HomeHome Metamath Proof Explorer
Theorem List (p. 165 of 325)
< 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
(1-22374)
  Hilbert Space Explorer  Hilbert Space Explorer
(22375-23897)
  Users' Mathboxes  Users' Mathboxes
(23898-32447)
 

Theorem List for Metamath Proof Explorer - 16401-16500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempsradd 16401 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+b  Y )  =  ( X  o F  .+  Y ) )
 
Theorempsraddcl 16402 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  B )
 
Theorempsrmulr 16403* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   =>    |-  .xb  =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( f `  x )  .x.  ( g `
  ( k  o F  -  x ) ) ) ) ) ) )
 
Theorempsrmulfval 16404* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F  .xb  G )  =  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
 y  e.  D  |  y  o R  <_  k }  |->  ( ( F `
  x )  .x.  ( G `  ( k  o F  -  x ) ) ) ) ) ) )
 
Theorempsrmulval 16405* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  (
 ( F  .xb  G ) `
  X )  =  ( R  gsumg  ( k  e.  {
 y  e.  D  |  y  o R  <_  X }  |->  ( ( F `
  k )  .x.  ( G `  ( X  o F  -  k
 ) ) ) ) ) )
 
Theorempsrmulcllem 16406* Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  B )
 
Theorempsrmulcl 16407 Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  B )
 
Theorempsrsca 16408 The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   =>    |-  ( ph  ->  R  =  (Scalar `  S )
 )
 
Theorempsrvscafval 16409* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   =>    |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
 )  o F  .x.  f ) )
 
Theorempsrvsca 16410* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( X  .xb  F )  =  ( ( D  X.  { X } )  o F  .x.  F )
 )
 
Theorempsrvscaval 16411* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  (
 ( X  .xb  F ) `
  Y )  =  ( X  .x.  ( F `  Y ) ) )
 
Theorempsrvscacl 16412 Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .x.  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( X  .x.  F )  e.  B )
 
Theorempsr0cl 16413* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  S )   =>    |-  ( ph  ->  ( D  X.  {  .0.  }
 )  e.  B )
 
Theorempsr0lid 16414* The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( D  X.  {  .0.  } )  .+  X )  =  X )
 
Theorempsrnegcl 16415* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( N  o.  X )  e.  B )
 
Theorempsrlinv 16416* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .+  =  ( +g  `  S )   =>    |-  ( ph  ->  (
 ( N  o.  X )  .+  X )  =  ( D  X.  {  .0.  } ) )
 
Theorempsrgrp 16417 The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  S  e.  Grp )
 
Theorempsr0 16418* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  O  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ph  ->  .0.  =  ( D  X.  { O } ) )
 
Theorempsrneg 16419* The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  M  =  ( inv g `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( M `  X )  =  ( N  o.  X ) )
 
Theorempsrlmod 16420 The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  S  e.  LMod
 )
 
Theorempsr1cl 16421* The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   =>    |-  ( ph  ->  U  e.  B )
 
Theorempsrlidm 16422* The identity element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( U  .x.  X )  =  X )
 
Theorempsrridm 16423* The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .x.  U )  =  X )
 
Theorempsrass1 16424* Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( ( X  .X.  Y )  .X.  Z )  =  ( X  .X.  ( Y  .X.  Z ) ) )
 
Theorempsrdi 16425* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ph  ->  ( X  .X.  ( Y  .+  Z ) )  =  ( ( X  .X.  Y )  .+  ( X  .X.  Z ) ) )
 
Theorempsrdir 16426* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ph  ->  ( ( X 
 .+  Y )  .X.  Z )  =  ( ( X  .X.  Z )  .+  ( Y  .X.  Z ) ) )
 
Theorempsrcom 16427* Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  ( X  .X.  Y )  =  ( Y  .X.  X ) )
 
Theorempsrass23 16428* Associative identities for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  K  =  (
 Base `  R )   &    |-  .x.  =  ( .s `  S )   &    |-  ( ph  ->  A  e.  K )   =>    |-  ( ph  ->  (
 ( ( A  .x.  X )  .X.  Y )  =  ( A  .x.  ( X  .X.  Y ) ) 
 /\  ( X  .X.  ( A  .x.  Y ) )  =  ( A 
 .x.  ( X  .X.  Y ) ) ) )
 
Theorempsrrng 16429 The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  S  e.  Ring
 )
 
Theorempsr1 16430* The identity element of the ring of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( 1r `  S )   =>    |-  ( ph  ->  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
 
Theorempsrcrng 16431 The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  S  e.  CRing
 )
 
Theorempsrassa 16432 The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  S  e. AssAlg )
 
Theoremresspsrbas 16433 A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ph  ->  B  =  ( Base `  P )
 )
 
Theoremresspsradd 16434 A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
 
Theoremresspsrmul 16435 A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( .r `  U ) Y )  =  ( X ( .r `  P ) Y ) )
 
Theoremresspsrvsca 16436 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  T  /\  Y  e.  B )
 )  ->  ( X ( .s `  U ) Y )  =  ( X ( .s `  P ) Y ) )
 
Theoremsubrgpsr 16437 A subring of the base ring induces a subring of power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   =>    |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R )
 )  ->  B  e.  (SubRing `  S ) )
 
Theoremmvridlem 16438* A bag containing one element is a finite bag. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   =>    |-  ( I  e.  V  ->  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) )  e.  D )
 
Theoremmvrfval 16439* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   =>    |-  ( ph  ->  V  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) )
 
Theoremmvrval 16440* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( V `  X )  =  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
 
Theoremmvrval2 16441* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  F  e.  D )   =>    |-  ( ph  ->  ( ( V `  X ) `  F )  =  if ( F  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
 )
 
Theoremmvrid 16442* The  X i-th coefficient of the term  X i is  1. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  (
 ( V `  X ) `  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) )  =  .1.  )
 
Theoremmvrf 16443 The power series variable function is a function from the index set to elements of the power series structure representing  X
i for each  i. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  V : I --> B )
 
Theoremmvrf1 16444 The power series variable function is injective if the base ring is nonzero. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  .1. 
 =/=  .0.  )   =>    |-  ( ph  ->  V : I -1-1-> B )
 
Theoremmvrcl2 16445 A power series variable is an element of the base set. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( V `  X )  e.  B )
 
Theoremreldmmpl 16446 The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- 
 Rel  dom mPoly
 
Theoremmplval 16447* Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  { f  e.  B  |  ( `' f " ( _V  \  {  .0.  } )
 )  e.  Fin }   =>    |-  P  =  ( Ss  U )
 
Theoremmplbas 16448* Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  ( Base `  P )   =>    |-  U  =  { f  e.  B  |  ( `' f " ( _V  \  {  .0.  } )
 )  e.  Fin }
 
Theoremmplelbas 16449 Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  ( Base `  P )   =>    |-  ( X  e.  U  <->  ( X  e.  B  /\  ( `' X " ( _V  \  {  .0.  } )
 )  e.  Fin )
 )
 
Theoremmplval2 16450 Self-referential expression for the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  U  =  (
 Base `  P )   =>    |-  P  =  ( Ss  U )
 
Theoremmplbasss 16451 The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  U  =  (
 Base `  P )   &    |-  B  =  ( Base `  S )   =>    |-  U  C_  B
 
Theoremmplelf 16452* A polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X : D --> K )
 
Theoremmplsubglem 16453* If  A is an ideal of sets (a nonempty collection closed under subset and binary union) of the set  D of finite bags (the primary applications being  A  =  Fin and  A  =  ~P B for some  B), then the set of all power series whose coefficient functions are supported on an element of  A is a subgroup of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  (/)  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  ( x  u.  y )  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  C_  x ) ) 
 ->  y  e.  A )   &    |-  ( ph  ->  U  =  { g  e.  B  |  ( `' g "
 ( _V  \  {  .0.  } ) )  e.  A } )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  U  e.  (SubGrp `  S )
 )
 
Theoremmpllsslem 16454* If  A is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set  D of finite bags (the primary applications being  A  =  Fin and  A  =  ~P B for some  B), then the set of all power series whose coefficient functions are supported on an element of  A is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  (/)  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  ( x  u.  y )  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  C_  x ) ) 
 ->  y  e.  A )   &    |-  ( ph  ->  U  =  { g  e.  B  |  ( `' g "
 ( _V  \  {  .0.  } ) )  e.  A } )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  e.  ( LSubSp `  S )
 )
 
Theoremmplsubg 16455 The set of polynomials is closed under addition, i.e. it is a subgroup of the set of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  U  e.  (SubGrp `  S )
 )
 
Theoremmpllss 16456 The set of polynomials is closed under scalar multiplication, i.e. it is a linear subspace of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  e.  ( LSubSp `  S )
 )
 
Theoremmplsubrglem 16457* Lemma for mplsubrg 16458. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  (  o F  +  " ( ( `' X " ( _V  \  {  .0.  } )
 )  X.  ( `' Y " ( _V  \  {  .0.  } ) ) ) )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X ( .r `  S ) Y )  e.  U )
 
Theoremmplsubrg 16458 The set of polynomials is closed under multiplication, i.e. it is a subring of the set of power series. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  e.  (SubRing `  S )
 )
 
Theoremmpl0 16459* The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  O  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  .0.  =  ( D  X.  { O } ) )
 
Theoremmpladd 16460 The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  P )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+b  Y )  =  ( X  o F  .+  Y ) )
 
Theoremmplmul 16461* The multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  P )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F  .xb  G )  =  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
 y  e.  D  |  y  o R  <_  k }  |->  ( ( F `
  x )  .x.  ( G `  ( k  o F  -  x ) ) ) ) ) ) )
 
Theoremmpl1 16462* The identity element of the ring of polynomials. (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 )   &    |-  U  =  ( 1r `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
 
Theoremmplsca 16463 The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   =>    |-  ( ph  ->  R  =  (Scalar `  P )
 )
 
Theoremmplvsca2 16464 The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  .x.  =  ( .s `  P )   =>    |-  .x.  =  ( .s `  S )
 
Theoremmplvsca 16465* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  .xb  =  ( .s `  P )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( X  .xb  F )  =  ( ( D  X.  { X } )  o F  .x.  F )
 )
 
Theoremmplvscaval 16466* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  .xb  =  ( .s `  P )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  (
 ( X  .xb  F ) `
  Y )  =  ( X  .x.  ( F `  Y ) ) )
 
Theoremmvrcl 16467 A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( V `  X )  e.  B )
 
Theoremmplgrp 16468 The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   =>    |-  ( ( I  e.  V  /\  R  e.  Grp )  ->  P  e.  Grp )
 
Theoremmpllmod 16469 The polynomial ring is a left module. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   =>    |-  ( ( I  e.  V  /\  R  e.  Ring
 )  ->  P  e.  LMod
 )
 
Theoremmplrng 16470 The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   =>    |-  ( ( I  e.  V  /\  R  e.  Ring
 )  ->  P  e.  Ring
 )
 
Theoremmplcrng 16471 The polynomial ring is a commutative ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   =>    |-  ( ( I  e.  V  /\  R  e.  CRing
 )  ->  P  e.  CRing
 )
 
Theoremmplassa 16472 The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   =>    |-  ( ( I  e.  V  /\  R  e.  CRing
 )  ->  P  e. AssAlg )
 
Theoremressmplbas2 16473 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  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  W  =  ( I mPwSer  H )   &    |-  C  =  (
 Base `  W )   &    |-  K  =  ( Base `  S )   =>    |-  ( ph  ->  B  =  ( C  i^i  K ) )
 
Theoremressmplbas 16474 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  P  =  ( Ss  B )   =>    |-  ( ph  ->  B  =  ( Base `  P )
 )
 
Theoremressmpladd 16475 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
 
Theoremressmplmul 16476 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( .r `  U ) Y )  =  ( X ( .r `  P ) Y ) )
 
Theoremressmplvsca 16477 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  T  /\  Y  e.  B )
 )  ->  ( X ( .s `  U ) Y )  =  ( X ( .s `  P ) Y ) )
 
Theoremsubrgmpl 16478 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   =>    |-  (
 ( I  e.  V  /\  T  e.  (SubRing `  R ) )  ->  B  e.  (SubRing `  S ) )
 
Theoremsubrgmvr 16479 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 16480 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 16481* 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 16482* 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 16483* 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 16484* 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 16485* 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 16486 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 16487* 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 16488* 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 16489 Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
 |- 
 Rel  dom ordPwSer
 
Theoremopsrval 16490* The value of the "ordered power series" function. This is the same as mPwSer psrval 16384, 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 16491* 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 16492 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 16493 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 16494 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 16495 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 16496 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 16497 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 16498 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 16499* Lemma for opsrtos 16501. (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 16500* Lemma for opsrtos 16501. (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 )
    < 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 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32447
  Copyright terms: Public domain < Previous  Next >