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Theorem List for Metamath Proof Explorer - 26401-26500   *Has distinct variable group(s)
TypeLabelDescription
Statement

19.13.8  Directed sets, nets

Theoremtailfval 26401* The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremtailval 26402 The tail of an element in a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremeltail 26403 An element of a tail. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremtailf 26404 The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremtailini 26405 A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.)

Theoremtailfb 26406 The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremfilnetlem1 26407* Lemma for filnet 26411. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremfilnetlem2 26408* Lemma for filnet 26411. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremfilnetlem3 26409* Lemma for filnet 26411. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremfilnetlem4 26410* Lemma for filnet 26411. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremfilnet 26411* A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

19.14.1  Logic and set theory

Theoremanim12da 26412 Conjoin antecedents and consequents in a deduction. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremsyldanl 26413 A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremunirep 26414* Define a quantity whose definition involves a choice of representative, but which is uniquely determined regardless of the choice. (Contributed by Jeff Madsen, 1-Jun-2011.)

Theoremcover2 26415* Two ways of expressing the statement "there is a cover of by elements of such that for each set in the cover, ." Note that and must be distinct. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremcover2g 26416* Two ways of expressing the statement "there is a cover of by elements of such that for each set in the cover, ." Note that and must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.)

Theorembrabg2 26417* Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremopelopab3 26418* Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)

Theoremcocanfo 26419 Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theorembrresi 26420 Restriction of a binary relation. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfnopabeqd 26421* Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremfvopabf4g 26422* Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremeqfnun 26423 Two functions on are equal if and only if they have equal restrictions to both and . (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremfnopabco 26424* Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremopropabco 26425* Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)

Theoremf1opr 26426* Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremcocnv 26427 Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremf1ocan1fv 26428 Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremf1ocan2fv 26429 Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminixp 26430* Intersection of Cartesian products over the same base set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremupixp 26431* Universal property of the indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremabrexdom 26432* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremabrexdom2 26433* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremac6gf 26434* Axiom of Choice. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremindexa 26435* If for every element of an indexing set there exists a corresponding element of another set , then there exists a subset of consisting only of those elements which are indexed by . Used to avoid the Axiom of Choice in situations where only the range of the choice function is needed. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremindexdom 26436* If for every element of an indexing set there exists a corresponding element of another set , then there exists a subset of consisting only of those elements which are indexed by , and which is dominated by the set . (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfrinfm 26437* A subset of a well-founded set has an infimum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremwelb 26438* A non-empty subset of a well-ordered set has a lower bound. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsupex2g 26439 Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsupclt 26440* Closure of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsupubt 26441* Upper bound property of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)

19.14.2  Real and complex numbers; integers

Theoremfilbcmb 26442* Combine a finite set of lower bounds. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremrdr 26443 Two ways of expressing the remainder when is divided by . (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremfzmul 26444 Membership of a product in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremfzadd2 26445 Membership of a sum in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)

19.14.3  Sequences and sums

Theoremsdclem2 26446* Lemma for sdc 26448. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsdclem1 26447* Lemma for sdc 26448. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsdc 26448* Strong dependent choice. Suppose we may choose an element of such that property holds, and suppose that if we have already chosen the first elements (represented here by a function from to ), we may choose another element so that all elements taken together have property . Then there exists an infinite sequence of elements of such that the first terms of this sequence satisfy for all . This theorem allows us to construct infinite seqeunces where each term depends on all the previous terms in the sequence. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.)

Theoremfdc 26449* Finite version of dependent choice. Construct a function whose value depends on the previous function value, except at a final point at which no new value can be chosen. The final hypothesis ensures that the process will terminate. The proof does not use the Axiom of Choice. (Contributed by Jeff Madsen, 18-Jun-2010.)

Theoremfdc1 26450* Variant of fdc 26449 with no specified base value. (Contributed by Jeff Madsen, 18-Jun-2010.)

Theoremseqpo 26451* Two ways to say that a sequence respects a partial order. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremincsequz 26452* An increasing sequence of natural numbers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremincsequz2 26453* An increasing sequence of natural numbers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnnubfi 26454* A bounded above set of natural numbers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)

Theoremnninfnub 26455* An infinite set of natural numbers is unbounded above. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)

Theoremcsbrn 26456* Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremtrirn 26457* Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)

19.14.4  Topology

Theoremsubspopn 26458 An open set is open in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremneificl 26459 Neighborhoods are closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Nov-2013.)

Theoremlpss2 26460 Limit points of a subset are limit points of the larger set. (Contributed by Jeff Madsen, 2-Sep-2009.)

19.14.5  Metric spaces

Theoremmetf1o 26461* Use a bijection with a metric space to construct a metric on a set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremblssp 26462 A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.)

Theoremmettrifi 26463* Generalized triangle inequality for arbitrary finite sums. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremlmclim2 26464* A sequence in a metric space converges to a point iff the distance between the point and the elements of the sequence converges to 0. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)

Theoremgeomcau 26465* If the distance between consecutive points in a sequence is bounded by a geometric sequence, then the sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)

Theoremcaures 26466 The restriction of a Cauchy sequence to a set of upper integers is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremcaushft 26467* A shifted Cauchy sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)

19.14.6  Continuous maps and homeomorphisms

Theoremconstcncf 26468* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved into main set.mm as cncfmptc 18941 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremidcncf 26469 The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 18942 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremsub1cncf 26470* Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremsub2cncf 26471* Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremcnres2 26472* The restriction of a continuous function to a subset is continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
t t

Theoremcnresima 26473 A continuous function is continuous onto its image. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremcncfres 26474* A continuous function on complex numbers restricted to a subset. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

19.14.7  Boundedness

Syntaxctotbnd 26475 Extend class notation with the class of totally bounded metric spaces.

Syntaxcbnd 26476 Extend class notation with the class of bounded metric spaces.

Definitiondf-totbnd 26477* Define the class of totally bounded metrics. A metric space is totally bounded iff it can be covered by a finite number of balls of any given radius. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremistotbnd 26478* The predicate "is a totally bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremistotbnd2 26479* The predicate "is a totally bounded metric space." (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremistotbnd3 26480* A metric space is totally bounded iff there is a finite ε-net for every positive ε. This differs from the definition in providing a finite set of ball centers rather than a finite set of balls. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremtotbndmet 26481 The predicate "totally bounded" implies is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorem0totbnd 26482 The metric (there is only one) on the empty set is totally bounded. (Contributed by Mario Carneiro, 16-Sep-2015.)

Theoremsstotbnd2 26483* Condition for a subset of a metric space to be totally bounded. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremsstotbnd 26484* Condition for a subset of a metric space to be totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremsstotbnd3 26485* Use a net that is not necessarily finite, but for which only finitely many balls meet the subset. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremtotbndss 26486 A subset of a totally bounded metric space is totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremequivtotbnd 26487* If the metric is "strongly finer" than (meaning that there is a positive real constant such that ), then total boundedness of implies total boundedness of . (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then one is totally bounded iff the other is.) (Contributed by Mario Carneiro, 14-Sep-2015.)

Definitiondf-bnd 26488* Define the class of bounded metrics. A metric space is bounded iff it can be covered by a single ball. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremisbnd 26489* The predicate "is a bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theorembndmet 26490 A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)

Theoremisbndx 26491* A "bounded extended metric" (meaning that it satisfies the same condition as a bounded metric, but with "metric" replaced with "extended metric") is a metric and thus is bounded in the conventional sense. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremisbnd2 26492* The predicate "is a bounded metric space". Uses a single point instead of an arbitrary point in the space. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremisbnd3 26493* A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 13-Sep-2015.)

Theoremisbnd3b 26494* A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 22-Sep-2015.)

Theorembndss 26495 A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremblbnd 26496 A ball is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 15-Jan-2014.)

Theoremssbnd 26497* A subset of a metric space is bounded iff it is contained in a ball around , for any in the larger space. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremtotbndbnd 26498 A totally bounded metric space is bounded. This theorem fails for extended metrics - a bounded extended metric is a metric, but there are totally bounded extended metrics that are not metrics (if we were to weaken istotbnd 26478 to only require that be an extended metric). A counterexample is the discrete extended metric (assigning distinct points distance ) on a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremequivbnd 26499* If the metric is "strongly finer" than (meaning that there is a positive real constant such that ), then boundedness of implies boundedness of . (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then one is bounded iff the other is.) (Contributed by Mario Carneiro, 14-Sep-2015.)

Theorembnd2lem 26500 Lemma for equivbnd2 26501 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.)

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