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Theorem List for Metamath Proof Explorer - 25401-25500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltrdom 25401* The domain of a left translation. The term is a constant. (Contributed by FL, 26-Apr-2012.)

Theoremltrset 25402* A left translation is a set. (Contributed by FL, 28-Apr-2012.)

Theoremltrran2 25403* The range of a left translation. The term is a constant. (Contributed by FL, 28-Apr-2012.)

Theoremltrooo 25404* A left translation is a bijection. The term is a constant. (Contributed by FL, 29-Apr-2012.)

Theoremltrcmp 25405* Left translation expressed as a composite. (Contributed by FL, 3-Jul-2012.)

Theoremltrinvlem 25406* The converse of a left translation. The term is a constant. (Contributed by FL, 30-Apr-2012.)

Theoremcmpltr2 25407* Composite of two left translations. The terms and are constant. (Contributed by FL, 2-Jul-2012.)

Theoremcmpltr 25408* Composite of two left translations. The terms and are constant. Don't use. See cmpltr2 25407. (Contributed by FL, 2-Jul-2012.) (Revised by Mario Carneiro, 2-Jun-2014.)

Theoremcmperltr 25409* A right and left translation expressed as a composite. Note that and can't be the same. (Contributed by FL, 2-Jul-2012.)

Theoremcmprltr 25410* Composite of two right and left translations. Note that and can't be the same. See cmprltr2 25411 for a more general version. (Contributed by FL, 2-Jul-2012.) (Proof shortened by Mario Carneiro, 26-Jul-2014.)

Theoremcmprltr2 25411* Composite of two right and left translations. No restriction: and can be equal. (Contributed by FL, 2-Jul-2012.)

Theoremrltrdom 25412* The domain of a right and left translation. (Contributed by FL, 2-Jul-2012.)

Theoremrltrset 25413* A right and left translation is a set. (Contributed by FL, 2-Jul-2012.)

Theoremrltrran 25414* The range of a right and left translation. Note that and are constant. (Contributed by FL, 2-Jul-2012.)

Theoremrltrooo 25415* A right and left translation is a bijection. (Contributed by FL, 2-Jul-2012.)

18.13.19  Fields and Rings

Theoremcom2i 25416* Property of a commutative structure with two operation. (Contributed by FL, 14-Feb-2010.)

Theoremrngmgmbs3 25417* The domain of the first variable of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.)

Theoremrngodmdmrn 25418 In a unital ring the range of the multiplication equals the domain of the first variable. (Contributed by FL, 24-Jan-2010.)

Theoremrngodmeqrn 25419 In a unital ring the domain of the first operand of the addition equals the domain of the second operand of the addition. (Contributed by FL, 11-Feb-2010.)

Theoremununr 25420* The unit of a unital ring is unique. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 23-Dec-2013.)

Theoremrngoinvcl 25421 The additive inverse of a unital ring element pertains to the unital ring. (Contributed by FL, 18-Apr-2010.)

Theoremmultinv 25422 Multiplication by an additive inverse. (Contributed by FL, 2-Sep-2009.)

Theoremmultinvb 25423 Multiplication by an additive inverse. (Contributed by FL, 6-Sep-2009.)

Theoremmult2inv 25424 Multiplication of two additive inverses. (Contributed by FL, 6-Sep-2009.)

Theoremrngounval2 25425* The value of the unit of a ring. (Contributed by FL, 12-Feb-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
GId

TheoremisfldOLD 25426* The predicate "is a field". (Contributed by FL, 6-Sep-2009.)

Theoremfldi 25427* The "axioms" of a field. (Contributed by FL, 15-Sep-2010.)
GId

Theoremfldax1 25428 1st "axiom" of a field. The addition is an abelian group. (Contributed by FL, 11-Jul-2010.)

Theoremfldax2 25429 2nd "axiom" of a field. The multiplication is an internal operation. (Contributed by FL, 11-Jul-2010.)

Theoremfldax3 25430* 3rd "axiom" of a field. The multiplication is associative. (Contributed by FL, 11-Jul-2010.)

Theoremfldax4 25431* 4th "axiom" of a field. The multiplication is distributive. (Contributed by FL, 11-Jul-2010.)

Theoremfldax5 25432* 5th "axiom" of a field. Existence of a neutral element. (Contributed by FL, 11-Jul-2010.)

Theoremfldax6 25433 6th "axiom" of a field. The multiplication is a group on the underlying set deprived from zero. (Contributed by FL, 11-Jul-2010.)
GId

Theoremfldax7 25434* 7th "axiom" of a field. The multiplication is commutative. (Contributed by FL, 11-Jul-2010.)

Theoremzrfld 25435 The zero ring is not a field. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)

Theoremzerdivemp1 25436* In a unitary ring a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.)
GId              GId

Theoremrngoridfz 25437* In a unitary ring a left invertible element is different from zero iff . (Contributed by FL, 18-Apr-2010.)
GId              GId

Theoremzintdom 25438 is a commutative ring. (Contributed by FL, 18-Apr-2010.)

Syntaxctofld 25439 Extend class notation with the class of all totally ordered fields.

Definitiondf-tofld 25440* Definition of a totally ordered field. Experimental. (Contributed by FL, 27-Jun-2011.)
GId

Syntaxczerodiv 25441 Extend class notation with the class of all the zero divisors.

Definitiondf-zd 25442* Definition of the zero divisors of a ring. Experimental. (Contributed by FL, 27-Jun-2011.)
GId GId GId

18.13.20  Ideals

Syntaxcidln 25443 Extend class notation with the class of ideals.
IdlNEW

Definitiondf-idlNEW 25444* 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 25445* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
IdlNEW

TheoremisidlNEW 25446* The predicate "is an ideal of the ring ." (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
IdlNEW

18.13.21  Generic modules and vector spaces (New Structure builder)

Syntaxcact 25447 Extend class notation to include actions.

Definitiondf-act 25448* 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

18.13.22  Generic modules and vector spaces

Syntaxcvec 25449 Extend class notation with the class of all generic vector spaces and modules.

Definitiondf-vec 25450* Definition of a vector space ( is a field ), or of a module ( is a ring ). (Contributed by FL, 12-Jul-2010.)
GId

Theoremvecval1b 25451* The predicate "is a vector space" or "is a module". (Contributed by FL, 12-Jul-2010.)
GId

Theoremvecval3b 25452* The "axioms" of a vector space or module. (Contributed by FL, 12-Jul-2010.)
GId

Theoremvecax1 25453 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 25454 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 25455* 3rd "axiom" of a vector space or module. Multiplication by 1. (Contributed by FL, 13-Sep-2010.)
GId

Theoremvecax4 25456* 4th "axiom" of a vector space or module. Multiplication by a scalar distributes over vector addition. (Contributed by FL, 13-Sep-2010.)

Theoremvecax5 25457* 5th "axiom" of a vector space or module. Multiplication by a sum of scalars. (Contributed by FL, 13-Sep-2010.)

Theoremvecax6 25458* 6th "axiom" of a vector space or module. Relation between scalar multiplication and vector multiplication. (Contributed by FL, 13-Sep-2010.)

Theoremvecax5b 25459 Multiplication by a sum of scalars. (Contributed by FL, 13-Sep-2010.)

Theoremcladdinvvec 25460 Closure of the additive inverse of a vector. (Contributed by FL, 13-Sep-2010.)

Theoremvec2inv 25461 Double inverse law for vector additive inverse. (Contributed by FL, 13-Sep-2010.)

Theoremsum2vv 25462 The sum of two vectors is a vector. (Contributed by FL, 13-Sep-2010.)

Theoremaddnull1 25463 Addition of the null vector. (Contributed by FL, 13-Sep-2010.)
GId

Theoremaddnull2 25464 Addition of the null vector. (Contributed by FL, 13-Sep-2010.)
GId

Theoremaddvecass 25465 The addition of vectors is associative. (Contributed by FL, 13-Sep-2010.)

Theoremaddvecom 25466 The addition of vectors is associative. (Contributed by FL, 13-Sep-2010.)

Theoreminvaddvec 25467 Additive inverse of a sum of vectors. (Contributed by FL, 13-Sep-2010.)

Theoremprodvs 25468 The product of a vector by a scalar is a vector. (Contributed by FL, 12-Sep-2010.)

Theoremvecsrcan 25469 Right cancellation law for vector subtraction. (Contributed by FL, 12-Sep-2010.)

Theoremvecslcan 25470 Left cancellation law for vector subtraction. (Contributed by FL, 12-Sep-2010.)

Theoremvwit 25471 A vector minus itself equals zero. (Contributed by FL, 12-Sep-2010.)
GId

Theoremsub2vec 25472 Definition of the subtraction of two vectors. (Contributed by FL, 12-Sep-2010.)
GId

Theoremmvecrtol 25473 Moving a vector from the right member of an equation into the left member. (Contributed by FL, 12-Sep-2010.)
GId

Theoremdblsubvec 25474 Double subtraction of vectors. (Contributed by FL, 12-Sep-2010.)
GId

Theoremvecrcan 25475 Right cancellation law for vector addition. (Contributed by FL, 12-Sep-2010.)
GId

Theoremveclcan 25476 Left cancellation law for vector addition. (Contributed by FL, 12-Sep-2010.)
GId

Theoremmvecrtol2 25477 Moving a vector from the right member of an equation into the left member. (Contributed by FL, 12-Sep-2010.)
GId

Theoremprvs 25478 The product of a vector by a scalar is a vector. (Contributed by FL, 12-Sep-2010.)

Theoremmulveczer 25479 Multiplication of a vector by zero. (Contributed by FL, 12-Sep-2010.)
GId                            GId

Theoremmulinvsca 25480 Multiplication by the inverse of a scalar. (Contributed by FL, 12-Sep-2010.)

Theoremmuldisc 25481* Multiplication by a difference of scalars. (Contributed by FL, 12-Sep-2010.)

Theoremglmrngo 25482 Generating a left module from a ring. (Contributed by FL, 29-May-2014.)

Theoremvecax5c 25483 Multiplication by a difference of scalars. (Contributed by FL, 12-Sep-2010.)

Theoremsvli2 25484* 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 25485 Extend class notation with the class of all generic subspace vector spaces and modules.

Definitiondf-svs 25486* 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 25487* 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 25488* A very concise definition of a subspace of a vector space. (Contributed by FL, 30-Dec-2010.)

18.13.23  Real vector spaces

Syntaxcvr 25489 Extend class notation with the class of all real vector spaces.

Definitiondf-vr 25490* 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 25491 The class of all real vector spaces is a relation. (Contributed by FL, 16-Nov-2008.)

Theoremvri 25492* 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.)

18.13.24  Matrices

Syntaxcmmat 25493 Addition of matrices.

Syntaxcsmat 25494 Scalar multiplication of matrices.

Syntaxcxmat 25495 Multiplication of matrices.

Definitiondf-amat 25496* Matrix addition. Meaningful if is a (a binary internal operation) at least. Experimental. (Contributed by FL, 29-Aug-2010.)

Definitiondf-smat 25497* Matrix left scalar multiplication. Meaningful if is a binary external operation. Experimental. (Contributed by FL, 29-Aug-2010.)

Definitiondf-mmat 25498* 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.)

18.13.25  Affine spaces

Syntaxcraffsp 25499 Extend class notation with the class of all R affine spaces.

Definitiondf-raffsp 25500* 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

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