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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcxrs 13401 Extend class notation with the extended real number structure.

Syntaxc0g 13402 Extend class notation with group identity element.

Syntaxcgsu 13403 Extend class notation to include finitely supported group sums.
g

Definitiondf-ordt 13404* Define the order topology, given an order , written as below. A closed subbasis for the order topology is given by the closed rays and , along with itself. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Definitiondf-xrs 13405* The extended real number structure. Unlike df-cnfld 16380, the extended real numbers do not have good algebraic properties, so this is not actually a group or anything higher, even though it has just as many operations as df-cnfld 16380. The main interest in this structure is in its ordering, which is complete and compact. The metric described here is an extension of the absolute value metric, but it is not itself a metric because is infinitely far from all other points. The topology is based on the order and not the extended metric (which would make an isolated point since there is nothing else in the -ball around it). All components of this structure agree with ℂfld when restricted to . (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet ordTop

Definitiondf-0g 13406* Define group identity element. (Contributed by NM, 20-Aug-2011.)

Definitiondf-gsum 13407* Define the group sum for the structure of a finite sequence of elements whose values are defined by the expression and whose set of indices is . It may be viewed as a product (if is a multiplication), a sum (if is an addition) or whatever. The variable is normally a free variable in ( i.e. can be thought of as ). The definition is meaningful in three contexts, depending on the size of the index set and each demanding different properties of .

1. If and has an identity element, then the sum equals this identity.

2. If and is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e. etc.

3. If is a finite set (or is non-zero for finitely many indices) and is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined.

4. If is an infinite set and is a Hausdorff topological group, then there is a meaningful sum, but g cannot handle this case. See df-tsms 17811. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)

g

Syntaxcqtop 13408 Extend class notation with the quotient topology function.
qTop

Syntaxcimas 13409 Image structure function.
s

Syntaxcqus 13410 Quotient structure function.
s

Syntaxcxps 13411 Binary product structure function.
s

Definitiondf-qtop 13412* Define the quotient topology given a function and topology on the domain of . (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Definitiondf-imas 13413* Define an image structure, which takes a structure and a function on the base set, and maps all the operations via the function. For this to work properly must either be injective or satisfy the well-definedness condition for each relevant operation.

Note that although we call this an "image" by association to df-ima 4704, in order to keep the definition simple we consider only the case when the domain of is equal to the base set of . Other cases can be achieved by restricting (with df-res 4703) and/or ( with df-ress 13157) to their common domain. (Contributed by Mario Carneiro, 23-Feb-2015.)

s Scalar Scalar Scalar TopSet qTop g

Definitiondf-divs 13414* Define a quotient ring (or quotient group), which is a special case of an image structure df-imas 13413 where the image function is . (Contributed by Mario Carneiro, 23-Feb-2015.)
s s

Definitiondf-xps 13415* Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.)
s s Scalars

Theoremimasval 13416* Value of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
s                             Scalar                                                               qTop        g                             Scalar TopSet

Theoremimasbas 13417 The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
s

Theoremimasds 13418* The distance function of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
s                                           g

Theoremimasdsfn 13419 The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremimasdsval 13420* The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
s                                                                g

Theoremimasdsval2 13421* The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
s                                                                       g

Theoremimasplusg 13422* The group operation in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
s

Theoremimasmulr 13423* The ring multiplication in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
s

Theoremimassca 13424 The scalar field of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s                             Scalar       Scalar

Theoremimasvsca 13425* The scalar multiplication operation of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s                             Scalar

Theoremimastset 13426 The topology of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s                                    TopSet       qTop

Theoremimasle 13427 The ordering of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremf1ocpbllem 13428 Lemma for f1ocpbl 13429. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremf1ocpbl 13429 An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremf1ovscpbl 13430 An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremf1olecpbl 13431 An injection is compatible with any relations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremimasaddfnlem 13432* The image structure operation is a function if the original operation is compatible with the function. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremimasaddvallem 13433* The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremimasaddflem 13434* The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremimasaddfn 13435* The image structure's group operation is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
s

Theoremimasaddval 13436* The value of an image structure's group operation. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremimasaddf 13437* The image structure's group operation is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremimasmulfn 13438* The image structure's ring multiplication is a function. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremimasmulval 13439* The value of an image structure's ring multiplication. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremimasmulf 13440* The image structure's ring multiplication is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremimasvscafn 13441* The image structure's scalar multiplication is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
s                             Scalar

Theoremimasvscaval 13442* The value of an image structure's scalar multiplication. (Contributed by Mario Carneiro, 24-Feb-2015.)
s                             Scalar

Theoremimasvscaf 13443* The image structure's scalar multiplication is closed in the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
s                             Scalar

Theoremimasless 13444 The order relation defined on an image set is a subset of the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremimasleval 13445* The value of the image structure's ordering when the order is compatible with the mapping function. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsval 13446* Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s                                    s

Theoremdivslem 13447* The function in divsval 13446 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremdivsin 13448 Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
s                                    s

Theoremdivsbas 13449 Base set of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremdivssca 13450 The scalar field of a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s                             Scalar       Scalar

Theoremdivsfval 13451* Value of the function in divsval 13446. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremercpbllem 13452* Lemma for ercpbl 13453. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremercpbl 13453* Translate the function compatiblity relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremerlecpbl 13454* Translate the relation compatiblity relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremdivsaddvallem 13455* Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsaddflem 13456* The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsaddval 13457* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsaddf 13458* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsmulval 13459* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsmulf 13460* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremxpsc 13461 A short expression for the pair function mapping to and to . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpscg 13462 A short expression for the pair function mapping to and to . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpscfn 13463 The pair function is a function on . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpsc0 13464 The pair function maps to . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpsc1 13465 The pair function maps to . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpscfv 13466 The value of the pair function at an element of . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpsfrnel 13467* Elementhood in the target space of the function appearing in xpsval 13476. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpsfeq 13468 A function on is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)

Theoremxpsfrnel2 13469* Elementhood in the target space of the function appearing in xpsval 13476. (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremxpscf 13470 Equivalent condition for the pair function to be a proper function on . (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxpsfval 13471* The value of the function appearing in xpsval 13476. (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremxpsff1o 13472* The function appearing in xpsval 13476 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair . (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremxpsfrn 13473* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremxpsfrn2 13474* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremxpsff1o2 13475* The function appearing in xpsval 13476 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremxpsval 13476* Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.)
s                                           Scalar       s        s

Theoremxpslem 13477* The indexed structure product that appears in xpsval 13476 has the same base as the target of the function . (Contributed by Mario Carneiro, 15-Aug-2015.)
s                                           Scalar       s

Theoremxpsbas 13478 The base set of the binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
s

Theoremxpsaddlem 13479* Lemma for xpsadd 13480 and xpsmul 13481. (Contributed by Mario Carneiro, 15-Aug-2015.)
s                                                                                                          Scalars

Theoremxpsadd 13480 Value of the addition operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
s

Theoremxpsmul 13481 Value of the multiplication operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
s

Theoremxpssca 13482 Value of the scalar field of a binary structure product. For concreteness, we choose the scalar field to match the left argument, but in most cases where this slot is meaningful both factors will have the same scalar field, so that it doesn't matter which factor is chosen. (Contributed by Mario Carneiro, 15-Aug-2015.)
s        Scalar                     Scalar

Theoremxpsvsca 13483 Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
s        Scalar

Theoremxpsless 13484 Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
s

Theoremxpsle 13485 Value of the ordering in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

7.2  Moore spaces

Syntaxcmre 13486 The class of Moore systems.
Moore

Syntaxcmrc 13487 The class function generating Moore closures.
mrCls

Syntaxcmri 13488 mrInd is a class function which takes a Moore system to its set of independent sets.
mrInd

Syntaxcacs 13489 The class of algebraic closure (Moore) systems.
ACS

Definitiondf-mre 13490* Define a Moore collection, which is a family of subsets of a base set which preserve arbitrary intersection. Elements of a Moore collection are termed closed; Moore collections generalize the notion of closedness from topologies (cldmre 16817) and vector spaces (lssmre 15725) to the most general setting in which such concepts make sense. Definition of Moore collection of sets in [Schechter] p. 78. A Moore collection may also be called a closure system (Section 0.6 in [Gratzer] p. 23.) The name Moore collection is after Eliakim Hastings Moore, who discussed these systems in Part I of [Moore] p. 53 to 76.

See ismre 13494, mresspw 13496, mre1cl 13498 and mreintcl 13499 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13504); as such the disjoint union of all Moore collections is sometimes considered as Moore, justified by mreunirn 13505. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)

Moore

Definitiondf-mrc 13491* Define the Moore closure of a generating set, which is the smallest closed set containing all generating elements. Definition of Moore closure in [Schechter] p. 79. This generalizes topological closure (mrccls 16818) and linear span (mrclsp 15748).

A Moore closure operation is (1) extensive, i.e., for all subsets of the base set (mrcssid 13521), (2) isotone, i.e., implies that for all subsets and of the base set (mrcss 13520), and (3) idempotent, i.e., for all subsets of the base set (mrcidm 13523.) Operators satisfying these three properties are in bijective correspondence with Moore collections, so these properties may be used to give an alternate characterization of a Moore collection by providing a closure operation on the set of subsets of a given base set which satisfies (1), (2), and (3); the closed sets can be recovered as those sets which equal their closures (Section 4.5 in [Schechter] p. 82.) (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by David Moews, 1-May-2017.)

mrCls Moore

Definitiondf-mri 13492* In a Moore system, a set is independent if no element of the set is in the closure of the set with the element removed (Section 0.6 in [Gratzer] p. 27; Definition 4.1.1 in [FaureFrolicher] p. 83.) mrInd is a class function which takes a Moore system to its set of independent sets. (Contributed by David Moews, 1-May-2017.)
mrInd Moore mrCls

Definitiondf-acs 13493* An important subclass of Moore systems are those which can be interpreted as closure under some collection of operators of finite arity (the collection itself is not required to be finite). These are termed algebraic closure systems; similar to definition (A) of an algebraic closure system in [Schechter] p. 84, but to avoid the complexity of an arbitrary mixed collection of functions of various arities (especially if the axiom of infinity omex 7346 is to be avoided), we consider a single function defined on finite sets instead. (Contributed by Stefan O'Rear, 2-Apr-2015.)
ACS Moore

Theoremismre 13494* Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremfnmre 13495 The Moore collection generator is a well-behaved function. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremmresspw 13496 A Moore collection is a subset of the power of the base set; each closed subset of the system is actually a subset of the base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremmress 13497 A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Moore

Theoremmre1cl 13498 In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremmreintcl 13499 A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremmreiincl 13500* A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Moore

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