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Theorem List for Metamath Proof Explorer - 24201-24300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrefld 24201 The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds        Field

Theoremreofld 24202 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
flds        oField

19.3.8  Topology

19.3.8.1  Pseudometrics

Syntaxcmetid 24203 Extend class notation with the class of metric identifications.
~Met

Syntaxcpstm 24204 Extend class notation with the metric induced by a pseudometric.
pstoMet

Definitiondf-metid 24205* Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
~Met PsMet

Definitiondf-pstm 24206* Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
pstoMet PsMet ~Met ~Met

Theoremmetidval 24207* Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met

Theoremmetidss 24208 As a relation, the metric identification is a subset of a cross product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met

Theoremmetidv 24209 and identify by the metric if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met

Theoremmetideq 24210 Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met ~Met

Theoremmetider 24211 The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet ~Met

Theorempstmval 24212* Value of the metric induced by a pseudometric . (Contributed by Thierry Arnoux, 7-Feb-2018.)
~Met       PsMet pstoMet

Theorempstmfval 24213 Function value of the metric induced by a pseudometric (Contributed by Thierry Arnoux, 11-Feb-2018.)
~Met       PsMet pstoMet

Theorempstmxmet 24214 The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.)
~Met       PsMet pstoMet

Theoremhauseqcn 24215 In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.)

19.3.8.3  Topology of the closed unit

Theoremunitsscn 24216 The closed unit is a subset of the set of the complex numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoremelunitrn 24217 The closed unit is a subset of the set of the real numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)

Theoremelunitcn 24218 The closed unit is a subset of the set of the complext numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)

Theoremelunitge0 24219 An element of the closed unit is positive Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 20-Dec-2016.)

Theoremunitssxrge0 24220 The closed unit is a subset of the set of the extended non-negative reals. Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoremunitdivcld 24221 Necessary conditions for a quotient to be in the closed unit. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.)

Theoremiistmd 24222 The closed unit monoid is a topological monoid. (Contributed by Thierry Arnoux, 25-Mar-2017.)
mulGrpflds        TopMnd

19.3.8.4  Topology of ` ( RR X. RR ) `

Theoremunicls 24223 The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.)

Theoremtpr2tp 24224 The usual topology on is the product topology of the usual topology on . (Contributed by Thierry Arnoux, 21-Sep-2017.)
TopOn

Theoremtpr2uni 24225 The usual topology on is the product topology of the usual topology on . (Contributed by Thierry Arnoux, 21-Sep-2017.)

Theoremxpinpreima 24226 Rewrite the cartesian product of two sets as the intersection of their preimage by and , the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Theoremxpinpreima2 24227 Rewrite the cartesian product of two sets as the intersection of their preimage by and , the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Theoremsqsscirc1 24228 The complex square of side is a subset of the complex circle of radius . (Contributed by Thierry Arnoux, 25-Sep-2017.)

Theoremsqsscirc2 24229 The complex square of side is a subset of the complex disc of radius . (Contributed by Thierry Arnoux, 25-Sep-2017.)

Theoremcnre2csqlem 24230* Lemma for cnre2csqima 24231 (Contributed by Thierry Arnoux, 27-Sep-2017.)

Theoremcnre2csqima 24231* Image of a centered square by the canonical bijection from to . (Contributed by Thierry Arnoux, 27-Sep-2017.)

Theoremtpr2rico 24232* For any point of an open set of the usual topology on there is an opened square which contains that point and is entirely in the open set. This is square is actually a ball by the norm . (Contributed by Thierry Arnoux, 21-Sep-2017.)

19.3.8.5  Order topology - misc. additions

Theoremcnvordtrestixx 24233* The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.)
ordTop t ordTop

19.3.8.6  Continuity in topological spaces - misc. additions

Theoremmndpluscn 24234* A mapping that is both a homeomorphism and a monoid homomorphism preserves the "continuousness" of the operation. (Contributed by Thierry Arnoux, 25-Mar-2017.)
TopOn       TopOn

Theoremmhmhmeotmd 24235 Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
MndHom               TopMnd              TopMnd

Theoremrmulccn 24236* Multiplication by a real constant is a continuous function (Contributed by Thierry Arnoux, 23-May-2017.)

Theoremraddcn 24237* Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)

Theoremxrmulc1cn 24238* The operation multiplying an extended real number by a non-negative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
ordTop

Theoremfmcncfil 24239 The image of a Cauchy filter by a continuous filter map is a Cauchy filter. (Contributed by Thierry Arnoux, 12-Nov-2017.)
CauFil CauFil

19.3.8.7  Topology of the extended non-negative real numbers monoid

Theoremxrge0hmph 24240 The extended non-negative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
ordTop t

Theoremxrge0iifcnv 24241* Define a bijection from to . (Contributed by Thierry Arnoux, 29-Mar-2017.)

Theoremxrge0iifcv 24242* The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.)

Theoremxrge0iifiso 24243* The defined bijection from the closed unit interval and the extended non-negative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)

Theoremxrge0iifhmeo 24244* Expose a homeomorphism from the closed unit interval and the extended non-negative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.)
ordTop t

Theoremxrge0iifhom 24245* The defined function from the closed unit interval and the extended non-negative reals is also a monoid homomorphism. (Contributed by Thierry Arnoux, 5-Apr-2017.)
ordTop t

Theoremxrge0iif1 24246* Condition for the defined function, to be a monoid homomorphism. (Contributed by Thierry Arnoux, 20-Jun-2017.)
ordTop t

Theoremxrge0iifmhm 24247* The defined function from the closed unit interval and the extended non-negative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
ordTop t        mulGrpflds MndHom s

Theoremxrge0pluscn 24248* The addition operation of the extended non-negative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.)
ordTop t

Theoremxrge0mulc1cn 24249* The operation multiplying a non-negative real numbers by a non-negative constant is continuous. (Contributed by Thierry Arnoux, 6-Jul-2017.)
ordTop t

Theoremxrge0tps 24250 The extended non-negative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.)
s

Theoremxrge0topn 24251 The topology of the extended non-negative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.)
s ordTop t

Theoremxrge0haus 24252 The topology of the extended non-negative real numbers is Hausdorff. (Contributed by Thierry Arnoux, 26-Jul-2017.)
s

Theoremxrge0tmdOLD 24253 The extended non-negative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
s TopMnd

Theoremxrge0tmd 24254 The extended non-negative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof Shortened by Thierry Arnoux, 21-Jun-2017.)
s TopMnd

Theoremlmlim 24255 Relate a limit in a given topology to a complex number limit, provided that topology agrees with the common topology on on the required subset. (Contributed by Thierry Arnoux, 11-Jul-2017.)
TopOn                     t fldt

Theoremlmlimxrge0 24256 Relate a limit in the non-negative extended reals to a complex limit, provided the considered function is a real function. (Contributed by Thierry Arnoux, 11-Jul-2017.)
s

Theoremrge0scvg 24257 Implication of convergence for a non-negative series. This could be used to shorten prmreclem6 13230 (Contributed by Thierry Arnoux, 28-Jul-2017.)

Theorempnfneige0 24258* A neighborhood of contains an unbounded interval based at a real number. See pnfnei 17224 (Contributed by Thierry Arnoux, 31-Jul-2017.)
s

Theoremlmxrge0 24259* Express "sequence converges to plus infinity" (i.e. diverges), for a sequence of non-negative extended real numbers. (Contributed by Thierry Arnoux, 2-Aug-2017.)
s

Theoremlmdvg 24260* If a monotonic sequence of real numbers diverges, it is unbounded. (Contributed by Thierry Arnoux, 4-Aug-2017.)

Theoremlmdvglim 24261* If a monotonic real number sequence diverges, it converges in the extended real numbers and its limit is plus infinity. (Contributed by Thierry Arnoux, 3-Aug-2017.)
s

19.3.9  Uniform Stuctures and Spaces

19.3.9.1  Hausdorff Completion

Syntaxchcmp 24262 Extend class notation with the Hausdorff completion relation.
HCmp

Definitiondf-hcmp 24263* Definition of the Hausdorff completion. In this definition, a structure is a Hausdorff completion of a uniform structure if is a complete uniform space, in which is dense, and which admits the same uniform structure. Theorem 3 of [BourbakiTop1] p. II.21. states the existence and unicity of such a completion. (Contributed by Thierry Arnoux, 5-Mar-2018.)
HCmp UnifOn CUnifSp UnifSt

19.3.10  Topology and algebraic structures

19.3.10.1  The norm on the ring of the integer numbers

Theoremzzsnm 24264 The norm of the ring of the integers (Contributed by Thierry Arnoux, 8-Nov-2017.)
flds

19.3.10.2  The complete ordered field of the real numbers

Theoremrecms 24265 The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds        CMetSp

Theoremreust 24266 The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.)
flds        UnifSt metUnif

Theoremrecusp 24267 The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
flds        CUnifSp

19.3.10.3  Topological ` ZZ ` -modules

Theoremzlm0 24268 Zero of a -module. (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod

Theoremzlm1 24269 Unit of a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod

Theoremzlmds 24270 Distance in a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod

Theoremzlmtset 24271 Topology in a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod       TopSet       TopSet

Theoremzlmnm 24272 Norm of a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod

Theoremzhmnrg 24273 The -module built from a normed ring is also a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod       NrmRing NrmRing

Theoremnmmulg 24274 The norm of a group product, provided the -module is normed. (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod       .g       NrmMod

Theoremzrhnm 24275 The norm of the image by RHom of an integer in a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod       RHom       NrmMod NrmRing NzRing

Theoremcnzh 24276 The -module of is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.)
Modfld NrmMod

Theoremrezh 24277 The -module of is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.)
flds        Mod NrmMod

19.3.10.4  The canonical embedding of the rational numbers into a division ring

Syntaxcqqh 24278 Map the rationals into a field.
QQHom

Definitiondf-qqh 24279* Define the canonical homomorphism from the rationals into any field. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.)
QQHom RHomUnit RHom/rRHom

Theoremqqhval 24280* Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.)
/r              RHom       QQHom Unit

Theoremzrhf1ker 24281 The kernel of the homomorphism from the integers to a ring, if it is injective. (Contributed by Thierry Arnoux, 26-Oct-2017.)
RHom

Theoremzrhchr 24282 The kernel of the homomorphism from the integers to a ring is injective if and only if the ring has characteristic 0 . (Contributed by Thierry Arnoux, 8-Nov-2017.)
RHom              chr

Theoremzrhker 24283 The kernel of the homomorphism from the integers to a ring with characteristic 0. (Contributed by Thierry Arnoux, 8-Nov-2017.)
RHom              chr

Theoremzrhunitpreima 24284 The preimage by RHom of the unit of a division ring is . (Contributed by Thierry Arnoux, 22-Oct-2017.)
RHom              chr Unit

Theoremelzrhunit 24285 Condition for the image by RHom to be a unit. (Contributed by Thierry Arnoux, 30-Oct-2017.)
RHom              chr Unit

Theoremelzdif0 24286 Lemma for qqhval2 24288 (Contributed by Thierry Arnoux, 29-Oct-2017.)

Theoremqqhval2lem 24287 Lemma for qqhval2 24288 (Contributed by Thierry Arnoux, 29-Oct-2017.)
/r       RHom       chr numer denom

Theoremqqhval2 24288* Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 26-Oct-2017.)
/r       RHom       chr QQHom numer denom

Theoremqqhvval 24289 Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
/r       RHom       chr QQHom numer denom

Theoremqqh0 24290 The image of by the QQHom homomorphism is the ring's zero. (Contributed by Thierry Arnoux, 22-Oct-2017.)
/r       RHom       chr QQHom

Theoremqqh1 24291 The image of by the QQHom homomorphism is the ring's unit. (Contributed by Thierry Arnoux, 22-Oct-2017.)
/r       RHom       chr QQHom

Theoremqqhf 24292 QQHom as a function. (Contributed by Thierry Arnoux, 28-Oct-2017.)
/r       RHom       chr QQHom

Theoremqqhvq 24293 The image of a quotient by the QQHom homomorphism. (Contributed by Thierry Arnoux, 28-Oct-2017.)
/r       RHom       chr QQHom

Theoremqqhghm 24294 The QQHom homomorphism is a group homomorphism if the target structure is a division ring. (Contributed by Thierry Arnoux, 9-Nov-2017.)
/r       RHom       flds        chr QQHom

Theoremqqhrhm 24295 The QQHom homomorphism is a ring homomorphism if the target structure is a field. If the target structure is a division ring, it is a group homomorphism, but not a ring homomorphism, because it does not preserve the ring multiplication operation. (Contributed by Thierry Arnoux, 29-Oct-2017.)
/r       RHom       flds        Field chr QQHom RingHom

Theoremqqhnm 24296 The norm of the image by QQHom of a rational number in a topological division ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod       NrmRing NrmMod chr QQHom

Theoremqqhcn 24297 The QQHom homomorphism is a continuous function. (Contributed by Thierry Arnoux, 9-Nov-2017.)
flds               Mod              NrmRing NrmMod chr QQHom

Theoremqqhucn 24298 The QQHom homomorphism is uniformly continuous. (Contributed by Thierry Arnoux, 28-Jan-2018.)
flds        UnifSt       metUnif        Mod       NrmRing              NrmMod       chr        QQHom Cnu

19.3.10.5  The canonical embedding of ` RR ` into a complete ordered field

Syntaxcrrh 24299 Map the real numbers into a complete field.
RRHom

Definitiondf-rrh 24300 Define the canonical homomorphism from the real numbers to any complete field, as the extension by continuity of the canonical homomorphism from the rational numbers. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.)
RRHom CnExtQQHom

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