxmssym 22080 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 ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))

Theoremxmstri2 22081 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 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))

Theoremmstri2 22082 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 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵)))

Theoremxmstri 22083 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵)))

Theoremmstri 22084 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐶𝐷𝐵)))

Theoremxmstri3 22085 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐵𝐷𝐶)))

Theoremmstri3 22086 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶)))

Theoremmsrtri 22087 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))

Theoremxmspropd 22088 Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))

Theoremmspropd 22089 Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))

Theoremsetsmsbas 22090 The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))       (𝜑𝑋 = (Base‘𝐾))

Theoremsetsmsds 22091 The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))       (𝜑 → (dist‘𝑀) = (dist‘𝐾))

Theoremsetsmstset 22092 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)       (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾))

Theoremsetsmstopn 22093 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)       (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾))

Theoremsetsxms 22094 The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)       (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ (∞Met‘𝑋)))

Theoremsetsms 22095 The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)       (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋)))

Theoremtmsval 22096 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)⟩))

Theoremtmslem 22097 Lemma for tmsbas 22098, tmsds 22099, and tmstopn 22100. (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)))

Theoremtmsbas 22098 The base set of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐾 = (toMetSp‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾))

Theoremtmsds 22099 The metric of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐾 = (toMetSp‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾))

Theoremtmstopn 22100 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))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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