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Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremply1bas 16201 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 16202 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 16203 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 16204 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 16205 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 16206 Obsolete version of elbasfv 13118 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 16207 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 16208 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 16209 Obsolete version of elbasfv 13118 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 16210 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 16211 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 16212 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 16213* 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 16214 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 16215 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 16216* 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 16217 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 16218* 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 16219 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 16220 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 16221 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 16222 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 16223 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 16224 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 16225 Obsolete version of mplvsca2 16117 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 16226 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 16227 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 16228 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 16229 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 16230 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 16231 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 16232 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 16233 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 16234 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 16235 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 16236 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 16237 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 16238 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 16239 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 16240* 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 16241* 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 16242 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 16243 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 16244 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 16245 Lemma for ply1basfvi 16246 and deg1fvi 19398. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  (/)  =  ( Base `  (Poly1 `  (/) ) )
 
Theoremply1basfvi 16246 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (  _I  `  R ) ) )
 
Theoremply1plusgfvi 16247 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (  _I  `  R )
 ) )
 
Theoremply1baspropd 16248* 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 16249* 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 16250 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 16251 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 16252 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  S  =  (PwSer1 `  R )   =>    |-  ( R  e.  Ring  ->  S  e.  Ring )
 
Theoremply1rng 16253 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  Ring )
 
Theorempsr1lmod 16254 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 16255 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 16256 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 16257 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  LMod )
 
Theoremply1sca 16258 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  V  ->  R  =  (Scalar `  P ) )
 
Theoremply1sca2 16259 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  (  _I  `  R )  =  (Scalar `  P )
 
Theoremply1mpl0 16260 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 16261 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 16262 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 16263 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 16264 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 ) )
 
Theoremsubrgvr1 16265 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
 |-  X  =  (var1 `  R )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  H  =  ( Rs  T )   =>    |-  ( ph  ->  X  =  (var1 `  H ) )
 
Theoremsubrgvr1cl 16266 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 5-Jul-2015.)
 |-  X  =  (var1 `  R )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   =>    |-  ( ph  ->  X  e.  B )
 
Theoremcoe1z 16267 The coefficient vector of 0. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  Y  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  (coe1 ` 
 .0.  )  =  (
 NN0  X.  { Y } ) )
 
Theoremcoe1add 16268 The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .+b  =  ( +g  `  Y )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .+b  G ) )  =  ( (coe1 `  F )  o F  .+  (coe1 `  G ) ) )
 
Theoremcoe1addfv 16269 A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .+b  =  ( +g  `  Y )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .+b  G ) ) `  X )  =  ( (
 (coe1 `
  F ) `  X )  .+  ( (coe1 `  G ) `  X ) ) )
 
Theoremcoe1subfv 16270 A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  N  =  ( -g `  R )   =>    |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .-  G ) ) `
  X )  =  ( ( (coe1 `  F ) `  X ) N ( (coe1 `  G ) `  X ) ) )
 
Theoremcoe1mul2lem1 16271 An equivalence for coe1mul2 16273. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  ( ( A  e.  NN0  /\  X  e.  ( NN0  ^m 
 1o ) )  ->  ( X  o R  <_  ( 1o  X.  { A } )  <->  ( X `  (/) )  e.  ( 0
 ... A ) ) )
 
Theoremcoe1mul2lem2 16272* An equivalence for coe1mul2 16273. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  H  =  { d  e.  ( NN0  ^m  1o )  |  d  o R  <_  ( 1o  X.  { k } ) }   =>    |-  (
 k  e.  NN0  ->  ( c  e.  H  |->  ( c `  (/) ) ) : H -1-1-onto-> ( 0 ... k
 ) )
 
Theoremcoe1mul2 16273* The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  S  =  (PwSer1 `  R )   &    |-  .xb  =  ( .r `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  S )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( k  e.  NN0  |->  ( R 
 gsumg  ( x  e.  (
 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  ( (coe1 `  G ) `  (
 k  -  x ) ) ) ) ) ) )
 
Theoremcoe1mul 16274* The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  .xb  =  ( .r `  Y )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  Y )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( k  e.  NN0  |->  ( R 
 gsumg  ( x  e.  (
 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  ( (coe1 `  G ) `  (
 k  -  x ) ) ) ) ) ) )
 
Theoremply1tmcl 16275 Closure of the expression for a univariate polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  ( C  .x.  ( D  .^  X ) )  e.  B )
 
Theoremcoe1tm 16276* Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  (coe1 `  ( C  .x.  ( D  .^  X ) ) )  =  ( x  e.  NN0  |->  if ( x  =  D ,  C ,  .0.  )
 ) )
 
Theoremcoe1tmfv1 16277 Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D )  =  C )
 
Theoremcoe1tmfv2 16278 Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  F  e.  NN0 )   &    |-  ( ph  ->  D  =/=  F )   =>    |-  ( ph  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  F )  =  .0.  )
 
Theoremcoe1tmmul2 16279* Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( A  .xb  ( C 
 .x.  ( D  .^  X ) ) ) )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  (
 ( (coe1 `  A ) `  ( x  -  D ) )  .X.  C ) ,  .0.  ) ) )
 
Theoremcoe1tmmul 16280* Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( ( C  .x.  ( D  .^  X ) )  .xb  A )
 )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  ( C  .X.  ( (coe1 `  A ) `  ( x  -  D ) ) ) ,  .0.  ) ) )
 
Theoremcoe1tmmul2fv 16281 Function value of a right-multiplication by a term in the shifted domain. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  Y  e.  NN0 )   =>    |-  ( ph  ->  (
 (coe1 `
  ( A  .xb  ( C  .x.  ( D 
 .^  X ) ) ) ) `  ( D  +  Y )
 )  =  ( ( (coe1 `  A ) `  Y )  .X.  C ) )
 
Theoremcoe1pwmul 16282* Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( ( D  .^  X )  .x.  A ) )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  (
 (coe1 `
  A ) `  ( x  -  D ) ) ,  .0.  ) ) )
 
Theoremcoe1pwmulfv 16283 Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  Y  e.  NN0 )   =>    |-  ( ph  ->  (
 (coe1 `
  ( ( D 
 .^  X )  .x.  A ) ) `  ( D  +  Y )
 )  =  ( (coe1 `  A ) `  Y ) )
 
Theoremply1scltm 16284 A scalar is a term with zero exponent. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  K ) 
 ->  ( A `  F )  =  ( F  .x.  ( 0  .^  X ) ) )
 
Theoremcoe1sclmul 16285 Coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  Y  e.  B )  ->  (coe1 `  ( ( A `
  X )  .xb  Y ) )  =  ( ( NN0  X.  { X } )  o F  .x.  (coe1 `  Y ) ) )
 
Theoremcoe1sclmulfv 16286 A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  K  /\  Y  e.  B ) 
 /\  .0.  e.  NN0 )  ->  ( (coe1 `  ( ( A `
  X )  .xb  Y ) ) `  .0.  )  =  ( X  .x.  ( (coe1 `  Y ) `  .0.  ) ) )
 
Theoremcoe1sclmul2 16287 Coefficient vector of a polynomial multiplied on the right by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  Y  e.  B )  ->  (coe1 `  ( Y  .xb  ( A `  X ) ) )  =  ( (coe1 `  Y )  o F  .x.  ( NN0  X. 
 { X } )
 ) )
 
Theoremply1sclf 16288 A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  A : K --> B )
 
Theoremcoe1scl 16289* Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K ) 
 ->  (coe1 `  ( A `  X ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  X ,  .0.  ) ) )
 
Theoremply1sclid 16290 Recover the base scalar from a scalar polynomial.. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K )  ->  X  =  ( (coe1 `  ( A `  X ) ) `  0 ) )
 
Theoremply1sclf1 16291 The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  A : K -1-1-> B )
 
Theoremply1scl0 16292 The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  Y  =  ( 0g `  P )   =>    |-  ( R  e.  Ring  ->  ( A `  .0.  )  =  Y )
 
Theoremply1scln0 16293 Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  Y  =  ( 0g `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  X  =/=  .0.  )  ->  ( A `  X )  =/=  Y )
 
Theoremply1scl1 16294 The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( 1r `  P )   =>    |-  ( R  e.  Ring  ->  ( A `  .1.  )  =  N )
 
Theoremply1coe 16295* Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .s `  P )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   &    |-  A  =  (coe1 `  K )   &    |-  R  e.  _V   =>    |-  (
 ( R  e.  CRing  /\  K  e.  B ) 
 ->  K  =  ( P 
 gsumg  ( k  e.  NN0  |->  ( ( A `  k ) 
 .x.  ( k  .^  X ) ) ) ) )
 
10.11  The complex numbers as an extensible structure
 
10.11.1  Definition and basic properties
 
Syntaxcxmt 16296 Extend class notation with the class of all extended metric spaces.
 class  * Met
 
Syntaxcme 16297 Extend class notation with the class of all metrics.
 class  Met
 
Syntaxcbl 16298 Extend class notation with the metric space ball function.
 class  ball
 
Syntaxcmopn 16299 Extend class notation with a function mapping each metric space to the family of its open sets.
 class  MetOpen
 
Definitiondf-xmet 16300* Define the set of all extended metrics on a given base set. The definition is similar to df-met 16301, but we also allow the metric to take on the value  +oo. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |- 
 * Met  =  ( x  e.  _V  |->  { d  e.  ( RR*  ^m  ( x  X.  x ) )  |  A. y  e.  x  A. z  e.  x  ( ( ( y d z )  =  0  <->  y  =  z
 )  /\  A. w  e.  x  ( y d z )  <_  (
 ( w d y ) + e ( w d z ) ) ) } )
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