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Theorem List for Metamath Proof Explorer - 16301-16400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdomnrrg 16301 In a domain, any nonzero element is a non-zero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
RLReg              Domn

Theoremopprdomn 16302 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
oppr       Domn Domn

Theoremabvn0b 16303 Another characterization of domains, hinted at in abvtriv 15870: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
AbsVal       Domn NzRing

Theoremdrngdomn 16304 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
Domn

Theoremisidom 16305 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn Domn

Theoremfldidom 16306 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
Field IDomn

Theoremfidomndrnglem 16307* Lemma for fidomndrng 16308. (Contributed by Mario Carneiro, 15-Jun-2015.)
r              Domn

Theoremfidomndrng 16308 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
Domn

Theoremfiidomfld 16309 A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.)
IDomn Field

10.9  Associative algebras

10.9.1  Definition and basic properties

Syntaxcasa 16310 Associative algebra.
AssAlg

Syntaxcasp 16311 Algebraic span function.
AlgSpan

Syntaxcascl 16312 Class of algebra scalar injection function.
algSc

Definitiondf-assa 16313* Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a (commutative) ring, coupled with a multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014.)
AssAlg Scalar

Definitiondf-asp 16314* Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan AssAlg SubRing

Definitiondf-ascl 16315* Every unital algebra contains a canonical homomorphic image of its ring of scalars as scalar multiples of the unit. This names the homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc Scalar

Theoremisassa 16316* The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassalem 16317 The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassaass 16318 Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassaassr 16319 Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassalmod 16320 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
AssAlg

Theoremassarng 16321 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
AssAlg

Theoremassasca 16322 An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Scalar       AssAlg

Theoremisassad 16323* Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)
Scalar                                                               AssAlg

Theoremissubassa 16324 The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)
s                             AssAlg AssAlg SubRing

Theoremsraassa 16325 The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
subringAlg        SubRing AssAlg

Theoremrlmassa 16326 The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod AssAlg

Theoremassapropd 16327* If two structures have the same components (properties), one is an associative algebra iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
Scalar       Scalar                     AssAlg AssAlg

Theoremaspval 16328* Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg SubRing

Theoremasplss 16329 The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg

Theoremaspid 16330 The algebraic span of a subalgebra is itself. (spanid 22769 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg SubRing

Theoremaspsubrg 16331 The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg SubRing

Theoremaspss 16332 Span preserves subset ordering. (spanss 22770 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg

Theoremaspssid 16333 A set of vectors is a subset of its span. (spanss2 22767 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg

Theoremasclfval 16334* Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar

Theoremasclval 16335 Value of a mapped algebra scalar. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar

Theoremasclfn 16336 Unconditional functionality of the algebra scalars function. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar

Theoremasclf 16337 The algebra scalars function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.)
algSc       Scalar

Theoremasclghm 16338 The algebra scalars function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)
algSc       Scalar

Theoremasclmul1 16339 Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar                                   AssAlg

Theoremasclmul2 16340 Right multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar                                   AssAlg

Theoremasclrhm 16341 The scalar injection is a ring homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar       AssAlg RingHom

Theoremrnascl 16342 The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc                     AssAlg

Theoremressascl 16343 The injection of scalars is invariant between subalgebras and superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       s        SubRing algSc

Theoremissubassa2 16344 A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc              AssAlg SubRing

Theoremasclpropd 16345* If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on can be discharged either by letting (if strong equality is known on ) or assuming is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
Scalar       Scalar                                          algSc algSc

Theoremaspval2 16346 The algebraic closure is the ring closure when the generating set is expanded to include all scalars. EDITORIAL : In light of this, is AlgSpan independently needed? (Contributed by Stefan O'Rear, 9-Mar-2015.)
AlgSpan       algSc       mrClsSubRing              AssAlg

10.10  Abstract multivariate polynomials

10.10.1  Definition and basic properties

Syntaxcmps 16347 Multivariate power series.
mPwSer

Syntaxcmvr 16348 Multivariate power series variables.
mVar

Syntaxcmpl 16349 Multivariate polynomials.
mPoly

Syntaxces 16350 Evaluation in a superring.
evalSub

Syntaxcevl 16351 Evaluation of a multivariate polynomial.
eval

Syntaxcmhp 16352 Multivariate polynomials.
mHomP

Syntaxcpsd 16353 Power series partial derivative function.
mPSDer

Syntaxcltb 16354 Ordering on terms of a multivariate polynomial.
bag

Syntaxcopws 16355 Ordered set of power series.
ordPwSer

Syntaxcslv 16356 Select a subset of variables in a multivariate polynomial.
selectVars

Syntaxcai 16357 Algebraically independent.
AlgInd

Definitiondf-psr 16358* Define the algebra of power series over the index set and with coefficients from the ring . (Contributed by Mario Carneiro, 21-Mar-2015.)
mPwSer g Scalar TopSet

Definitiondf-mvr 16359* Define the generating elements of the power series algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Definitiondf-mpl 16360* Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree). (Contributed by Mario Carneiro, 7-Jan-2015.)
mPoly mPwSer s

Definitiondf-evls 16361* Define the evaluation map for the polynomial algebra. The function evalSub makes sense when is an index set, is a ring, is a subring of , and where is the set of polynomials in mPoly . This function maps an element of the formal polynomial algebra (with coefficients in ) to a function from assignments of the variables to elements of formed by evaluating the polynomial with the given assignments. (Contributed by Stefan O'Rear, 11-Mar-2015.)
evalSub SubRing mPoly s RingHom s algSc mVar s

Definitiondf-evl 16362* A simplication of evalSub when the evaluation ring is the same as the coefficient ring. (Contributed by Stefan O'Rear, 19-Mar-2015.)
eval evalSub

Definitiondf-mhp 16363* Define the subspaces of order- homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mHomP mPoly

Definitiondf-psd 16364* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPSDer mPwSer .g

Definitiondf-ltbag 16365* Define a well-order on the set of all finite bags from the index set given a wellordering of . (Contributed by Mario Carneiro, 8-Feb-2015.)
bag

Definitiondf-opsr 16366* Define a total order on the set of all power series in from the index set given a wellordering of and a totally ordered base ring . (Contributed by Mario Carneiro, 8-Feb-2015.)
ordPwSer mPwSer sSet bag

Definitiondf-selv 16367* Define the "variable selection" function. The function selectVars maps elements of mPoly bijectively onto mPoly mPoly in the natural way, for example if and it would map mPoly to mPoly mPoly . This, for example, allows one to treat a multivariate polynomial as a univariate polynomial with coefficients in a polynomial ring with one less variable. (Contributed by Mario Carneiro, 21-Mar-2015.)
selectVars mPoly mPoly Scalar evalSub s mVar mPoly mVar

Definitiondf-algind 16368* Define the predicate "the set is algebraically independent in the algebra ". A collection of vectors is algebraically independent if no nontrivial polynomial with elements from the subset evaluates to zero. (Contributed by Mario Carneiro, 21-Mar-2015.)
AlgInd mPoly s evalSub

Theoremreldmpsr 16369 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPwSer

Theorempsrval 16370* Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer                                                         g                                    Scalar TopSet

Theorempsrvalstr 16371 The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
Scalar TopSet Struct

Theorempsrbag 16372* Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagf 16373* A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglesupp 16374* The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglecl 16375* The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagaddcl 16376* The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.)

Theorempsrbagcon 16377* The analogue of the statement " implies " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglefi 16378* There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.)

Theorempsrbagconcl 16379* The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagconf1o 16380* Bag complementation is a bijection on the set of bags dominated by a given bag . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremgsumbagdiaglem 16381* Lemma for gsumbagdiag 16382. (Contributed by Mario Carneiro, 5-Jan-2015.)

Theoremgsumbagdiag 16382* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 12502 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.)
CMnd              g g

Theorempsrass1lem 16383* A group sum commutation used by psrass1 16410. (Contributed by Mario Carneiro, 5-Jan-2015.)
CMnd                     g g g g

Theorempsrbas 16384* The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer

Theorempsrelbas 16385* An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrplusg 16386 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer

Theorempsradd 16387 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsraddcl 16388 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrmulr 16389* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer                                    g

Theorempsrmulfval 16390* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                                                  g

Theorempsrmulval 16391* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                                                         g

Theorempsrmulcllem 16392* Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrmulcl 16393 Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrsca 16394 The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                      Scalar

Theorempsrvscafval 16395* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer

Theorempsrvsca 16396* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrvscaval 16397* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrvscacl 16398 Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0cl 16399* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0lid 16400* The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

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