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Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfdc 26301* 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 26302* Variant of fdc 26301 with no specified base value. (Contributed by Jeff Madsen, 18-Jun-2010.)

Theoremseqpo 26303* Two ways to say that a sequence respects a partial order. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremincsequz 26304* An increasing sequence of natural numbers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremincsequz2 26305* An increasing sequence of natural numbers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnnubfi 26306* A bounded above set of natural numbers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)

Theoremnninfnub 26307* An infinite set of natural numbers is unbounded above. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)

Theoremcsbrn 26308* Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremtrirn 26309* Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)

19.14.4  Topology

Theoremsubspopn 26310 An open set is open in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
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Theoremneificl 26311 Neighborhoods are closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Nov-2013.)

Theoremlpss2 26312 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 26313* Use a bijection with a metric space to construct a metric on a set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremblssp 26314 A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.)

Theoremmettrifi 26315* Generalized triangle inequality for arbitrary finite sums. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremlmclim2 26316* 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 26317* 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 26318 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 26319* 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 26320* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved into main set.mm as cncfmptc 18880 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremidcncf 26321 The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 18881 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremsub1cncf 26322* Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremsub2cncf 26323* Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremcnres2 26324* 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.)
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Theoremcnresima 26325 A continuous function is continuous onto its image. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
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Theoremcncfres 26326* 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 26327 Extend class notation with the class of totally bounded metric spaces.

Syntaxcbnd 26328 Extend class notation with the class of bounded metric spaces.

Definitiondf-totbnd 26329* 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 26330* The predicate "is a totally bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremistotbnd2 26331* The predicate "is a totally bounded metric space." (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremistotbnd3 26332* 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 26333 The predicate "totally bounded" implies is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorem0totbnd 26334 The metric (there is only one) on the empty set is totally bounded. (Contributed by Mario Carneiro, 16-Sep-2015.)

Theoremsstotbnd2 26335* Condition for a subset of a metric space to be totally bounded. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremsstotbnd 26336* 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 26337* 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 26338 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 26339* 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 26340* 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 26341* The predicate "is a bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theorembndmet 26342 A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)

Theoremisbndx 26343* 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 26344* 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 26345* A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 13-Sep-2015.)

Theoremisbnd3b 26346* A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 22-Sep-2015.)

Theorembndss 26347 A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremblbnd 26348 A ball is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 15-Jan-2014.)

Theoremssbnd 26349* 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 26350 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 26330 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 26351* 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 26352 Lemma for equivbnd2 26353 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.)

Theoremequivbnd2 26353* If balls are totally bounded in the metric , then balls are totally bounded in the equivalent metric . (Contributed by Mario Carneiro, 15-Sep-2015.)

Theoremprdsbnd 26354* The product metric over finite index set is bounded if all the factors are bounded. (Contributed by Mario Carneiro, 13-Sep-2015.)
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Theoremprdstotbnd 26355* The product metric over finite index set is totally bounded if all the factors are totally bounded. (Contributed by Mario Carneiro, 20-Sep-2015.)
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Theoremprdsbnd2 26356* If balls are totally bounded in each factor, then balls are bounded in a metric product. (Contributed by Mario Carneiro, 16-Sep-2015.)
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Theoremcntotbnd 26357 A subset of the complexes is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremcnpwstotbnd 26358 A subset of , where , is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)
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19.14.8  Isometries

Syntaxcismty 26359 Extend class notation with the class of metric space isometries.

Definitiondf-ismty 26360* Define a function which takes two metric spaces and returns the set of isometries between the spaces. An isometry is a bijection which preserves distance. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremismtyval 26361* The set of isometries between two metric spaces. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremisismty 26362* The condition "is an isometry". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremismtycnv 26363 The inverse of an isometry is an isometry. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremismtyima 26364 The image of a ball under an isometry is another ball. (Contributed by Jeff Madsen, 31-Jan-2014.)

Theoremismtyhmeolem 26365 Lemma for ismtyhmeo 26366. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremismtyhmeo 26366 An isometry is a homeomorphism on the induced topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremismtybndlem 26367 Lemma for ismtybnd 26368. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 19-Jan-2014.)

Theoremismtybnd 26368 Isometries preserve boundedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 19-Jan-2014.)

Theoremismtyres 26369 A restriction of an isometry is an isometry. The condition is not necessary but makes the proof easier. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

19.14.9  Heine-Borel Theorem

Theoremheibor1lem 26370 Lemma for heibor1 26371. A compact metric space is complete. This proof works by considering the collection for each , which has the finite intersection property because any finite intersection of upper integer sets is another upper integer set, so any finite intersection of the image closures will contain for some . Thus, by compactness, the intersection contains a point , which must then be the convergent point of . (Contributed by Jeff Madsen, 17-Jan-2014.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremheibor1 26371 One half of heibor 26382, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 19209 and total boundedness here, which follows trivially from the fact that the set of all -balls is an open cover of , so finitely many cover . (Contributed by Jeff Madsen, 16-Jan-2014.)

Theoremheiborlem1 26372* Lemma for heibor 26382. We work with a fixed open cover throughout. The set is the set of all subsets of that admit no finite subcover of . (We wish to prove that is empty.) If a set has no finite subcover, then any finite cover of must contain a set that also has no finite subcover. (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem2 26373* Lemma for heibor 26382. Substitutions for the set . (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem3 26374* Lemma for heibor 26382. Using countable choice ax-cc 8262, we have fixed in advance a collection of finite nets for (note that an -net is a set of points in whose -balls cover ). The set is the subset of these points whose corresponding balls have no finite subcover (i.e. in the set ). If the theorem was false, then would be in , and so some ball at each level would also be in . But we can say more than this; given a ball on level , since level covers the space and thus also , using heiborlem1 26372 there is a ball on the next level whose intersection with also has no finite subcover. Now since the set is a countable union of finite sets, it is countable (which needs ax-cc 8262 via iunctb 8396), and so we can apply ax-cc 8262 to directly to get a function from to itself, which points from each ball in to a ball on the next level in , and such that the intersection between these balls is also in . (Contributed by Jeff Madsen, 18-Jan-2014.)

Theoremheiborlem4 26375* Lemma for heibor 26382. Using the function constructed in heiborlem3 26374, construct an infinite path in . (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem5 26376* Lemma for heibor 26382. The function is a set of point-and-radius pairs suitable for application to caubl 19199. (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem6 26377* Lemma for heibor 26382. Since the sequence of balls connected by the function ensures that each ball nontrivially intersects with the next (since the empty set has a finite subcover, the intersection of any two successive balls in the sequence is nonempty), and each ball is half the size of the previous one, the distance between the centers is at most times the size of the larger, and so if we expand each ball by a factor of we get a nested sequence of balls. (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem7 26378* Lemma for heibor 26382. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem8 26379* Lemma for heibor 26382. The previous lemmas establish that the sequence is Cauchy, so using completeness we now consider the convergent point . By assumption, is an open cover, so is an element of some , and some ball centered at is contained in . But the sequence contains arbitrarily small balls close to , so some element of the sequence is contained in . And finally we arrive at a contradiction, because is a finite subcover of that covers , yet . For convenience, we write this contradiction as where is all the accumulated hypotheses and is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.)

Theoremheiborlem9 26380* Lemma for heibor 26382. Discharge the hypotheses of heiborlem8 26379 by applying caubl 19199 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-Jan-2014.)

Theoremheiborlem10 26381* Lemma for heibor 26382. The last remaining piece of the proof is to find an element such that , i.e. is an element of that has no finite subcover, which is true by heiborlem1 26372, since is a finite cover of , which has no finite subcover. Thus, the rest of the proof follows to a contradiction, and thus there must be a finite subcover of that covers , i.e. is compact. (Contributed by Jeff Madsen, 22-Jan-2014.)

Theoremheibor 26382 Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 26371 and heiborlem1 26372 for a description of the proof. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jan-2014.)

19.14.10  Banach Fixed Point Theorem

Theorembfplem1 26383* Lemma for bfp 26385. The sequence , which simply starts from any point in the space and iterates , satisfies the property that the distance from to decreases by at least after each step. Thus, the total distance from any to is bounded by a geometric series, and the sequence is Cauchy. Therefore, it converges to a point since the space is complete. (Contributed by Jeff Madsen, 17-Jun-2014.)

Theorembfplem2 26384* Lemma for bfp 26385. Using the point found in bfplem1 26383, we show that this convergent point is a fixed point of . Since for any positive , the sequence is in for all (where ), we have and , so is in every neighborhood of and is a fixed point of . (Contributed by Jeff Madsen, 5-Jun-2014.)

Theorembfp 26385* Banach fixed point theorem, also known as contraction mapping theorem. A contraction on a complete metric space has a unique fixed point. We show existence in the lemmas, and uniqueness here - if has two fixed points, then the distance between them is less than times itself, a contradiction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)

19.14.11  Euclidean space

Syntaxcrrn 26386 Extend class notation with the n-dimensional Euclidean space.

Definitiondf-rrn 26387* Define n-dimensional Euclidean space as a metric space with the standard Euclidean norm given by the quadratic mean. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremrrnval 26388* The n-dimensional Euclidean space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrnmval 26389* The value of the Euclidean metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrnmet 26390 Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)

Theoremrrndstprj1 26391 The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrndstprj2 26392* Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 26391 can be used to show that the supremum norm and Euclidean norm are equivalent. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrncmslem 26393* Lemma for rrncms 26394. (Contributed by Jeff Madsen, 6-Jun-2014.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrncms 26394 Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrepwsmet 26395 The supremum metric on is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
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Theoremrrnequiv 26396 The supremum metric on is equivalent to the metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
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Theoremrrntotbnd 26397 A set in Euclidean space is totally bounded iff its is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)

Theoremrrnheibor 26398 Heine-Borel theorem for Euclidean space. A subset of Euclidean space is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)

19.14.12  Intervals (continued)

Theoremismrer1 26399* An isometry between and . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)

Theoremreheibor 26400 Heine-Borel theorem for real numbers. A subset of is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)

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