HomeHome Metamath Proof Explorer
Theorem List (p. 187 of 389)
< 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
(1-26235)
  Hilbert Space Explorer  Hilbert Space Explorer
(26236-27759)
  Users' Mathboxes  Users' Mathboxes
(27760-38841)
 

Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremply1plusgfvi 18601 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (  _I  `  R )
 ) )
 
Theoremply1baspropd 18602* Property deduction for univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  S ) ) )
 
Theoremply1plusgpropd 18603* Property deduction for univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  S ) ) )
 
Theoremopsrring 18604 Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  O  e.  Ring )
 
Theoremopsrlmod 18605 Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  O  e.  LMod )
 
Theorempsr1ring 18606 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  S  =  (PwSer1 `  R )   =>    |-  ( R  e.  Ring  ->  S  e.  Ring )
 
Theoremply1ring 18607 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  Ring )
 
Theorempsr1lmod 18608 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (PwSer1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  LMod )
 
Theorempsr1sca 18609 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
 |-  P  =  (PwSer1 `  R )   =>    |-  ( R  e.  V  ->  R  =  (Scalar `  P ) )
 
Theorempsr1sca2 18610 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
 |-  P  =  (PwSer1 `  R )   =>    |-  (  _I  `  R )  =  (Scalar `  P )
 
Theoremply1lmod 18611 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  LMod )
 
Theoremply1sca 18612 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  V  ->  R  =  (Scalar `  P ) )
 
Theoremply1sca2 18613 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  (  _I  `  R )  =  (Scalar `  P )
 
Theoremply1mpl0 18614 The univariate polynomial ring has the same zero as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
 |-  M  =  ( 1o mPoly  R )   &    |-  P  =  (Poly1 `  R )   &    |-  .0.  =  ( 0g `  P )   =>    |-  .0.  =  ( 0g `  M )
 
Theoremply10s0 18615 Zero times a univariate polynomial is the zero polynomial (lmod0vs 17863 analog.) (Contributed by AV, 2-Dec-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .*  =  ( .s `  P )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  M  e.  B ) 
 ->  (  .0.  .*  M )  =  ( 0g `  P ) )
 
Theoremply1mpl1 18616 The univariate polynomial ring has the same one as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
 |-  M  =  ( 1o mPoly  R )   &    |-  P  =  (Poly1 `  R )   &    |-  .1.  =  ( 1r `  P )   =>    |-  .1.  =  ( 1r `  M )
 
Theoremply1ascl 18617 The univariate polynomial ring inherits the multivariate ring's scalar function. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by Fan Zheng, 26-Jun-2016.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   =>    |-  A  =  (algSc `  ( 1o mPoly  R ) )
 
Theoremsubrg1ascl 18618 The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  C  =  (algSc `  U )   =>    |-  ( ph  ->  C  =  ( A  |`  T ) )
 
Theoremsubrg1asclcl 18619 The scalars in a polynomial algebra are in the subring algebra iff the scalar value is in the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  B  =  (
 Base `  U )   &    |-  K  =  ( Base `  R )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  (
 ( A `  X )  e.  B  <->  X  e.  T ) )
 
Theoremsubrgvr1 18620 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
 |-  X  =  (var1 `  R )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  H  =  ( Rs  T )   =>    |-  ( ph  ->  X  =  (var1 `  H ) )
 
Theoremsubrgvr1cl 18621 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 5-Jul-2015.)
 |-  X  =  (var1 `  R )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   =>    |-  ( ph  ->  X  e.  B )
 
Theoremcoe1z 18622 The coefficient vector of 0. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  Y  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  (coe1 ` 
 .0.  )  =  (
 NN0  X.  { Y } ) )
 
Theoremcoe1add 18623 The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .+b  =  ( +g  `  Y )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .+b  G ) )  =  ( (coe1 `  F )  oF  .+  (coe1 `  G ) ) )
 
Theoremcoe1addfv 18624 A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .+b  =  ( +g  `  Y )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .+b  G ) ) `  X )  =  ( (
 (coe1 `
  F ) `  X )  .+  ( (coe1 `  G ) `  X ) ) )
 
Theoremcoe1subfv 18625 A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  N  =  ( -g `  R )   =>    |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .-  G ) ) `
  X )  =  ( ( (coe1 `  F ) `  X ) N ( (coe1 `  G ) `  X ) ) )
 
Theoremcoe1mul2lem1 18626 An equivalence for coe1mul2 18628. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  ( ( A  e.  NN0  /\  X  e.  ( NN0  ^m 
 1o ) )  ->  ( X  oR  <_  ( 1o  X.  { A } )  <->  ( X `  (/) )  e.  ( 0
 ... A ) ) )
 
Theoremcoe1mul2lem2 18627* An equivalence for coe1mul2 18628. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  H  =  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }   =>    |-  (
 k  e.  NN0  ->  ( c  e.  H  |->  ( c `  (/) ) ) : H -1-1-onto-> ( 0 ... k
 ) )
 
Theoremcoe1mul2 18628* The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  S  =  (PwSer1 `  R )   &    |-  .xb  =  ( .r `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  S )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( k  e.  NN0  |->  ( R 
 gsumg  ( x  e.  (
 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  ( (coe1 `  G ) `  (
 k  -  x ) ) ) ) ) ) )
 
Theoremcoe1mul 18629* The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  .xb  =  ( .r `  Y )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  Y )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( k  e.  NN0  |->  ( R 
 gsumg  ( x  e.  (
 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  ( (coe1 `  G ) `  (
 k  -  x ) ) ) ) ) ) )
 
Theoremply1moncl 18630 Closure of the expression for a univariate primitive monomial. (Contributed by AV, 14-Aug-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   =>    |-  ( ( R  e.  Ring  /\  D  e.  NN0 )  ->  ( D  .^  X )  e.  B )
 
Theoremply1tmcl 18631 Closure of the expression for a univariate polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 25-Nov-2019.)
 |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  ( C  .x.  ( D  .^  X ) )  e.  B )
 
Theoremcoe1tm 18632* Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  (coe1 `  ( C  .x.  ( D  .^  X ) ) )  =  ( x  e.  NN0  |->  if ( x  =  D ,  C ,  .0.  )
 ) )
 
Theoremcoe1tmfv1 18633 Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D )  =  C )
 
Theoremcoe1tmfv2 18634 Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  F  e.  NN0 )   &    |-  ( ph  ->  D  =/=  F )   =>    |-  ( ph  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  F )  =  .0.  )
 
Theoremcoe1tmmul2 18635* Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( A  .xb  ( C 
 .x.  ( D  .^  X ) ) ) )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  (
 ( (coe1 `  A ) `  ( x  -  D ) )  .X.  C ) ,  .0.  ) ) )
 
Theoremcoe1tmmul 18636* Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( ( C  .x.  ( D  .^  X ) )  .xb  A )
 )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  ( C  .X.  ( (coe1 `  A ) `  ( x  -  D ) ) ) ,  .0.  ) ) )
 
Theoremcoe1tmmul2fv 18637 Function value of a right-multiplication by a term in the shifted domain. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  Y  e.  NN0 )   =>    |-  ( ph  ->  (
 (coe1 `
  ( A  .xb  ( C  .x.  ( D 
 .^  X ) ) ) ) `  ( D  +  Y )
 )  =  ( ( (coe1 `  A ) `  Y )  .X.  C ) )
 
Theoremcoe1pwmul 18638* Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( ( D  .^  X )  .x.  A ) )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  (
 (coe1 `
  A ) `  ( x  -  D ) ) ,  .0.  ) ) )
 
Theoremcoe1pwmulfv 18639 Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  Y  e.  NN0 )   =>    |-  ( ph  ->  (
 (coe1 `
  ( ( D 
 .^  X )  .x.  A ) ) `  ( D  +  Y )
 )  =  ( (coe1 `  A ) `  Y ) )
 
Theoremply1scltm 18640 A scalar is a term with zero exponent. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  K ) 
 ->  ( A `  F )  =  ( F  .x.  ( 0  .^  X ) ) )
 
Theoremcoe1sclmul 18641 Coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  Y  e.  B )  ->  (coe1 `  ( ( A `
  X )  .xb  Y ) )  =  ( ( NN0  X.  { X } )  oF  .x.  (coe1 `  Y ) ) )
 
Theoremcoe1sclmulfv 18642 A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  K  /\  Y  e.  B ) 
 /\  .0.  e.  NN0 )  ->  ( (coe1 `  ( ( A `
  X )  .xb  Y ) ) `  .0.  )  =  ( X  .x.  ( (coe1 `  Y ) `  .0.  ) ) )
 
Theoremcoe1sclmul2 18643 Coefficient vector of a polynomial multiplied on the right by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  Y  e.  B )  ->  (coe1 `  ( Y  .xb  ( A `  X ) ) )  =  ( (coe1 `  Y )  oF  .x.  ( NN0  X. 
 { X } )
 ) )
 
Theoremply1sclf 18644 A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  A : K --> B )
 
Theoremply1sclcl 18645 The value of the algebra scalars function for (univariate) polynomials applied to a scalar results in a constant polynomial. (Contributed by AV, 27-Nov-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   =>    |-  (
 ( R  e.  Ring  /\  S  e.  K ) 
 ->  ( A `  S )  e.  B )
 
Theoremcoe1scl 18646* Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K ) 
 ->  (coe1 `  ( A `  X ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  X ,  .0.  ) ) )
 
Theoremply1sclid 18647 Recover the base scalar from a scalar polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K )  ->  X  =  ( (coe1 `  ( A `  X ) ) `  0 ) )
 
Theoremply1sclf1 18648 The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  A : K -1-1-> B )
 
Theoremply1scl0 18649 The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  Y  =  ( 0g `  P )   =>    |-  ( R  e.  Ring  ->  ( A `  .0.  )  =  Y )
 
Theoremply1scln0 18650 Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  Y  =  ( 0g `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  X  =/=  .0.  )  ->  ( A `  X )  =/=  Y )
 
Theoremply1scl1 18651 The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( 1r `  P )   =>    |-  ( R  e.  Ring  ->  ( A `  .1.  )  =  N )
 
Theoremply1idvr1 18652 The identity of a polynomial ring expressed as power of the polynomial variable. (Contributed by AV, 14-Aug-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( R  e.  Ring  ->  ( 0  .^  X )  =  ( 1r `  P ) )
 
Theoremcply1mul 18653* The product of two constant polynomials is a constant polynomial. (Contributed by AV, 18-Nov-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  .X. 
 =  ( .r `  P )   =>    |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B ) )  ->  ( A. c  e.  NN  (
 ( (coe1 `  F ) `  c )  =  .0.  /\  ( (coe1 `  G ) `  c )  =  .0.  )  ->  A. c  e.  NN  ( (coe1 `  ( F  .X.  G ) ) `  c
 )  =  .0.  )
 )
 
Theoremply1coefsupp 18654* The decomposition of a univariate polynomial is finitely supported. Formerly part of proof for ply1coe 18655. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .s `  P )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   &    |-  A  =  (coe1 `  K )   =>    |-  ( ( R  e.  Ring  /\  K  e.  B ) 
 ->  ( k  e.  NN0  |->  ( ( A `  k )  .x.  ( k 
 .^  X ) ) ) finSupp  ( 0g `  P ) )
 
Theoremply1coe 18655* Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 7-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .s `  P )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   &    |-  A  =  (coe1 `  K )   =>    |-  ( ( R  e.  Ring  /\  K  e.  B ) 
 ->  K  =  ( P 
 gsumg  ( k  e.  NN0  |->  ( ( A `  k ) 
 .x.  ( k  .^  X ) ) ) ) )
 
Theoremply1coeOLD 18656* Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) Obsolete as of 28-Sep-2019. Use ply1coe 18655 instead. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .s `  P )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   &    |-  A  =  (coe1 `  K )   &    |-  R  e.  _V   =>    |-  (
 ( R  e.  CRing  /\  K  e.  B ) 
 ->  K  =  ( P 
 gsumg  ( k  e.  NN0  |->  ( ( A `  k ) 
 .x.  ( k  .^  X ) ) ) ) )
 
Theoremeqcoe1ply1eq 18657* Two polynomials over the same ring are equal if they have identical coefficients. (Contributed by AV, 7-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  A  =  (coe1 `  K )   &    |-  C  =  (coe1 `  L )   =>    |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( A. k  e. 
 NN0  ( A `  k )  =  ( C `  k )  ->  K  =  L )
 )
 
Theoremply1coe1eq 18658* Two polynomials over the same ring are equal iff they have identical coefficients. (Contributed by AV, 13-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  A  =  (coe1 `  K )   &    |-  C  =  (coe1 `  L )   =>    |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( A. k  e. 
 NN0  ( A `  k )  =  ( C `  k )  <->  K  =  L ) )
 
Theoremcply1coe0 18659* All but the first coefficient of a constant polynomial ( i.e. a "lifted scalar") are zero. (Contributed by AV, 16-Nov-2019.)
 |-  K  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  S  e.  K ) 
 ->  A. n  e.  NN  ( (coe1 `  ( A `  S ) ) `  n )  =  .0.  )
 
Theoremcply1coe0bi 18660* A polynomial is constant (i.e. a "lifted scalar") iff all but the first coefficient are zero. (Contributed by AV, 16-Nov-2019.)
 |-  K  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  M  e.  B ) 
 ->  ( E. s  e.  K  M  =  ( A `  s )  <->  A. n  e.  NN  ( (coe1 `  M ) `  n )  =  .0.  ) )
 
Theoremcoe1fzgsumdlem 18661* Lemma for coe1fzgsumd 18662 (induction step). (Contributed by AV, 8-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ( m  e. 
 Fin  /\  -.  a  e.  m  /\  ph )  ->  ( ( A. x  e.  m  M  e.  B  ->  ( (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `
  K )  =  ( R  gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) ) )  ->  ( A. x  e.  ( m  u.  { a } ) M  e.  B  ->  ( (coe1 `  ( P  gsumg  ( x  e.  ( m  u.  { a } )  |->  M ) ) ) `  K )  =  ( R  gsumg  ( x  e.  ( m  u.  { a }
 )  |->  ( (coe1 `  M ) `  K ) ) ) ) ) )
 
Theoremcoe1fzgsumd 18662* Value of an evaluated coefficient in a finite group sum of polynomials. (Contributed by AV, 8-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  A. x  e.  N  M  e.  B )   &    |-  ( ph  ->  N  e.  Fin )   =>    |-  ( ph  ->  ( (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) `  K )  =  ( R  gsumg  ( x  e.  N  |->  ( (coe1 `  M ) `  K ) ) ) )
 
Theoremgsumsmonply1 18663* A finite group sum of scaled monomials is a univariate polynomial. (Contributed by AV, 8-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  K  =  ( Base `  R )   &    |-  .*  =  ( .s `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  A. k  e.  NN0  A  e.  K )   &    |-  ( ph  ->  ( k  e.  NN0  |->  A ) finSupp  .0.  )   =>    |-  ( ph  ->  ( P  gsumg  ( k  e.  NN0  |->  ( A  .*  (
 k  .^  X )
 ) ) )  e.  B )
 
Theoremgsummoncoe1 18664* A coefficient of the polynomial represented as sum of scaled monomials is the coefficient of the corresponding scaled monomial. (Contributed by AV, 13-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  K  =  ( Base `  R )   &    |-  .*  =  ( .s `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  A. k  e.  NN0  A  e.  K )   &    |-  ( ph  ->  ( k  e.  NN0  |->  A ) finSupp  .0.  )   &    |-  ( ph  ->  L  e.  NN0 )   =>    |-  ( ph  ->  (
 (coe1 `
  ( P  gsumg  ( k  e.  NN0  |->  ( A  .*  ( k  .^  X ) ) ) ) ) `  L )  =  [_ L  /  k ]_ A )
 
Theoremgsumply1eq 18665* Two univariate polynomials given as (finitely supported) sum of scaled monomials are equal iff the corresponding coefficients are equal. (Contributed by AV, 21-Nov-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  K  =  ( Base `  R )   &    |-  .*  =  ( .s `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  A. k  e.  NN0  A  e.  K )   &    |-  ( ph  ->  ( k  e.  NN0  |->  A ) finSupp  .0.  )   &    |-  ( ph  ->  A. k  e.  NN0  B  e.  K )   &    |-  ( ph  ->  ( k  e.  NN0  |->  B ) finSupp  .0.  )   &    |-  ( ph  ->  O  =  ( P  gsumg  ( k  e.  NN0  |->  ( A  .*  ( k  .^  X ) ) ) ) )   &    |-  ( ph  ->  Q  =  ( P  gsumg  ( k  e.  NN0  |->  ( B  .*  ( k  .^  X ) ) ) ) )   =>    |-  ( ph  ->  ( O  =  Q  <->  A. k  e.  NN0  A  =  B ) )
 
Theoremlply1binom 18666* The binomial theorem for linear polynomials (monic polynomials of degree 1) over commutative rings:  ( X  +  A
) ^ N is the sum from  k  =  0 to  N of  ( N  _C  k )  x.  (
( A ^ ( N  -  k )
)  x.  ( X ^ k ) ). (Contributed by AV, 25-Aug-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |- 
 .+  =  ( +g  `  P )   &    |-  .X.  =  ( .r `  P )   &    |-  .x.  =  (.g `  P )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  B  =  (
 Base `  P )   =>    |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  B ) 
 ->  ( N  .^  ( X  .+  A ) )  =  ( P  gsumg  ( k  e.  ( 0 ...
 N )  |->  ( ( N  _C  k ) 
 .x.  ( ( ( N  -  k ) 
 .^  A )  .X.  ( k  .^  X ) ) ) ) ) )
 
Theoremlply1binomsc 18667* The binomial theorem for linear polynomials (monic polynomials of degree 1) over commutative rings, expressed by an element of this ring:  ( X  +  A ) ^ N is the sum from  k  =  0 to  N of  ( N  _C  k )  x.  (
( A ^ ( N  -  k )
)  x.  ( X ^ k ) ). (Contributed by AV, 25-Aug-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |- 
 .+  =  ( +g  `  P )   &    |-  .X.  =  ( .r `  P )   &    |-  .x.  =  (.g `  P )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  K  =  (
 Base `  R )   &    |-  S  =  (algSc `  P )   &    |-  H  =  (mulGrp `  R )   &    |-  E  =  (.g `  H )   =>    |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K ) 
 ->  ( N  .^  ( X  .+  ( S `  A ) ) )  =  ( P  gsumg  ( k  e.  ( 0 ...
 N )  |->  ( ( N  _C  k ) 
 .x.  ( ( S `
  ( ( N  -  k ) E A ) )  .X.  ( k  .^  X ) ) ) ) ) )
 
10.10.5  Univariate polynomial evaluation
 
Syntaxces1 18668 Evaluation of a univariate polynomial in a subring.
 class evalSub1
 
Syntaxce1 18669 Evaluation of a univariate polynomial.
 class eval1
 
Definitiondf-evls1 18670* Define the evaluation map for the univariate polynomial algebra. The function  ( S evalSub1  R ) : V --> ( S  ^m  S ) makes sense when  S is a ring and  R is a subring of  S, and where  V is the set of polynomials in  (Poly1 `  R ). This function maps an element of the formal polynomial algebra (with coefficients in  R) to a function from assignments to the variable from  S into an element of  S formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |- evalSub1  =  ( s  e.  _V ,  r  e.  ~P ( Base `  s )  |-> 
 [_ ( Base `  s
 )  /  b ]_ ( ( x  e.  ( b  ^m  (
 b  ^m  1o )
 )  |->  ( x  o.  ( y  e.  b  |->  ( 1o  X.  {
 y } ) ) ) )  o.  (
 ( 1o evalSub  s ) `  r ) ) )
 
Definitiondf-evl1 18671* Define the evaluation map for the univariate polynomial algebra. The function  (eval1 `  R ) : V --> ( R  ^m  R ) makes sense when  R is a ring, and  V is the set of polynomials in  (Poly1 `  R ). This function maps an element of the formal polynomial algebra (with coefficients in  R) to a function from assignments to the variable from  R into an element of  R formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |- eval1  =  ( r  e.  _V  |->  [_ ( Base `  r )  /  b ]_ ( ( x  e.  ( b 
 ^m  ( b  ^m  1o ) )  |->  ( x  o.  ( y  e.  b  |->  ( 1o  X.  { y } ) ) ) )  o.  ( 1o eval  r ) ) )
 
Theoremreldmevls1 18672 Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.)
 |- 
 Rel  dom evalSub1
 
Theoremply1frcl 18673 Reverse closure for the set of univariate polynomial functions. (Contributed by AV, 9-Sep-2019.)
 |-  Q  =  ran  ( S evalSub1  R )   =>    |-  ( X  e.  Q  ->  ( S  e.  _V  /\  R  e.  ~P ( Base `  S ) ) )
 
Theoremevls1fval 18674* Value of the univariate polynomial evaluation map function. (Contributed by AV, 7-Sep-2019.)
 |-  Q  =  ( S evalSub1  R )   &    |-  E  =  ( 1o evalSub  S )   &    |-  B  =  (
 Base `  S )   =>    |-  ( ( S  e.  V  /\  R  e.  ~P B )  ->  Q  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( E `  R ) ) )
 
Theoremevls1val 18675* Value of the univariate polynomial evaluation map. (Contributed by AV, 10-Sep-2019.)
 |-  Q  =  ( S evalSub1  R )   &    |-  E  =  ( 1o evalSub  S )   &    |-  B  =  (
 Base `  S )   &    |-  M  =  ( 1o mPoly  ( Ss  R ) )   &    |-  K  =  (
 Base `  M )   =>    |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K )  ->  ( Q `  A )  =  ( (
 ( E `  R ) `  A )  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )
 
Theoremevls1rhmlem 18676* Lemma for evl1rhm 18686 and evls1rhm 18677 (formerly part of the proof of evl1rhm 18686): The first function of the composition forming the univariate polynomial evaluation map function for a (sub)ring is a ring homomorphism. (Contributed by AV, 11-Sep-2019.)
 |-  B  =  ( Base `  R )   &    |-  T  =  ( R  ^s  B )   &    |-  F  =  ( x  e.  ( B 
 ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )   =>    |-  ( R  e.  CRing  ->  F  e.  ( ( R  ^s  ( B  ^m  1o ) ) RingHom  T ) )
 
Theoremevls1rhm 18677 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by AV, 11-Sep-2019.)
 |-  Q  =  ( S evalSub1  R )   &    |-  B  =  (
 Base `  S )   &    |-  T  =  ( S  ^s  B )   &    |-  U  =  ( Ss  R )   &    |-  W  =  (Poly1 `  U )   =>    |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S ) )  ->  Q  e.  ( W RingHom  T ) )
 
Theoremevls1sca 18678 Univariate polynomial evaluation maps scalars to constant functions. (Contributed by AV, 8-Sep-2019.)
 |-  Q  =  ( S evalSub1  R )   &    |-  W  =  (Poly1 `  U )   &    |-  U  =  ( Ss  R )   &    |-  B  =  (
 Base `  S )   &    |-  A  =  (algSc `  W )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  X  e.  R )   =>    |-  ( ph  ->  ( Q `  ( A `  X ) )  =  ( B  X.  { X } ) )
 
Theoremevls1gsumadd 18679* Univariate polynomial evaluation maps (additive) group sums to group sums. (Contributed by AV, 14-Sep-2019.)
 |-  Q  =  ( S evalSub1  R )   &    |-  K  =  (
 Base `  S )   &    |-  W  =  (Poly1 `  U )   &    |-  .0.  =  ( 0g `  W )   &    |-  U  =  ( Ss  R )   &    |-  P  =  ( S  ^s  K )   &    |-  B  =  (
 Base `  W )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ( ph  /\  x  e.  N )  ->  Y  e.  B )   &    |-  ( ph  ->  N 
 C_  NN0 )   &    |-  ( ph  ->  ( x  e.  N  |->  Y ) finSupp  .0.  )   =>    |-  ( ph  ->  ( Q `  ( W  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( P  gsumg  ( x  e.  N  |->  ( Q `  Y ) ) ) )
 
Theoremevls1gsummul 18680* Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 14-Sep-2019.)
 |-  Q  =  ( S evalSub1  R )   &    |-  K  =  (
 Base `  S )   &    |-  W  =  (Poly1 `  U )   &    |-  G  =  (mulGrp `  W )   &    |-  .1.  =  ( 1r `  W )   &    |-  U  =  ( Ss  R )   &    |-  P  =  ( S  ^s  K )   &    |-  H  =  (mulGrp `  P )   &    |-  B  =  (
 Base `  W )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ( ph  /\  x  e.  N )  ->  Y  e.  B )   &    |-  ( ph  ->  N 
 C_  NN0 )   &    |-  ( ph  ->  ( x  e.  N  |->  Y ) finSupp  .1.  )   =>    |-  ( ph  ->  ( Q `  ( G  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( H  gsumg  ( x  e.  N  |->  ( Q `  Y ) ) ) )
 
Theoremevls1varpw 18681 Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by AV, 14-Sep-2019.)
 |-  Q  =  ( S evalSub1  R )   &    |-  U  =  ( Ss  R )   &    |-  W  =  (Poly1 `  U )   &    |-  G  =  (mulGrp `  W )   &    |-  X  =  (var1 `  U )   &    |-  B  =  (
 Base `  S )   &    |-  .^  =  (.g `  G )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( Q `  ( N  .^  X ) )  =  ( N (.g `  (mulGrp `  ( S  ^s  B )
 ) ) ( Q `
  X ) ) )
 
Theoremevl1fval 18682* Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  Q  =  ( 1o eval  R )   &    |-  B  =  (
 Base `  R )   =>    |-  O  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  Q )
 
Theoremevl1val 18683* Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  Q  =  ( 1o eval  R )   &    |-  B  =  (
 Base `  R )   &    |-  M  =  ( 1o mPoly  R )   &    |-  K  =  ( Base `  M )   =>    |-  (
 ( R  e.  CRing  /\  A  e.  K ) 
 ->  ( O `  A )  =  ( ( Q `  A )  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )
 
Theoremevl1fval1lem 18684 Lemma for evl1fval1 18685. (Contributed by AV, 11-Sep-2019.)
 |-  Q  =  (eval1 `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  _V  ->  Q  =  ( R evalSub1  B ) )
 
Theoremevl1fval1 18685 Value of the simple/same ring evaluation map function for univariate polynomials. (Contributed by AV, 11-Sep-2019.)
 |-  Q  =  (eval1 `  R )   &    |-  B  =  ( Base `  R )   =>    |-  Q  =  ( R evalSub1  B )
 
Theoremevl1rhm 18686 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.) (Proof shortened by AV, 13-Sep-2019.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  T  =  ( R 
 ^s 
 B )   &    |-  B  =  (
 Base `  R )   =>    |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  T ) )
 
Theoremfveval1fvcl 18687 The function value of the evaluation function of a polynomial is an element of the underlying ring. (Contributed by AV, 17-Sep-2019.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M  e.  U )   =>    |-  ( ph  ->  ( ( O `  M ) `  Y )  e.  B )
 
Theoremevl1sca 18688 Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  CRing  /\  X  e.  B ) 
 ->  ( O `  ( A `  X ) )  =  ( B  X.  { X } ) )
 
Theoremevl1scad 18689 Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( ( A `  X )  e.  U  /\  ( ( O `  ( A `  X ) ) `  Y )  =  X ) )
 
Theoremevl1var 18690 Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  CRing  ->  ( O `  X )  =  (  _I  |`  B ) )
 
Theoremevl1vard 18691 Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  e.  U  /\  ( ( O `  X ) `  Y )  =  Y )
 )
 
Theoremevls1var 18692 Univariate polynomial evaluation for subrings maps the variable to the identity function. (Contributed by AV, 13-Sep-2019.)
 |-  Q  =  ( S evalSub1  R )   &    |-  X  =  (var1 `  U )   &    |-  U  =  ( Ss  R )   &    |-  B  =  (
 Base `  S )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   =>    |-  ( ph  ->  ( Q `  X )  =  (  _I  |`  B )
 )
 
Theoremevls1scasrng 18693 The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 13-Sep-2019.)
 |-  Q  =  ( S evalSub1  R )   &    |-  O  =  (eval1 `  S )   &    |-  W  =  (Poly1 `  U )   &    |-  U  =  ( Ss  R )   &    |-  P  =  (Poly1 `  S )   &    |-  B  =  (
 Base `  S )   &    |-  A  =  (algSc `  W )   &    |-  C  =  (algSc `  P )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  X  e.  R )   =>    |-  ( ph  ->  ( Q `  ( A `  X ) )  =  ( O `  ( C `  X ) ) )
 
Theoremevls1varsrng 18694 The evaluation of the variable of univariate polynomials over subring yields the same result as evaluated as variable of the polynomials over the ring itself. (Contributed by AV, 12-Sep-2019.)
 |-  Q  =  ( S evalSub1  R )   &    |-  O  =  (eval1 `  S )   &    |-  V  =  (var1 `  U )   &    |-  U  =  ( Ss  R )   &    |-  B  =  (
 Base `  S )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   =>    |-  ( ph  ->  ( Q `  V )  =  ( O `  V ) )
 
Theoremevl1addd 18695 Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y )  =  V )
 )   &    |-  ( ph  ->  ( N  e.  U  /\  ( ( O `  N ) `  Y )  =  W )
 )   &    |-  .+b  =  ( +g  `  P )   &    |- 
 .+  =  ( +g  `  R )   =>    |-  ( ph  ->  (
 ( M  .+b  N )  e.  U  /\  (
 ( O `  ( M  .+b  N ) ) `
  Y )  =  ( V  .+  W ) ) )
 
Theoremevl1subd 18696 Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y )  =  V )
 )   &    |-  ( ph  ->  ( N  e.  U  /\  ( ( O `  N ) `  Y )  =  W )
 )   &    |-  .-  =  ( -g `  P )   &    |-  D  =  (
 -g `  R )   =>    |-  ( ph  ->  ( ( M 
 .-  N )  e.  U  /\  ( ( O `  ( M 
 .-  N ) ) `
  Y )  =  ( V D W ) ) )
 
Theoremevl1muld 18697 Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y )  =  V )
 )   &    |-  ( ph  ->  ( N  e.  U  /\  ( ( O `  N ) `  Y )  =  W )
 )   &    |-  .xb  =  ( .r `  P )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ph  ->  ( ( M  .xb  N )  e.  U  /\  (
 ( O `  ( M  .xb  N ) ) `
  Y )  =  ( V  .x.  W ) ) )
 
Theoremevl1vsd 18698 Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y )  =  V )
 )   &    |-  ( ph  ->  N  e.  B )   &    |-  .xb  =  ( .s `  P )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ph  ->  ( ( N  .xb  M )  e.  U  /\  ( ( O `  ( N 
 .xb  M ) ) `  Y )  =  ( N  .x.  V ) ) )
 
Theoremevl1expd 18699 Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y )  =  V )
 )   &    |-  .xb  =  (.g `  (mulGrp `  P ) )   &    |-  .^  =  (.g `  (mulGrp `  R ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( N  .xb  M )  e.  U  /\  (
 ( O `  ( N  .xb  M ) ) `
  Y )  =  ( N  .^  V ) ) )
 
Theorempf1const 18700 Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  Q  =  ran  (eval1 `  R )   =>    |-  ( ( R  e.  CRing  /\  X  e.  B ) 
 ->  ( B  X.  { X } )  e.  Q )
    < 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 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38841
  Copyright terms: Public domain < Previous  Next >