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Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmplasclf 16201 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 16202 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 16203 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 16204* 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 16205* 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 16206* 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 16207* 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 16208* 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 16209* 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 16210* 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 16211* 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 16212* 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 16213 Univariate power series.
 class PwSer1
 
Syntaxcv1 16214 The base variable of a univariate power series.
 class var1
 
Syntaxcpl1 16215 Univariate polynomials.
 class Poly1
 
Syntaxces1 16216 Evaluation in a subring.
 class evalSub1
 
Syntaxce1 16217 Evaluation of a univariate polynomial.
 class eval1
 
Syntaxcco1 16218 Convert a multivariate polynomial representation to univariate.
 class coe1
 
Syntaxctp1 16219 Convert a univariate polynomial representation to multivariate.
 class toPoly1
 
Definitiondf-psr1 16220 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |- PwSer1  =  ( r  e.  _V  |->  ( ( 1o ordPwSer  r ) `
  (/) ) )
 
Definitiondf-vr1 16221 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 16222 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 16223* 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 16224* 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 16225* 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 16226* 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 16227 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 16228 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  S  =  (PwSer1 `  R )   =>    |-  S  =  ( ( 1o ordPwSer  R ) `  (/) )
 
Theorempsr1crng 16229 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 16230 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 16231 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 16232 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 16233 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 16234 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 16235 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 16236 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 16237 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 16238 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 16239 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 16240 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 16241 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 16242 Obsolete version of elbasfv 13154 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 16243 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 16244 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 16245 Obsolete version of elbasfv 13154 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 16246 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 16247 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 16248 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 16249* 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 16250 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 16251 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 16252* 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 16253 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 16254* 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 16255 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 16256 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 16257 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 16258 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 16259 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 16260 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 16261 Obsolete version of mplvsca2 16153 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 16262 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 16263 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 16264 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 16265 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 16266 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 16267 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 )
 
Theoremply1mulr 16268 Value of 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.  =  ( .r `  Y )   =>    |-  .x.  =  ( .r `  S )
 
Theoremressply1bas2 16269 The base set of a restricted polynomial algebra consists of power series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  W  =  (PwSer1 `  H )   &    |-  C  =  ( Base `  W )   &    |-  K  =  ( Base `  S )   =>    |-  ( ph  ->  B  =  ( C  i^i  K ) )
 
Theoremressply1bas 16270 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  P  =  ( Ss  B )   =>    |-  ( ph  ->  B  =  ( Base `  P )
 )
 
Theoremressply1add 16271 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
 
Theoremressply1mul 16272 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( .r `  U ) Y )  =  ( X ( .r `  P ) Y ) )
 
Theoremressply1vsca 16273 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  T  /\  Y  e.  B )
 )  ->  ( X ( .s `  U ) Y )  =  ( X ( .s `  P ) Y ) )
 
Theoremsubrgply1 16274 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   =>    |-  ( T  e.  (SubRing `  R )  ->  B  e.  (SubRing `  S ) )
 
Theorempsrbaspropd 16275 Property deduction for power series base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  ( Base `  R )  =  ( Base `  S )
 )   =>    |-  ( ph  ->  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  S ) ) )
 
Theorempsrplusgpropd 16276* Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  ( I mPwSer  S ) ) )
 
Theoremmplbaspropd 16277* Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  S ) ) )
 
Theoremstrov2rcl 16278 Reverse closure for polynomial-resembling things. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  S  =  ( I F R )   &    |-  B  =  ( Base `  S )   &    |-  Rel  dom 
 F   =>    |-  ( X  e.  B  ->  I  e.  _V )
 
Theorempsropprmul 16279 Reversing multiplication in a ring reverses multiplication in the power series ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  Y  =  ( I mPwSer  R )   &    |-  S  =  (oppr `  R )   &    |-  Z  =  ( I mPwSer  S )   &    |-  .x.  =  ( .r `  Y )   &    |-  .xb  =  ( .r `  Z )   &    |-  B  =  ( Base `  Y )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .xb  G )  =  ( G  .x.  F ) )
 
Theoremply1opprmul 16280 Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  S  =  (oppr `  R )   &    |-  Z  =  (Poly1 `  S )   &    |- 
 .x.  =  ( .r `  Y )   &    |-  .xb  =  ( .r `  Z )   &    |-  B  =  ( Base `  Y )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .xb  G )  =  ( G  .x.  F ) )
 
Theorem00ply1bas 16281 Lemma for ply1basfvi 16282 and deg1fvi 19434. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  (/)  =  ( Base `  (Poly1 `  (/) ) )
 
Theoremply1basfvi 16282 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (  _I  `  R ) ) )
 
Theoremply1plusgfvi 16283 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (  _I  `  R )
 ) )
 
Theoremply1baspropd 16284* Property deduction for univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  S ) ) )
 
Theoremply1plusgpropd 16285* Property deduction for univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  S ) ) )
 
Theoremopsrrng 16286 Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  O  e.  Ring )
 
Theoremopsrlmod 16287 Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  O  e.  LMod )
 
Theorempsr1rng 16288 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  S  =  (PwSer1 `  R )   =>    |-  ( R  e.  Ring  ->  S  e.  Ring )
 
Theoremply1rng 16289 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  Ring )
 
Theorempsr1lmod 16290 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (PwSer1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  LMod )
 
Theorempsr1sca 16291 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
 |-  P  =  (PwSer1 `  R )   =>    |-  ( R  e.  V  ->  R  =  (Scalar `  P ) )
 
Theorempsr1sca2 16292 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
 |-  P  =  (PwSer1 `  R )   =>    |-  (  _I  `  R )  =  (Scalar `  P )
 
Theoremply1lmod 16293 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  LMod )
 
Theoremply1sca 16294 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  V  ->  R  =  (Scalar `  P ) )
 
Theoremply1sca2 16295 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  (  _I  `  R )  =  (Scalar `  P )
 
Theoremply1mpl0 16296 The univariate polynomial ring has the same zero as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
 |-  M  =  ( 1o mPoly  R )   &    |-  P  =  (Poly1 `  R )   &    |-  .0.  =  ( 0g `  P )   =>    |-  .0.  =  ( 0g `  M )
 
Theoremply1mpl1 16297 The univariate polynomial ring has the same one as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
 |-  M  =  ( 1o mPoly  R )   &    |-  P  =  (Poly1 `  R )   &    |-  .1.  =  ( 1r `  P )   =>    |-  .1.  =  ( 1r `  M )
 
Theoremply1ascl 16298 The univariate polynomial ring inherits the multivariate ring's scalar function. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by Fan Zheng, 26-Jun-2016.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   =>    |-  A  =  (algSc `  ( 1o mPoly  R ) )
 
Theoremsubrg1ascl 16299 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  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  C  =  (algSc `  U )   =>    |-  ( ph  ->  C  =  ( A  |`  T ) )
 
Theoremsubrg1asclcl 16300 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  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  B  =  (
 Base `  U )   &    |-  K  =  ( Base `  R )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  (
 ( A `  X )  e.  B  <->  X  e.  T ) )
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