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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmet2ndc 17901* A metric space is second-countable iff it is separable (has a countable dense subset). (Contributed by Mario Carneiro, 13-Apr-2015.)

Theoremmetrest 17902 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 17903 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
s

Theoremressms 17904 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
s

Theoremprdsmslem1 17905 Lemma for prdsms 17909. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsxmslem1 17906 Lemma for prdsms 17909. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsxmslem2 17907* Lemma for prdsxms 17908. 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 17908 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 17909 The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theorempwsxms 17910 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theorempwsms 17911 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremxpsxms 17912 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremxpsms 17913 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremtmsxps 17914 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsmopn 17915 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsval 17916 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsval2 17917 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

11.3.4  Continuity in metric spaces

Theoremmetcnp3 17918* 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 17919* 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 17920* Two ways to say a mapping from metric to metric is continuous at point . The distance arguments are swapped compared to metcnp 17919 (and Munkres' metcn 17921) for compatibility with df-lm 16791. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcn 17921* 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 17922* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 17919. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpi2 17923* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 17920. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpi3 17924* Epsilon-delta property of a metric space function continuous at . A variation of metcnpi2 17923 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremtxmetcnp 17925* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxmetcn 17926* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

11.3.5  Examples of metric spaces

Theoremdscmet 17927* The discrete metric on any set . Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)

Theoremdscopn 17928* The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.)

Theoremnrmmetd 17929* Show that a group norm generates a metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremabvmet 17930 An absolute value generates a metric defined by , analogously to cnmet 18113. (In fact, the ring structure is not needed at all; the group properties abveq0 15426 and abvtri 15430, abvneg 15434 are sufficient.) (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
AbsVal

11.3.6  Normed algebraic structures

Syntaxcnm 17931 Norm of a normed ring.

Syntaxcngp 17932 The class of all normed groups.
NrmGrp

Syntaxctng 17933 Make a normed group from a norm and a group.
toNrmGrp

Syntaxcnrg 17934 Normed ring.
NrmRing

Syntaxcnlm 17935 Normed module.
NrmMod

Syntaxcnvc 17936 Normed vector space.
NrmVec

Definitiondf-nm 17937* 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 17938 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 17939* 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 17940 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 17941* 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 17942 A normed vector space is a normed module which is also a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec NrmMod

Theoremnmfval 17943* The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremnmval 17944 The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremnmfval2 17945* The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremnmval2 17946 The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremnmf2 17947 The norm is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremnmpropd 17948 Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremnmpropd2 17949* Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremisngp 17950 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremisngp2 17951 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremisngp3 17952* The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremngpgrp 17953 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngpms 17954 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngpxms 17955 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngptps 17956 A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmGrp

Theoremngpds 17957 Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngpdsr 17958 Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngpds2 17959 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngpds2r 17960 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngpds3 17961 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngpds3r 17962 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngprcan 17963 Cancel right addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremngplcan 17964 Cancel left addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp

Theoremisngp4 17965* Express the property of being a normed group purely in terms of right-translation invariance of the metric instead of using the definition of norm (which itself uses the metric). (Contributed by Mario Carneiro, 29-Oct-2015.)
NrmGrp

Theoremngpinvds 17966 Two elements are the same distance apart as their inverses. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremngpsubcan 17967 Cancel right subtraction inside a distance calculation. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnmf 17968 The norm on a normed group is a function into the reals. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnmcl 17969 The norm of a normed group is closed in the reals. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnmge0 17970 The norm of a normed group is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnmeq0 17971 The identity is the only element of the group with zero norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnmne0 17972 The norm of a nonzero element is nonzero. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnmrpcl 17973 The norm of a nonzero element is a positive real. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnminv 17974 The norm of a negated element is the same as the norm of the original element. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnmmtri 17975 The triangle inequality for the norm of a subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnmsub 17976 The norm of the difference between two elements. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnmrtri 17977 Reverse triangle inequality for the norm of a subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnm2dif 17978 Inequality for the difference of norms. (Contributed by Mario Carneiro, 6-Oct-2015.)
NrmGrp

Theoremnmtri 17979 The triangle inequality for the norm of a sum. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnm0 17980 Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremsubgnm 17981 The norm in a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
s                      SubGrp

Theoremsubgnm2 17982 A substructure assigns the same values to the norms of elements of a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
s                      SubGrp

Theoremsubgngp 17983 A normed group restricted to a subgroup is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
s        NrmGrp SubGrp NrmGrp

Theoremngptgp 17984 A normed abelian group is a topological group (with the topology induced by the metric induced by the norm). (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremngppropd 17985* Property deduction for a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp NrmGrp

Theoremreldmtng 17986 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
toNrmGrp

Theoremtngval 17987 Value of the function which augments a given structure with a norm . (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp                             sSet sSet TopSet

Theoremtnglem 17988 Lemma for tngbas 17989 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp        Slot

Theoremtngbas 17989 The base set of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp

Theoremtngplusg 17990 The group addition of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp

Theoremtng0 17991 The group identity of a structure augmented with a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp

Theoremtngmulr 17992 The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp

Theoremtngsca 17993 The scalar ring of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp        Scalar       Scalar

Theoremtngvsca 17994 The scalar multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp

Theoremtngip 17995 The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp

Theoremtngds 17996 The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.)
toNrmGrp

Theoremtngtset 17997 The topology generated by a normed structure. (Contributed by Mario Carneiro, 3-Oct-2015.)
toNrmGrp                      TopSet

Theoremtngtopn 17998 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp

Theoremtngnm 17999 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp

Theoremtngngp2 18000 A norm turns a group into a normed group iff the generated metric is in fact a metric. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp                      NrmGrp

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