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Theorem List for Metamath Proof Explorer - 16001-16100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremislpir 16001 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e. LPIR  <->  ( R  e.  Ring  /\  U  =  P ) )
 
Theoremlpiss 16002 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e.  Ring  ->  P  C_  U )
 
Theoremislpir2 16003 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e. LPIR  <->  ( R  e.  Ring  /\  U  C_  P )
 )
 
Theoremlpirrng 16004 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( R  e. LPIR  ->  R  e.  Ring )
 
Theoremdrnglpir 16005 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  ( R  e.  DivRing  ->  R  e. LPIR )
 
Theoremrspsn 16006* Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  K  =  (RSpan `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  G  e.  B ) 
 ->  ( K `  { G } )  =  { x  |  G  .||  x }
 )
 
Theoremlidldvgen 16007* 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 16008* 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 16009 The class of nonzero rings.
 class NzRing
 
Definitiondf-nzr 16010 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 16011 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 16012 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 16013 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( R  e. NzRing  ->  R  e.  Ring )
 
Theoremdrngnzr 16014 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( R  e.  DivRing  ->  R  e. NzRing )
 
Theoremisnzr2 16015 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 16016 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 16017 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 16018 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 16019 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 rings
 
Syntaxcrlreg 16020 Set of left-regular elements in a ring.
 class RLReg
 
Syntaxcdomn 16021 Class of (ring theoretic) domains.
 class Domn
 
Syntaxcidom 16022 Class of integral domains.
 class IDomn
 
Syntaxcpid 16023 Class of principal ideal domains.
 class PID
 
Definitiondf-rlreg 16024* 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 16025* 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 16026 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |- IDomn  =  ( CRing  i^i Domn )
 
Definitiondf-pid 16027 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 16028* 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 16029* 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 16030 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 16031 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 16032 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 16033 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 16034 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 16035* 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 16036 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  ( R  e. Domn  ->  R  e. NzRing )
 
Theoremdomnrng 16037 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  ( R  e. Domn  ->  R  e.  Ring )
 
Theoremdomneq0 16038 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 16039 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 16040 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 16041 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 16042 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 16043 Another characterization of domains, hinted at in abvtriv 15606: 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 16044 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( R  e.  DivRing  ->  R  e. Domn )
 
Theoremisidom 16045 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  ( R  e. IDomn  <->  ( R  e.  CRing  /\  R  e. Domn ) )
 
Theoremfldidom 16046 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( R  e. Field  ->  R  e. IDomn )
 
Theoremfidomndrnglem 16047* Lemma for fidomndrng 16048. (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 16048 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 16049 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 16050 Associative algebra.
 class AssAlg
 
Syntaxcasp 16051 Algebraic span function.
 class AlgSpan
 
Syntaxcascl 16052 Class of algebra scalar injection function.
 class algSc
 
Definitiondf-assa 16053* 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 16054* 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 16055* 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 16056* 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 16057 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 16058 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 16059 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 16060 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  ( W  e. AssAlg  ->  W  e.  LMod )
 
Theoremassarng 16061 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  ( W  e. AssAlg  ->  W  e.  Ring )
 
Theoremassasca 16062 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 16063* 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 16064 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 16065 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 16066 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 16067* 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 16068* 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 16069 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 16070 The algebraic span of a subalgebra is itself. (spanid 21926 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 16071 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 16072 Span preserves subset ordering. (spanss 21927 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 16073 A set of vectors is a subset of its span. (spanss2 21924 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 16074* 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 16075 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 16076 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 16077 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 16078 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 16079 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 16080 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 16081 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 16082 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 16083 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 16084 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 16085* 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 16086 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 16087 Multivariate power series.
 class mPwSer
 
Syntaxcmvr 16088 Multivariate power series variables.
 class mVar
 
Syntaxcmpl 16089 Multivariate polynomials.
 class mPoly
 
Syntaxces 16090 Evaluation in a superring.
 class evalSub
 
Syntaxcevl 16091 Evaluation of a multivariate polynomial.
 class eval
 
Syntaxcmhp 16092 Multivariate polynomials.
 class mHomP
 
Syntaxcpsd 16093 Power series partial derivative function.
 class mPSDer
 
Syntaxcltb 16094 Ordering on terms of a multivariate polynomial.
 class  <bag
 
Syntaxcopws 16095 Ordered set of power series.
 class ordPwSer
 
Syntaxcslv 16096 Select a subset of variables in a multivariate polynomial.
 class selectVars
 
Syntaxcai 16097 Algebraically independent.
 class AlgInd
 
Definitiondf-psr 16098* 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 16099* 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 16100* 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 }
 ) )
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