HomeHome Metamath Proof Explorer
Theorem List (p. 163 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21494)
  Hilbert Space Explorer  Hilbert Space Explorer
(21495-23017)
  Users' Mathboxes  Users' Mathboxes
(23018-31433)
 

Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubrgmvr 16201 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 16202 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 16203* 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 16204* 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 16205* 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 16206* 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 16207* 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 16208 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 16209* 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 16210* 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 16211 Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
 |- 
 Rel  dom ordPwSer
 
Theoremopsrval 16212* The value of the "ordered power series" function. This is the same as mPwSer psrval 16106, 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 16213* 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 16214 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 16215 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 16216 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 16217 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 16218 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 16219 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 16220 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 16221* Lemma for opsrtos 16223. (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 16222* Lemma for opsrtos 16223. (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 16223 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 16224 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 16225 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 16226 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 16227 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 16228 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 16229 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 16230* 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 16231* 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 16232* 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 16233* 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 16234 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 16235 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 16236 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 16237* 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 16238* 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 16239* 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 16240* 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 16241* 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 16242* 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 16243* 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 16244* 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 16245* 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 16246 Univariate power series.
 class PwSer1
 
Syntaxcv1 16247 The base variable of a univariate power series.
 class var1
 
Syntaxcpl1 16248 Univariate polynomials.
 class Poly1
 
Syntaxces1 16249 Evaluation in a subring.
 class evalSub1
 
Syntaxce1 16250 Evaluation of a univariate polynomial.
 class eval1
 
Syntaxcco1 16251 Convert a multivariate polynomial representation to univariate.
 class coe1
 
Syntaxctp1 16252 Convert a univariate polynomial representation to multivariate.
 class toPoly1
 
Definitiondf-psr1 16253 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |- PwSer1  =  ( r  e.  _V  |->  ( ( 1o ordPwSer  r ) `
  (/) ) )
 
Definitiondf-vr1 16254 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 16255 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 16256* 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 16257* 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 16258* 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 16259* 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 16260 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 16261 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  S  =  (PwSer1 `  R )   =>    |-  S  =  ( ( 1o ordPwSer  R ) `  (/) )
 
Theorempsr1crng 16262 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 16263 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 16264 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 16265 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 16266 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 16267 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 16268 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 16269 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 16270 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 16271 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 16272 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 16273 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 16274 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 16275 Obsolete version of elbasfv 13187 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 16276 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 16277 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 16278 Obsolete version of elbasfv 13187 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 16279 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 16280 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 16281 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 16282* 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 16283 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 16284 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 16285* 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 16286 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 16287* 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 16288 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 16289 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 16290 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 16291 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 16292 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 16293 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 16294 Obsolete version of mplvsca2 16186 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 16295 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 16296 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 )
 
Theorempsr1vsca 16297 Value of scalar multiplication 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 )   &    |-  .x.  =  ( .s `  Y )   =>    |-  .x.  =  ( .s `  S )
 
Theorempsr1mulr 16298 Value of multiplication 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 )   &    |-  .x.  =  ( .r `  Y )   =>    |-  .x.  =  ( .r `  S )
 
Theoremply1plusg 16299 Value of addition in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  S  =  ( 1o mPoly  R )   &    |-  .+  =  ( +g  `  Y )   =>    |-  .+  =  ( +g  `  S )
 
Theoremply1vsca 16300 Value of scalar multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  S  =  ( 1o mPoly  R )   &    |-  .x.  =  ( .s `  Y )   =>    |-  .x.  =  ( .s `  S )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31433
  Copyright terms: Public domain < Previous  Next >