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

Definitiondf-frsmgrp 25401* Definition of a free semigroup. The definition is somewhat cryptic. Let's say it guarantees the elements of the semigroup can be decomposed into elementary components and that the decomposition is unique. As a consequence you define the elements of the semigroup with nice recursive function by giving the value for every elementary component and the recursive equation . This is not true in every semigroup. For intance if you take the semigroup of strings generated by the elementary components "ab", "c", "a", "bc", the string "abc" is equal to "ab" "c" or to "a" "bc" and those beautiful recursive function can't exist. (See a nice explanation in Gallier p. 20.) Experimental. (Contributed by FL, 15-Jul-2012.)

18.13.18  Translations

Theoremtrdom2 25402* The domain of a right translation. The term is a constant: is not present. (Contributed by FL, 21-Jun-2010.)

Theoremtrset 25403* A right translation is a set. (Contributed by FL, 19-Sep-2010.)

Theoremtrran2 25404* The range of a right translation. The term is a constant: is not present. (Contributed by FL, 21-Jun-2010.)

Theoremtrooo 25405* A right translation is a bijection. The term is a constant. (Contributed by FL, 21-Jun-2010.)

Theoremtrinv 25406* The converse of a right translation. The term is a constant. (Contributed by FL, 21-Jun-2010.)

Theoremcmprtr 25407* Composite of two right translations. The terms and are constant. Don't use. See cmprtr2 25408. (Contributed by FL, 17-Oct-2010.)

Theoremcmprtr2 25408* Composite of two right translations. (cmprtr 25407 with a distinct variable condition relaxed.) (Contributed by FL, 1-Jan-2011.)

Theoremimtr 25409* The image of a set through a translation. (Contributed by FL, 30-Dec-2010.)

Theoremprsubrtr 25410* The product of a subset of by an element of is the image of by a right translation. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)

Theoremcaytr 25411* "It follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor, the effect is simply to reproduce the group... ." Cayley, On the theory of groups, as depending on the symbolic equation th^n = 1, 1854. (it is the original paper where the axiomatic definition of a group was given for the first time.) (Contributed by FL, 15-Oct-2012.)

Theoremltrdom 25412* The domain of a left translation. The term is a constant. (Contributed by FL, 26-Apr-2012.)

Theoremltrset 25413* A left translation is a set. (Contributed by FL, 28-Apr-2012.)

Theoremltrran2 25414* The range of a left translation. The term is a constant. (Contributed by FL, 28-Apr-2012.)

Theoremltrooo 25415* A left translation is a bijection. The term is a constant. (Contributed by FL, 29-Apr-2012.)

Theoremltrcmp 25416* Left translation expressed as a composite. (Contributed by FL, 3-Jul-2012.)

Theoremltrinvlem 25417* The converse of a left translation. The term is a constant. (Contributed by FL, 30-Apr-2012.)

Theoremcmpltr2 25418* Composite of two left translations. The terms and are constant. (Contributed by FL, 2-Jul-2012.)

Theoremcmpltr 25419* Composite of two left translations. The terms and are constant. Don't use. See cmpltr2 25418. (Contributed by FL, 2-Jul-2012.) (Revised by Mario Carneiro, 2-Jun-2014.)

Theoremcmperltr 25420* A right and left translation expressed as a composite. Note that and can't be the same. (Contributed by FL, 2-Jul-2012.)

Theoremcmprltr 25421* Composite of two right and left translations. Note that and can't be the same. See cmprltr2 25422 for a more general version. (Contributed by FL, 2-Jul-2012.) (Proof shortened by Mario Carneiro, 26-Jul-2014.)

Theoremcmprltr2 25422* Composite of two right and left translations. No restriction: and can be equal. (Contributed by FL, 2-Jul-2012.)

Theoremrltrdom 25423* The domain of a right and left translation. (Contributed by FL, 2-Jul-2012.)

Theoremrltrset 25424* A right and left translation is a set. (Contributed by FL, 2-Jul-2012.)

Theoremrltrran 25425* The range of a right and left translation. Note that and are constant. (Contributed by FL, 2-Jul-2012.)

Theoremrltrooo 25426* A right and left translation is a bijection. (Contributed by FL, 2-Jul-2012.)

18.13.19  Fields and Rings

Theoremcom2i 25427* Property of a commutative structure with two operation. (Contributed by FL, 14-Feb-2010.)

Theoremrngmgmbs3 25428* 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 25429 In a unital ring the range of the multiplication equals the domain of the first variable. (Contributed by FL, 24-Jan-2010.)

Theoremrngodmeqrn 25430 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 25431* The unit of a unital ring is unique. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 23-Dec-2013.)

Theoremrngoinvcl 25432 The additive inverse of a unital ring element pertains to the unital ring. (Contributed by FL, 18-Apr-2010.)

Theoremmultinv 25433 Multiplication by an additive inverse. (Contributed by FL, 2-Sep-2009.)

Theoremmultinvb 25434 Multiplication by an additive inverse. (Contributed by FL, 6-Sep-2009.)

Theoremmult2inv 25435 Multiplication of two additive inverses. (Contributed by FL, 6-Sep-2009.)

Theoremrngounval2 25436* The value of the unit of a ring. (Contributed by FL, 12-Feb-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
GId

TheoremisfldOLD 25437* The predicate "is a field". (Contributed by FL, 6-Sep-2009.)

Theoremfldi 25438* The "axioms" of a field. (Contributed by FL, 15-Sep-2010.)
GId

Theoremfldax1 25439 1st "axiom" of a field. The addition is an abelian group. (Contributed by FL, 11-Jul-2010.)

Theoremfldax2 25440 2nd "axiom" of a field. The multiplication is an internal operation. (Contributed by FL, 11-Jul-2010.)

Theoremfldax3 25441* 3rd "axiom" of a field. The multiplication is associative. (Contributed by FL, 11-Jul-2010.)

Theoremfldax4 25442* 4th "axiom" of a field. The multiplication is distributive. (Contributed by FL, 11-Jul-2010.)

Theoremfldax5 25443* 5th "axiom" of a field. Existence of a neutral element. (Contributed by FL, 11-Jul-2010.)

Theoremfldax6 25444 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 25445* 7th "axiom" of a field. The multiplication is commutative. (Contributed by FL, 11-Jul-2010.)

Theoremzrfld 25446 The zero ring is not a field. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)

Theoremzerdivemp1 25447* In a unitary ring a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.)
GId              GId

Theoremrngoridfz 25448* In a unitary ring a left invertible element is different from zero iff . (Contributed by FL, 18-Apr-2010.)
GId              GId

Theoremzintdom 25449 is a commutative ring. (Contributed by FL, 18-Apr-2010.)

Syntaxctofld 25450 Extend class notation with the class of all totally ordered fields.

Definitiondf-tofld 25451* Definition of a totally ordered field. Experimental. (Contributed by FL, 27-Jun-2011.)
GId

Syntaxczerodiv 25452 Extend class notation with the class of all the zero divisors.

Definitiondf-zd 25453* Definition of the zero divisors of a ring. Experimental. (Contributed by FL, 27-Jun-2011.)
GId GId GId

18.13.20  Ideals

Syntaxcidln 25454 Extend class notation with the class of ideals.
IdlNEW

Definitiondf-idlNEW 25455* 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 25456* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
IdlNEW

TheoremisidlNEW 25457* 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 25458 Extend class notation to include actions.

Definitiondf-act 25459* 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 25460 Extend class notation with the class of all generic vector spaces and modules.

Definitiondf-vec 25461* Definition of a vector space ( is a field ), or of a module ( is a ring ). (Contributed by FL, 12-Jul-2010.)
GId

Theoremvecval1b 25462* The predicate "is a vector space" or "is a module". (Contributed by FL, 12-Jul-2010.)
GId

Theoremvecval3b 25463* The "axioms" of a vector space or module. (Contributed by FL, 12-Jul-2010.)
GId

Theoremvecax1 25464 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 25465 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 25466* 3rd "axiom" of a vector space or module. Multiplication by 1. (Contributed by FL, 13-Sep-2010.)
GId

Theoremvecax4 25467* 4th "axiom" of a vector space or module. Multiplication by a scalar distributes over vector addition. (Contributed by FL, 13-Sep-2010.)

Theoremvecax5 25468* 5th "axiom" of a vector space or module. Multiplication by a sum of scalars. (Contributed by FL, 13-Sep-2010.)

Theoremvecax6 25469* 6th "axiom" of a vector space or module. Relation between scalar multiplication and vector multiplication. (Contributed by FL, 13-Sep-2010.)

Theoremvecax5b 25470 Multiplication by a sum of scalars. (Contributed by FL, 13-Sep-2010.)

Theoremcladdinvvec 25471 Closure of the additive inverse of a vector. (Contributed by FL, 13-Sep-2010.)

Theoremvec2inv 25472 Double inverse law for vector additive inverse. (Contributed by FL, 13-Sep-2010.)

Theoremsum2vv 25473 The sum of two vectors is a vector. (Contributed by FL, 13-Sep-2010.)

GId

GId

Theoremaddvecass 25476 The addition of vectors is associative. (Contributed by FL, 13-Sep-2010.)

Theoremaddvecom 25477 The addition of vectors is associative. (Contributed by FL, 13-Sep-2010.)

Theoreminvaddvec 25478 Additive inverse of a sum of vectors. (Contributed by FL, 13-Sep-2010.)

Theoremprodvs 25479 The product of a vector by a scalar is a vector. (Contributed by FL, 12-Sep-2010.)

Theoremvecsrcan 25480 Right cancellation law for vector subtraction. (Contributed by FL, 12-Sep-2010.)

Theoremvecslcan 25481 Left cancellation law for vector subtraction. (Contributed by FL, 12-Sep-2010.)

Theoremvwit 25482 A vector minus itself equals zero. (Contributed by FL, 12-Sep-2010.)
GId

Theoremsub2vec 25483 Definition of the subtraction of two vectors. (Contributed by FL, 12-Sep-2010.)
GId

Theoremmvecrtol 25484 Moving a vector from the right member of an equation into the left member. (Contributed by FL, 12-Sep-2010.)
GId

Theoremdblsubvec 25485 Double subtraction of vectors. (Contributed by FL, 12-Sep-2010.)
GId

Theoremvecrcan 25486 Right cancellation law for vector addition. (Contributed by FL, 12-Sep-2010.)
GId

Theoremveclcan 25487 Left cancellation law for vector addition. (Contributed by FL, 12-Sep-2010.)
GId

Theoremmvecrtol2 25488 Moving a vector from the right member of an equation into the left member. (Contributed by FL, 12-Sep-2010.)
GId

Theoremprvs 25489 The product of a vector by a scalar is a vector. (Contributed by FL, 12-Sep-2010.)

Theoremmulveczer 25490 Multiplication of a vector by zero. (Contributed by FL, 12-Sep-2010.)
GId                            GId

Theoremmulinvsca 25491 Multiplication by the inverse of a scalar. (Contributed by FL, 12-Sep-2010.)

Theoremmuldisc 25492* Multiplication by a difference of scalars. (Contributed by FL, 12-Sep-2010.)

Theoremglmrngo 25493 Generating a left module from a ring. (Contributed by FL, 29-May-2014.)

Theoremvecax5c 25494 Multiplication by a difference of scalars. (Contributed by FL, 12-Sep-2010.)

Theoremsvli2 25495* 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 25496 Extend class notation with the class of all generic subspace vector spaces and modules.

Definitiondf-svs 25497* 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 25498* 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 25499* A very concise definition of a subspace of a vector space. (Contributed by FL, 30-Dec-2010.)

18.13.23  Real vector spaces

Syntaxcvr 25500 Extend class notation with the class of all real vector spaces.

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