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Theorem List for Intuitionistic Logic Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxmeteq0 12601 The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmeteq0 12602 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 12603 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmettri2 12604 Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremxmet0 12605 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 12606 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 12607 The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetge0 12608 The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetlecl 12609 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetsym 12610 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetpsmet 12611 An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theoremxmettpos 12612 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
tpos

Theoremmetsym 12613 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 12614 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 12615 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 12616 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmettri3 12617 Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)

Theoremxmetrtri 12618 One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremmetrtri 12619 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 12620 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 12621 Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetreslem 12622 Lemma for metres 12625. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremmetres2 12623 Lemma for metres 12625. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)

Theoremxmetres 12624 A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremmetres 12625 A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theorem0met 12626 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

7.2.3  Metric space balls

Theoremblfvalps 12627* 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 12628* 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 12629 A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.)

Theoremblvalps 12630* 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 12631* 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 12632 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 12633 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremelbl2ps 12634 Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremelbl2 12635 Membership in a ball. (Contributed by NM, 9-Mar-2007.)

Theoremelbl3ps 12636 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
PsMet

Theoremelbl3 12637 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)

Theoremblcomps 12638 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremblcom 12639 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)

Theoremxblpnfps 12640 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 12641 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 12642 The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theorembldisj 12643 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 12644 A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theorembl2in 12645 Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremxblss2ps 12646 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 12649 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 12647 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 12649 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 12648 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 12649 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 12650 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 12651 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 12652 Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremblrnps 12653* 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 12654* 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 12655 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 12656 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblcntrps 12657 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 12658 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremxblm 12659* A ball is inhabited iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theorembln0 12660 A ball is not empty. It is also inhabited, as seen at blcntr 12658. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblelrnps 12661 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 12662 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 12663 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 12664 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 12665 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 12666 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 12667 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 12668 The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremblssps 12669* 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 12670* 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 12671* 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 12672* Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremssblex 12673* 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 12674* 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 12675 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 12676 A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)

Theoremxmeterval 12677 Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxmeter 12678 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxmetec 12679 The equivalence classes under the finite separation equivalence relation are infinity balls. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremblssec 12680 A ball centered at is contained in the set of points finitely separated from . This is just an application of ssbl 12668 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremblpnfctr 12681 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 12682 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 12679, 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 12683 The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)

Theoremmopnval 12684 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 12686, the open sets of a metric space form a topology , whose base set is by mopnuni 12687. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremmopntopon 12685 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 12686 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 12687 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 12688* 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 12689 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 12690 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 12691* 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 12692 An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.)

Theoremisxms 12693 Express the predicate " is an extended metric space" with underlying set and distance function . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremisxms2 12694 Express the predicate " is an extended metric space" with underlying set and distance function . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremisms 12695 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 12696 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 12697 The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmstopn 12698 The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremxmstps 12699 An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmsxms 12700 A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)

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