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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremasplss 15901 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 15902 The algebraic span of a subalgebra is itself. (spanid 21756 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 15903 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 15904 Span preserves subset ordering. (spanss 21757 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 15905 A set of vectors is a subset of its span. (spanss2 21754 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 15906* 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 15907 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 15908 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 15909 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 15910 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 15911 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 15912 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 15913 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 15914 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 15915 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 15916 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 15917* 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 15918 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 15919 Multivariate power series.
 class mPwSer
 
Syntaxcmvr 15920 Multivariate power series variables.
 class mVar
 
Syntaxcmpl 15921 Multivariate polynomials.
 class mPoly
 
Syntaxces 15922 Evaluation in a superring.
 class evalSub
 
Syntaxcevl 15923 Evaluation of a multivariate polynomial.
 class eval
 
Syntaxcmhp 15924 Multivariate polynomials.
 class mHomP
 
Syntaxcpsd 15925 Power series partial derivative function.
 class mPSDer
 
Syntaxcltb 15926 Ordering on terms of a multivariate polynomial.
 class  <bag
 
Syntaxcopws 15927 Ordered set of power series.
 class ordPwSer
 
Syntaxcslv 15928 Select a subset of variables in a multivariate polynomial.
 class selectVars
 
Syntaxcai 15929 Algebraically independent.
 class AlgInd
 
Definitiondf-psr 15930* 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 15931* 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 15932* 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 15933* 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 15934* 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 15935* 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 15936* 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 15937* 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 15938* 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 ) ) } >. ) ) )
 
Definitiondf-selv 15939* Define the "variable selection" function. The function  ( (
I selectVars  R ) `  J
) maps elements of  ( I mPoly  R ) bijectively onto  ( J mPoly  ( ( I  \  J ) mPoly 
R ) ) in the natural way, for example if  I  =  { x ,  y } and  J  =  { y } it would map  1  +  x  +  y  +  x
y  e.  ( { x ,  y } mPoly 
ZZ ) to  ( 1  +  x )  +  ( 1  +  x ) y  e.  ( { y } mPoly  ( {
x } mPoly  ZZ )
). 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  =  ( i  e.  _V ,  r  e.  _V  |->  ( j  e.  ~P i  |->  ( f  e.  ( i mPoly  r ) 
 |->  [_ ( ( i 
 \  j ) mPoly  r
 )  /  s ]_ [_ ( x  e.  (Scalar `  s )  |->  ( x ( .s `  s
 ) ( 1r `  s ) ) ) 
 /  c ]_ (
 ( ( ( i evalSub  s ) `  (
 c  "s  r ) ) `  ( c  o.  f
 ) ) `  ( x  e.  i  |->  if ( x  e.  j ,  ( ( j mVar  (
 ( i  \  j
 ) mPoly  r ) ) `  x ) ,  (
 c  o.  ( ( ( i  \  j
 ) mVar  r ) `  x ) ) ) ) ) ) ) )
 
Definitiondf-algind 15940* Define the predicate "the set  v is algebraically independent in the algebra  w". 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  =  ( w  e.  _V ,  k  e.  ~P ( Base `  w )  |->  { v  e.  ~P ( Base `  w )  |  Fun  `' ( f  e.  ( Base `  (
 v mPoly  ( ws  k ) ) ) 
 |->  ( ( ( ( v evalSub  w ) `  k
 ) `  f ) `  (  _I  |`  v ) ) ) } )
 
Theoremreldmpsr 15941 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- 
 Rel  dom mPwSer
 
Theorempsrval 15942* Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  ( TopOpen `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  B  =  ( K  ^m  D ) )   &    |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )   &    |-  .X. 
 =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( f `  x )  .x.  ( g `
  ( k  o F  -  x ) ) ) ) ) ) )   &    |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F  .x.  f ) )   &    |-  ( ph  ->  J  =  (
 Xt_ `  ( D  X.  { O } )
 ) )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  X )   =>    |-  ( ph  ->  S  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s
 `  ndx ) ,  .xb  >. ,  <. (TopSet `  ndx ) ,  J >. } ) )
 
Theorempsrvalstr 15943 The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s
 `  ndx ) ,  .x.  >. ,  <. (TopSet `  ndx ) ,  J >. } ) Struct  <. 1 ,  9 >.
 
Theorempsrbag 15944* Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( I  e.  V  ->  ( F  e.  D  <->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) ) )
 
Theorempsrbagf 15945* A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  F  e.  D )  ->  F : I
 --> NN0 )
 
Theorempsrbaglesupp 15946* The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F ) ) 
 ->  ( `' G " NN )  C_  ( `' F " NN )
 )
 
Theorempsrbaglecl 15947* The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F ) ) 
 ->  G  e.  D )
 
Theorempsrbagaddcl 15948* The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o F  +  G )  e.  D )
 
Theorempsrbagcon 15949* The analogue of the statement " 0  <_  G  <_  F implies  0  <_  F  -  G  <_  F " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F ) ) 
 ->  ( ( F  o F  -  G )  e.  D  /\  ( F  o F  -  G )  o R  <_  F ) )
 
Theorempsrbaglefi 15950* There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  F  e.  D )  ->  { y  e.  D  |  y  o R  <_  F }  e.  Fin )
 
Theorempsrbagconcl 15951* The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   =>    |-  ( ( I  e.  V  /\  F  e.  D  /\  X  e.  S )  ->  ( F  o F  -  X )  e.  S )
 
Theorempsrbagconf1o 15952* Bag complementation is a bijection on the set of bags dominated by a given bag  F. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   =>    |-  ( ( I  e.  V  /\  F  e.  D )  ->  ( x  e.  S  |->  ( F  o F  -  x ) ) : S -1-1-onto-> S )
 
Theoremgsumbagdiaglem 15953* Lemma for gsumbagdiag 15954. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   =>    |-  ( ( ph  /\  ( X  e.  S  /\  Y  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  X ) }
 ) )  ->  ( Y  e.  S  /\  X  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  Y ) }
 ) )
 
Theoremgsumbagdiag 15954* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 12117 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  (
 ( ph  /\  ( j  e.  S  /\  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) } )
 )  ->  X  e.  B )   =>    |-  ( ph  ->  ( G  gsumg  ( j  e.  S ,  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) }  |->  X ) )  =  ( G  gsumg  ( k  e.  S ,  j  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  k ) }  |->  X ) ) )
 
Theorempsrass1lem 15955* A group sum commutation used by psrass1 15982. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  (
 ( ph  /\  ( j  e.  S  /\  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) } )
 )  ->  X  e.  B )   &    |-  ( k  =  ( n  o F  -  j )  ->  X  =  Y )   =>    |-  ( ph  ->  ( G  gsumg  ( n  e.  S  |->  ( G  gsumg  ( j  e.  { x  e.  D  |  x  o R  <_  n }  |->  Y ) ) ) )  =  ( G  gsumg  ( j  e.  S  |->  ( G  gsumg  ( k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j
 ) }  |->  X ) ) ) ) )
 
Theorempsrbas 15956* The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  V )   =>    |-  ( ph  ->  B  =  ( K  ^m  D ) )
 
Theorempsrelbas 15957* An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X : D --> K )
 
Theorempsrplusg 15958 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  S )   =>    |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )
 
Theorempsradd 15959 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+b  Y )  =  ( X  o F  .+  Y ) )
 
Theorempsraddcl 15960 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  B )
 
Theorempsrmulr 15961* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   =>    |-  .xb  =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( f `  x )  .x.  ( g `
  ( k  o F  -  x ) ) ) ) ) ) )
 
Theorempsrmulfval 15962* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F  .xb  G )  =  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
 y  e.  D  |  y  o R  <_  k }  |->  ( ( F `
  x )  .x.  ( G `  ( k  o F  -  x ) ) ) ) ) ) )
 
Theorempsrmulval 15963* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  (
 ( F  .xb  G ) `
  X )  =  ( R  gsumg  ( k  e.  {
 y  e.  D  |  y  o R  <_  X }  |->  ( ( F `
  k )  .x.  ( G `  ( X  o F  -  k
 ) ) ) ) ) )
 
Theorempsrmulcllem 15964* Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  B )
 
Theorempsrmulcl 15965 Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  B )
 
Theorempsrsca 15966 The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   =>    |-  ( ph  ->  R  =  (Scalar `  S )
 )
 
Theorempsrvscafval 15967* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   =>    |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
 )  o F  .x.  f ) )
 
Theorempsrvsca 15968* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( X  .xb  F )  =  ( ( D  X.  { X } )  o F  .x.  F )
 )
 
Theorempsrvscaval 15969* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  (
 ( X  .xb  F ) `
  Y )  =  ( X  .x.  ( F `  Y ) ) )
 
Theorempsrvscacl 15970 Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .x.  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( X  .x.  F )  e.  B )
 
Theorempsr0cl 15971* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  S )   =>    |-  ( ph  ->  ( D  X.  {  .0.  }
 )  e.  B )
 
Theorempsr0lid 15972* The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( D  X.  {  .0.  } )  .+  X )  =  X )
 
Theorempsrnegcl 15973* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( N  o.  X )  e.  B )
 
Theorempsrlinv 15974* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .+  =  ( +g  `  S )   =>    |-  ( ph  ->  (
 ( N  o.  X )  .+  X )  =  ( D  X.  {  .0.  } ) )
 
Theorempsrgrp 15975 The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  S  e.  Grp )
 
Theorempsr0 15976* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  O  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ph  ->  .0.  =  ( D  X.  { O } ) )
 
Theorempsrneg 15977* The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  M  =  ( inv g `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( M `  X )  =  ( N  o.  X ) )
 
Theorempsrlmod 15978 The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  S  e.  LMod
 )
 
Theorempsr1cl 15979* The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   =>    |-  ( ph  ->  U  e.  B )
 
Theorempsrlidm 15980* The identity element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( U  .x.  X )  =  X )
 
Theorempsrridm 15981* The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .x.  U )  =  X )
 
Theorempsrass1 15982* Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( ( X  .X.  Y )  .X.  Z )  =  ( X  .X.  ( Y  .X.  Z ) ) )
 
Theorempsrdi 15983* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ph  ->  ( X  .X.  ( Y  .+  Z ) )  =  ( ( X  .X.  Y )  .+  ( X  .X.  Z ) ) )
 
Theorempsrdir 15984* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ph  ->  ( ( X 
 .+  Y )  .X.  Z )  =  ( ( X  .X.  Z )  .+  ( Y  .X.  Z ) ) )
 
Theorempsrcom 15985* Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  ( X  .X.  Y )  =  ( Y  .X.  X ) )
 
Theorempsrass23 15986* Associative identities for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  K  =  (
 Base `  R )   &    |-  .x.  =  ( .s `  S )   &    |-  ( ph  ->  A  e.  K )   =>    |-  ( ph  ->  (
 ( ( A  .x.  X )  .X.  Y )  =  ( A  .x.  ( X  .X.  Y ) ) 
 /\  ( X  .X.  ( A  .x.  Y ) )  =  ( A 
 .x.  ( X  .X.  Y ) ) ) )
 
Theorempsrrng 15987 The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  S  e.  Ring
 )
 
Theorempsr1 15988* The identity element of the ring of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( 1r `  S )   =>    |-  ( ph  ->  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
 
Theorempsrcrng 15989 The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  S  e.  CRing
 )
 
Theorempsrassa 15990 The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  S  e. AssAlg )
 
Theoremresspsrbas 15991 A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ph  ->  B  =  ( Base `  P )
 )
 
Theoremresspsradd 15992 A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
 
Theoremresspsrmul 15993 A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( .r `  U ) Y )  =  ( X ( .r `  P ) Y ) )
 
Theoremresspsrvsca 15994 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  T  /\  Y  e.  B )
 )  ->  ( X ( .s `  U ) Y )  =  ( X ( .s `  P ) Y ) )
 
Theoremsubrgpsr 15995 A subring of the base ring induces a subring of power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   =>    |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R )
 )  ->  B  e.  (SubRing `  S ) )
 
Theoremmvridlem 15996* A bag containing one element is a finite bag. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   =>    |-  ( I  e.  V  ->  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) )  e.  D )
 
Theoremmvrfval 15997* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   =>    |-  ( ph  ->  V  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) )
 
Theoremmvrval 15998* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( V `  X )  =  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
 
Theoremmvrval2 15999* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  F  e.  D )   =>    |-  ( ph  ->  ( ( V `  X ) `  F )  =  if ( F  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
 )
 
Theoremmvrid 16000* The  X i-th coefficient of the term  X i is  1. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  (
 ( V `  X ) `  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) )  =  .1.  )
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