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Theorem List for Metamath Proof Explorer - 18501-18600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremstdbdmopn 18501* The standard bounded metric corresponding to generates the same topology as . (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmopnex 18502* The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmethaus 18503 The topology generated by a metric space is Hausdorff. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)

Theoremmet1stc 18504 The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremmet2ndci 18505 A separable metric space (a metric space with a countable dense subset) is second-countable. (Contributed by Mario Carneiro, 13-Apr-2015.)

Theoremmet2ndc 18506* A metric space is second-countable iff it is separable (has a countable dense subset). (Contributed by Mario Carneiro, 13-Apr-2015.)

Theoremmetrest 18507 Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
t

Theoremressxms 18508 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
s

Theoremressms 18509 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
s

Theoremprdsmslem1 18510 Lemma for prdsms 18514. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsxmslem1 18511 Lemma for prdsms 18514. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsxmslem2 18512* Lemma for prdsxms 18513. The topology generated by the supremum metric is the same as the product topology, when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsxms 18513 The indexed product structure is an extended metric space when the index set is finite. (Although the extended metric is still valid when the index set is infinite, it no longer agrees with the product topology, which is not metrizable in any case.) (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsms 18514 The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theorempwsxms 18515 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theorempwsms 18516 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremxpsxms 18517 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremxpsms 18518 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremtmsxps 18519 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsmopn 18520 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsval 18521 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsval2 18522 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

11.4.5  Continuity in metric spaces

Theoremmetcnp3 18523* Two ways to express that is continuous at for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Theoremmetcnp 18524* Two ways to say a mapping from metric to metric is continuous at point . (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Theoremmetcnp2 18525* Two ways to say a mapping from metric to metric is continuous at point . The distance arguments are swapped compared to metcnp 18524 (and Munkres' metcn 18526) for compatibility with df-lm 17247. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcn 18526* Two ways to say a mapping from metric to metric is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon" there is a positive "delta" such that a distance less than delta in maps to a distance less than epsilon in . (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Theoremmetcnpi 18527* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 18524. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpi2 18528* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 18525. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpi3 18529* Epsilon-delta property of a metric space function continuous at . A variation of metcnpi2 18528 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremtxmetcnp 18530* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxmetcn 18531* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

11.4.6  The uniform structure generated by a metric

TheoremmetuvalOLD 18532* Value of the uniform structure generated by metric . (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
metUnifOLD

Theoremmetuval 18533* Value of the uniform structure generated by metric . (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet metUnif

TheoremmetustelOLD 18534* Define a filter base generated by a metric . (Contributed by Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustel 18535* Define a filter base generated by a metric . (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustssOLD 18536* Range of the elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustss 18537* Range of the elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustrelOLD 18538* Elements of the filter base generated by the metric are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustrel 18539* Elements of the filter base generated by the metric are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetusttoOLD 18540* Any two elements of the filter base generated by the metric can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustto 18541* Any two elements of the filter base generated by the metric can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustidOLD 18542* The identity diagonal is included in all elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustid 18543* The identity diagonal is included in all elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustsymOLD 18544* Elements of the filter base generated by the metric are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustsym 18545* Elements of the filter base generated by the metric are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustexhalfOLD 18546* For any element of the filter base generated by the metric , the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustexhalf 18547* For any element of the filter base generated by the metric , the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustfbasOLD 18548* The filter base generated by a metric . (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustfbas 18549* The filter base generated by a metric . (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustOLD 18550* The uniform structure generated by a metric (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
UnifOn

Theoremmetust 18551* The uniform structure generated by a metric (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet UnifOn

TheoremcfilucfilOLD 18552* Given a metric and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 19171. (Contributed by Thierry Arnoux, 29-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
CauFilu

Theoremcfilucfil 18553* Given a metric and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 19171. (Contributed by Thierry Arnoux, 29-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet CauFilu

TheoremmetuustOLD 18554 The uniform structure generated by metric is a uniform structure. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
metUnifOLD UnifOn

Theoremmetuust 18555 The uniform structure generated by metric is a uniform structure. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet metUnif UnifOn

Theoremcfilucfil2OLD 18556* Given a metric and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 19171. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
CauFilumetUnifOLD

Theoremcfilucfil2 18557* Given a metric and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 19171. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet CauFilumetUnif

Theoremblval2 18558 The ball around a point , alternative definition. (Contributed by Thierry Arnoux, 7-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremelbl4 18559 Membership in a ball, alternative definition. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

TheoremmetuelOLD 18560* Elementhood in the uniform structure generated by a metric (Contributed by Thierry Arnoux, 8-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
metUnifOLD

Theoremmetuel 18561* Elementhood in the uniform structure generated by a metric (Contributed by Thierry Arnoux, 8-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet metUnif

Theoremmetuel2 18562* Elementhood in the uniform structure generated by a metric (Contributed by Thierry Arnoux, 24-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
metUnif       PsMet

TheoremmetustblOLD 18563* The "section" image of an entourage at a point always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
metUnifOLD

Theoremmetustbl 18564* The "section" image of an entourage at a point always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.)
PsMet metUnif

TheoremmetutopOLD 18565 The topology induced by a uniform structure generated by a metric is that metric's open sets. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
unifTopmetUnifOLD

Theorempsmetutop 18566 The topology induced by a uniform structure generated by a metric is generated by that metric's open balls. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet unifTopmetUnif

Theoremxmetutop 18567 The topology induced by a uniform structure generated by an extended metric is that metric's open sets. (Contributed by Thierry Arnoux, 11-Mar-2018.)
unifTopmetUnif

TheoremxmsuspOLD 18568 If the uniform set of a metric space is the uniform structure generated by its metric, then it is a uniform space. (Contributed by Thierry Arnoux, 14-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
UnifSt       metUnifOLD UnifSp

Theoremxmsusp 18569 If the uniform set of a metric space is the uniform structure generated by its metric, then it is a uniform space. (Contributed by Thierry Arnoux, 14-Dec-2017.)
UnifSt       metUnif UnifSp

Theoremrestmetu 18570 The uniform structure generated by the restriction of a metric is its trace. (Contributed by Thierry Arnoux, 18-Dec-2017.)
PsMet metUnift metUnif

TheoremmetucnOLD 18571* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 18526. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
metUnifOLD       metUnifOLD                                   Cnu

Theoremmetucn 18572* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 18526. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
metUnif       metUnif                     PsMet       PsMet       Cnu

11.4.7  Examples of metric spaces

Theoremdscmet 18573* The discrete metric on any set . Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)

Theoremdscopn 18574* The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.)

Theoremnrmmetd 18575* Show that a group norm generates a metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremabvmet 18576 An absolute value generates a metric defined by , analogously to cnmet 18759. (In fact, the ring structure is not needed at all; the group properties abveq0 15869 and abvtri 15873, abvneg 15877 are sufficient.) (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
AbsVal

11.4.8  Normed algebraic structures

Syntaxcnm 18577 Norm of a normed ring.

Syntaxcngp 18578 The class of all normed groups.
NrmGrp

Syntaxctng 18579 Make a normed group from a norm and a group.
toNrmGrp

Syntaxcnrg 18580 Normed ring.
NrmRing

Syntaxcnlm 18581 Normed module.
NrmMod

Syntaxcnvc 18582 Normed vector space.
NrmVec

Definitiondf-nm 18583* Define the norm on a group or ring (when it makes sense) in terms of the distance to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)

Definitiondf-ngp 18584 Define a normed group, which is a group with a right-translation-invariant metric. This is not a standard notion, but is helpful as the most general context in which a metric-like norm makes sense. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Definitiondf-tng 18585* Define a function that fills in the topology and metric components of a structure given a group and a norm on it. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp sSet sSet TopSet

Definitiondf-nrg 18586 A normed ring is a ring with an induced topology and metric such that the metric is translation-invariant and the norm (distance from 0) is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing NrmGrp AbsVal

Definitiondf-nlm 18587* A normed (left) module is a module which is also a normed group over a normed ring, such that the norm distributes over scalar multiplication. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmMod NrmGrp Scalar NrmRing

Definitiondf-nvc 18588 A normed vector space is a normed module which is also a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec NrmMod

Theoremnmfval 18589* The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremnmval 18590 The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremnmfval2 18591* The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremnmval2 18592 The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremnmf2 18593 The norm is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremnmpropd 18594 Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremnmpropd2 18595* Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremisngp 18596 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremisngp2 18597 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremisngp3 18598* The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremngpgrp 18599 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngpms 18600 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

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