HomeHome Metamath Proof Explorer
Theorem List (p. 231 of 402)
< 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
(1-26569)
  Hilbert Space Explorer  Hilbert Space Explorer
(26570-28092)
  Users' Mathboxes  Users' Mathboxes
(28093-40161)
 

Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcdg1 23001 Univariate polynomial degree.
 class deg1
 
Definitiondf-mdeg 23002* Define the degree of a polynomial. Note (SO): as an experiment I am using a definition which makes the degree of the zero polynomial -oo, contrary to the convention used in df-dgr 23143. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
 |- mDeg  =  ( i  e.  _V ,  r  e.  _V  |->  ( f  e.  ( Base `  ( i mPoly  r
 ) )  |->  sup ( ran  ( h  e.  (
 f supp  ( 0g `  r ) )  |->  (fld  gsumg  h ) ) ,  RR* ,  <  ) ) )
 
Definitiondf-deg1 23003 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |- deg1  =  ( r  e.  _V  |->  ( 1o mDeg  r )
 )
 
Theoremreldmmdeg 23004 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |- 
 Rel  dom mDeg
 
Theoremtdeglem1 23005* Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
 |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( I  e.  V  ->  H : A --> NN0 )
 
Theoremtdeglem3 23006* Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
 |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( ( I  e.  V  /\  X  e.  A  /\  Y  e.  A )  ->  ( H `  ( X  oF  +  Y ) )  =  ( ( H `  X )  +  ( H `  Y ) ) )
 
Theoremtdeglem4 23007* There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( ( H `  X )  =  0  <->  X  =  ( I  X.  { 0 } ) ) )
 
Theoremtdeglem2 23008 Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  ( h  e.  ( NN0  ^m  1o )  |->  ( h `  (/) ) )  =  ( h  e.  ( NN0  ^m  1o )  |->  (fld 
 gsumg  h ) )
 
Theoremmdegfval 23009* Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  D  =  ( f  e.  B  |->  sup (
 ( H " (
 f supp  .0.  ) ) , 
 RR* ,  <  ) )
 
Theoremmdegval 23010* Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H " ( F supp 
 .0.  ) ) , 
 RR* ,  <  ) )
 
Theoremmdegleb 23011* Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ( ( D `
  F )  <_  G 
 <-> 
 A. x  e.  A  ( G  <  ( H `
  x )  ->  ( F `  x )  =  .0.  ) ) )
 
Theoremmdeglt 23012* If there is an upper limit on the degree of a polynomial that is lower than the degree of some exponent bag, then that exponent bag is unrepresented in the polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  ( D `  F )  < 
 ( H `  X ) )   =>    |-  ( ph  ->  ( F `  X )  =  .0.  )
 
Theoremmdegldg 23013* A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   &    |-  Y  =  ( 0g `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  Y )  ->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `  x )  =  ( D `  F ) ) )
 
Theoremmdegxrcl 23014 Closure of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  ( D `  F )  e.  RR* )
 
Theoremmdegxrf 23015 Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   =>    |-  D : B --> RR*
 
Theoremmdegcl 23016 Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  ( D `  F )  e.  ( NN0  u.  { -oo } ) )
 
Theoremmdeg0 23017 Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  .0.  =  ( 0g `  P )   =>    |-  ( ( I  e.  V  /\  R  e.  Ring
 )  ->  ( D ` 
 .0.  )  = -oo )
 
Theoremmdegnn0cl 23018 Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  .0.  =  ( 0g `  P )   &    |-  B  =  ( Base `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  ->  ( D `  F )  e.  NN0 )
 
Theoremdegltlem1 23019 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  ( ( X  e.  ( NN0  u.  { -oo } )  /\  Y  e.  ZZ )  ->  ( X  <  Y  <->  X  <_  ( Y  -  1 ) ) )
 
Theoremdegltp1le 23020 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  ( ( X  e.  ( NN0  u.  { -oo } )  /\  Y  e.  ZZ )  ->  ( X  <  ( Y  +  1 )  <->  X  <_  Y ) )
 
Theoremmdegaddle 23021 The degree of a sum is at most the maximum of the degrees of the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  Y  =  ( I mPoly  R )   &    |-  D  =  ( I mDeg  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( D `  ( F  .+  G ) )  <_  if ( ( D `  F )  <_  ( D `
  G ) ,  ( D `  G ) ,  ( D `  F ) ) )
 
Theoremmdegvscale 23022 The degree of a scalar multiple of a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  Y  =  ( I mPoly  R )   &    |-  D  =  ( I mDeg  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  ( ph  ->  F  e.  K )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( D `  ( F 
 .x.  G ) )  <_  ( D `  G ) )
 
Theoremmdegvsca 23023 The degree of a scalar multiple of a polynomial is exactly the degree of the original polynomial when the multiple is a non-zero-divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
 |-  Y  =  ( I mPoly  R )   &    |-  D  =  ( I mDeg  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  E  =  (RLReg `  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  ( ph  ->  F  e.  E )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( D `  ( F 
 .x.  G ) )  =  ( D `  G ) )
 
Theoremmdegle0 23024 A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  Y  =  ( I mPoly  R )   &    |-  D  =  ( I mDeg  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  A  =  (algSc `  Y )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  (
 ( D `  F )  <_  0  <->  F  =  ( A `  ( F `  ( I  X.  { 0 } ) ) ) ) )
 
Theoremmdegmullem 23025* Lemma for mdegmulle2 23026. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  Y  =  ( I mPoly  R )   &    |-  D  =  ( I mDeg  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .r `  Y )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  J  e.  NN0 )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  ( D `  F ) 
 <_  J )   &    |-  ( ph  ->  ( D `  G ) 
 <_  K )   &    |-  A  =  {
 a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }   &    |-  H  =  ( b  e.  A  |->  (fld  gsumg  b ) )   =>    |-  ( ph  ->  ( D `  ( F  .x.  G ) )  <_  ( J  +  K )
 )
 
Theoremmdegmulle2 23026 The multivariate degree of a product of polynomials is at most the sum of the degrees of the polynomials. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  Y  =  ( I mPoly  R )   &    |-  D  =  ( I mDeg  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .r `  Y )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  J  e.  NN0 )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  ( D `  F ) 
 <_  J )   &    |-  ( ph  ->  ( D `  G ) 
 <_  K )   =>    |-  ( ph  ->  ( D `  ( F  .x.  G ) )  <_  ( J  +  K )
 )
 
Theoremdeg1fval 23027 Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  D  =  ( deg1  `  R )   =>    |-  D  =  ( 1o mDeg  R )
 
Theoremdeg1xrf 23028 Functionality of univariate polynomial degree, weak range. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  D : B --> RR*
 
Theoremdeg1xrcl 23029 Closure of univariate polynomial degree in extended reals. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  ( D `  F )  e.  RR* )
 
Theoremdeg1cl 23030 Sharp closure of univariate polynomial degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  ( D `  F )  e.  ( NN0  u. 
 { -oo } ) )
 
Theoremmdegpropd 23031* Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( I mDeg  R )  =  ( I mDeg 
 S ) )
 
Theoremdeg1fvi 23032 Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  ( deg1  `  R )  =  ( deg1  `  (  _I  `  R ) )
 
Theoremdeg1propd 23033* Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-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  ->  ( deg1  `  R )  =  ( deg1  `  S ) )
 
Theoremdeg1z 23034 Degree of the zero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   =>    |-  ( R  e.  Ring  ->  ( D `  .0.  )  = -oo )
 
Theoremdeg1nn0cl 23035 Degree of a nonzero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  B  =  (
 Base `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  ->  ( D `  F )  e.  NN0 )
 
Theoremdeg1n0ima 23036 Degree image of a set of polynomials which does not include zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  B  =  (
 Base `  P )   =>    |-  ( R  e.  Ring 
 ->  ( D " ( B  \  {  .0.  }
 ) )  C_  NN0 )
 
Theoremdeg1nn0clb 23037 A polynomial is nonzero iff it has definite degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  B  =  (
 Base `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B )  ->  ( F  =/=  .0.  <->  ( D `  F )  e.  NN0 )
 )
 
Theoremdeg1lt0 23038 A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  B  =  (
 Base `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B )  ->  ( ( D `  F )  <  0  <->  F  =  .0.  ) )
 
Theoremdeg1ldg 23039 A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  B  =  (
 Base `  P )   &    |-  Y  =  ( 0g `  R )   &    |-  A  =  (coe1 `  F )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  ->  ( A `  ( D `
  F ) )  =/=  Y )
 
Theoremdeg1ldgn 23040 An index at which a polynomial is zero, cannot be its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  B  =  (
 Base `  P )   &    |-  Y  =  ( 0g `  R )   &    |-  A  =  (coe1 `  F )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  X  e.  NN0 )   &    |-  ( ph  ->  ( A `  X )  =  Y )   =>    |-  ( ph  ->  ( D `  F )  =/= 
 X )
 
Theoremdeg1ldgdomn 23041 A nonzero univariate polynomial over a domain always has a non-zero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  B  =  (
 Base `  P )   &    |-  E  =  (RLReg `  R )   &    |-  A  =  (coe1 `  F )   =>    |-  ( ( R  e. Domn  /\  F  e.  B  /\  F  =/=  .0.  )  ->  ( A `  ( D `  F ) )  e.  E )
 
Theoremdeg1leb 23042* Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  (coe1 `  F )   =>    |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ( ( D `
  F )  <_  G 
 <-> 
 A. x  e.  NN0  ( G  <  x  ->  ( A `  x )  =  .0.  ) ) )
 
Theoremdeg1val 23043 Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  (coe1 `  F )   =>    |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( A supp  .0.  ) ,  RR*
 ,  <  ) )
 
Theoremdeg1lt 23044 If the degree of a univariate polynomial is less than some index, then that coefficient must be zero. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  (coe1 `  F )   =>    |-  ( ( F  e.  B  /\  G  e.  NN0  /\  ( D `  F )  <  G )  ->  ( A `  G )  =  .0.  )
 
Theoremdeg1ge 23045 Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  (coe1 `  F )   =>    |-  ( ( F  e.  B  /\  G  e.  NN0  /\  ( A `  G )  =/=  .0.  )  ->  G  <_  ( D `  F ) )
 
Theoremcoe1mul3 23046 The coefficient vector of multiplication in the univariate polynomial ring, at indices high enough that at most one component can be active in the sum. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  .xb  =  ( .r `  Y )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  Y )   &    |-  D  =  ( deg1  `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  I  e.  NN0 )   &    |-  ( ph  ->  ( D `  F ) 
 <_  I )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  J  e.  NN0 )   &    |-  ( ph  ->  ( D `  G )  <_  J )   =>    |-  ( ph  ->  (
 (coe1 `
  ( F  .xb  G ) ) `  ( I  +  J )
 )  =  ( ( (coe1 `  F ) `  I )  .x.  ( (coe1 `  G ) `  J ) ) )
 
Theoremcoe1mul4 23047 Value of the "leading" coefficient of a product of two nonzero polynomials. This will fail to actually be the leading coefficient only if it is zero (requiring the basic ring to contain zero divisors). (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  .xb  =  ( .r `  Y )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  Y )   &    |-  D  =  ( deg1  `  R )   &    |-  .0.  =  ( 0g `  Y )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  F  =/=  .0.  )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  G  =/=  .0.  )   =>    |-  ( ph  ->  ( (coe1 `  ( F  .xb  G ) ) `  (
 ( D `  F )  +  ( D `  G ) ) )  =  ( ( (coe1 `  F ) `  ( D `  F ) ) 
 .x.  ( (coe1 `  G ) `  ( D `  G ) ) ) )
 
Theoremdeg1addle 23048 The degree of a sum is at most the maximum of the degrees of the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( D `  ( F  .+  G ) )  <_  if ( ( D `  F )  <_  ( D `
  G ) ,  ( D `  G ) ,  ( D `  F ) ) )
 
Theoremdeg1addle2 23049 If both factors have degree bounded by  L, then the sum of the polynomials also has degree bounded by  L. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  L  e.  RR* )   &    |-  ( ph  ->  ( D `  F ) 
 <_  L )   &    |-  ( ph  ->  ( D `  G ) 
 <_  L )   =>    |-  ( ph  ->  ( D `  ( F  .+  G ) )  <_  L )
 
Theoremdeg1add 23050 Exact degree of a sum of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  ( D `  G )  <  ( D `  F ) )   =>    |-  ( ph  ->  ( D `  ( F 
 .+  G ) )  =  ( D `  F ) )
 
Theoremdeg1vscale 23051 The degree of a scalar times a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  ( ph  ->  F  e.  K )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( D `  ( F 
 .x.  G ) )  <_  ( D `  G ) )
 
Theoremdeg1vsca 23052 The degree of a scalar times a polynomial is exactly the degree of the original polynomial when the scalar is not a zero divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  E  =  (RLReg `  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  ( ph  ->  F  e.  E )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( D `  ( F 
 .x.  G ) )  =  ( D `  G ) )
 
Theoremdeg1invg 23053 The degree of the negated polynomial is the same as the original. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  N  =  ( invg `  Y )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( D `  ( N `
  F ) )  =  ( D `  F ) )
 
Theoremdeg1suble 23054 The degree of a difference of polynomials is bounded by the maximum of degrees. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( D `  ( F 
 .-  G ) ) 
 <_  if ( ( D `
  F )  <_  ( D `  G ) ,  ( D `  G ) ,  ( D `  F ) ) )
 
Theoremdeg1sub 23055 Exact degree of a difference of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  ( D `  G )  <  ( D `
  F ) )   =>    |-  ( ph  ->  ( D `  ( F  .-  G ) )  =  ( D `  F ) )
 
Theoremdeg1mulle2 23056 Produce a bound on the product of two univariate polynomials given bounds on the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .r `  Y )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  J  e.  NN0 )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  ( D `  F ) 
 <_  J )   &    |-  ( ph  ->  ( D `  G ) 
 <_  K )   =>    |-  ( ph  ->  ( D `  ( F  .x.  G ) )  <_  ( J  +  K )
 )
 
Theoremdeg1sublt 23057 Subtraction of two polynomials limited to the same degree with the same leading coefficient gives a polynomial with a smaller degree. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .-  =  ( -g `  P )   &    |-  ( ph  ->  L  e.  NN0 )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  ( D `  F )  <_  L )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  ( D `  G ) 
 <_  L )   &    |-  A  =  (coe1 `  F )   &    |-  C  =  (coe1 `  G )   &    |-  ( ph  ->  ( (coe1 `  F ) `  L )  =  (
 (coe1 `
  G ) `  L ) )   =>    |-  ( ph  ->  ( D `  ( F 
 .-  G ) )  <  L )
 
Theoremdeg1le0 23058 A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B ) 
 ->  ( ( D `  F )  <_  0  <->  F  =  ( A `  ( (coe1 `  F ) `  0 ) ) ) )
 
Theoremdeg1sclle 23059 A scalar polynomial has nonpositive degree. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  K  =  ( Base `  R )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  K ) 
 ->  ( D `  ( A `  F ) ) 
 <_  0 )
 
Theoremdeg1scl 23060 A nonzero scalar polynomial has zero degree. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  K  =  ( Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  F  e.  K  /\  F  =/=  .0.  )  ->  ( D `  ( A `
  F ) )  =  0 )
 
Theoremdeg1mul2 23061 Degree of multiplication of two nonzero polynomials when the first leads with a non-zero-divisor coefficient. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  .0.  =  ( 0g `  P )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  F  =/=  .0.  )   &    |-  ( ph  ->  (
 (coe1 `
  F ) `  ( D `  F ) )  e.  E )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  G  =/=  .0.  )   =>    |-  ( ph  ->  ( D `  ( F 
 .x.  G ) )  =  ( ( D `  F )  +  ( D `  G ) ) )
 
Theoremdeg1mul3 23062 Degree of multiplication of a polynomial on the left by a non-zero-dividing scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Jul-2019.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  E  /\  G  e.  B )  ->  ( D `  (
 ( A `  F )  .x.  G ) )  =  ( D `  G ) )
 
Theoremdeg1mul3le 23063 Degree of multiplication of a polynomial on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  K  =  ( Base `  R )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  (
 ( A `  F )  .x.  G ) ) 
 <_  ( D `  G ) )
 
Theoremdeg1tmle 23064 Limiting degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  F  e.  NN0 )  ->  ( D `  ( C 
 .x.  ( F  .^  X ) ) ) 
 <_  F )
 
Theoremdeg1tm 23065 Exact degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  ( C  e.  K  /\  C  =/=  .0.  )  /\  F  e.  NN0 )  ->  ( D `  ( C  .x.  ( F  .^  X ) ) )  =  F )
 
Theoremdeg1pwle 23066 Limiting degree of a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e.  Ring  /\  F  e.  NN0 )  ->  ( D `  ( F  .^  X ) ) 
 <_  F )
 
Theoremdeg1pw 23067 Exact degree of a variable power over a nontrivial ring. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e. NzRing  /\  F  e.  NN0 )  ->  ( D `  ( F  .^  X ) )  =  F )
 
Theoremply1nz 23068 Univariate polynomials over a nonzero ring are a nonzero ring. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e. NzRing  ->  P  e. NzRing )
 
Theoremply1nzb 23069 Univariate polynomials are nonzero iff the base is nonzero. Or in contraposition, the univariate polynomials over the zero ring are also zero. (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  Ring  ->  ( R  e. NzRing  <->  P  e. NzRing ) )
 
Theoremply1domn 23070 Corollary of deg1mul2 23061: the univariate polynomials over a domain are a domain. This is true for multivariate but with a much more complicated proof. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e. Domn  ->  P  e. Domn )
 
Theoremply1idom 23071 The ring of univariate polynomials over an integral domain is itself an integral domain. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e. IDomn  ->  P  e. IDomn )
 
14.1.2  The division algorithm for univariate polynomials
 
Syntaxcmn1 23072 Monic polynomials.
 class Monic1p
 
Syntaxcuc1p 23073 Unitic polynomials.
 class Unic1p
 
Syntaxcq1p 23074 Univariate polynomial quotient.
 class quot1p
 
Syntaxcr1p 23075 Univariate polynomial remainder.
 class rem1p
 
Syntaxcig1p 23076 Univariate polynomial ideal generator.
 class idlGen1p
 
Syntaxcig1pold 23077 Univariate polynomial ideal generator.
 class idlGen1p
 
Definitiondf-mon1 23078* Define the set of monic univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |- Monic1p  =  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  =  ( 1r `  r
 ) ) } )
 
Definitiondf-uc1p 23079* Define the set of unitic univariate polynomials, as the polynomials with an invertible leading coefficient. This is not a standard concept but is useful to us as the set of polynomials which can be used as the divisor in the polynomial division theorem ply1divalg 23086. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |- Unic1p  =  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  e.  (Unit `  r )
 ) } )
 
Definitiondf-q1p 23080* Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 23086. We actually use the reversed version for better harmony with our divisibility df-dvdsr 17868. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |- quot1p  =  ( r  e.  _V  |->  [_ (Poly1 `  r )  /  p ]_ [_ ( Base `  p )  /  b ]_ ( f  e.  b ,  g  e.  b  |->  ( iota_ q  e.  b  ( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p ) g ) ) )  <  ( ( deg1  `  r ) `  g
 ) ) ) )
 
Definitiondf-r1p 23081* Define the remainder after dividing two univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |- rem1p  =  ( r  e.  _V  |->  [_ ( Base `  (Poly1 `  r
 ) )  /  b ]_ ( f  e.  b ,  g  e.  b  |->  ( f ( -g `  (Poly1 `  r ) ) ( ( f (quot1p `  r ) g ) ( .r `  (Poly1 `  r ) ) g ) ) ) )
 
Definitiondf-ig1p 23082* Define a choice function for generators of ideals over a division ring; this is the unique monic polynomial of minimal degree in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)
 |- idlGen1p  =  ( r  e.  _V  |->  ( i  e.  (LIdeal `  (Poly1 `  r ) ) 
 |->  if ( i  =  { ( 0g `  (Poly1 `  r ) ) } ,  ( 0g `  (Poly1 `  r ) ) ,  ( iota_ g  e.  (
 i  i^i  (Monic1p `  r
 ) ) ( ( deg1  `  r ) `  g
 )  = inf ( ( ( deg1  `  r ) "
 ( i  \  {
 ( 0g `  (Poly1 `  r ) ) }
 ) ) ,  RR ,  <  ) ) ) ) )
 
Definitiondf-ig1pOLD 23083* Define a choice function for generators of ideals over a division ring; this is the unique monic polynomial of minimal degree in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) Obsolete version of df-ig1p 23082 as of 25-Sep-2020. (New usage is discouraged.)
 |- idlGen1p  =  ( r  e.  _V  |->  ( i  e.  (LIdeal `  (Poly1 `  r ) ) 
 |->  if ( i  =  { ( 0g `  (Poly1 `  r ) ) } ,  ( 0g `  (Poly1 `  r ) ) ,  ( iota_ g  e.  (
 i  i^i  (Monic1p `  r
 ) ) ( ( deg1  `  r ) `  g
 )  =  sup (
 ( ( deg1  `  r ) " ( i  \  {
 ( 0g `  (Poly1 `  r ) ) }
 ) ) ,  RR ,  `'  <  ) ) ) ) )
 
Theoremply1divmo 23084* Uniqueness of a quotient in a polynomial division. For polynomials  F ,  G such that  G  =/=  0 and the leading coefficient of  G is not a zero divisor, there is at most one polynomial  q which satisfies  F  =  ( G  x.  q )  +  r where the degree of  r is less than the degree of  G. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by NM, 17-Jun-2017.)
 |-  P  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  B  =  ( Base `  P )   &    |-  .-  =  ( -g `  P )   &    |-  .0.  =  ( 0g `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  G  =/=  .0.  )   &    |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  E )   &    |-  E  =  (RLReg `  R )   =>    |-  ( ph  ->  E* q  e.  B  ( D `  ( F  .-  ( G 
 .xb  q ) ) )  <  ( D `
  G ) )
 
Theoremply1divex 23085* Lemma for ply1divalg 23086: existence part. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  B  =  ( Base `  P )   &    |-  .-  =  ( -g `  P )   &    |-  .0.  =  ( 0g `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  G  =/=  .0.  )   &    |-  .1.  =  ( 1r `  R )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  I  e.  K )   &    |-  ( ph  ->  (
 ( (coe1 `  G ) `  ( D `  G ) )  .x.  I )  =  .1.  )   =>    |-  ( ph  ->  E. q  e.  B  ( D `  ( F  .-  ( G 
 .xb  q ) ) )  <  ( D `
  G ) )
 
Theoremply1divalg 23086* The division algorithm for univariate polynomials over a ring. For polynomials  F ,  G such that  G  =/=  0 and the leading coefficient of  G is a unit, there are unique polynomials  q and  r  =  F  -  ( G  x.  q ) such that the degree of  r is less than the degree of  G. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  B  =  ( Base `  P )   &    |-  .-  =  ( -g `  P )   &    |-  .0.  =  ( 0g `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  G  =/=  .0.  )   &    |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  U )   &    |-  U  =  (Unit `  R )   =>    |-  ( ph  ->  E! q  e.  B  ( D `  ( F  .-  ( G  .xb  q ) ) )  <  ( D `  G ) )
 
Theoremply1divalg2 23087* Reverse the order of multiplication in ply1divalg 23086 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  B  =  ( Base `  P )   &    |-  .-  =  ( -g `  P )   &    |-  .0.  =  ( 0g `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  G  =/=  .0.  )   &    |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  U )   &    |-  U  =  (Unit `  R )   =>    |-  ( ph  ->  E! q  e.  B  ( D `  ( F  .-  ( q  .xb  G ) ) )  <  ( D `  G ) )
 
Theoremuc1pval 23088* Value of the set of unitic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  C  =  (Unic1p `  R )   &    |-  U  =  (Unit `  R )   =>    |-  C  =  { f  e.  B  |  ( f  =/=  .0.  /\  (
 (coe1 `
  f ) `  ( D `  f ) )  e.  U ) }
 
Theoremisuc1p 23089 Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  C  =  (Unic1p `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( F  e.  C  <->  ( F  e.  B  /\  F  =/=  .0.  /\  (
 (coe1 `
  F ) `  ( D `  F ) )  e.  U ) )
 
Theoremmon1pval 23090* Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  M  =  (Monic1p `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  M  =  { f  e.  B  |  ( f  =/=  .0.  /\  (
 (coe1 `
  f ) `  ( D `  f ) )  =  .1.  ) }
 
Theoremismon1p 23091 Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  M  =  (Monic1p `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( F  e.  M  <->  ( F  e.  B  /\  F  =/=  .0.  /\  (
 (coe1 `
  F ) `  ( D `  F ) )  =  .1.  )
 )
 
Theoremuc1pcl 23092 Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  C  =  (Unic1p `  R )   =>    |-  ( F  e.  C  ->  F  e.  B )
 
Theoremmon1pcl 23093 Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  M  =  (Monic1p `  R )   =>    |-  ( F  e.  M  ->  F  e.  B )
 
Theoremuc1pn0 23094 Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  C  =  (Unic1p `  R )   =>    |-  ( F  e.  C  ->  F  =/=  .0.  )
 
Theoremmon1pn0 23095 Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  M  =  (Monic1p `  R )   =>    |-  ( F  e.  M  ->  F  =/=  .0.  )
 
Theoremuc1pdeg 23096 Unitic polynomials have nonnegative degrees. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  C  =  (Unic1p `  R )   =>    |-  ( ( R  e.  Ring  /\  F  e.  C ) 
 ->  ( D `  F )  e.  NN0 )
 
Theoremuc1pldg 23097 Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  U  =  (Unit `  R )   &    |-  C  =  (Unic1p `  R )   =>    |-  ( F  e.  C  ->  ( (coe1 `  F ) `  ( D `  F ) )  e.  U )
 
Theoremmon1pldg 23098 Unitic polynomials have one leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  M  =  (Monic1p `  R )   =>    |-  ( F  e.  M  ->  ( (coe1 `  F ) `  ( D `  F ) )  =  .1.  )
 
Theoremmon1puc1p 23099 Monic polynomials are unitic. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  C  =  (Unic1p `  R )   &    |-  M  =  (Monic1p `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  M ) 
 ->  X  e.  C )
 
Theoremuc1pmon1p 23100 Make a unitic polynomial monic by multiplying a factor to normalize the leading coefficient. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  C  =  (Unic1p `  R )   &    |-  M  =  (Monic1p `  R )   &    |-  P  =  (Poly1 `  R )   &    |- 
 .x.  =  ( .r `  P )   &    |-  A  =  (algSc `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  I  =  (
 invr `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( ( A `  ( I `
  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X )  e.  M )
    < 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 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40161
  Copyright terms: Public domain < Previous  Next >