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Type | Label | Description |
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Statement | ||
Theorem | pwfi2en 36001* | Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.) |
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Theorem | frlmpwfi 36002 | Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Proof shortened by AV, 14-Jun-2020.) |
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Theorem | gicabl 36003 | Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) |
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Theorem | imasgim 36004 | A relabeling of the elements of a group induces an isomorphism to the relabeled group. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) (Revised by Mario Carneiro, 11-Aug-2015.) |
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Theorem | basfn 36005 | Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) |
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Theorem | isnumbasgrplem1 36006 | A set which is equipollent to the base set of a definable Abelian group is the base set of some (relabeled) Abelian group. (Contributed by Stefan O'Rear, 8-Jul-2015.) |
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Theorem | harn0 36007 | The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.) |
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Theorem | numinfctb 36008 | A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.) |
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Theorem | isnumbasgrplem2 36009 | If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
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Theorem | isnumbasgrplem3 36010 | Every nonempty numerable set can be given the structure of an Abelian group, either a finite cyclic group or a vector space over Z/2Z. (Contributed by Stefan O'Rear, 10-Jul-2015.) |
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Theorem | isnumbasabl 36011 | A set is numerable iff it and its Hartogs number can be jointly given the structure of an Abelian group. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
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Theorem | isnumbasgrp 36012 | A set is numerable iff it and its Hartogs number can be jointly given the structure of a group. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
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Theorem | dfacbasgrp 36013 | A choice equivalent in abstract algebra: All nonempty sets admit a group structure. From http://mathoverflow.net/a/12988. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
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Syntax | clnr 36014 | Extend class notation with the class of left Noetherian rings. |
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Definition | df-lnr 36015 | A ring is left-Noetherian iff it is Noetherian as a left module over itself. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
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Theorem | islnr 36016 | Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
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Theorem | lnrring 36017 | Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
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Theorem | lnrlnm 36018 | Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
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Theorem | islnr2 36019* | Property of being a left-Noetherian ring in terms of finite generation of ideals (the usual "pure ring theory" definition). (Contributed by Stefan O'Rear, 24-Jan-2015.) |
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Theorem | islnr3 36020 | Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | lnr2i 36021* | Given an ideal in a left-Noetherian ring, there is a finite subset which generates it. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
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Theorem | lpirlnr 36022 | Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
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Theorem | lnrfrlm 36023 | Finite-dimensional free modules over a Noetherian ring are Noetherian. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
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Theorem | lnrfg 36024 | Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.) |
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Theorem | lnrfgtr 36025 | A submodule of a finitely generated module over a Noetherian ring is finitely generated. Often taken as the definition of Noetherian ring. (Contributed by Stefan O'Rear, 7-Feb-2015.) |
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Syntax | cldgis 36026 | The leading ideal sequence used in the Hilbert Basis Theorem. |
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Definition | df-ldgis 36027* |
Define a function which carries polynomial ideals to the sequence of
coefficient ideals of leading coefficients of degree- ![]() |
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Theorem | hbtlem1 36028* | Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
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Theorem | hbtlem2 36029 | Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
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Theorem | hbtlem7 36030 | Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | hbtlem4 36031 | The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
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Theorem | hbtlem3 36032 | The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
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Theorem | hbtlem5 36033* | The leading ideal function is strictly monotone. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
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Theorem | hbtlem6 36034* | There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
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Theorem | hbt 36035 | The Hilbert Basis Theorem - the ring of univariate polynomials over a Noetherian ring is a Noetherian ring. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Syntax | cmnc 36036 | Extend class notation with the class of monic polynomials. |
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Syntax | cplylt 36037 | Extend class notatin with the class of limited-degree polynomials. |
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Definition | df-mnc 36038* | Define the class of monic polynomials. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
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Definition | df-plylt 36039* | Define the class of limited-degree polynomials. (Contributed by Stefan O'Rear, 8-Dec-2014.) |
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Theorem | dgrsub2 36040 | Subtracting two polynomials with the same degree and top coefficient gives a polynomial of strictly lower degree. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | elmnc 36041 | Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
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Theorem | mncply 36042 | A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
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Theorem | mnccoe 36043 | A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
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Theorem | mncn0 36044 | A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
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Syntax | cdgraa 36045 | Extend class notation to include the degree function for algebraic numbers. |
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Syntax | cdgraaold 36046 | Extend class notation to include the degree function for algebraic numbers (old version). |
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Syntax | cmpaa 36047 | Extend class notation to include the minimal polynomial for an algebraic number. |
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Definition | df-dgraa 36048* | Define the degree of an algebraic number as the smallest degree of any nonzero polynomial which has said number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.) |
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Definition | df-dgraaOLD 36049* | Define the degree of an algebraic number as the smallest degree of any nonzero polynomial which has said number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of df-dgraa 36048 as of 29-Sep-2020. (New usage is discouraged.) |
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Definition | df-mpaa 36050* | Define the minimal polynomial of an algebraic number as the unique monic polynomial which achieves the minimum of degAA. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | dgraaval 36051* | Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.) |
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Theorem | dgraavalOLD 36052* | Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraaval 36051 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | dgraalem 36053* | Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.) |
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Theorem | dgraalemOLD 36054* | Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraalem 36053 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | dgraacl 36055 | Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | dgraaclOLD 36056 | Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraacl 36055 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | dgraaf 36057 | Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.) |
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Theorem | dgraafOLD 36058 | Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraaf 36057 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | dgraaub 36059 | Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.) |
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Theorem | dgraaubOLD 36060 | Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraaub 36059 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | dgraa0p 36061 | A rational polynomial of degree less than an algebraic number cannot be zero at that number unless it is the zero polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | mpaaeu 36062* | An algebraic number has exactly one monic polynomial of the least degree. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | mpaaval 36063* | Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | mpaalem 36064 | Properties of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | mpaacl 36065 | Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | mpaadgr 36066 | Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | mpaaroot 36067 | The minimal polynomial of an algebraic number has the number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | mpaamn 36068 | Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Syntax | citgo 36069 | Extend class notation with the integral-over predicate. |
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Syntax | cza 36070 | Extend class notation with the class of algebraic integers. |
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Definition | df-itgo 36071* |
A complex number is said to be integral over a subset if it is the root
of a monic polynomial with coefficients from the subset. This
definition is typically not used for fields but it works there, see
aaitgo 36074. This definition could work for subsets of
an arbitrary ring
with a more general definition of polynomials. TODO: use ![]() |
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Definition | df-za 36072 | Define an algebraic integer as a complex number which is the root of a monic integer polynomial. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
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Theorem | itgoval 36073* | Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
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Theorem | aaitgo 36074 |
The standard algebraic numbers ![]() |
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Theorem | itgoss 36075 | An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
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Theorem | itgocn 36076 | All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
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Theorem | cnsrexpcl 36077 | Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
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Theorem | fsumcnsrcl 36078* | Finite sums are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
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Theorem | cnsrplycl 36079 | Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
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Theorem | rgspnval 36080* | Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
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Theorem | rgspncl 36081 | The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
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Theorem | rgspnssid 36082 | The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
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Theorem | rgspnmin 36083 | The ring-span is contained in all subspaces which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
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Theorem | rgspnid 36084 | The span of a subring is itself. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
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Theorem | rngunsnply 36085* | Adjoining one element to a ring results in a set of polynomial evaluations. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
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Theorem | flcidc 36086* | Finite linear combinations with an indicator function. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
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Syntax | cmend 36087 | Syntax for module endomorphism algebra. |
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Definition | df-mend 36088* | Define the endomorphism algebra of a module. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
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Theorem | algstr 36089 | Lemma to shorten proofs of algbase 36090 through algvsca 36094. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) |
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Theorem | algbase 36090 | The base set of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) |
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Theorem | algaddg 36091 | The additive operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) |
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Theorem | algmulr 36092 | The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) |
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Theorem | algsca 36093 | The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) |
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Theorem | algvsca 36094 | The scalar product operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) |
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Theorem | mendval 36095* | Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
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Theorem | mendbas 36096 | Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
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Theorem | mendplusgfval 36097* | Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
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Theorem | mendplusg 36098 | A specific addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.) |
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Theorem | mendmulrfval 36099* | Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
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Theorem | mendmulr 36100 | A specific multiplication in the module endormoprhism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.) |
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