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Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcmt 18301 Extend class notation with the class of all metric spaces.

Syntaxctmt 18302 Extend class notation with the function mapping a metric to a metric space.
toMetSp

Definitiondf-xms 18303 Define the (proper) class of all extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Definitiondf-ms 18304 Define the (proper) class of all metric spaces. (Contributed by NM, 27-Aug-2006.)

Definitiondf-tms 18305 Define the function mapping a metric to a metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp sSet TopSet

Theoremismet 18306* Express the predicate " is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremisxmet 18307* Express the predicate " is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremismeti 18308* Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremisxmetd 18309* Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremisxmet2d 18310* It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample: satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetflem 18311* Lemma for metf 18313 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetf 18312 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetf 18313 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)

Theoremxmetcl 18314 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)

Theoremmetcl 18315 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)

Theoremismet2 18316 An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetxmet 18317 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetdmdm 18318 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremmetdmdm 18319 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxmetunirn 18320 Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremxmeteq0 18321 The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmeteq0 18322 The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4. (Contributed by NM, 30-Aug-2006.)

Theoremxmettri2 18323 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmettri2 18324 Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremxmet0 18325 The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmet0 18326 The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by NM, 30-Aug-2006.)

Theoremxmetge0 18327 The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetge0 18328 The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetlecl 18329 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetsym 18330 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetpsmet 18331 An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theoremxmettpos 18332 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
tpos

Theoremmetsym 18333 The distance function of a metric space is symmetric. Definition 14-1.1(c) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremxmettri 18334 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmettri 18335 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.)

Theoremxmettri3 18336 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmettri3 18337 Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)

Theoremxmetrtri 18338 One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremxmetrtri2 18339 The reverse triangle inequality for the distance function of an extended metric. In order to express the "extended absolute value function", we use the distance function xrsdsval 16697 defined on the extended real structure. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremmetrtri 18340 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.)

Theoremxmetgt0 18341 The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetgt0 18342 The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of [Gleason] p. 223 and its converse. (Contributed by NM, 27-Aug-2006.)

Theoremmetn0 18343 A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetres2 18344 Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetreslem 18345 Lemma for metres 18348. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremmetres2 18346 Lemma for metres 18348. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)

Theoremxmetres 18347 A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremmetres 18348 A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theorem0met 18349 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremprdsdsf 18350* The product metric is a function into the nonnegative extended reals. In general this means that it is not a metric but rather an *extended* metric (even when all the factors are metrics), but it will be a metric when restricted to regions where it does not take infinite values. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremprdsxmetlem 18351* The product metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremprdsxmet 18352* The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 18351. (Contributed by Mario Carneiro, 26-Sep-2015.)
s

Theoremprdsmet 18353* The product metric is a metric when the index set is finite. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremressprdsds 18354* Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
s        s s

Theoremresspwsds 18355 Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
s        s s

Theoremimasdsf1olem 18356* Lemma for imasdsf1o 18357. (Contributed by Mario Carneiro, 21-Aug-2015.)
s                                                                s               g

Theoremimasdsf1o 18357 The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremimasf1oxmet 18358 The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremimasf1omet 18359 The image of a metric is a metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
s

Theoremxpsdsfn 18360 Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremxpsdsfn2 18361 Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremxpsxmetlem 18362* Lemma for xpsxmet 18363. (Contributed by Mario Carneiro, 21-Aug-2015.)
s                                                                       Scalars

Theoremxpsxmet 18363 A product metric of extended metrics is an extended metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
s

Theoremxpsdsval 18364 Value of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremxpsmet 18365 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 18366* 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 18367* 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 18368* 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 18369* 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 18370 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 18371 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremelbl2ps 18372 Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremelbl2 18373 Membership in a ball. (Contributed by NM, 9-Mar-2007.)

Theoremelbl3ps 18374 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
PsMet

Theoremelbl3 18375 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)

Theoremblcomps 18376 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremblcom 18377 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)

Theoremxblpnfps 18378 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 18379 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 18380 The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theorembldisj 18381 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 18382 A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theorembl2in 18383 Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremxblss2ps 18384 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 18387 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 18385 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 18387 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 18386 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 18387 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 18388 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 18389 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 18390 Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremblrnps 18391* 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 18392* 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 18393 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 18394 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblcntrps 18395 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 18396 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremxbln0 18397 A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theorembln0 18398 A ball is not empty. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblelrnps 18399 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 18400 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.)

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