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Theorem List for Metamath Proof Explorer - 16001-16100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlidldvgen 16001* An element generates an ideal iff it is contained in the ideal and all elements are right-divided by it. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (LIdeal `  R )   &    |-  K  =  (RSpan `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U  /\  G  e.  B ) 
 ->  ( I  =  ( K `  { G } )  <->  ( G  e.  I  /\  A. x  e.  I  G  .||  x ) ) )
 
Theoremlpigen 16002* An ideal is principal iff it contains an element which right-divides all elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  P  =  (LPIdeal `  R )   &    |-  .||  =  ( ||r
 `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  ( I  e.  P  <->  E. x  e.  I  A. y  e.  I  x  .||  y ) )
 
10.8.4  Nonzero rings
 
Syntaxcnzr 16003 The class of nonzero rings.
 class NzRing
 
Definitiondf-nzr 16004 A nonzero or nontrivial ring is a ring with at least two values, or equivalently where 1 and 0 are different. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- NzRing  =  { r  e.  Ring  |  ( 1r `  r
 )  =/=  ( 0g `  r ) }
 
Theoremisnzr 16005 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- 
 .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. NzRing  <->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
 )
 
Theoremnzrnz 16006 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- 
 .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. NzRing  ->  .1.  =/=  .0.  )
 
Theoremnzrrng 16007 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( R  e. NzRing  ->  R  e.  Ring )
 
Theoremdrngnzr 16008 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( R  e.  DivRing  ->  R  e. NzRing )
 
Theoremisnzr2 16009 Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e. NzRing  <->  ( R  e.  Ring  /\  2o  ~<_  B ) )
 
Theoremopprnzr 16010 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e. NzRing  ->  O  e. NzRing )
 
Theoremrngelnzr 16011 A ring is nonzero if it has a nonzero element. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 13-Jun-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  ( B  \  {  .0.  } ) )  ->  R  e. NzRing )
 
Theoremnzrunit 16012 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. NzRing  /\  A  e.  U ) 
 ->  A  =/=  .0.  )
 
Theoremsubrgnzr 16013 A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  S  =  ( Rs  A )   =>    |-  ( ( R  e. NzRing  /\  A  e.  (SubRing `  R ) )  ->  S  e. NzRing )
 
10.8.5  Left regular elements. More kinds of ring
 
Syntaxcrlreg 16014 Set of left-regular elements in a ring.
 class RLReg
 
Syntaxcdomn 16015 Class of (ring theoretic) domains.
 class Domn
 
Syntaxcidom 16016 Class of integral domains.
 class IDomn
 
Syntaxcpid 16017 Class of principal ideal domains.
 class PID
 
Definitiondf-rlreg 16018* Define the set of left-regular elements in a ring as those elements which are not left zero divisors, meaning that multiplying a nonzero element on the left by a left-regular element gives a nonzero product. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |- RLReg  =  ( r  e.  _V  |->  { x  e.  ( Base `  r )  |  A. y  e.  ( Base `  r ) ( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r
 ) ) } )
 
Definitiondf-domn 16019* A domain is a nonzero ring in which there are no nontrivial zero divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |- Domn  =  { r  e. NzRing  |  [. ( Base `  r )  /  b ]. [. ( 0g `  r )  /  z ]. A. x  e.  b  A. y  e.  b  ( ( x ( .r `  r
 ) y )  =  z  ->  ( x  =  z  \/  y  =  z ) ) }
 
Definitiondf-idom 16020 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |- IDomn  =  ( CRing  i^i Domn )
 
Definitiondf-pid 16021 A principal ideal domain is an integral domain satisfying the left principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |- PID 
 =  (IDomn  i^i LPIR )
 
Theoremrrgval 16022* Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x 
 .x.  y )  =  .0.  ->  y  =  .0.  ) }
 
Theoremisrrg 16023* Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  ( ( X  .x.  y
 )  =  .0.  ->  y  =  .0.  ) ) )
 
Theoremrrgeq0i 16024 Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( X  e.  E  /\  Y  e.  B )  ->  ( ( X 
 .x.  Y )  =  .0. 
 ->  Y  =  .0.  )
 )
 
Theoremrrgeq0 16025 Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  E  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  =  .0.  <->  Y  =  .0.  ) )
 
Theoremrrgsupp 16026 Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  E )   &    |-  ( ph  ->  Y : I --> B )   =>    |-  ( ph  ->  ( `' ( ( I  X.  { X } )  o F  .x.  Y ) " ( _V  \  {  .0.  } ) )  =  ( `' Y "
 ( _V  \  {  .0.  } ) ) )
 
Theoremrrgss 16027 Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  E  C_  B
 
Theoremunitrrg 16028 Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e.  Ring  ->  U  C_  E )
 
Theoremisdomn 16029* Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. Domn  <->  ( R  e. NzRing  /\ 
 A. x  e.  B  A. y  e.  B  ( ( x  .x.  y
 )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
 ) ) )
 
Theoremdomnnzr 16030 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  ( R  e. Domn  ->  R  e. NzRing )
 
Theoremdomnrng 16031 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  ( R  e. Domn  ->  R  e.  Ring )
 
Theoremdomneq0 16032 In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  ) ) )
 
Theoremdomnmuln0 16033 In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/=  .0.  )
 )  ->  ( X  .x.  Y )  =/=  .0.  )
 
Theoremisdomn2 16034 A ring is a domain iff all nonzero elements are non-zero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  E  =  (RLReg `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. Domn  <->  ( R  e. NzRing  /\  ( B  \  {  .0.  } )  C_  E ) )
 
Theoremdomnrrg 16035 In a domain, any nonzero element is a non-zero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  E  =  (RLReg `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  e.  E )
 
Theoremopprdomn 16036 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e. Domn  ->  O  e. Domn )
 
Theoremabvn0b 16037 Another characterization of domains, hinted at in abvtriv 15600: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  A  =  (AbsVal `  R )   =>    |-  ( R  e. Domn  <->  ( R  e. NzRing  /\  A  =/=  (/) ) )
 
Theoremdrngdomn 16038 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( R  e.  DivRing  ->  R  e. Domn )
 
Theoremisidom 16039 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  ( R  e. IDomn  <->  ( R  e.  CRing  /\  R  e. Domn ) )
 
Theoremfldidom 16040 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( R  e. Field  ->  R  e. IDomn )
 
Theoremfidomndrnglem 16041* Lemma for fidomndrng 16042. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e. Domn )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A  e.  ( B  \  {  .0.  }
 ) )   &    |-  F  =  ( x  e.  B  |->  ( x  .x.  A )
 )   =>    |-  ( ph  ->  A  .|| 
 .1.  )
 
Theoremfidomndrng 16042 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( B  e.  Fin  ->  ( R  e. Domn  <->  R  e.  DivRing ) )
 
Theoremfiidomfld 16043 A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( B  e.  Fin  ->  ( R  e. IDomn  <->  R  e. Field ) )
 
10.9  Associative algebras
 
10.9.1  Definition and basic properties
 
Syntaxcasa 16044 Associative algebra.
 class AssAlg
 
Syntaxcasp 16045 Algebraic span function.
 class AlgSpan
 
Syntaxcascl 16046 Class of algebra scalar injection function.
 class algSc
 
Definitiondf-assa 16047* Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a (commutative) ring, coupled with an 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  =  { w  e.  ( LMod  i^i  Ring )  |  [. (Scalar `  w )  /  f ]. ( f  e. 
 CRing  /\  A. r  e.  ( Base `  f ) A. x  e.  ( Base `  w ) A. y  e.  ( Base `  w ) [. ( .s `  w )  /  s ]. [. ( .r
 `  w )  /  t ]. ( ( ( r s x ) t y )  =  ( r s ( x t y ) )  /\  ( x t ( r s y ) )  =  ( r s ( x t y ) ) ) ) }
 
Definitiondf-asp 16048* Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- AlgSpan  =  ( w  e. AssAlg  |->  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  ( (SubRing `  w )  i^i  ( LSubSp `  w ) )  |  s  C_  t } ) )
 
Definitiondf-ascl 16049* 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  =  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w ) )  |->  ( x ( .s `  w ) ( 1r `  w ) ) ) )
 
Theoremisassa 16050* The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  B  =  (
 Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  W )   =>    |-  ( W  e. AssAlg  <->  ( ( W  e.  LMod  /\  W  e.  Ring  /\  F  e.  CRing )  /\  A. r  e.  B  A. x  e.  V  A. y  e.  V  ( ( ( r  .x.  x )  .X.  y )  =  ( r  .x.  ( x  .X.  y ) )  /\  ( x  .X.  ( r 
 .x.  y ) )  =  ( r  .x.  ( x  .X.  y ) ) ) ) )
 
Theoremassalem 16051 The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  B  =  (
 Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  W )   =>    |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
 )  ->  ( (
 ( A  .x.  X )  .X.  Y )  =  ( A  .x.  ( X  .X.  Y ) ) 
 /\  ( X  .X.  ( A  .x.  Y ) )  =  ( A 
 .x.  ( X  .X.  Y ) ) ) )
 
Theoremassaass 16052 Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  B  =  (
 Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  W )   =>    |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
 )  ->  ( ( A  .x.  X )  .X.  Y )  =  ( A 
 .x.  ( X  .X.  Y ) ) )
 
Theoremassaassr 16053 Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  B  =  (
 Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  W )   =>    |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
 )  ->  ( X  .X.  ( A  .x.  Y ) )  =  ( A  .x.  ( X  .X.  Y ) ) )
 
Theoremassalmod 16054 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  ( W  e. AssAlg  ->  W  e.  LMod )
 
Theoremassarng 16055 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  ( W  e. AssAlg  ->  W  e.  Ring )
 
Theoremassasca 16056 An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. AssAlg  ->  F  e.  CRing )
 
Theoremisassad 16057* Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  ( ph  ->  V  =  ( Base `  W )
 )   &    |-  ( ph  ->  F  =  (Scalar `  W )
 )   &    |-  ( ph  ->  B  =  ( Base `  F )
 )   &    |-  ( ph  ->  .x.  =  ( .s `  W ) )   &    |-  ( ph  ->  .X. 
 =  ( .r `  W ) )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  W  e.  Ring
 )   &    |-  ( ph  ->  F  e.  CRing )   &    |-  ( ( ph  /\  ( r  e.  B  /\  x  e.  V  /\  y  e.  V ) )  ->  ( ( r  .x.  x )  .X.  y )  =  ( r  .x.  ( x  .X.  y ) ) )   &    |-  ( ( ph  /\  (
 r  e.  B  /\  x  e.  V  /\  y  e.  V )
 )  ->  ( x  .X.  ( r  .x.  y
 ) )  =  ( r  .x.  ( x  .X.  y ) ) )   =>    |-  ( ph  ->  W  e. AssAlg )
 
Theoremissubassa 16058 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  =  ( Ws  A )   &    |-  L  =  (
 LSubSp `  W )   &    |-  V  =  ( Base `  W )   &    |-  .1.  =  ( 1r `  W )   =>    |-  ( ( W  e. AssAlg  /\ 
 .1.  e.  A  /\  A  C_  V )  ->  ( S  e. AssAlg  <->  ( A  e.  (SubRing `  W )  /\  A  e.  L )
 ) )
 
Theoremsraassa 16059 The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  A  =  ( ( subringAlg  `  W ) `  S )   =>    |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W ) )  ->  A  e. AssAlg )
 
Theoremrlmassa 16060 The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( R  e.  CRing  ->  (ringLMod `  R )  e. AssAlg )
 
Theoremassapropd 16061* 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.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   &    |-  ( ph  ->  F  =  (Scalar `  K ) )   &    |-  ( ph  ->  F  =  (Scalar `  L ) )   &    |-  P  =  (
 Base `  F )   &    |-  (
 ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s `  L ) y ) )   =>    |-  ( ph  ->  ( K  e. AssAlg  <->  L  e. AssAlg ) )
 
Theoremaspval 16062* Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  A  =  (AlgSpan `  W )   &    |-  V  =  ( Base `  W )   &    |-  L  =  (
 LSubSp `  W )   =>    |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W )  i^i  L )  |  S  C_  t } )
 
Theoremasplss 16063 The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  A  =  (AlgSpan `  W )   &    |-  V  =  ( Base `  W )   &    |-  L  =  (
 LSubSp `  W )   =>    |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  e.  L )
 
Theoremaspid 16064 The algebraic span of a subalgebra is itself. (spanid 21918 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  A  =  (AlgSpan `  W )   &    |-  V  =  ( Base `  W )   &    |-  L  =  (
 LSubSp `  W )   =>    |-  ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )  /\  S  e.  L )  ->  ( A `  S )  =  S )
 
Theoremaspsubrg 16065 The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  A  =  (AlgSpan `  W )   &    |-  V  =  ( Base `  W )   =>    |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  e.  (SubRing `  W ) )
 
Theoremaspss 16066 Span preserves subset ordering. (spanss 21919 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  A  =  (AlgSpan `  W )   &    |-  V  =  ( Base `  W )   =>    |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S )  ->  ( A `  T ) 
 C_  ( A `  S ) )
 
Theoremaspssid 16067 A set of vectors is a subset of its span. (spanss2 21916 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  A  =  (AlgSpan `  W )   &    |-  V  =  ( Base `  W )   =>    |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  C_  ( A `  S ) )
 
Theoremasclfval 16068* Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .1.  =  ( 1r `  W )   =>    |-  A  =  ( x  e.  K  |->  ( x 
 .x.  .1.  ) )
 
Theoremasclval 16069 Value of a mapped algebra scalar. (Contributed by Mario Carneiro, 8-Mar-2015.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .1.  =  ( 1r `  W )   =>    |-  ( X  e.  K  ->  ( A `  X )  =  ( X  .x.  .1.  ) )
 
Theoremasclfn 16070 Unconditional functionality of the algebra scalars function. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   =>    |-  A  Fn  K
 
Theoremasclf 16071 The algebra scalars function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e.  Ring )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  K  =  (
 Base `  F )   &    |-  B  =  ( Base `  W )   =>    |-  ( ph  ->  A : K --> B )
 
Theoremasclghm 16072 The algebra scalars function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e.  Ring )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  A  e.  ( F  GrpHom  W ) )
 
Theoremasclmul1 16073 Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  V  =  ( Base `  W )   &    |-  .X.  =  ( .r `  W )   &    |-  .x. 
 =  ( .s `  W )   =>    |-  ( ( W  e. AssAlg  /\  R  e.  K  /\  X  e.  V )  ->  ( ( A `  R )  .X.  X )  =  ( R  .x.  X ) )
 
Theoremasclmul2 16074 Right multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  V  =  ( Base `  W )   &    |-  .X.  =  ( .r `  W )   &    |-  .x. 
 =  ( .s `  W )   =>    |-  ( ( W  e. AssAlg  /\  R  e.  K  /\  X  e.  V )  ->  ( X  .X.  ( A `  R ) )  =  ( R  .x.  X ) )
 
Theoremasclrhm 16075 The scalar injection is a ring homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   =>    |-  ( W  e. AssAlg  ->  A  e.  ( F RingHom  W ) )
 
Theoremrnascl 16076 The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  A  =  (algSc `  W )   &    |-  .1.  =  ( 1r `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( W  e. AssAlg  ->  ran 
 A  =  ( N `
  {  .1.  }
 ) )
 
Theoremressascl 16077 The injection of scalars is invariant between subalgebras and superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  A  =  (algSc `  W )   &    |-  X  =  ( Ws  S )   =>    |-  ( S  e.  (SubRing `  W )  ->  A  =  (algSc `  X )
 )
 
Theoremissubassa2 16078 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.)
 |-  A  =  (algSc `  W )   &    |-  L  =  (
 LSubSp `  W )   =>    |-  ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  ->  ( S  e.  L  <->  ran 
 A  C_  S )
 )
 
Theoremasclpropd 16079* If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on  1r can be discharged either by letting  W  =  _V (if strong equality is known on  .s) or assuming  K is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
 |-  F  =  (Scalar `  K )   &    |-  G  =  (Scalar `  L )   &    |-  ( ph  ->  P  =  ( Base `  F )
 )   &    |-  ( ph  ->  P  =  ( Base `  G )
 )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  W )
 )  ->  ( x ( .s `  K ) y )  =  ( x ( .s `  L ) y ) )   &    |-  ( ph  ->  ( 1r `  K )  =  ( 1r `  L ) )   &    |-  ( ph  ->  ( 1r `  K )  e.  W )   =>    |-  ( ph  ->  (algSc `  K )  =  (algSc `  L ) )
 
Theoremaspval2 16080 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.)
 |-  A  =  (AlgSpan `  W )   &    |-  C  =  (algSc `  W )   &    |-  R  =  (mrCls `  (SubRing `  W )
 )   &    |-  V  =  ( Base `  W )   =>    |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  ( R `  ( ran  C  u.  S ) ) )
 
10.10  Abstract Multivariate Polynomials
 
10.10.1  Definition and basic properties
 
Syntaxcmps 16081 Multivariate power series.
 class mPwSer
 
Syntaxcmvr 16082 Multivariate power series variables.
 class mVar
 
Syntaxcmpl 16083 Multivariate polynomials.
 class mPoly
 
Syntaxces 16084 Evaluation in a superring.
 class evalSub
 
Syntaxcevl 16085 Evaluation of a multivariate polynomial.
 class eval
 
Syntaxcmhp 16086 Multivariate polynomials.
 class mHomP
 
Syntaxcpsd 16087 Power series partial derivative function.
 class mPSDer
 
Syntaxcltb 16088 Ordering on terms of a multivariate polynomial.
 class  <bag
 
Syntaxcopws 16089 Ordered set of power series.
 class ordPwSer
 
Syntaxcslv 16090 Select a subset of variables in a multivariate polynomial.
 class selectVars
 
Syntaxcai 16091 Algebraically independent.
 class AlgInd
 
Definitiondf-psr 16092* Define the algebra of power series over the index set  i and with coefficients from the ring  r. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- mPwSer  =  ( i  e.  _V ,  r  e.  _V  |->  [_
 { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin } 
 /  d ]_ [_ (
 ( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base ` 
 ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r
 )  |`  ( b  X.  b ) ) >. , 
 <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `  x ) ( .r
 `  r ) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  r >. , 
 <. ( .s `  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r ) f ) ) >. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
 ( TopOpen `  r ) } ) ) >. } ) )
 
Definitiondf-mvr 16093* Define the generating elements of the power series algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- mVar  =  ( i  e.  _V ,  r  e.  _V  |->  ( x  e.  i  |->  ( f  e.  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  |->  if (
 f  =  ( y  e.  i  |->  if (
 y  =  x , 
 1 ,  0 ) ) ,  ( 1r
 `  r ) ,  ( 0g `  r
 ) ) ) ) )
 
Definitiondf-mpl 16094* 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  =  ( i  e.  _V ,  r  e.  _V  |->  [_ ( i mPwSer  r ) 
 /  w ]_ ( ws  { f  e.  ( Base `  w )  |  ( `' f " ( _V  \  { ( 0g `  r ) } )
 )  e.  Fin }
 ) )
 
Definitiondf-evls 16095* Define the evaluation map for the polynomial algebra. The function  ( (
I evalSub  S ) `  R
) : V --> ( S  ^m  ( S  ^m  I ) ) makes sense when  I is an index set,  S is a ring,  R is a subring of  S, and where  V is the set of polynomials in  ( I mPoly  R
). This function maps an element of the formal polynomial algebra (with coefficients in  R) to a function from assignments  I --> S of the variables to elements of  S formed by evaluating the polynomial with the given assignments. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 |- evalSub  =  ( i  e.  _V ,  s  e.  CRing  |->  [_ ( Base `  s )  /  b ]_ ( r  e.  (SubRing `  s )  |-> 
 [_ ( i mPoly  (
 ss  r ) )  /  w ]_ ( iota_ f  e.  ( w RingHom  ( s  ^s  ( b  ^m  i ) ) ) ( ( f  o.  (algSc `  w ) )  =  ( x  e.  r  |->  ( ( b  ^m  i )  X.  { x } ) )  /\  ( f  o.  (
 i mVar  ( ss  r ) ) )  =  ( x  e.  i  |->  ( g  e.  ( b 
 ^m  i )  |->  ( g `  x ) ) ) ) ) ) )
 
Definitiondf-evl 16096* A simplication of evalSub when the evaluation ring is the same as the coefficient ring. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |- eval  =  ( i  e.  _V ,  r  e.  _V  |->  ( ( i evalSub  r
 ) `  ( Base `  r ) ) )
 
Definitiondf-mhp 16097* Define the subspaces of order-  n homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- mHomP  =  ( i  e.  _V ,  r  e.  _V  |->  ( n  e.  NN0  |->  { f  e.  ( Base `  ( i mPoly  r ) )  |  ( `' f " ( _V  \  { ( 0g `  r ) } )
 )  C_  { g  e.  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  |  sum_ j  e.  NN0  ( g `  j
 )  =  n } } ) )
 
Definitiondf-psd 16098* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- mPSDer  =  ( i  e.  _V ,  r  e.  _V  |->  ( x  e.  i  |->  ( f  e.  ( Base `  ( i mPwSer  r
 ) )  |->  ( k  e.  { h  e.  ( NN0  ^m  i
 )  |  ( `' h " NN )  e.  Fin }  |->  ( ( ( k `  x )  +  1 )
 (.g `  r ) ( f `  ( k  o F  +  (
 y  e.  i  |->  if ( y  =  x ,  1 ,  0 ) ) ) ) ) ) ) ) )
 
Definitiondf-ltbag 16099* Define a well-order on the set of all finite bags from the index set  i given a wellordering  r of  i. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |- 
 <bag 
 =  ( r  e. 
 _V ,  i  e. 
 _V  |->  { <. x ,  y >.  |  ( { x ,  y }  C_  { h  e.  ( NN0  ^m  i
 )  |  ( `' h " NN )  e.  Fin }  /\  E. z  e.  i  (
 ( x `  z
 )  <  ( y `  z )  /\  A. w  e.  i  (
 z r w  ->  ( x `  w )  =  ( y `  w ) ) ) ) } )
 
Definitiondf-opsr 16100* Define a total order on the set of all power series in  s from the index set  i given a wellordering  r of  i and a totally ordered base ring  s. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |- ordPwSer  =  ( i  e.  _V ,  s  e.  _V  |->  ( r  e.  ~P ( i  X.  i
 )  |->  [_ ( i mPwSer  s
 )  /  p ]_ ( p sSet  <. ( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  p )  /\  ( [. { h  e.  ( NN0  ^m  i
 )  |  ( `' h " NN )  e.  Fin }  /  d ]. E. z  e.  d  ( ( x `  z ) ( lt `  s ) ( y `
  z )  /\  A. w  e.  d  ( w ( r  <bag  i ) z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) ) )
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