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Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembndss 26101 A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremblbnd 26102 A ball is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 15-Jan-2014.)

Theoremssbnd 26103* 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 26104 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 26084 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 26105* 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 26106 Lemma for equivbnd2 26107 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.)

Theoremequivbnd2 26107* 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 26108* The product metric over finite index set is bounded if all the factors are bounded. (Contributed by Mario Carneiro, 13-Sep-2015.)
s

Theoremprdstotbnd 26109* The product metric over finite index set is totally bounded if all the factors are totally bounded. (Contributed by Mario Carneiro, 20-Sep-2015.)
s

Theoremprdsbnd2 26110* If balls are totally bounded in each factor, then balls are bounded in a metric product. (Contributed by Mario Carneiro, 16-Sep-2015.)
s

Theoremcntotbnd 26111 A subset of the complexes is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremcnpwstotbnd 26112 A subset of , where , is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)
flds s

19.14.9  Isometries

Syntaxcismty 26113 Extend class notation with the class of metric space isometries.

Definitiondf-ismty 26114* 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 26115* The set of isometries between two metric spaces. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremisismty 26116* The condition "is an isometry". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremismtycnv 26117 The inverse of an isometry is an isometry. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremismtyima 26118 The image of a ball under an isometry is another ball. (Contributed by Jeff Madsen, 31-Jan-2014.)

Theoremismtyhmeolem 26119 Lemma for ismtyhmeo 26120. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremismtyhmeo 26120 An isometry is a homeomorphism on the induced topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremismtybndlem 26121 Lemma for ismtybnd 26122. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 19-Jan-2014.)

Theoremismtybnd 26122 Isometries preserve boundedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 19-Jan-2014.)

Theoremismtyres 26123 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.)

19.14.10  Heine-Borel Theorem

Theoremheibor1lem 26124 Lemma for heibor1 26125. 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 26125 One half of heibor 26136, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 18958 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 26126* Lemma for heibor 26136. 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 26127* Lemma for heibor 26136. Substitutions for the set . (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem3 26128* Lemma for heibor 26136. Using countable choice ax-cc 8208, 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 26126 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 8208 via iunctb 8343), and so we can apply ax-cc 8208 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 26129* Lemma for heibor 26136. Using the function constructed in heiborlem3 26128, construct an infinite path in . (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem5 26130* Lemma for heibor 26136. The function is a set of point-and-radius pairs suitable for application to caubl 18948. (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem6 26131* Lemma for heibor 26136. 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 26132* Lemma for heibor 26136. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem8 26133* Lemma for heibor 26136. 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 26134* Lemma for heibor 26136. Discharge the hypotheses of heiborlem8 26133 by applying caubl 18948 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-Jan-2014.)

Theoremheiborlem10 26135* Lemma for heibor 26136. 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 26126, 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 26136 Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 26125 and heiborlem1 26126 for a description of the proof. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jan-2014.)

19.14.11  Banach Fixed Point Theorem

Theorembfplem1 26137* Lemma for bfp 26139. 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 26138* Lemma for bfp 26139. Using the point found in bfplem1 26137, 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 26139* 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.)

19.14.12  Euclidean space

Syntaxcrrn 26140 Extend class notation with the n-dimensional Euclidean space.

Definitiondf-rrn 26141* 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 26142* The n-dimensional Euclidean space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrnmval 26143* The value of the Euclidean metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrnmet 26144 Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)

Theoremrrndstprj1 26145 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 26146* Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 26145 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 26147* Lemma for rrncms 26148. (Contributed by Jeff Madsen, 6-Jun-2014.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrncms 26148 Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrepwsmet 26149 The supremum metric on is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
flds s

Theoremrrnequiv 26150 The supremum metric on is equivalent to the metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
flds s

Theoremrrntotbnd 26151 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 26152 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.)

19.14.13  Intervals (continued)

Theoremismrer1 26153* An isometry between and . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)

Theoremreheibor 26154 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.)

Theoremiccbnd 26155 A closed interval in is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Sep-2015.)

TheoremicccmpALT 26156 A closed interval in is compact. Alternate proof of icccmp 18544 using the Heine-Borel theorem heibor 26136. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Aug-2014.)

19.14.14  Groups and related structures

Theoremexidcl 26157 Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremexidreslem 26158* Lemma for exidres 26159 and exidresid 26160. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
GId

Theoremexidres 26159 The restriction of a binary operation with identity to a subset containing the identity has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
GId

Theoremexidresid 26160 The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
GId              GId

Theoremablo4pnp 26161 A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.)

Theoremgrpoeqdivid 26162 Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId

Theoremghomf 26163 Mapping property of a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.)
GrpOpHom

Theoremghomco 26164 The composition of two group homomorphisms is a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
GrpOpHom GrpOpHom GrpOpHom

Theoremghomdiv 26165 Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)
GrpOpHom

Theoremgrpokerinj 26166 A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
GId              GId       GrpOpHom

19.14.15  Rings

Theoremrngonegcl 26167 A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)

GId

GId

Theoremrngosub 26170 Subtraction in a ring, in terms of addition and negation. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngonegmn1l 26171 Negation in a ring is the same as left multiplication by . (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremrngonegmn1r 26172 Negation in a ring is the same as right multiplication by . (Contributed by Jeff Madsen, 19-Jun-2010.)
GId

Theoremrngoneglmul 26173 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngonegrmul 26174 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngosubdi 26175 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngosubdir 26176 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremzerdivemp1x 26177* In a unitary ring a left invertible element is not a zero divisor. Generalization of zerdivemp1 21533 by Frederic Line. (Contributed by Jeff Madsen, 18-Apr-2010.)
GId              GId

Theoremisdrngo1 26178 The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
GId

Theoremdivrngcl 26179 The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.)
GId

Theoremisdrngo2 26180* A division ring is a ring in which and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.)
GId              GId

Theoremisdrngo3 26181* A division ring is a ring in which and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId              GId

19.14.16  Ring homomorphisms

Syntaxcrnghom 26182 Extend class notation with the class of ring homomorphisms.

Syntaxcrngiso 26183 Extend class notation with the class of ring isomorphisms.

Syntaxcrisc 26184 Extend class notation with the ring isomorphism relation.

Definitiondf-rngohom 26185* Define the function which gives the set of ring homomorphisms between two given rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
GId GId

Theoremrngohomval 26186* The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)
GId                            GId

Theoremisrngohom 26187* The predicate "is a ring homomorphism from to ." (Contributed by Jeff Madsen, 19-Jun-2010.)
GId                            GId

Theoremrngohomf 26188 A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngohomcl 26189 Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)

Theoremrngohom1 26190 A ring homomorphism preserves . (Contributed by Jeff Madsen, 24-Jun-2011.)
GId              GId

Theoremrngohommul 26192 Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)

Theoremrngogrphom 26193 A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
GrpOpHom

Theoremrngohom0 26194 A ring homomorphism preserves . (Contributed by Jeff Madsen, 2-Jan-2011.)
GId              GId

Theoremrngohomsub 26195 Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)

Theoremrngohomco 26196 The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngokerinj 26197 A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
GId                     GId

Definitiondf-rngoiso 26198* Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoisoval 26199* The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremisrngoiso 26200 The predicate "is a ring isomorphism between and ." (Contributed by Jeff Madsen, 16-Jun-2011.)

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