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Type | Label | Description |
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Statement | ||
Theorem | prdsms 21601 | The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.) |
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Theorem | pwsxms 21602 | The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
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Theorem | pwsms 21603 | The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
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Theorem | xpsxms 21604 | A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
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Theorem | xpsms 21605 | A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
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Theorem | tmsxps 21606 | Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.) |
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Theorem | tmsxpsmopn 21607 | Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.) |
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Theorem | tmsxpsval 21608 | Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.) |
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Theorem | tmsxpsval2 21609 | Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.) |
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Theorem | metcnp3 21610* |
Two ways to express that ![]() ![]() |
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Theorem | metcnp 21611* |
Two ways to say a mapping from metric ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metcnp2 21612* |
Two ways to say a mapping from metric ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metcn 21613* |
Two ways to say a mapping from metric ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | metcnpi 21614* | Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 21611. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.) |
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Theorem | metcnpi2 21615* | Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 21612. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metcnpi3 21616* |
Epsilon-delta property of a metric space function continuous at ![]() |
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Theorem | txmetcnp 21617* | Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) |
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Theorem | txmetcn 21618* | Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) |
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Theorem | metuval 21619* |
Value of the uniform structure generated by metric ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metustel 21620* |
Define a filter base ![]() ![]() |
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Theorem | metustss 21621* |
Range of the elements of the filter base generated by the metric
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metustrel 21622* |
Elements of the filter base generated by the metric ![]() |
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Theorem | metustto 21623* |
Any two elements of the filter base generated by the metric ![]() |
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Theorem | metustid 21624* |
The identity diagonal is included in all elements of the filter base
generated by the metric ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metustsym 21625* |
Elements of the filter base generated by the metric ![]() |
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Theorem | metustexhalf 21626* |
For any element ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metustfbas 21627* |
The filter base generated by a metric ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metust 21628* |
The uniform structure generated by a metric ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cfilucfil 21629* |
Given a metric ![]() |
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Theorem | metuust 21630 |
The uniform structure generated by metric ![]() |
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Theorem | cfilucfil2 21631* |
Given a metric ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | blval2 21632 |
The ball around a point ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | elbl4 21633 | Membership in a ball, alternative definition. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metuel 21634* |
Elementhood in the uniform structure generated by a metric ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metuel2 21635* |
Elementhood in the uniform structure generated by a metric ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | metustbl 21636* |
The "section" image of an entourage at a point ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | psmetutop 21637 |
The topology induced by a uniform structure generated by a metric ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | xmetutop 21638 |
The topology induced by a uniform structure generated by an extended
metric ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | xmsusp 21639 | 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | restmetu 21640 | The uniform structure generated by the restriction of a metric is its trace. (Contributed by Thierry Arnoux, 18-Dec-2017.) |
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Theorem | metucn 21641* | Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 21613. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dscmet 21642* |
The discrete metric on any set ![]() |
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Theorem | dscopn 21643* | The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.) |
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Theorem | nrmmetd 21644* | Show that a group norm generates a metric. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | abvmet 21645 |
An absolute value ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Syntax | cnm 21646 | Norm of a normed ring. |
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Syntax | cngp 21647 | The class of all normed groups. |
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Syntax | ctng 21648 | Make a normed group from a norm and a group. |
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Syntax | cnrg 21649 | Normed ring. |
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Syntax | cnlm 21650 | Normed module. |
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Syntax | cnvc 21651 | Normed vector space. |
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Definition | df-nm 21652* | 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.) |
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Definition | df-ngp 21653 | 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.) |
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Definition | df-tng 21654* | 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.) |
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Definition | df-nrg 21655 | 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.) |
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Definition | df-nlm 21656* | 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.) |
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Definition | df-nvc 21657 | A normed vector space is a normed module which is also a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmfval 21658* | The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | nmval 21659 | The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | nmfval2 21660* | The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | nmval2 21661 | The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | nmf2 21662 | The norm is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | nmpropd 21663 | Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmpropd2 21664* | Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | isngp 21665 | The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | isngp2 21666 | The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | isngp3 21667* | The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | ngpgrp 21668 | A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ngpms 21669 | A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ngpxms 21670 | A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ngptps 21671 | A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
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Theorem | ngpds 21672 | Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ngpdsr 21673 | Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ngpds2 21674 | Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ngpds2r 21675 | Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ngpds3 21676 | Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ngpds3r 21677 | Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ngprcan 21678 | Cancel right addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ngplcan 21679 | Cancel left addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | isngp4 21680* | 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.) |
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Theorem | ngpinvds 21681 | Two elements are the same distance apart as their inverses. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | ngpsubcan 21682 | Cancel right subtraction inside a distance calculation. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmf 21683 | The norm on a normed group is a function into the reals. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmcl 21684 | The norm of a normed group is closed in the reals. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmge0 21685 | The norm of a normed group is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmeq0 21686 | The identity is the only element of the group with zero norm. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmne0 21687 | The norm of a nonzero element is nonzero. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmrpcl 21688 | The norm of a nonzero element is a positive real. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nminv 21689 | The norm of a negated element is the same as the norm of the original element. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmmtri 21690 | The triangle inequality for the norm of a subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmsub 21691 | The norm of the difference between two elements. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nmrtri 21692 | Reverse triangle inequality for the norm of a subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nm2dif 21693 | Inequality for the difference of norms. (Contributed by Mario Carneiro, 6-Oct-2015.) |
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Theorem | nmtri 21694 | The triangle inequality for the norm of a sum. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | nm0 21695 | Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | subgnm 21696 | The norm in a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | subgnm2 21697 | A substructure assigns the same values to the norms of elements of a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | subgngp 21698 | A normed group restricted to a subgroup is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | ngptgp 21699 | 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.) |
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Theorem | ngppropd 21700* | Property deduction for a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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