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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremzrfld 24601 The zero ring is not a field. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)

Theoremzerdivemp1 24602* In a unitary ring a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.)
GId              GId

Theoremrngoridfz 24603* In a unitary ring a left invertible element is different from zero iff . (Contributed by FL, 18-Apr-2010.)
GId              GId

Theoremzintdom 24604 is a commutative ring. (Contributed by FL, 18-Apr-2010.)

Syntaxctofld 24605 Extend class notation with the class of all totally ordered fields.

Definitiondf-tofld 24606* Definition of a totally ordered field. Experimental. (Contributed by FL, 27-Jun-2011.)
GId

Syntaxczerodiv 24607 Extend class notation with the class of all the zero divisors.

Definitiondf-zd 24608* Definition of the zero divisors of a ring. Experimental. (Contributed by FL, 27-Jun-2011.)
GId GId GId

16.11.20  Ideals

Syntaxcidln 24609 Extend class notation with the class of ideals.
IdlNEW

Definitiondf-idlNEW 24610* Define the class of (two-sided) ideals of a ring . A subset of is an ideal if it contains , is closed under addition, and is closed under multiplication on either side by any element of . (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
IdlNEW

TheoremidlvalNEW 24611* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
IdlNEW

TheoremisidlNEW 24612* The predicate "is an ideal of the ring ." (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
IdlNEW

16.11.21  Generic modules and vector spaces (New Structure builder)

Syntaxcact 24613 Extend class notation to include actions.

Definitiondf-act 24614* Definition of an action law. The action is the function ( k ^m ( v ^m v ). Definitions equivalent through currying. (Contributed by FL, 24-Dec-2013.)
Scalar

16.11.22  Generic modules and vector spaces

Syntaxcvec 24615 Extend class notation with the class of all generic vector spaces and modules.

Definitiondf-vec 24616* Definition of a vector space ( is a field ), or of a module ( is a ring ). (Contributed by FL, 12-Jul-2010.)
GId

Theoremvecval1b 24617* The predicate "is a vector space" or "is a module". (Contributed by FL, 12-Jul-2010.)
GId

Theoremvecval3b 24618* The "axioms" of a vector space or module. (Contributed by FL, 12-Jul-2010.)
GId

Theoremvecax1 24619 1st "axiom" of a vector space or module. The vector addition is an abelian group. (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by FL, 14-Sep-2010.)

Theoremvecax2 24620 2nd "axiom" of a vector space or module. Domain, codomain and functionality of the multiplication of a vector by a scalar. (Contributed by FL, 14-Sep-2010.)

Theoremvecax3 24621* 3rd "axiom" of a vector space or module. Multiplication by 1. (Contributed by FL, 13-Sep-2010.)
GId

Theoremvecax4 24622* 4th "axiom" of a vector space or module. Multiplication by a scalar distributes over vector addition. (Contributed by FL, 13-Sep-2010.)

Theoremvecax5 24623* 5th "axiom" of a vector space or module. Multiplication by a sum of scalars. (Contributed by FL, 13-Sep-2010.)

Theoremvecax6 24624* 6th "axiom" of a vector space or module. Relation between scalar multiplication and vector multiplication. (Contributed by FL, 13-Sep-2010.)

Theoremvecax5b 24625 Multiplication by a sum of scalars. (Contributed by FL, 13-Sep-2010.)

Theoremcladdinvvec 24626 Closure of the additive inverse of a vector. (Contributed by FL, 13-Sep-2010.)

Theoremvec2inv 24627 Double inverse law for vector additive inverse. (Contributed by FL, 13-Sep-2010.)

Theoremsum2vv 24628 The sum of two vectors is a vector. (Contributed by FL, 13-Sep-2010.)

Theoremaddnull1 24629 Addition of the null vector. (Contributed by FL, 13-Sep-2010.)
GId

Theoremaddnull2 24630 Addition of the null vector. (Contributed by FL, 13-Sep-2010.)
GId

Theoremaddvecass 24631 The addition of vectors is associative. (Contributed by FL, 13-Sep-2010.)

Theoremaddvecom 24632 The addition of vectors is associative. (Contributed by FL, 13-Sep-2010.)

Theoreminvaddvec 24633 Additive inverse of a sum of vectors. (Contributed by FL, 13-Sep-2010.)

Theoremprodvs 24634 The product of a vector by a scalar is a vector. (Contributed by FL, 12-Sep-2010.)

Theoremvecsrcan 24635 Right cancellation law for vector subtraction. (Contributed by FL, 12-Sep-2010.)

Theoremvecslcan 24636 Left cancellation law for vector subtraction. (Contributed by FL, 12-Sep-2010.)

Theoremvwit 24637 A vector minus itself equals zero. (Contributed by FL, 12-Sep-2010.)
GId

Theoremsub2vec 24638 Definition of the subtraction of two vectors. (Contributed by FL, 12-Sep-2010.)
GId

Theoremmvecrtol 24639 Moving a vector from the right member of an equation into the left member. (Contributed by FL, 12-Sep-2010.)
GId

Theoremdblsubvec 24640 Double subtraction of vectors. (Contributed by FL, 12-Sep-2010.)
GId

Theoremvecrcan 24641 Right cancellation law for vector addition. (Contributed by FL, 12-Sep-2010.)
GId

Theoremveclcan 24642 Left cancellation law for vector addition. (Contributed by FL, 12-Sep-2010.)
GId

Theoremmvecrtol2 24643 Moving a vector from the right member of an equation into the left member. (Contributed by FL, 12-Sep-2010.)
GId

Theoremprvs 24644 The product of a vector by a scalar is a vector. (Contributed by FL, 12-Sep-2010.)

Theoremmulveczer 24645 Multiplication of a vector by zero. (Contributed by FL, 12-Sep-2010.)
GId                            GId

Theoremmulinvsca 24646 Multiplication by the inverse of a scalar. (Contributed by FL, 12-Sep-2010.)

Theoremmuldisc 24647* Multiplication by a difference of scalars. (Contributed by FL, 12-Sep-2010.)

Theoremglmrngo 24648 Generating a left module from a ring. (Contributed by FL, 29-May-2014.)

Theoremvecax5c 24649 Multiplication by a difference of scalars. (Contributed by FL, 12-Sep-2010.)

Theoremsvli2 24650* If a finite sequence of vectors are linearly independant, two combinations of those vectors are equal iff the scalars are equal. (Contributed by FL, 9-Nov-2010.)
GId                     GId

Syntaxcsvec 24651 Extend class notation with the class of all generic subspace vector spaces and modules.

Definitiondf-svs 24652* A sub-vector space of a vector space is a vector space that has the same scalar set than , whose addition and whose multiplication are restrictions of those of . (Contributed by FL, 30-Dec-2010.)

Theoremsvs2 24653* A textbook definition. A sub-vector space of a vector space is a subset that is itself a vector space under the inherited operations. (Contributed by FL, 31-Dec-2010.)

Theoremsvs3 24654* A very concise definition of a subspace of a vector space. (Contributed by FL, 30-Dec-2010.)

16.11.23  Real vector spaces

Syntaxcvr 24655 Extend class notation with the class of all real vector spaces.

Definitiondf-vr 24656* Define the class of all real vector spaces. The definition of a and a don't much differ. There may be a way to get both in only one definition. A seems mandatory if one wants to do classical cartesian geometry. We can't use a instead. Changing the field changes important properties such as the dimension. (Contributed by FL, 16-Nov-2008.)

Theoremvrrel 24657 The class of all real vector spaces is a relation. (Contributed by FL, 16-Nov-2008.)

Theoremvri 24658* The properties of a real vector space, which is an abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of real numbers. The variable was chosen because is already used for the universal class. (Contributed by FL, 16-Nov-2008.)

16.11.24  Matrices

Syntaxcmmat 24659 Addition of matrices.

Syntaxcsmat 24660 Scalar multiplication of matrices.

Syntaxcxmat 24661 Multiplication of matrices.

Definitiondf-amat 24662* Matrix addition. Meaningful if is a (a binary internal operation) at least. Experimental. (Contributed by FL, 29-Aug-2010.)

Definitiondf-smat 24663* Matrix left scalar multiplication. Meaningful if is a binary external operation. Experimental. (Contributed by FL, 29-Aug-2010.)

Definitiondf-mmat 24664* Matrix multiplication. Meaningful if is a ring at least. here should be (to be traditional). But in set.mm is oriented and has a limit definition embedded and thus doesn't fit the needs of this generic definition. Experimental. (Contributed by FL, 29-Aug-2010.)

16.11.25  Affine spaces

Syntaxcraffsp 24665 Extend class notation with the class of all R affine spaces.

Definitiondf-raffsp 24666* Define a affine space id est a vector space ( called the free vectors class ) together with a function . associates to each vector a bijection from a set (called the space) to itself (here is retrieved from the operation.) Technically speaking, is a faithful (i.e. injective) and transitive group action (id est a group homomorphism whose range is the underlying set of a symmetry group ). Informally speaking the aim of all of that is to associate to each point of a unique point of through the "action" of a vector of and thus to formalize the idea of translation. When we have embedded the idea of translation it is easy to define a repere and thus all the cartesian geometry is available. (Contributed by FL, 29-Aug-2010.)
GrpOpHom

16.11.26  Intervals of reals and extended reals

Theorembsi 24667* Membership to the set of open intervals implied the existence of two bounds in the set of the extended reals. (Contributed by FL, 31-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo1t3 24668 If an open interval has an element, then . (Contributed by FL, 14-Aug-2007.)

Theoremoisbmi 24669 An open interval with its upper bound equal to is empty. (Contributed by FL, 12-Sep-2007.)

Theoremoisbmj 24670 An open interval with its lower bound equal to is empty. (Contributed by FL, 12-Sep-2007.)

Theoremtruni1 24671 Closure of translation in a half-infinite interval. (Contributed by FL, 11-Sep-2007.)

Theoremtruni2 24672 Closure of translation in a half-infinite interval. (Contributed by FL, 26-Jan-2009.)

Theoremtruni3 24673 Closure of translation in a half-infinite interval. (Contributed by FL, 26-Jan-2009.)

Theoremcbci 24674 The center belongs to a centered interval. (Contributed by FL, 5-Jan-2009.)

Theoremoibbi1 24675 An open interval is included in a bound below interval. (Contributed by FL, 26-Jan-2009.) (Revised by Mario Carneiro, 3-May-2015.)

Theoremoibbi2 24676 An open interval is included in a bound above interval. (Contributed by FL, 26-Jan-2009.)

Theoremnelioo5 24677 Membership in an open interval of extended reals. (Contributed by FL, 7-Dec-2010.)

16.11.27  Topology

Theoremtopnem 24678 A topology is not empty. (Contributed by FL, 1-Jun-2008.)

Theoremclsint 24679 The closure of an intersection is included in the intersection of the closures. (Contributed by FL, 23-Feb-2009.)

Theoremislp3 24680* The predicate " is a limit point of " in terms of open sets. see islp2 16709, elcls 16642, islp 16704. (Contributed by FL, 31-Jul-2009.)

Theoreminttop2 24681* The intersection of a family of topologies is a topology. (Contributed by FL, 19-Sep-2011.)

Theoreminttop3 24682 The intersection of a family of topologies is a topology. (Contributed by FL, 19-Sep-2011.)

Theoreminttop4 24683 The intersection of two topologies is a topology. (Contributed by FL, 19-Sep-2011.)

Theoremunint2t 24684 The intersection of two topologies over the same underlying set is a topology over . compare uniin 3747. (Contributed by FL, 27-Nov-2011.)

Theoremintfmu2 24685* The intersection of a family of topologies over the same underying set is a topology over . (Contributed by FL, 27-Nov-2011.)

Theoremapnei 24686* Any point has a neighborhood. (Contributed by FL, 15-Oct-2012.)

Theoremnpmp 24687 A neighborhood of a point can't be empty. (Contributed by FL, 15-Oct-2012.)

Theorembasexre 24688 A basis for the standard topology over the extended reals. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)

Theoremstovr 24689 The standard topology over . (Contributed by FL, 15-Sep-2013.)

Theoremcldifemp 24690 The closure of a class is empty iff is empty. (Contributed by FL, 15-Sep-2013.)

16.11.28  Continuous functions

Theoremcnrsfin 24691 A mapping remains continuous when the topology associated to its domain is replaced by a finer one. (Contributed by FL, 22-May-2008.)

Theoremcnrscoa 24692 A mapping remains continuous when the topology associated to its range is replaced by a coarser one. (Contributed by FL, 1-Jun-2008.)

Theoremmapdiscn 24693 Any mapping whose domain is associated to the discrete topology is continuous. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Mario Carneiro, 7-Apr-2015.)

Theoremmapudiscn 24694 Any mapping whose range is associated to the undiscrete topology is continuous. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremsallnei 24695* Two ways to state the set of all the neighborhoods. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremnsn 24696* The neighborhoods of the singletons are neighborhoods. (Contributed by FL, 2-Aug-2009.)

Theoremosneisi 24697* The non empty open sets are neighborhoods of the singletons. (Contributed by FL, 16-Jul-2009.)

Theoremelsubops 24698 The elements of a subbase are open sets. (Contributed by FL, 16-Apr-2012.) (Revised by Mario Carneiro, 14-Dec-2013.)

16.11.29  Homeomorphisms

Theoremdmhmph 24699 is a relation whose domain is included in . (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremrnhmph 24700 is a relation whose range is included in . (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)

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