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

Theoremxrstos 24201 The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.)
Toset

Theoremxrsclat 24202 The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.)

Theoremxrsp0 24203 The poset 0 of the extended real numbers is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)

Theoremxrsp1 24204 The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)

Theoremressmulgnn 24205 Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 12-Jun-2017.)
s               .g              .g

Theoremressmulgnn0 24206 Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 14-Jun-2017.)
s               .g                     .g

19.3.6.5  The extended non-negative real numbers monoid

Theoremxrge0base 24207 The base of the extended non-negative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
s

Theoremxrge00 24208 The zero of the extended non-negative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
s

Theoremxrge0plusg 24209 The additive law of the extended non-negative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.)
s

Theoremxrge0mulgnn0 24210 The group multiple function in the extended non-negative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
.gs

Theoremxrge0addass 24211 Associativity of extended non-negative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)

Theoremxrge0neqmnf 24212 An extended non-negative real cannot be minus infinity. (Contributed by Thierry Arnoux, 9-Jun-2017.)

Theoremxrge0nre 24213 An extended real which is not a real is plus infinity. (Contributed by Thierry Arnoux, 16-Oct-2017.)

Theoremxrge0addgt0 24214 The sum of nonnegative and positive numbers is positive. See addgtge0 9516 (Contributed by Thierry Arnoux, 6-Jul-2017.)

Theoremxrge0adddir 24215 Distributivity of extended non-negative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)

Theoremxrge0npcan 24216 Extended non-negative real version of npcan 9314. (Contributed by Thierry Arnoux, 9-Jun-2017.)

Theoremfsumrp0cl 24217* Closure of a finite sum of positive integers. (Contributed by Thierry Arnoux, 25-Jun-2017.)

19.3.7  Algebra

19.3.7.1  Finitely supported group sums - misc additions

Theoremsumpr 24218* A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoremgsumsn2 24219* Group sum of a singleton. (Contributed by Thierry Arnoux, 30-Jan-2017.)
g

Theoremgsumpropd2lem 24220* Lemma for gsumpropd2 24221 (Contributed by Thierry Arnoux, 28-Jun-2017.)
g g

Theoremgsumpropd2 24221* A stronger version of gsumpropd 14776, working for magma, where only the closure of the addition operation on a common base is required. (Contributed by Thierry Arnoux, 28-Jun-2017.)
g g

Theoremgsumconstf 24222* Sum of a constant series (Contributed by Thierry Arnoux, 5-Jul-2017.)
.g       g

Theoremxrge0tsmsd 24223* Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 30-Jan-2017.)
s                      g        tsums

Theoremxrge0tsmsbi 24224 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.)
s                      tsums tsums

Theoremxrge0tsmseq 24225 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 24-Mar-2017.)
s                      tsums        tsums

Theoremdvrdir 24226 Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
Unit              /r

Theoremrdivmuldivd 24227 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
Unit              /r

Theoremrnginvval 24228* The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.)
Unit

Theoremdvrcan5 24229 Cancellation law for common factor in ratio. (divcan5 9716 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)
Unit       /r

Theoremsubrgchr 24230 If is a subring of , then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.)
SubRing chrs chr

19.3.7.3  Ordered groups

Syntaxcogrp 24231 Extend class notation with the class of all ordered groups.
oGrp

Definitiondf-ogrp 24232* Define class of all ordered groups. An ordered group is a group with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.)
oGrp Toset

19.3.7.4  Ordered fields

Syntaxcofld 24233 Extend class notation with the class of all ordered fields.
oField

Definitiondf-ofld 24234* Define class of all ordered fields. An ordered field is a field with a total ordering compatible with the operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
oField Field Toset

Theoremisofld 24235* An ordered field is a field with a total ordering compatible with the operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
oField Field Toset

Theoremofldfld 24236 An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField Field

Theoremofldtos 24237 An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField Toset

Theoremofldadd 24238 In an ordered field, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField

Theoremofldmul 24239 In an ordered field, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField

Theoremofldsqr 24240 In an ordered field, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField

Theoremofldaddlt 24241 In an ordered field, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField

Theoremofld0le1 24242 In an ordered field, the ring unit is positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
oField

Theoremofldlt1 24243 In an ordered field, the ring unit is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
oField

Theoremofldchr 24244 The characteristic of an ordered field is zero. (Contributed by Thierry Arnoux, 21-Jan-2018.)
oField chr

Theoremsubofld 24245 Every subfield of an ordered field is also an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
oField s Field s oField

19.3.7.5  The Archimedean property for generic algebraic structures

Syntaxcinftm 24246 Class notation for the infinitesimal relation.
<<<

Syntaxcarchi 24247 Class notation for the Archimedean property.
Archi

Definitiondf-inftm 24248* Define the relation " is infinitesimal with respect to " for a structure . (Contributed by Thierry Arnoux, 30-Jan-2018.)
<<< .g

Definitiondf-archi 24249 A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Archi <<<

Theoreminftmrel 24250 The infinitesimal relation for a structure (Contributed by Thierry Arnoux, 30-Jan-2018.)
<<<

Theoremisinftm 24251* Express is infinitesimal with respect to for a structure . (Contributed by Thierry Arnoux, 30-Jan-2018.)
.g              <<<

Theoremisarchi 24252* Express the predicate " is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
<<<       Archi

Theorempnfinf 24253 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 24254 The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
Archi

Theoremisarchi2 24255* 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 24256 A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
r       r       RingHom

Theoremrhmopp 24257 A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
RingHom oppr RingHom oppr

Theoremelrhmunit 24258 Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
RingHom Unit Unit

Theoremrhmdvd 24259 A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.)
Unit              /r              RingHom

Theoremrhmunitinv 24260 Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
RingHom Unit

Theoremkerunit 24261 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 24262 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 24263 The base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.)
flds

Theoremzzsplusg 24264 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.)
flds

Theoremzzsmulg 24265 The multiplication (group power) opereation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.)
flds        .g

Theoremzzsmulr 24266 The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremzzs0 24267 The neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremzzs1 24268 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 24269 The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremremulg 24270 The multiplication (group power) operation of the group of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds        .g

Theoremreplusg 24271 The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
flds

Theoremremulr 24272 The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremre0g 24273 The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremre1r 24274 The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds

Theoremrele2 24275 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
flds

Theoremrelt 24276 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
flds

Theoremredvr 24277 The division operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds        /r

Theoremretos 24278 The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.)
flds        Toset

Theoremrefld 24279 The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.)
flds        Field

Theoremreofld 24280 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 24281 Extend class notation with the class of metric identifications.
~Met

Syntaxcpstm 24282 Extend class notation with the metric induced by a pseudometric.
pstoMet

Definitiondf-metid 24283* Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
~Met PsMet

Definitiondf-pstm 24284* Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
pstoMet PsMet ~Met ~Met

Theoremmetidval 24285* Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met

Theoremmetidss 24286 As a relation, the metric identification is a subset of a cross product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met

Theoremmetidv 24287 and identify by the metric if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met

Theoremmetideq 24288 Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet ~Met ~Met

Theoremmetider 24289 The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet ~Met

Theorempstmval 24290* Value of the metric induced by a pseudometric . (Contributed by Thierry Arnoux, 7-Feb-2018.)
~Met       PsMet pstoMet

Theorempstmfval 24291 Function value of the metric induced by a pseudometric (Contributed by Thierry Arnoux, 11-Feb-2018.)
~Met       PsMet pstoMet

Theorempstmxmet 24292 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 24293 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 24294 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 24295 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 24296 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 24297 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 24298 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 24299 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 24300 The closed unit monoid is a topological monoid. (Contributed by Thierry Arnoux, 25-Mar-2017.)
mulGrpflds        TopMnd

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