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Theorem List for Metamath Proof Explorer - 16401-16500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempsrbagconcl 16401* The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   =>    |-  ( ( I  e.  V  /\  F  e.  D  /\  X  e.  S )  ->  ( F  o F  -  X )  e.  S )
 
Theorempsrbagconf1o 16402* Bag complementation is a bijection on the set of bags dominated by a given bag  F. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   =>    |-  ( ( I  e.  V  /\  F  e.  D )  ->  ( x  e.  S  |->  ( F  o F  -  x ) ) : S -1-1-onto-> S )
 
Theoremgsumbagdiaglem 16403* Lemma for gsumbagdiag 16404. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   =>    |-  ( ( ph  /\  ( X  e.  S  /\  Y  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  X ) }
 ) )  ->  ( Y  e.  S  /\  X  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  Y ) }
 ) )
 
Theoremgsumbagdiag 16404* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 12524 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  (
 ( ph  /\  ( j  e.  S  /\  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) } )
 )  ->  X  e.  B )   =>    |-  ( ph  ->  ( G  gsumg  ( j  e.  S ,  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) }  |->  X ) )  =  ( G  gsumg  ( k  e.  S ,  j  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  k ) }  |->  X ) ) )
 
Theorempsrass1lem 16405* A group sum commutation used by psrass1 16432. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  (
 ( ph  /\  ( j  e.  S  /\  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) } )
 )  ->  X  e.  B )   &    |-  ( k  =  ( n  o F  -  j )  ->  X  =  Y )   =>    |-  ( ph  ->  ( G  gsumg  ( n  e.  S  |->  ( G  gsumg  ( j  e.  { x  e.  D  |  x  o R  <_  n }  |->  Y ) ) ) )  =  ( G  gsumg  ( j  e.  S  |->  ( G  gsumg  ( k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j
 ) }  |->  X ) ) ) ) )
 
Theorempsrbas 16406* The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  V )   =>    |-  ( ph  ->  B  =  ( K  ^m  D ) )
 
Theorempsrelbas 16407* An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X : D --> K )
 
Theorempsrplusg 16408 The addition 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 )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  S )   =>    |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )
 
Theorempsradd 16409 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 16410 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 16411* 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 16412* 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 16413* 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 16414* 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 16415 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 16416 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 16417* 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 16418* 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 16419* 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 16420 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 16421* 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 16422* 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 16423* 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 16424* 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 16425 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 16426* 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 16427* 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 16428 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 16429* 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 16430* 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 16431* 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 16432* 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 16433* 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 16434* 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 16435* 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 16436* 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 16437 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 16438* 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 16439 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 16440 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 16441 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 16442 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 16443 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 16444 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 16445 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 16446* 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 16447* 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 16448* 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 16449* 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 16450* 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 16451 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 16452 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 16453 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 16454 The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- 
 Rel  dom mPoly
 
Theoremmplval 16455* 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 16456* 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 16457 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 16458 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 16459 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 16460* 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 16461* 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 16462* 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 16463 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 16464 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 16465* Lemma for mplsubrg 16466. (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 16466 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 16467* 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 16468 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 16469* 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 16470* 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 16471 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 16472 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 16473* 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 16474* 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 16475 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 16476 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 16477 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 16478 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 16479 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 16480 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 16481 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 16482 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 16483 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 16484 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 16485 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 16486 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 16487 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 16488 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 16489* 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 16490* 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 16491* 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 16492* 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 16493* 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 16494 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 16495* 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 16496* 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 16497 Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
 |- 
 Rel  dom ordPwSer
 
Theoremopsrval 16498* The value of the "ordered power series" function. This is the same as mPwSer psrval 16392, 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 16499* 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 16500 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 ) ,  .<_  >.
 ) )
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