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Type | Label | Description |
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Statement | ||
Theorem | tsetid 12701 | Utility theorem: index-independent form of df-tset 12611. (Contributed by NM, 20-Oct-2012.) |
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Theorem | tsetslid 12702 | Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.) |
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Theorem | tsetndxnn 12703 | The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.) |
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Theorem | basendxlttsetndx 12704 | The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.) |
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Theorem | tsetndxnbasendx 12705 | The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.) |
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Theorem | tsetndxnplusgndx 12706 | The slot for the topology is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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Theorem | tsetndxnmulrndx 12707 | The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.) |
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Theorem | tsetndxnstarvndx 12708 | The slot for the topology is not the slot for the involution in an extensible structure. (Contributed by AV, 11-Nov-2024.) |
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Theorem | slotstnscsi 12709 |
The slots Scalar, ![]() ![]() |
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Theorem | topgrpstrd 12710 | A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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Theorem | topgrpbasd 12711 | The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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Theorem | topgrpplusgd 12712 | The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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Theorem | topgrptsetd 12713 | The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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Theorem | plendx 12714 | Index value of the df-ple 12612 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) |
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Theorem | pleid 12715 | Utility theorem: self-referencing, index-independent form of df-ple 12612. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.) |
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Theorem | pleslid 12716 |
Slot property of ![]() |
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Theorem | plendxnn 12717 | The index value of the order slot is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 30-Oct-2024.) |
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Theorem | basendxltplendx 12718 |
The index value of the ![]() ![]() |
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Theorem | plendxnbasendx 12719 | The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.) |
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Theorem | plendxnplusgndx 12720 | The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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Theorem | plendxnmulrndx 12721 | The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 1-Nov-2024.) |
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Theorem | plendxnscandx 12722 | The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. (Contributed by AV, 1-Nov-2024.) |
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Theorem | plendxnvscandx 12723 | The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. (Contributed by AV, 1-Nov-2024.) |
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Theorem | slotsdifplendx 12724 | The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.) |
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Theorem | dsndx 12725 | Index value of the df-ds 12614 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | dsid 12726 | Utility theorem: index-independent form of df-ds 12614. (Contributed by Mario Carneiro, 23-Dec-2013.) |
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Theorem | dsslid 12727 |
Slot property of ![]() |
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Theorem | dsndxnn 12728 | The index of the slot for the distance in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.) |
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Theorem | basendxltdsndx 12729 | The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. (Contributed by AV, 28-Oct-2024.) |
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Theorem | dsndxnbasendx 12730 | The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.) |
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Theorem | dsndxnplusgndx 12731 | The slot for the distance function is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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Theorem | dsndxnmulrndx 12732 | The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.) |
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Theorem | slotsdnscsi 12733 |
The slots Scalar, ![]() ![]() ![]() |
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Theorem | dsndxntsetndx 12734 | The slot for the distance function is not the slot for the topology in an extensible structure. (Contributed by AV, 29-Oct-2024.) |
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Theorem | slotsdifdsndx 12735 | The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.) |
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Theorem | unifndx 12736 | Index value of the df-unif 12615 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.) (New usage is discouraged.) |
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Theorem | unifid 12737 | Utility theorem: index-independent form of df-unif 12615. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
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Theorem | unifndxnn 12738 | The index of the slot for the uniform set in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.) |
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Theorem | basendxltunifndx 12739 | The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. (Contributed by AV, 28-Oct-2024.) |
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Theorem | unifndxnbasendx 12740 | The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) |
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Theorem | unifndxntsetndx 12741 | The slot for the uniform set is not the slot for the topology in an extensible structure. (Contributed by AV, 28-Oct-2024.) |
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Theorem | slotsdifunifndx 12742 | The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.) |
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Theorem | homid 12743 | Utility theorem: index-independent form of df-hom 12616. (Contributed by Mario Carneiro, 7-Jan-2017.) |
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Theorem | homslid 12744 |
Slot property of ![]() |
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Theorem | ccoid 12745 | Utility theorem: index-independent form of df-cco 12617. (Contributed by Mario Carneiro, 7-Jan-2017.) |
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Theorem | ccoslid 12746 | Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.) |
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Syntax | crest 12747 | Extend class notation with the function returning a subspace topology. |
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Syntax | ctopn 12748 | Extend class notation with the topology extractor function. |
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Definition | df-rest 12749* |
Function returning the subspace topology induced by the topology ![]() ![]() |
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Definition | df-topn 12750 | Define the topology extractor function. This differs from df-tset 12611 when a structure has been restricted using df-iress 12523; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.) |
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Theorem | restfn 12751 | The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.) |
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Theorem | topnfn 12752 | The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.) |
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Theorem | restval 12753* |
The subspace topology induced by the topology ![]() ![]() |
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Theorem | elrest 12754* | The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
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Theorem | elrestr 12755 | Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
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Theorem | restid2 12756 | The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
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Theorem | restsspw 12757 | The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
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Theorem | restid 12758 | The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.) |
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Theorem | topnvalg 12759 | Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.) |
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Theorem | topnidg 12760 | Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.) |
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Theorem | topnpropgd 12761 | The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.) |
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Syntax | ctg 12762 | Extend class notation with a function that converts a basis to its corresponding topology. |
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Syntax | cpt 12763 | Extend class notation with a function whose value is a product topology. |
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Syntax | c0g 12764 | Extend class notation with group identity element. |
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Syntax | cgsu 12765 | Extend class notation to include finitely supported group sums. |
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Definition | df-0g 12766* |
Define group identity element. Remark: this definition is required here
because the symbol ![]() |
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Definition | df-igsum 12767* |
Define a finite group sum (also called "iterated sum") of a
structure.
Given
1. If
2. If 3. This definition does not handle other cases. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 27-Jun-2025.) |
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Definition | df-topgen 12768* | Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78. (Contributed by NM, 16-Jul-2006.) |
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Definition | df-pt 12769* | Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.) |
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Theorem | tgval 12770* | The topology generated by a basis. See also tgval2 14028 and tgval3 14035. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
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Theorem | tgvalex 12771 | The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.) |
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Theorem | ptex 12772 | Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.) |
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Syntax | cprds 12773 | The function constructing structure products. |
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Syntax | cpws 12774 | The function constructing structure powers. |
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Definition | df-prds 12775* | Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) |
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Theorem | reldmprds 12776 | The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) |
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Theorem | prdsex 12777 | Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.) |
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Definition | df-pws 12778* | Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.) |
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Syntax | cimas 12779 | Image structure function. |
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Syntax | cqus 12780 | Quotient structure function. |
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Syntax | cxps 12781 | Binary product structure function. |
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Definition | df-iimas 12782* |
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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
Note that although we call this an "image" by association to
df-ima 4657,
in order to keep the definition simple we consider only the case when
the domain of |
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Definition | df-qus 12783* |
Define a quotient ring (or quotient group), which is a special case of
an image structure df-iimas 12782 where the image function is
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Definition | df-xps 12784* | Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.) |
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Theorem | imasex 12785 | Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.) |
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Theorem | imasival 12786* | Value of an image structure. The is a lemma for the theorems imasbas 12787, imasplusg 12788, and imasmulr 12789 and should not be needed once they are proved. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Jim Kingdon, 11-Mar-2025.) (New usage is discouraged.) |
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Theorem | imasbas 12787 | The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.) |
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Theorem | imasplusg 12788* | The group operation in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
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Theorem | imasmulr 12789* | The ring multiplication in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
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Theorem | f1ocpbllem 12790 | Lemma for f1ocpbl 12791. (Contributed by Mario Carneiro, 24-Feb-2015.) |
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Theorem | f1ocpbl 12791 | An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.) |
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Theorem | f1ovscpbl 12792 | An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.) |
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Theorem | f1olecpbl 12793 | An injection is compatible with any relations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.) |
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Theorem | imasaddfnlemg 12794* | The image structure operation is a function if the original operation is compatible with the function. (Contributed by Mario Carneiro, 23-Feb-2015.) |
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Theorem | imasaddvallemg 12795* | The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.) |
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Theorem | imasaddflemg 12796* | The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.) |
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Theorem | imasaddfn 12797* | The image structure's group operation is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.) |
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Theorem | imasaddval 12798* | The value of an image structure's group operation. (Contributed by Mario Carneiro, 23-Feb-2015.) |
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Theorem | imasaddf 12799* | The image structure's group operation is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
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Theorem | imasmulfn 12800* | The image structure's ring multiplication is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) |
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