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Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremressmpladd 16201 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
 
Theoremressmplmul 16202 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( .r `  U ) Y )  =  ( X ( .r `  P ) Y ) )
 
Theoremressmplvsca 16203 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  T  /\  Y  e.  B )
 )  ->  ( X ( .s `  U ) Y )  =  ( X ( .s `  P ) Y ) )
 
Theoremsubrgmpl 16204 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   =>    |-  (
 ( I  e.  V  /\  T  e.  (SubRing `  R ) )  ->  B  e.  (SubRing `  S ) )
 
Theoremsubrgmvr 16205 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  H  =  ( Rs  T )   =>    |-  ( ph  ->  V  =  ( I mVar  H ) )
 
Theoremsubrgmvrf 16206 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   =>    |-  ( ph  ->  V : I --> B )
 
Theoremmplmon 16207* A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  (
 y  e.  D  |->  if ( y  =  X ,  .1.  ,  .0.  )
 )  e.  B )
 
Theoremmplmonmul 16208* The product of two monomials adds the exponent vectors together. For example, the product of  ( x ^ 2 ) ( y ^
2 ) with  ( y ^ 1 ) ( z ^ 3 ) is  ( x ^ 2 ) ( y ^
3 ) ( z ^ 3 ), where the exponent vectors  <. 2 ,  2 ,  0 >. and  <. 0 ,  1 ,  3
>. are added to give  <. 2 ,  3 ,  3 >.. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  D )   &    |-  .x.  =  ( .r `  P )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  (
 ( y  e.  D  |->  if ( y  =  X ,  .1.  ,  .0.  )
 )  .x.  ( y  e.  D  |->  if ( y  =  Y ,  .1.  ,  .0.  ) ) )  =  ( y  e.  D  |->  if ( y  =  ( X  o F  +  Y ) ,  .1.  ,  .0.  ) ) )
 
Theoremmplcoe1 16209* Decompose a polynomial into a finite sum of monomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .s `  P )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X  =  ( P  gsumg  ( k  e.  D  |->  ( ( X `  k )  .x.  ( y  e.  D  |->  if (
 y  =  k ,  .1.  ,  .0.  )
 ) ) ) ) )
 
Theoremmplcoe3 16210* Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  V  =  ( I mVar  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( N  .^  ( V `  X ) ) )
 
Theoremmplcoe2 16211* Decompose a monomial into a finite product of powers of variables. (The assumption that  R is a commutative ring is not strictly necessary, because the submonoid of monomials is in the center of the multiplicative monoid of polynomials, but it simplifies the proof.) (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  V  =  ( I mVar  R )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  (
 y  e.  D  |->  if ( y  =  Y ,  .1.  ,  .0.  )
 )  =  ( G 
 gsumg  ( k  e.  I  |->  ( ( Y `  k )  .^  ( V `
  k ) ) ) ) )
 
Theoremmplbas2 16212 An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  V  =  ( I mVar  R )   &    |-  A  =  (AlgSpan `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  ( A `  ran  V )  =  ( Base `  P ) )
 
Theoremltbval 16213* Value of the well-order on finite bags. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  C  =  ( T  <bag  I )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  W )   =>    |-  ( ph  ->  C  =  { <. x ,  y >.  |  ( { x ,  y }  C_  D  /\  E. z  e.  I  ( ( x `  z )  <  ( y `
  z )  /\  A. w  e.  I  ( z T w  ->  ( x `  w )  =  ( y `  w ) ) ) ) } )
 
Theoremltbwe 16214* The finite bag order is a well-order, given a well order of the index set. (Contributed by Mario Carneiro, 2-Jun-2015.)
 |-  C  =  ( T  <bag  I )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  W )   &    |-  ( ph  ->  T  We  I )   =>    |-  ( ph  ->  C  We  D )
 
Theoremreldmopsr 16215 Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
 |- 
 Rel  dom ordPwSer
 
Theoremopsrval 16216* The value of the "ordered power series" function. This is the same as mPwSer psrval 16110, but with the addition of a well-order on  I we can turn a strict order on 
R into a strict order on the power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  B  =  (
 Base `  S )   &    |-  .<  =  ( lt `  R )   &    |-  C  =  ( T  <bag  I )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  ( y `
  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  O  =  ( S sSet  <. ( le ` 
 ndx ) ,  .<_  >.
 ) )
 
Theoremopsrle 16217* An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  B  =  (
 Base `  S )   &    |-  .<  =  ( lt `  R )   &    |-  C  =  ( T  <bag  I )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .<_  =  ( le `  O )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `
  z )  .<  ( y `  z ) 
 /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } )
 
Theoremopsrval2 16218 Self-referential expression for the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  .<_  =  ( le `  O )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  ( ph  ->  T  C_  ( I  X.  I
 ) )   =>    |-  ( ph  ->  O  =  ( S sSet  <. ( le ` 
 ndx ) ,  .<_  >.
 ) )
 
Theoremopsrbaslem 16219 Get a component of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  <  10   =>    |-  ( ph  ->  ( E `  S )  =  ( E `  O ) )
 
Theoremopsrbas 16220 The base set of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  (
 Base `  S )  =  ( Base `  O )
 )
 
Theoremopsrplusg 16221 The addition operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  (
 +g  `  S )  =  ( +g  `  O ) )
 
Theoremopsrmulr 16222 The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  ( .r `  S )  =  ( .r `  O ) )
 
Theoremopsrvsca 16223 The scalar product operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  ( .s `  S )  =  ( .s `  O ) )
 
Theoremopsrsca 16224 The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   =>    |-  ( ph  ->  R  =  (Scalar `  O )
 )
 
Theoremopsrtoslem1 16225* Lemma for opsrtos 16227. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e. Toset )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   &    |-  ( ph  ->  T  We  I
 )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .<  =  ( lt `  R )   &    |-  C  =  ( T  <bag  I )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ps  <->  E. z  e.  D  ( ( x `  z )  .<  ( y `
  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) ) )   &    |-  .<_  =  ( le `  O )   =>    |-  ( ph  ->  .<_  =  ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  u.  (  _I  |`  B ) ) )
 
Theoremopsrtoslem2 16226* Lemma for opsrtos 16227. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e. Toset )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   &    |-  ( ph  ->  T  We  I
 )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .<  =  ( lt `  R )   &    |-  C  =  ( T  <bag  I )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ps  <->  E. z  e.  D  ( ( x `  z )  .<  ( y `
  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) ) )   &    |-  .<_  =  ( le `  O )   =>    |-  ( ph  ->  O  e. Toset )
 
Theoremopsrtos 16227 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e. Toset )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   &    |-  ( ph  ->  T  We  I
 )   =>    |-  ( ph  ->  O  e. Toset )
 
Theoremopsrso 16228 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e. Toset )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   &    |-  ( ph  ->  T  We  I
 )   &    |- 
 .<_  =  ( lt `  O )   &    |-  B  =  ( Base `  O )   =>    |-  ( ph  ->  .<_  Or  B )
 
Theoremopsrcrng 16229 The ring of ordered power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  O  e.  CRing )
 
Theoremopsrassa 16230 The ring of ordered power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  O  e. AssAlg )
 
Theoremmplrcl 16231 Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   =>    |-  ( X  e.  B  ->  I  e.  _V )
 
Theoremmplelsfi 16232 A polynomial treated as a coefficient function has finitely many nonzero terms. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  R  e.  V )   =>    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } )
 )  e.  Fin )
 
Theoremmvrf2 16233 The power series/polynomial variable function maps indices to polynomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  V : I --> B )
 
Theoremmplmon2 16234* Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  .x.  =  ( .s `  P )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  K  e.  D )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .x.  ( y  e.  D  |->  if ( y  =  K ,  .1.  ,  .0.  ) ) )  =  ( y  e.  D  |->  if ( y  =  K ,  X ,  .0.  )
 ) )
 
Theorempsrbag0 16235* The empty bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( I  e.  V  ->  ( I  X.  {
 0 } )  e.  D )
 
Theorempsrbagsn 16236* A singleton bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( I  e.  V  ->  ( x  e.  I  |->  if ( x  =  K ,  1 ,  0 ) )  e.  D )
 
Theoremmplascl 16237* Value of the scalar injection into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( A `  X )  =  ( y  e.  D  |->  if ( y  =  ( I  X.  { 0 } ) ,  X ,  .0.  ) ) )
 
Theoremmplasclf 16238 The scalar injection is a function into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  K  =  ( Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  A : K --> B )
 
Theoremsubrgascl 16239 The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  A  =  (algSc `  P )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  C  =  (algSc `  U )   =>    |-  ( ph  ->  C  =  ( A  |`  T ) )
 
Theoremsubrgasclcl 16240 The scalars in a polynomial algebra are in the subring algebra iff the scalar value is in the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  A  =  (algSc `  P )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  B  =  ( Base `  U )   &    |-  K  =  (
 Base `  R )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  (
 ( A `  X )  e.  B  <->  X  e.  T ) )
 
Theoremmplmon2cl 16241* A scaled monomial is a polynomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  ( Base `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  X  e.  C )   &    |-  ( ph  ->  K  e.  D )   =>    |-  ( ph  ->  (
 y  e.  D  |->  if ( y  =  K ,  X ,  .0.  )
 )  e.  B )
 
Theoremmplmon2mul 16242* Product of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  ( Base `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  .xb 
 =  ( .r `  P )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  Y  e.  D )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  G  e.  C )   =>    |-  ( ph  ->  (
 ( y  e.  D  |->  if ( y  =  X ,  F ,  .0.  )
 )  .xb  ( y  e.  D  |->  if ( y  =  Y ,  G ,  .0.  ) ) )  =  ( y  e.  D  |->  if ( y  =  ( X  o F  +  Y ) ,  ( F  .x.  G ) ,  .0.  ) ) )
 
Theoremmplind 16243* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. The commutativity condition is stronger than strictly needed. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 |-  K  =  ( Base `  R )   &    |-  V  =  ( I mVar  R )   &    |-  Y  =  ( I mPoly  R )   &    |-  .+  =  ( +g  `  Y )   &    |- 
 .x.  =  ( .r `  Y )   &    |-  C  =  (algSc `  Y )   &    |-  B  =  (
 Base `  Y )   &    |-  (
 ( ph  /\  ( x  e.  H  /\  y  e.  H ) )  ->  ( x  .+  y )  e.  H )   &    |-  (
 ( ph  /\  ( x  e.  H  /\  y  e.  H ) )  ->  ( x  .x.  y )  e.  H )   &    |-  (
 ( ph  /\  x  e.  K )  ->  ( C `  x )  e.  H )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  ( V `  x )  e.  H )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  X  e.  H )
 
Theoremmplcoe4 16244* Decompose a polynomial into a finite sum of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X  =  ( P  gsumg  ( k  e.  D  |->  ( y  e.  D  |->  if ( y  =  k ,  ( X `  k ) ,  .0.  ) ) ) ) )
 
10.10.2  Polynomial evaluation
 
Theoremevlslem4 16245* The support of a tensor product of ring element families is contained in the product of the supports. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .x. 
 =  ( .r `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  X  e.  B )   &    |-  ( ( ph  /\  y  e.  J )  ->  Y  e.  B )   =>    |-  ( ph  ->  ( `' ( x  e.  I ,  y  e.  J  |->  ( X  .x.  Y ) ) " ( _V  \  {  .0.  } )
 )  C_  ( ( `' ( x  e.  I  |->  X ) " ( _V  \  {  .0.  }
 ) )  X.  ( `' ( y  e.  J  |->  Y ) " ( _V  \  {  .0.  }
 ) ) ) )
 
Theorempsrbagsuppfi 16246* Finite bags have finite nonzero-support. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   =>    |-  ( ( X  e.  D  /\  I  e.  V )  ->  ( `' X " ( _V  \  {
 0 } ) )  e.  Fin )
 
Theorempsrbagev1 16247* A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  C  =  (
 Base `  T )   &    |-  .x.  =  (.g `  T )   &    |-  .0.  =  ( 0g `  T )   &    |-  ( ph  ->  T  e. CMnd )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  G : I --> C )   &    |-  ( ph  ->  I  e.  _V )   =>    |-  ( ph  ->  (
 ( B  o F  .x.  G ) : I --> C  /\  ( `' ( B  o F  .x.  G ) " ( _V  \  {  .0.  } ) )  e. 
 Fin ) )
 
Theorempsrbagev2 16248* Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  C  =  (
 Base `  T )   &    |-  .x.  =  (.g `  T )   &    |-  .0.  =  ( 0g `  T )   &    |-  ( ph  ->  T  e. CMnd )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  G : I --> C )   &    |-  ( ph  ->  I  e.  _V )   =>    |-  ( ph  ->  ( T  gsumg  ( B  o F  .x.  G ) )  e.  C )
 
Theoremevlslem2 16249* A linear function on the polynomial ring which is multiplicative on scaled monomials is generally multiplicative. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  S  e.  CRing
 )   &    |-  ( ph  ->  E  e.  ( P  GrpHom  S ) )   &    |-  ( ( ph  /\  ( ( x  e.  B  /\  y  e.  B )  /\  (
 j  e.  D  /\  i  e.  D )
 ) )  ->  ( E `  ( k  e.  D  |->  if ( k  =  ( j  o F  +  i ) ,  (
 ( x `  j
 ) ( .r `  R ) ( y `
  i ) ) ,  .0.  ) ) )  =  ( ( E `  ( k  e.  D  |->  if (
 k  =  j ,  ( x `  j
 ) ,  .0.  )
 ) )  .x.  ( E `  ( k  e.  D  |->  if ( k  =  i ,  ( y `
  i ) ,  .0.  ) ) ) ) )   =>    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( E `  ( x ( .r
 `  P ) y ) )  =  ( ( E `  x )  .x.  ( E `  y ) ) )
 
10.10.3  Univariate polynomials
 
Syntaxcps1 16250 Univariate power series.
 class PwSer1
 
Syntaxcv1 16251 The base variable of a univariate power series.
 class var1
 
Syntaxcpl1 16252 Univariate polynomials.
 class Poly1
 
Syntaxces1 16253 Evaluation in a subring.
 class evalSub1
 
Syntaxce1 16254 Evaluation of a univariate polynomial.
 class eval1
 
Syntaxcco1 16255 Convert a multivariate polynomial representation to univariate.
 class coe1
 
Syntaxctp1 16256 Convert a univariate polynomial representation to multivariate.
 class toPoly1
 
Definitiondf-psr1 16257 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |- PwSer1  =  ( r  e.  _V  |->  ( ( 1o ordPwSer  r ) `
  (/) ) )
 
Definitiondf-vr1 16258 Define the base element of a univariate power series (the  X element of the set  R [ X ] of polynomials and also the  X in the set  R [ [ X ] ] of power series). (Contributed by Mario Carneiro, 8-Feb-2015.)
 |- var1  =  ( r  e.  _V  |->  ( ( 1o mVar  r
 ) `  (/) ) )
 
Definitiondf-ply1 16259 Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |- Poly1  =  ( r  e.  _V  |->  ( (PwSer1 `  r )s  ( Base `  ( 1o mPoly  r )
 ) ) )
 
Definitiondf-evls1 16260* Define the evaluation map for the univariate polynomial algebra. The function  ( S evalSub1  R ) : V --> ( S  ^m  S ) makes sense when  S is a ring and  R is a subring of  S, and where  V is the set of polynomials in  (Poly1 `  R ). This function maps an element of the formal polynomial algebra (with coefficients in  R) to a function from assignments to the variable from  S into an element of  S formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |- evalSub1  =  ( s  e.  _V ,  r  e.  ~P ( Base `  s )  |-> 
 [_ ( Base `  s
 )  /  b ]_ ( ( x  e.  ( b  ^m  (
 b  ^m  1o )
 )  |->  ( x  o.  ( y  e.  b  |->  ( 1o  X.  {
 y } ) ) ) )  o.  (
 ( 1o evalSub  s ) `  r ) ) )
 
Definitiondf-evl1 16261* Define the evaluation map for the univariate polynomial algebra. The function  (eval1 `  R ) : V --> ( R  ^m  R ) makes sense when  R is a ring, and  V is the set of polynomials in  (Poly1 `  R ). This function maps an element of the formal polynomial algebra (with coefficients in  R) to a function from assignments to the variable from  R into an element of  R formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |- eval1  =  ( r  e.  _V  |->  [_ ( Base `  r )  /  b ]_ ( ( x  e.  ( b 
 ^m  ( b  ^m  1o ) )  |->  ( x  o.  ( y  e.  b  |->  ( 1o  X.  { y } ) ) ) )  o.  ( 1o eval  r ) ) )
 
Definitiondf-coe1 16262* Define the coefficient function for a univariate polynomial. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |- coe1  =  ( f  e.  _V  |->  ( n  e.  NN0  |->  ( f `
  ( 1o  X.  { n } ) ) ) )
 
Definitiondf-toply1 16263* Define a function which maps a coefficient function for a univariate polynomial to the corresponding polynomial object. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |- toPoly1  =  ( f  e.  _V  |->  ( n  e.  ( NN0  ^m  1o )  |->  ( f `  ( n `
  (/) ) ) ) )
 
Theorempsr1baslem 16264 The set of finite bags on  1o is just the set of all functions from  1o to  NN0. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  ( NN0  ^m  1o )  =  { f  e.  ( NN0  ^m  1o )  |  ( `' f " NN )  e. 
 Fin }
 
Theorempsr1val 16265 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  S  =  (PwSer1 `  R )   =>    |-  S  =  ( ( 1o ordPwSer  R ) `  (/) )
 
Theorempsr1crng 16266 The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  S  =  (PwSer1 `  R )   =>    |-  ( R  e.  CRing  ->  S  e.  CRing )
 
Theorempsr1assa 16267 The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  S  =  (PwSer1 `  R )   =>    |-  ( R  e.  CRing  ->  S  e. AssAlg )
 
Theorempsr1tos 16268 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.)
 |-  S  =  (PwSer1 `  R )   =>    |-  ( R  e. Toset  ->  S  e. Toset )
 
Theorempsr1bas2 16269 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  S )   &    |-  O  =  ( 1o mPwSer  R )   =>    |-  B  =  ( Base `  O )
 
Theorempsr1bas 16270 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  S  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  S )   &    |-  K  =  (
 Base `  R )   =>    |-  B  =  ( K  ^m  ( NN0  ^m 
 1o ) )
 
Theoremvr1val 16271 The value of the generator of the power series algebra (the  X in  R [ [ X ] ]). Since all univariate polynomial rings over a fixed base ring  R are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and  1o  =  { (/) } is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
 |-  X  =  (var1 `  R )   =>    |-  X  =  ( ( 1o mVar  R ) `  (/) )
 
Theoremvr1cl2 16272 The variable  X is a member of the power series algebra  R [ [ X ] ]. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  X  =  (var1 `  R )   &    |-  S  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  S )   =>    |-  ( R  e.  Ring  ->  X  e.  B )
 
Theoremply1val 16273 The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  S  =  (PwSer1 `  R )   =>    |-  P  =  ( Ss  (
 Base `  ( 1o mPoly  R ) ) )
 
Theoremply1bas 16274 The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  S  =  (PwSer1 `  R )   &    |-  U  =  ( Base `  P )   =>    |-  U  =  ( Base `  ( 1o mPoly  R )
 )
 
Theoremply1lss 16275 Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  S  =  (PwSer1 `  R )   &    |-  U  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  U  e.  ( LSubSp `  S ) )
 
Theoremply1subrg 16276 Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  S  =  (PwSer1 `  R )   &    |-  U  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  U  e.  (SubRing `  S ) )
 
Theoremply1crng 16277 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  CRing  ->  P  e.  CRing )
 
Theoremply1assa 16278 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  CRing  ->  P  e. AssAlg )
 
Theorempsr1rclOLD 16279 Obsolete version of elbasfv 13191 as of 5-Apr-2016. Reverse closure for ring existence from the univariate power series base set. (Contributed by Stefan O'Rear, 25-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  P  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  R  e.  _V )
 
Theorempsr1bascl 16280 A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  P  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  F  e.  ( Base `  ( 1o mPwSer  R )
 ) )
 
Theorempsr1basf 16281 Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  P  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( F  e.  B  ->  F : (
 NN0  ^m  1o ) --> K )
 
Theoremply1rclOLD 16282 Obsolete version of elbasfv 13191 as of 5-Apr-2016. Reverse closure for ring existence from the univariate polynomial base set. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  R  e.  _V )
 
Theoremply1basf 16283 Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( F  e.  B  ->  F : (
 NN0  ^m  1o ) --> K )
 
Theoremply1bascl 16284 A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  F  e.  ( Base `  (PwSer1 `  R ) ) )
 
Theoremply1bascl2 16285 A univariate polynomial is a multivariate polynomial on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  F  e.  ( Base `  ( 1o mPoly  R )
 ) )
 
Theoremcoe1fval 16286* Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   =>    |-  ( F  e.  V  ->  A  =  ( n  e.  NN0  |->  ( F `
  ( 1o  X.  { n } ) ) ) )
 
Theoremcoe1fv 16287 Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   =>    |-  ( ( F  e.  V  /\  N  e.  NN0 )  ->  ( A `  N )  =  ( F `  ( 1o  X.  { N } ) ) )
 
Theoremfvcoe1 16288 Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   =>    |-  ( ( F  e.  V  /\  X  e.  ( NN0  ^m  1o ) ) 
 ->  ( F `  X )  =  ( A `  ( X `  (/) ) ) )
 
Theoremcoe1fval3 16289* Univariate power series coeffecient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  A  =  (coe1 `  F )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (PwSer1 `  R )   &    |-  G  =  ( y  e.  NN0  |->  ( 1o 
 X.  { y } )
 )   =>    |-  ( F  e.  B  ->  A  =  ( F  o.  G ) )
 
Theoremcoe1f2 16290 Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  A  =  (coe1 `  F )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (PwSer1 `  R )   &    |-  K  =  (
 Base `  R )   =>    |-  ( F  e.  B  ->  A : NN0 --> K )
 
Theoremcoe1fval2 16291* Univariate polynomial coeffecient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (Poly1 `  R )   &    |-  G  =  ( y  e.  NN0  |->  ( 1o 
 X.  { y } )
 )   =>    |-  ( F  e.  B  ->  A  =  ( F  o.  G ) )
 
Theoremcoe1f 16292 Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (Poly1 `  R )   &    |-  K  =  (
 Base `  R )   =>    |-  ( F  e.  B  ->  A : NN0 --> K )
 
Theoremcoe1sfi 16293 Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (Poly1 `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( F  e.  B  ->  ( `' A "
 ( _V  \  {  .0.  } ) )  e. 
 Fin )
 
Theoremvr1cl 16294 The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  X  =  (var1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  X  e.  B )
 
Theoremopsr0 16295 Zero in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  ( 0g `  S )  =  ( 0g `  O ) )
 
Theoremopsr1 16296 One in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  ( 1r `  S )  =  ( 1r `  O ) )
 
Theoremmplplusg 16297 Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  Y  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  .+  =  ( +g  `  Y )   =>    |-  .+  =  ( +g  `  S )
 
TheoremmplvscafvalOLD 16298 Obsolete version of mplvsca2 16190 as of 5-Apr-2016. Value of scalar multiplication in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  Y  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  .x.  =  ( .s `  Y )   =>    |-  .x.  =  ( .s `  S )
 
Theoremmplmulr 16299 Value of multiplication in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  Y  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  .x.  =  ( .r `  Y )   =>    |-  .x.  =  ( .r `  S )
 
Theorempsr1plusg 16300 Value of addition in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  Y  =  (PwSer1 `  R )   &    |-  S  =  ( 1o mPwSer  R )   &    |-  .+  =  ( +g  `  Y )   =>    |-  .+  =  ( +g  `  S )
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