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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremprdsxmet 18401* The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 18400. (Contributed by Mario Carneiro, 26-Sep-2015.)
s

Theoremprdsmet 18402* The product metric is a metric when the index set is finite. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremressprdsds 18403* Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
s        s s

Theoremresspwsds 18404 Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
s        s s

Theoremimasdsf1olem 18405* Lemma for imasdsf1o 18406. (Contributed by Mario Carneiro, 21-Aug-2015.)
s                                                                s               g

Theoremimasdsf1o 18406 The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremimasf1oxmet 18407 The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremimasf1omet 18408 The image of a metric is a metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
s

Theoremxpsdsfn 18409 Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremxpsdsfn2 18410 Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremxpsxmetlem 18411* Lemma for xpsxmet 18412. (Contributed by Mario Carneiro, 21-Aug-2015.)
s                                                                       Scalars

Theoremxpsxmet 18412 A product metric of extended metrics is an extended metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
s

Theoremxpsdsval 18413 Value of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremxpsmet 18414 The direct product of two metric spaces. Definition 14-1.5 of [Gleason] p. 225. (Contributed by NM, 20-Jun-2007.) (Revised by Mario Carneiro, 20-Aug-2015.)
s

11.4.3  Metric space balls

Theoremblfvalps 18415* The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

Theoremblfval 18416* The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Proof shortened by Thierry Arnoux, 11-Feb-2018.)

Theoremblvalps 18417* The ball around a point is the set of all points whose distance from is less than the ball's radius . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremblval 18418* The ball around a point is the set of all points whose distance from is less than the ball's radius . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremelblps 18419 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremelbl 18420 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremelbl2ps 18421 Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremelbl2 18422 Membership in a ball. (Contributed by NM, 9-Mar-2007.)

Theoremelbl3ps 18423 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
PsMet

Theoremelbl3 18424 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)

Theoremblcomps 18425 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremblcom 18426 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)

Theoremxblpnfps 18427 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.)
PsMet

Theoremxblpnf 18428 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.)

Theoremblpnf 18429 The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theorembldisj 18430 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.)

Theoremblgt0 18431 A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theorembl2in 18432 Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremxblss2ps 18433 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 18436 for extended metrics, we have to assume the balls are a finite distance apart, or else will not even be in the infinity ball around . (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremxblss2 18434 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 18436 for extended metrics, we have to assume the balls are a finite distance apart, or else will not even be in the infinity ball around . (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremblss2ps 18435 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.)
PsMet

Theoremblss2 18436 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.)

Theoremblhalf 18437 A ball of radius is contained in a ball of radius centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.)

Theoremblfps 18438 Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremblf 18439 Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremblrnps 18440* Membership in the range of the ball function. Note that is the collection of all balls for metric . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremblrn 18441* Membership in the range of the ball function. Note that is the collection of all balls for metric . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremxblcntrps 18442 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.)
PsMet

Theoremxblcntr 18443 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblcntrps 18444 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.)
PsMet

Theoremblcntr 18445 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremxbln0 18446 A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theorembln0 18447 A ball is not empty. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblelrnps 18448 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.)
PsMet

Theoremblelrn 18449 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.)

Theoremblssm 18450 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.)

Theoremunirnblps 18451 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.)
PsMet

Theoremunirnbl 18452 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.)

Theoremblin 18453 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.)

Theoremssblps 18454 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.)
PsMet

Theoremssbl 18455 The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremblssps 18456* Any point in a ball can be centered in another ball that is a subset of . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremblss 18457* Any point in a ball can be centered in another ball that is a subset of . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremblssexps 18458* 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.)
PsMet

Theoremblssex 18459* Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremssblex 18460* 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.)

Theoremblin2 18461* 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.)

Theoremblbas 18462 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.)

Theoremblres 18463 A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)

Theoremxmeterval 18464 Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxmeter 18465 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxmetec 18466 The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 6954, 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.)

Theoremblssec 18467 A ball centered at is contained in the set of points finitely separated from . This is just an application of ssbl 18455 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremblpnfctr 18468 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.)

Theoremxmetresbl 18469 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 18466, 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.)

11.4.4  Open sets of a metric space

Theoremmopnval 18470 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 is the family of all open sets in the metric space determined by the metric . By mopntop 18472, the open sets of a metric space form a topology , whose base set is by mopnuni 18473. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremmopntopon 18471 The set of open sets of a metric space is a topology on . 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.)
TopOn

Theoremmopntop 18472 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.)

Theoremmopnuni 18473 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.)

Theoremelmopn 18474* The defining property of an open set of a metric space. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremmopnfss 18475 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.)

Theoremmopnm 18476 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.)

Theoremelmopn2 18477* A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremmopnss 18478 An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.)

Theoremisxms 18479 Express the predicate " is an extended metric space" with underlying set and distance function . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremisxms2 18480 Express the predicate " is an extended metric space" with underlying set and distance function . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremisms 18481 Express the predicate " is a metric space" with underlying set and distance function . (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremisms2 18482 Express the predicate " is a metric space" with underlying set and distance function . (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremxmstopn 18483 The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmstopn 18484 The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremxmstps 18485 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmsxms 18486 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmstps 18487 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremxmsxmet 18488 The distance function, suitably truncated, is a metric on . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremmsmet 18489 The distance function, suitably truncated, is a metric on . (Contributed by Mario Carneiro, 12-Nov-2013.)

Theoremmsf 18490 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremxmsxmet2 18491 The distance function, suitably truncated, is a metric on . (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremmsmet2 18492 The distance function, suitably truncated, is a metric on . (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremmscl 18493 Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)

Theoremxmscl 18494 Closure of the distance function of an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremxmsge0 18495 The distance function in an extended metric space is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremxmseq0 18496 The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremxmssym 18497 The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremxmstri2 18498 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremmstri2 18499 Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremxmstri 18500 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.)

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