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

Definitiondf-inftm 24201* Define the relation " is infinitesimal with respect to " for a structure . (Contributed by Thierry Arnoux, 30-Jan-2018.)
<<< .g

Definitiondf-archi 24202 A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Archi <<<

Theoreminftmrel 24203 The infinitesimal relation for a structure (Contributed by Thierry Arnoux, 30-Jan-2018.)
<<<

Theoremisinftm 24204* Express is infinitesimal with respect to for a structure . (Contributed by Thierry Arnoux, 30-Jan-2018.)
.g              <<<

Theoremisarchi 24205* Express the predicate " is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
<<<       Archi

Theorempnfinf 24206 Plus infinity is an infinite for the completed real line, as any real number is infinitesimal compared to it. (Contributed by Thierry Arnoux, 1-Feb-2018.)
<<<

Theoremxrnarchi 24207 The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
Archi

Theoremisarchi2 24208* Alternative way to express the predicate " is Archimedean ", for Tosets. (Contributed by Thierry Arnoux, 30-Jan-2018.)
.g                     Toset Archi

19.3.7.6  Ring homomorphisms - misc additions

Theoremrhmdvdsr 24209 A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
r       r       RingHom

Theoremrhmopp 24210 A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
RingHom oppr RingHom oppr

Theoremelrhmunit 24211 Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
RingHom Unit Unit

Theoremrhmdvd 24212 A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.)
Unit              /r              RingHom

Theoremrhmunitinv 24213 Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
RingHom Unit

Theoremkerunit 24214 If a unit element lies in the kernel of a ring homomorphism, then , i.e. the target ring is the zero ring. (Contributed by Thierry Arnoux, 24-Oct-2017.)
Unit                     RingHom

Theoremkerf1hrm 24215 A ring homomorphism is injective if and only if its kernel is the singleton . (Contributed by Thierry Arnoux, 27-Oct-2017.)
RingHom

19.3.7.7  The ring of integers

Theoremzzsbase 24216 The base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.)
flds

Theoremzzsplusg 24217 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.)
flds

Theoremzzsmulg 24218 The multiplication (group power) opereation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.)
flds        .g

Theoremzzsmulr 24219 The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremzzs0 24220 The neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremzzs1 24221 The multiplicative neutral element of the ring of integers (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

19.3.7.8  The ordered field of reals

Theoremrebase 24222 The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremremulg 24223 The multiplication (group power) operation of the group of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds        .g

Theoremreplusg 24224 The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
flds

Theoremremulr 24225 The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremre0g 24226 The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremre1r 24227 The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremrele2 24228 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
flds

Theoremrelt 24229 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
flds

Theoremredvr 24230 The division operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds        /r

Theoremretos 24231 The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.)
flds        Toset

Theoremrefld 24232 The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds        Field

Theoremreofld 24233 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 24234 Extend class notation with the class of metric identifications.
~Met

Syntaxcpstm 24235 Extend class notation with the metric induced by a pseudometric.
pstoMet

Definitiondf-metid 24236* Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
~Met PsMet

Definitiondf-pstm 24237* Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
pstoMet PsMet ~Met ~Met

Theoremmetidval 24238* Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met

Theoremmetidss 24239 As a relation, the metric identification is a subset of a cross product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met

Theoremmetidv 24240 and identify by the metric if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met

Theoremmetideq 24241 Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met ~Met

Theoremmetider 24242 The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet ~Met

Theorempstmval 24243* Value of the metric induced by a pseudometric . (Contributed by Thierry Arnoux, 7-Feb-2018.)
~Met       PsMet pstoMet

Theorempstmfval 24244 Function value of the metric induced by a pseudometric (Contributed by Thierry Arnoux, 11-Feb-2018.)
~Met       PsMet pstoMet

Theorempstmxmet 24245 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

19.3.8.2  Continuity - misc additions

Theoremhauseqcn 24246 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 24247 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 24248 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 24249 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 24250 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 24251 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 24252 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 24253 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 24254 The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.)

Theoremtpr2tp 24255 The usual topology on is the product topology of the usual topology on . (Contributed by Thierry Arnoux, 21-Sep-2017.)
TopOn

Theoremtpr2uni 24256 The usual topology on is the product topology of the usual topology on . (Contributed by Thierry Arnoux, 21-Sep-2017.)

Theoremxpinpreima 24257 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 24258 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 24259 The complex square of side is a subset of the complex circle of radius . (Contributed by Thierry Arnoux, 25-Sep-2017.)

Theoremsqsscirc2 24260 The complex square of side is a subset of the complex disc of radius . (Contributed by Thierry Arnoux, 25-Sep-2017.)

Theoremcnre2csqlem 24261* Lemma for cnre2csqima 24262 (Contributed by Thierry Arnoux, 27-Sep-2017.)

Theoremcnre2csqima 24262* Image of a centered square by the canonical bijection from to . (Contributed by Thierry Arnoux, 27-Sep-2017.)

Theoremtpr2rico 24263* 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 24264* 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 24265* 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 24266 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 24267* Multiplication by a real constant is a continuous function (Contributed by Thierry Arnoux, 23-May-2017.)

Theoremraddcn 24268* Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)

Theoremxrmulc1cn 24269* The operation multiplying an extended real number by a non-negative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
ordTop

Theoremfmcncfil 24270 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 24271 The extended non-negative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
ordTop t

Theoremxrge0iifcnv 24272* Define a bijection from to . (Contributed by Thierry Arnoux, 29-Mar-2017.)

Theoremxrge0iifcv 24273* The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.)

Theoremxrge0iifiso 24274* 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 24275* Expose a homeomorphism from the closed unit interval and the extended non-negative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.)
ordTop t

Theoremxrge0iifhom 24276* 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 24277* Condition for the defined function, to be a monoid homomorphism. (Contributed by Thierry Arnoux, 20-Jun-2017.)
ordTop t

Theoremxrge0iifmhm 24278* 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 24279* The addition operation of the extended non-negative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.)
ordTop t

Theoremxrge0mulc1cn 24280* 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 24281 The extended non-negative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.)
s

Theoremxrge0topn 24282 The topology of the extended non-negative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.)
s ordTop t

Theoremxrge0haus 24283 The topology of the extended non-negative real numbers is Hausdorff. (Contributed by Thierry Arnoux, 26-Jul-2017.)
s

Theoremxrge0tmdOLD 24284 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 24285 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

19.3.8.8  Limits - misc additions

Theoremlmlim 24286 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 24287 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 24288 Implication of convergence for a non-negative series. This could be used to shorten prmreclem6 13244 (Contributed by Thierry Arnoux, 28-Jul-2017.)

Theorempnfneige0 24289* A neighborhood of contains an unbounded interval based at a real number. See pnfnei 17238 (Contributed by Thierry Arnoux, 31-Jul-2017.)
s

Theoremlmxrge0 24290* 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 24291* If a monotonic sequence of real numbers diverges, it is unbounded. (Contributed by Thierry Arnoux, 4-Aug-2017.)

Theoremlmdvglim 24292* 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 24293 Extend class notation with the Hausdorff completion relation.
HCmp

Definitiondf-hcmp 24294* 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 24295 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 24296 The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds        CMetSp

Theoremreust 24297 The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.)
flds        UnifSt metUnif

Theoremrecusp 24298 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 24299 Zero of a -module. (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod

Theoremzlm1 24300 Unit of a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
Mod

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