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Theorem List for Metamath Proof Explorer - 32201-32300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremismtyhmeo 32201 An isometry is a homeomorphism on the induced topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremismtybndlem 32202 Lemma for ismtybnd 32203. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 19-Jan-2014.)

Theoremismtybnd 32203 Isometries preserve boundedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 19-Jan-2014.)

Theoremismtyres 32204 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.)

21.18.9  Heine-Borel Theorem

Theoremheibor1lem 32205 Lemma for heibor1 32206. 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 32206 One half of heibor 32217, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 22365 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 32207* Lemma for heibor 32217. 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 32208* Lemma for heibor 32217. Substitutions for the set . (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem3 32209* Lemma for heibor 32217. Using countable choice ax-cc 8883, 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 32207 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 8883 via iunctb 9017), and so we can apply ax-cc 8883 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 32210* Lemma for heibor 32217. Using the function constructed in heiborlem3 32209, construct an infinite path in . (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem5 32211* Lemma for heibor 32217. The function is a set of point-and-radius pairs suitable for application to caubl 22355. (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem6 32212* Lemma for heibor 32217. 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 32213* Lemma for heibor 32217. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014.)

Theoremheiborlem8 32214* Lemma for heibor 32217. 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 32215* Lemma for heibor 32217. Discharge the hypotheses of heiborlem8 32214 by applying caubl 22355 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-Jan-2014.)

Theoremheiborlem10 32216* Lemma for heibor 32217. 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 32207, 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 32217 Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 32206 and heiborlem1 32207 for a description of the proof. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jan-2014.)

21.18.10  Banach Fixed Point Theorem

Theorembfplem1 32218* Lemma for bfp 32220. 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 32219* Lemma for bfp 32220. Using the point found in bfplem1 32218, 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 32220* 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.)

21.18.11  Euclidean space

Syntaxcrrn 32221 Extend class notation with the n-dimensional Euclidean space.

Definitiondf-rrn 32222* 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 32223* The n-dimensional Euclidean space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrnmval 32224* The value of the Euclidean metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrnmet 32225 Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)

Theoremrrndstprj1 32226 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 32227* Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 32226 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 32228* Lemma for rrncms 32229. (Contributed by Jeff Madsen, 6-Jun-2014.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrrncms 32229 Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

Theoremrepwsmet 32230 The supremum metric on is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
flds s

Theoremrrnequiv 32231 The supremum metric on is equivalent to the metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
flds s

Theoremrrntotbnd 32232 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 32233 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.)

21.18.12  Intervals (continued)

Theoremismrer1 32234* An isometry between and . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)

Theoremreheibor 32235 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 32236 A closed interval in is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Sep-2015.)

TheoremicccmpALT 32237 A closed interval in is compact. Alternate proof of icccmp 21921 using the Heine-Borel theorem heibor 32217. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Aug-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

21.18.13  Groups and related structures

Theoremexidcl 32238 Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremexidreslem 32239* Lemma for exidres 32240 and exidresid 32241. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
GId

Theoremexidres 32240 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 32241 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 32242 A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.)

Theoremgrpoeqdivid 32243 Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId

Theoremghomf 32244 Mapping property of a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.)
GrpOpHom

Theoremghomco 32245 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 32246 Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)
GrpOpHom

Theoremgrpokerinj 32247 A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
GId              GId       GrpOpHom

21.18.14  Rings

Theoremrngonegcl 32248 A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)

GId

GId

Theoremrngosub 32251 Subtraction in a ring, in terms of addition and negation. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngonegmn1l 32252 Negation in a ring is the same as left multiplication by . (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremrngonegmn1r 32253 Negation in a ring is the same as right multiplication by . (Contributed by Jeff Madsen, 19-Jun-2010.)
GId

Theoremrngoneglmul 32254 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngonegrmul 32255 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngosubdi 32256 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngosubdir 32257 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremzerdivemp1x 32258* In a unitary ring a left invertible element is not a zero divisor. Generalization of zerdivemp1 26243 by Frederic Line. (Contributed by Jeff Madsen, 18-Apr-2010.)
GId              GId

Theoremisdrngo1 32259 The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
GId

Theoremdivrngcl 32260 The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.)
GId

Theoremisdrngo2 32261* A division ring is a ring in which and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.)
GId              GId

Theoremisdrngo3 32262* A division ring is a ring in which and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId              GId

21.18.15  Ring homomorphisms

Syntaxcrnghom 32263 Extend class notation with the class of ring homomorphisms.

Syntaxcrngiso 32264 Extend class notation with the class of ring isomorphisms.

Syntaxcrisc 32265 Extend class notation with the ring isomorphism relation.

Definitiondf-rngohom 32266* Define the function which gives the set of ring homomorphisms between two given rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
GId GId

Theoremrngohomval 32267* The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)
GId                            GId

Theoremisrngohom 32268* The predicate "is a ring homomorphism from to ." (Contributed by Jeff Madsen, 19-Jun-2010.)
GId                            GId

Theoremrngohomf 32269 A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngohomcl 32270 Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)

Theoremrngohom1 32271 A ring homomorphism preserves . (Contributed by Jeff Madsen, 24-Jun-2011.)
GId              GId

Theoremrngohommul 32273 Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)

Theoremrngogrphom 32274 A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
GrpOpHom

Theoremrngohom0 32275 A ring homomorphism preserves . (Contributed by Jeff Madsen, 2-Jan-2011.)
GId              GId

Theoremrngohomsub 32276 Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)

Theoremrngohomco 32277 The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngokerinj 32278 A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
GId                     GId

Definitiondf-rngoiso 32279* Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoisoval 32280* The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremisrngoiso 32281 The predicate "is a ring isomorphism between and ." (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoiso1o 32282 A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoisohom 32283 A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoisocnv 32284 The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoisoco 32285 The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

Definitiondf-risc 32286* Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremisriscg 32287* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremisrisc 32288* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrisc 32289* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrisci 32290 Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremriscer 32291 Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)

21.18.16  Commutative rings

Syntaxccring 32292 Extend class notation with the class of commutative rings.
CRingOps

Definitiondf-crngo 32293 Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps

Theoremiscrngo 32294 The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps

Theoremiscrngo2 32295* The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps

Theoremiscringd 32296* Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.)
CRingOps

Theoremcrngorngo 32297 A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
CRingOps

Theoremcrngocom 32298 The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps

Theoremcrngm23 32299 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
CRingOps

Theoremcrngm4 32300 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
CRingOps

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