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Theorem List for Intuitionistic Logic Explorer - 12701-12800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremismet2 12701 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 12702 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetdmdm 12703 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremmetdmdm 12704 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxmetunirn 12705 Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremxmeteq0 12706 The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmeteq0 12707 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 12708 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmettri2 12709 Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremxmet0 12710 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 12711 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 12712 The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetge0 12713 The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetlecl 12714 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetsym 12715 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetpsmet 12716 An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theoremxmettpos 12717 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
tpos

Theoremmetsym 12718 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 12719 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 12720 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 12721 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmettri3 12722 Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)

Theoremxmetrtri 12723 One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremmetrtri 12724 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 21-Apr-2023.)

Theoremmetn0 12725 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 12726 Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetreslem 12727 Lemma for metres 12730. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremmetres2 12728 Lemma for metres 12730. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)

Theoremxmetres 12729 A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremmetres 12730 A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theorem0met 12731 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

7.2.3  Metric space balls

Theoremblfvalps 12732* 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 12733* 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.)

Theoremblex 12734 A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.)

Theoremblvalps 12735* 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 12736* 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 12737 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 12738 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremelbl2ps 12739 Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremelbl2 12740 Membership in a ball. (Contributed by NM, 9-Mar-2007.)

Theoremelbl3ps 12741 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
PsMet

Theoremelbl3 12742 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)

Theoremblcomps 12743 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremblcom 12744 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)

Theoremxblpnfps 12745 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 12746 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 12747 The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theorembldisj 12748 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 12749 A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theorembl2in 12750 Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremxblss2ps 12751 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 12754 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 12752 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 12754 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 12753 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 12754 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 12755 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 12756 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 12757 Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremblrnps 12758* 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 12759* 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 12760 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 12761 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblcntrps 12762 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 12763 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremxblm 12764* A ball is inhabited iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theorembln0 12765 A ball is not empty. It is also inhabited, as seen at blcntr 12763. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblelrnps 12766 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 12767 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 12768 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 12769 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 12770 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.)

Theoremblininf 12771 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.)
inf

Theoremssblps 12772 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 12773 The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremblssps 12774* 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 12775* 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 12776* 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 12777* Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremssblex 12778* 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 12779* 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 12780 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 12781 A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)

Theoremxmeterval 12782 Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxmeter 12783 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxmetec 12784 The equivalence classes under the finite separation equivalence relation are infinity balls. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremblssec 12785 A ball centered at is contained in the set of points finitely separated from . This is just an application of ssbl 12773 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremblpnfctr 12786 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 12787 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 12784, 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.)

7.2.4  Open sets of a metric space

Theoremmopnrel 12788 The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)

Theoremmopnval 12789 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 12791, the open sets of a metric space form a topology , whose base set is by mopnuni 12792. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremmopntopon 12790 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 12791 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 12792 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 12793* 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 12794 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 12795 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 12796* 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 12797 An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.)

Theoremisxms 12798 Express the predicate " is an extended metric space" with underlying set and distance function . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremisxms2 12799 Express the predicate " is an extended metric space" with underlying set and distance function . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremisms 12800 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.)

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