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Theorem List for Metamath Proof Explorer - 17701-17800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtlmtmd 17701 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod TopMnd

Theoremtlmtps 17702 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmlmod 17703 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmtrg 17704 The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremtlmscatps 17705 The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremistvc 17706 A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod TopDRing

Theoremtvctdrg 17707 The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopDRing

Theoremcnmpt1vsca 17708* Continuity of scalar multiplication; analogue of cnmpt12f 17192 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn

Theoremcnmpt2vsca 17709* Continuity of scalar multiplication; analogue of cnmpt22f 17201 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn       TopOn

Theoremtlmtgp 17710 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvctlm 17711 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvclmod 17712 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtvclvec 17713 A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)

11.3  Metric spaces

11.3.1  Basic metric space properties

Syntaxcxme 17714 Extend class notation with the class of all extended metric spaces.

Syntaxcmt 17715 Extend class notation with the class of all metric spaces.

Syntaxctmt 17716 Extend class notation with the function mapping a metric to a metric space.
toMetSp

Definitiondf-xms 17717 Define the (proper) class of all extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Definitiondf-ms 17718 Define the (proper) class of all metric spaces. (Contributed by NM, 27-Aug-2006.)

Definitiondf-tms 17719 Define the function mapping a metric to a metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp sSet TopSet

Theoremismet 17720* Express the predicate " is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremisxmet 17721* Express the predicate " is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremismeti 17722* Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremisxmetd 17723* Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremisxmet2d 17724* 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 17725* Lemma for metf 17727 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetf 17726 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetf 17727 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)

Theoremxmetcl 17728 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)

Theoremmetcl 17729 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)

Theoremismet2 17730 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 17731 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetdmdm 17732 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremmetdmdm 17733 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxmetunirn 17734 Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremxmeteq0 17735 The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmeteq0 17736 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 17737 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmettri2 17738 Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremxmet0 17739 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 17740 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 17741 The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetge0 17742 The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetlecl 17743 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetsym 17744 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmettpos 17745 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
tpos

Theoremmetsym 17746 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 17747 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 17748 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 17749 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmettri3 17750 Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)

Theoremxmetrtri 17751 One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremxmetrtri2 17752 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 16247 defined on the extended real structure. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremmetrtri 17753 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.)

Theoremxmetgt0 17754 The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetgt0 17755 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 17756 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 17757 Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetreslem 17758 Lemma for metres 17761. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremmetres2 17759 Lemma for metres 17761. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)

Theoremxmetres 17760 A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremmetres 17761 A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theorem0met 17762 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremprdsdsf 17763* 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 17764* The product metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremprdsxmet 17765* The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 17764. (Contributed by Mario Carneiro, 26-Sep-2015.)
s

Theoremprdsmet 17766* The product metric is a metric when the index set is finite. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremressprdsds 17767* Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
s        s s

Theoremresspwsds 17768 Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
s        s s

Theoremimasdsf1olem 17769* Lemma for imasdsf1o 17770. (Contributed by Mario Carneiro, 21-Aug-2015.)
s                                                                s               g

Theoremimasdsf1o 17770 The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremimasf1oxmet 17771 The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremimasf1omet 17772 The image of a metric is a metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
s

Theoremxpsdsfn 17773 Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremxpsdsfn2 17774 Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremxpsxmetlem 17775* Lemma for xpsxmet 17776. (Contributed by Mario Carneiro, 21-Aug-2015.)
s                                                                       Scalars

Theoremxpsxmet 17776 A product metric of extended metrics is an extended metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
s

Theoremxpsdsval 17777 Value of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremxpsmet 17778 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

11.3.2  Metric space balls

Theoremblfval 17779* The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremblval 17780* 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.)

Theoremelbl 17781 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremelbl2 17782 Membership in a ball. (Contributed by NM, 9-Mar-2007.)

Theoremelbl3 17783 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)

Theoremblcom 17784 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)

Theoremxblpnf 17785 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 17786 The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theorembldisj 17787 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 17788 A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theorembl2in 17789 Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremxblss2 17790 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 17791 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.)

Theoremblss2 17791 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 17792 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.)

Theoremblf 17793 Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremblrn 17794* 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.)

Theoremxblcntr 17795 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblcntr 17796 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremxbln0 17797 A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theorembln0 17798 A ball is not empty. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblelrn 17799 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 17800 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.)

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