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Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremprdsleval 13701* Value of the product ordering in a structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
s

Theoremprdsdsval 13702* Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremprdsvscaval 13703* Scalar multiplication in a structure product is pointwise. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theoremprdsvscafval 13704 Scalar multiplication of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theoremprdsbas3 13705* The base set of an indexed structure product. (Contributed by Mario Carneiro, 13-Sep-2015.)
s

Theoremprdsbasmpt2 13706* A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Mario Carneiro, 3-Jul-2015.) (Revised by Mario Carneiro, 13-Sep-2015.)
s

Theoremprdsbascl 13707* An element of the base has projections closed in the factors. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremprdsdsval2 13708* Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremprdsdsval3 13709* Value of the metric in a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theorempwsval 13710 Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
s        Scalar       s

Theorempwsbas 13711 Base set of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempwselbasb 13712 Membership in the base set of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s

Theorempwselbas 13713 An element of a structure power is a function from the index set to the base set of the structure. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
s

Theorempwsplusgval 13714 Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempwsmulrval 13715 Value of multiplication in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempwsle 13716 Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.)
s

Theorempwsleval 13717* Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.)
s

Theorempwsvscafval 13718 Scalar multiplication in a structure power is pointwise. (Contributed by Mario Carneiro, 11-Jan-2015.)
s                             Scalar

Theorempwsvscaval 13719 Scalar multiplication of a single coordinate in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
s                             Scalar

Theorempwssca 13720 The ring of scalars of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        Scalar       Scalar

Theorempwsdiagel 13721 Membership of diagonal elements in the structure power base set. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s

Theorempwssnf1o 13722* Triviality of singleton powers: set equipollence. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s

7.1.4  Definition of the structure quotient

Syntaxcordt 13723 Extend class notation with the order topology.
ordTop

Syntaxcxrs 13724 Extend class notation with the extended real number structure.

Syntaxc0g 13725 Extend class notation with group identity element.

Syntaxcgsu 13726 Extend class notation to include finitely supported group sums.
g

Definitiondf-ordt 13727* 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 13728* The extended real number structure. Unlike df-cnfld 16706, 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 16706. 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 13729* Define group identity element. (Contributed by NM, 20-Aug-2011.)

Definitiondf-gsum 13730* 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 18158. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)

g

Syntaxcqtop 13731 Extend class notation with the quotient topology function.
qTop

Syntaxcimas 13732 Image structure function.
s

Syntaxcqus 13733 Quotient structure function.
s

Syntaxcxps 13734 Binary product structure function.
s

Definitiondf-qtop 13735* Define the quotient topology given a function and topology on the domain of . (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Definitiondf-imas 13736* 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 4893, 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 4892) and/or ( with df-ress 13478) to their common domain. (Contributed by Mario Carneiro, 23-Feb-2015.)

s Scalar Scalar Scalar TopSet qTop g

Definitiondf-divs 13737* Define a quotient ring (or quotient group), which is a special case of an image structure df-imas 13736 where the image function is . (Contributed by Mario Carneiro, 23-Feb-2015.)
s s

Definitiondf-xps 13738* Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.)
s s Scalars

Theoremimasval 13739* 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 13740 The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
s

Theoremimasds 13741* The distance function of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
s                                           g

Theoremimasdsfn 13742 The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremimasdsval 13743* The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
s                                                                g

Theoremimasdsval2 13744* The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
s                                                                       g

Theoremimasplusg 13745* The group operation in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
s

Theoremimasmulr 13746* The ring multiplication in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
s

Theoremimassca 13747 The scalar field of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s                             Scalar       Scalar

Theoremimasvsca 13748* The scalar multiplication operation of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s                             Scalar

Theoremimastset 13749 The topology of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s                                    TopSet       qTop

Theoremimasle 13750 The ordering of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremf1ocpbllem 13751 Lemma for f1ocpbl 13752. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremf1ocpbl 13752 An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremf1ovscpbl 13753 An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremf1olecpbl 13754 An injection is compatible with any relations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremimasaddfnlem 13755* The image structure operation is a function if the original operation is compatible with the function. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremimasaddvallem 13756* The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremimasaddflem 13757* The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremimasaddfn 13758* 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 13759* The value of an image structure's group operation. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremimasaddf 13760* The image structure's group operation is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremimasmulfn 13761* The image structure's ring multiplication is a function. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremimasmulval 13762* The value of an image structure's ring multiplication. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremimasmulf 13763* The image structure's ring multiplication is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremimasvscafn 13764* The image structure's scalar multiplication is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
s                             Scalar

Theoremimasvscaval 13765* The value of an image structure's scalar multiplication. (Contributed by Mario Carneiro, 24-Feb-2015.)
s                             Scalar

Theoremimasvscaf 13766* The image structure's scalar multiplication is closed in the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
s                             Scalar

Theoremimasless 13767 The order relation defined on an image set is a subset of the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremimasleval 13768* 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 13769* Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s                                    s

Theoremdivslem 13770* The function in divsval 13769 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremdivsin 13771 Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
s                                    s

Theoremdivsbas 13772 Base set of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
s

Theoremdivssca 13773 The scalar field of a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s                             Scalar       Scalar

Theoremdivsfval 13774* Value of the function in divsval 13769. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremercpbllem 13775* Lemma for ercpbl 13776. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremercpbl 13776* Translate the function compatiblity relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremerlecpbl 13777* Translate the relation compatiblity relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremdivsaddvallem 13778* Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsaddflem 13779* The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsaddval 13780* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsaddf 13781* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsmulval 13782* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremdivsmulf 13783* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremxpsc 13784 A short expression for the pair function mapping to and to . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpscg 13785 A short expression for the pair function mapping to and to . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpscfn 13786 The pair function is a function on . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpsc0 13787 The pair function maps to . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpsc1 13788 The pair function maps to . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpscfv 13789 The value of the pair function at an element of . (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpsfrnel 13790* Elementhood in the target space of the function appearing in xpsval 13799. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremxpsfeq 13791 A function on is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)

Theoremxpsfrnel2 13792* Elementhood in the target space of the function appearing in xpsval 13799. (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremxpscf 13793 Equivalent condition for the pair function to be a proper function on . (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxpsfval 13794* The value of the function appearing in xpsval 13799. (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremxpsff1o 13795* The function appearing in xpsval 13799 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair . (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremxpsfrn 13796* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremxpsfrn2 13797* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)

Theoremxpsff1o2 13798* The function appearing in xpsval 13799 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremxpsval 13799* Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.)
s                                           Scalar       s        s

Theoremxpslem 13800* The indexed structure product that appears in xpsval 13799 has the same base as the target of the function . (Contributed by Mario Carneiro, 15-Aug-2015.)
s                                           Scalar       s

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