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Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremftc1lem3 19401* Lemma for ftc1 19405. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `  C ) )   &    |-  J  =  ( Lt  RR )   &    |-  K  =  ( Lt  D )   &    |-  L  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  F : D --> CC )
 
Theoremftc1lem4 19402* Lemma for ftc1 19405. (Contributed by Mario Carneiro, 31-Aug-2014.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `  C ) )   &    |-  J  =  ( Lt  RR )   &    |-  K  =  ( Lt  D )   &    |-  L  =  (
 TopOpen ` fld )   &    |-  H  =  ( z  e.  ( ( A [,] B ) 
 \  { C }
 )  |->  ( ( ( G `  z )  -  ( G `  C ) )  /  ( z  -  C ) ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  y  e.  D ) 
 ->  ( ( abs `  (
 y  -  C ) )  <  R  ->  ( abs `  ( ( F `  y )  -  ( F `  C ) ) )  <  E ) )   &    |-  ( ph  ->  X  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( X  -  C ) )  <  R )   &    |-  ( ph  ->  Y  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( Y  -  C ) )  <  R )   =>    |-  ( ( ph  /\  X  <  Y )  ->  ( abs `  ( ( ( ( G `  Y )  -  ( G `  X ) )  /  ( Y  -  X ) )  -  ( F `  C ) ) )  <  E )
 
Theoremftc1lem5 19403* Lemma for ftc1 19405. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `  C ) )   &    |-  J  =  ( Lt  RR )   &    |-  K  =  ( Lt  D )   &    |-  L  =  (
 TopOpen ` fld )   &    |-  H  =  ( z  e.  ( ( A [,] B ) 
 \  { C }
 )  |->  ( ( ( G `  z )  -  ( G `  C ) )  /  ( z  -  C ) ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  y  e.  D ) 
 ->  ( ( abs `  (
 y  -  C ) )  <  R  ->  ( abs `  ( ( F `  y )  -  ( F `  C ) ) )  <  E ) )   &    |-  ( ph  ->  X  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( X  -  C ) )  <  R )   =>    |-  ( ( ph  /\  X  =/=  C )  ->  ( abs `  ( ( H `
  X )  -  ( F `  C ) ) )  <  E )
 
Theoremftc1lem6 19404* Lemma for ftc1 19405. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `  C ) )   &    |-  J  =  ( Lt  RR )   &    |-  K  =  ( Lt  D )   &    |-  L  =  (
 TopOpen ` fld )   &    |-  H  =  ( z  e.  ( ( A [,] B ) 
 \  { C }
 )  |->  ( ( ( G `  z )  -  ( G `  C ) )  /  ( z  -  C ) ) )   =>    |-  ( ph  ->  ( F `  C )  e.  ( H lim CC  C ) )
 
Theoremftc1 19405* The Fundamental Theorem of Calculus, part one. The function formed by varying the right endpoint of an integral is differentiable at  C with derivative  F ( C ) if the original function is continuous at  C. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `  C ) )   &    |-  J  =  ( Lt  RR )   &    |-  K  =  ( Lt  D )   &    |-  L  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  C ( RR  _D  G ) ( F `  C ) )
 
Theoremftc1cn 19406* Strengthen the assumptions of ftc1 19405 to when the function  F is continuous on the entire interval  ( A ,  B ); in this case we can calculate  _D  G exactly. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  F  e.  L ^1 )   =>    |-  ( ph  ->  ( RR  _D  G )  =  F )
 
Theoremftc2 19407* The Fundamental Theorem of Calculus, part two. If  F is a function continuous on  [ A ,  B ] and continuously differentiable on  ( A ,  B ), then the integral of the derivative of  F is equal to  F ( B )  -  F ( A ). (Contributed by Mario Carneiro, 2-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( RR  _D  F )  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  L ^1 )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> CC ) )   =>    |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `  t
 )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
 
Theoremftc2ditglem 19408* Lemma for ftc2ditg 19409. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  A  e.  ( X [,] Y ) )   &    |-  ( ph  ->  B  e.  ( X [,] Y ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  ( ( X (,) Y ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  L ^1 )   &    |-  ( ph  ->  F  e.  (
 ( X [,] Y ) -cn-> CC ) )   =>    |-  ( ( ph  /\  A  <_  B )  ->  S__ [ A  ->  B ] ( ( RR 
 _D  F ) `  t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
 
Theoremftc2ditg 19409* Directed integral analog of ftc2 19407. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  A  e.  ( X [,] Y ) )   &    |-  ( ph  ->  B  e.  ( X [,] Y ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  ( ( X (,) Y ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  L ^1 )   &    |-  ( ph  ->  F  e.  (
 ( X [,] Y ) -cn-> CC ) )   =>    |-  ( ph  ->  S__
 [ A  ->  B ] ( ( RR 
 _D  F ) `  t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
 
Theoremitgparts 19410* Integration by parts. If  B ( x ) is the derivative of  A ( x ) and  D ( x ) is the derivative of  C ( x ), and  E  =  ( A  x.  B ) ( X ) and  F  =  ( A  x.  B ) ( Y ), then under suitable integrability and differentiability assumptions, the integral of  A  x.  D from  X to  Y is equal to  F  -  E minus the integral of  B  x.  C. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  X  <_  Y )   &    |-  ( ph  ->  ( x  e.  ( X [,] Y )  |->  A )  e.  ( ( X [,] Y ) -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  ( X [,] Y )  |->  C )  e.  ( ( X [,] Y )
 -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( X (,) Y ) -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  D )  e.  ( ( X (,) Y ) -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  ( A  x.  D ) )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  ( B  x.  C ) )  e.  L ^1 )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( X [,] Y )  |->  C ) )  =  ( x  e.  ( X (,) Y )  |->  D ) )   &    |-  ( ( ph  /\  x  =  X )  ->  ( A  x.  C )  =  E )   &    |-  ( ( ph  /\  x  =  Y ) 
 ->  ( A  x.  C )  =  F )   =>    |-  ( ph  ->  S. ( X (,) Y ) ( A  x.  D )  _d x  =  ( ( F  -  E )  -  S. ( X (,) Y ) ( B  x.  C )  _d x ) )
 
Theoremitgsubstlem 19411* Lemma for itgsubst 19412. (Contributed by Mario Carneiro, 12-Sep-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  X  <_  Y )   &    |-  ( ph  ->  Z  e.  RR* )   &    |-  ( ph  ->  W  e.  RR* )   &    |-  ( ph  ->  ( x  e.  ( X [,] Y )  |->  A )  e.  ( ( X [,] Y )
 -cn-> ( Z (,) W ) ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( ( X (,) Y ) -cn-> CC )  i^i  L ^1 )
 )   &    |-  ( ph  ->  ( u  e.  ( Z (,) W )  |->  C )  e.  ( ( Z (,) W ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )   &    |-  ( u  =  A  ->  C  =  E )   &    |-  ( x  =  X  ->  A  =  K )   &    |-  ( x  =  Y  ->  A  =  L )   &    |-  ( ph  ->  M  e.  ( Z (,) W ) )   &    |-  ( ph  ->  N  e.  ( Z (,) W ) )   &    |-  ( ( ph  /\  x  e.  ( X [,] Y ) ) 
 ->  A  e.  ( M (,) N ) )   =>    |-  ( ph  ->  S__ [ K  ->  L ] C  _d u  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
 
Theoremitgsubst 19412* Integration by  u-substitution. If  A ( x ) is a continuous, differentiable function from  [ X ,  Y ] to  ( Z ,  W ), whose derivative is continuous and integrable, and  C ( u ) is a continuous function on  ( Z ,  W ), then the integral of  C ( u ) from  K  =  A ( X ) to  L  =  A ( Y ) is equal to the integral of  C ( A ( x ) )  _D  A ( x ) from  X to  Y. In this part of the proof we discharge the assumptions in itgsubstlem 19411, which use the fact that  ( Z ,  W ) is open to shrink the interval a little to  ( M ,  N ) where  Z  <  M  <  N  <  W- this is possible because  A ( x ) is a continuous function on a closed interval, so its range is in fact a closed interval, and we have some wiggle room on the edges. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  X  <_  Y )   &    |-  ( ph  ->  Z  e.  RR* )   &    |-  ( ph  ->  W  e.  RR* )   &    |-  ( ph  ->  ( x  e.  ( X [,] Y )  |->  A )  e.  ( ( X [,] Y )
 -cn-> ( Z (,) W ) ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( ( X (,) Y ) -cn-> CC )  i^i  L ^1 )
 )   &    |-  ( ph  ->  ( u  e.  ( Z (,) W )  |->  C )  e.  ( ( Z (,) W ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )   &    |-  ( u  =  A  ->  C  =  E )   &    |-  ( x  =  X  ->  A  =  K )   &    |-  ( x  =  Y  ->  A  =  L )   =>    |-  ( ph  ->  S__ [ K  ->  L ] C  _d u  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
 
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
 
13.1  Polynomials
 
13.1.1  Abstract polynomials, continued
 
Theoremevlslem6 19413* Lemma for evlseu 19416. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  C  =  ( Base `  S )   &    |-  K  =  ( Base `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  T  =  (mulGrp `  S )   &    |-  .^  =  (.g `  T )   &    |- 
 .x.  =  ( .r `  S )   &    |-  V  =  ( I mVar  R )   &    |-  E  =  ( p  e.  B  |->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( p `  b ) )  .x.  ( T  gsumg  (
 b  o F  .^  G ) ) ) ) ) )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  F  e.  ( R RingHom  S ) )   &    |-  ( ph  ->  G : I --> C )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T  gsumg  (
 b  o F  .^  G ) ) ) ) : D --> C  /\  ( `' ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T  gsumg  (
 b  o F  .^  G ) ) ) ) " ( _V  \  { ( 0g `  S ) } )
 )  e.  Fin )
 )
 
Theoremevlslem3 19414* Lemma for evlseu 19416. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  C  =  ( Base `  S )   &    |-  K  =  ( Base `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  T  =  (mulGrp `  S )   &    |-  .^  =  (.g `  T )   &    |- 
 .x.  =  ( .r `  S )   &    |-  V  =  ( I mVar  R )   &    |-  E  =  ( p  e.  B  |->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( p `  b ) )  .x.  ( T  gsumg  (
 b  o F  .^  G ) ) ) ) ) )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  F  e.  ( R RingHom  S ) )   &    |-  ( ph  ->  G : I --> C )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  H  e.  K )   =>    |-  ( ph  ->  ( E `  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) )  =  ( ( F `  H )  .x.  ( T 
 gsumg  ( A  o F  .^  G ) ) ) )
 
Theoremevlslem1 19415* Lemma for evlseu 19416, give a formula for (the unique) polynomial evaluation homomorphism. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  C  =  ( Base `  S )   &    |-  K  =  ( Base `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  T  =  (mulGrp `  S )   &    |-  .^  =  (.g `  T )   &    |- 
 .x.  =  ( .r `  S )   &    |-  V  =  ( I mVar  R )   &    |-  E  =  ( p  e.  B  |->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( p `  b ) )  .x.  ( T  gsumg  (
 b  o F  .^  G ) ) ) ) ) )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  F  e.  ( R RingHom  S ) )   &    |-  ( ph  ->  G : I --> C )   &    |-  A  =  (algSc `  P )   =>    |-  ( ph  ->  ( E  e.  ( P RingHom  S )  /\  ( E  o.  A )  =  F  /\  ( E  o.  V )  =  G ) )
 
Theoremevlseu 19416* For a given intepretation of the variables  G and of the scalars  F, this extends to a homomorphic interpretation of the polynomial ring in exactly one way. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  C  =  (
 Base `  S )   &    |-  A  =  (algSc `  P )   &    |-  V  =  ( I mVar  R )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  F  e.  ( R RingHom  S ) )   &    |-  ( ph  ->  G : I --> C )   =>    |-  ( ph  ->  E! m  e.  ( P RingHom  S )
 ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G )
 )
 
Theoremreldmevls 19417 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |- 
 Rel  dom evalSub
 
Theoremmpfrcl 19418 Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  Q  =  ran  (
 ( I evalSub  S ) `  R )   =>    |-  ( X  e.  Q  ->  ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
 ) )
 
Theoremevlsval 19419* Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 |-  Q  =  ( ( I evalSub  S ) `  R )   &    |-  W  =  ( I mPoly  U )   &    |-  V  =  ( I mVar  U )   &    |-  U  =  ( Ss  R )   &    |-  T  =  ( S  ^s  ( B  ^m  I
 ) )   &    |-  B  =  (
 Base `  S )   &    |-  A  =  (algSc `  W )   &    |-  X  =  ( x  e.  R  |->  ( ( B  ^m  I )  X.  { x } ) )   &    |-  Y  =  ( x  e.  I  |->  ( g  e.  ( B  ^m  I )  |->  ( g `  x ) ) )   =>    |-  ( ( I  e. 
 _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  ->  Q  =  ( iota_ f  e.  ( W RingHom  T ) ( ( f  o.  A )  =  X  /\  (
 f  o.  V )  =  Y ) ) )
 
Theoremevlsval2 19420* Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.)
 |-  Q  =  ( ( I evalSub  S ) `  R )   &    |-  W  =  ( I mPoly  U )   &    |-  V  =  ( I mVar  U )   &    |-  U  =  ( Ss  R )   &    |-  T  =  ( S  ^s  ( B  ^m  I
 ) )   &    |-  B  =  (
 Base `  S )   &    |-  A  =  (algSc `  W )   &    |-  X  =  ( x  e.  R  |->  ( ( B  ^m  I )  X.  { x } ) )   &    |-  Y  =  ( x  e.  I  |->  ( g  e.  ( B  ^m  I )  |->  ( g `  x ) ) )   =>    |-  ( ( I  e. 
 _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  ->  ( Q  e.  ( W RingHom  T ) 
 /\  ( ( Q  o.  A )  =  X  /\  ( Q  o.  V )  =  Y ) ) )
 
Theoremevlsrhm 19421 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Stefan O'Rear, 12-Mar-2015.)
 |-  Q  =  ( ( I evalSub  S ) `  R )   &    |-  W  =  ( I mPoly  U )   &    |-  U  =  ( Ss  R )   &    |-  T  =  ( S  ^s  ( B  ^m  I
 ) )   &    |-  B  =  (
 Base `  S )   =>    |-  ( ( I  e.  V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) ) 
 ->  Q  e.  ( W RingHom  T ) )
 
Theoremevlssca 19422 Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.)
 |-  Q  =  ( ( I evalSub  S ) `  R )   &    |-  W  =  ( I mPoly  U )   &    |-  U  =  ( Ss  R )   &    |-  B  =  (
 Base `  S )   &    |-  A  =  (algSc `  W )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  X  e.  R )   =>    |-  ( ph  ->  ( Q `  ( A `
  X ) )  =  ( ( B 
 ^m  I )  X.  { X } ) )
 
Theoremevlsvar 19423* Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.)
 |-  Q  =  ( ( I evalSub  S ) `  R )   &    |-  V  =  ( I mVar 
 U )   &    |-  U  =  ( Ss  R )   &    |-  B  =  (
 Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( Q `  ( V `
  X ) )  =  ( g  e.  ( B  ^m  I
 )  |->  ( g `  X ) ) )
 
Theoremevlval 19424 Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ( I eval 
 R )   &    |-  B  =  (
 Base `  R )   =>    |-  Q  =  ( ( I evalSub  R ) `  B )
 
Theoremevlrhm 19425 The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ( I eval 
 R )   &    |-  B  =  (
 Base `  R )   &    |-  W  =  ( I mPoly  R )   &    |-  T  =  ( R  ^s  ( B  ^m  I ) )   =>    |-  ( ( I  e.  V  /\  R  e.  CRing
 )  ->  Q  e.  ( W RingHom  T ) )
 
Theoremevl1fval 19426* Value of the simple/same ring evalutation 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 19427* Value of the simple/same ring evalutation 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 } ) ) ) )
 
Theoremevl1rhm 19428 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  T  =  ( R 
 ^s 
 B )   &    |-  B  =  (
 Base `  R )   =>    |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  T ) )
 
Theoremevl1sca 19429 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 19430 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 19431 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 19432 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 )
 )
 
Theoremevl1addd 19433 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 19434 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 19435 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 19436 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 19437 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 ) ) )
 
Theoremmpfconst 19438 Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  B  =  ( Base `  S )   &    |-  Q  =  ran  ( ( I evalSub  S ) `  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  X  e.  R )   =>    |-  ( ph  ->  ( ( B  ^m  I
 )  X.  { X } )  e.  Q )
 
Theoremmpfproj 19439* Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  B  =  ( Base `  S )   &    |-  Q  =  ran  ( ( I evalSub  S ) `  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  J  e.  I )   =>    |-  ( ph  ->  ( f  e.  ( B 
 ^m  I )  |->  ( f `  J ) )  e.  Q )
 
Theoremmpfsubrg 19440 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  Q  =  ran  (
 ( I evalSub  S ) `  R )   =>    |-  ( ( I  e. 
 _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  ->  Q  e.  (SubRing `  ( S  ^s  (
 ( Base `  S )  ^m  I ) ) ) )
 
Theoremmpff 19441 Polynomial functions are functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  Q  =  ran  (
 ( I evalSub  S ) `  R )   &    |-  B  =  (
 Base `  S )   =>    |-  ( F  e.  Q  ->  F : ( B  ^m  I ) --> B )
 
Theoremmpfaddcl 19442 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  Q  =  ran  (
 ( I evalSub  S ) `  R )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ( F  e.  Q  /\  G  e.  Q )  ->  ( F  o F  .+  G )  e.  Q )
 
Theoremmpfmulcl 19443 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  Q  =  ran  (
 ( I evalSub  S ) `  R )   &    |-  .x.  =  ( .r `  S )   =>    |-  ( ( F  e.  Q  /\  G  e.  Q )  ->  ( F  o F  .x.  G )  e.  Q )
 
Theoremmpfind 19444* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  B  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  Q  =  ran  ( ( I evalSub  S ) `  R )   &    |-  ( ( ph  /\  (
 ( f  e.  Q  /\  ta )  /\  (
 g  e.  Q  /\  et ) ) )  ->  ze )   &    |-  ( ( ph  /\  ( ( f  e.  Q  /\  ta )  /\  ( g  e.  Q  /\  et ) ) ) 
 ->  si )   &    |-  ( x  =  ( ( B  ^m  I )  X.  { f } )  ->  ( ps  <->  ch ) )   &    |-  ( x  =  ( g  e.  ( B  ^m  I )  |->  ( g `  f ) )  ->  ( ps  <->  th ) )   &    |-  ( x  =  f  ->  ( ps  <->  ta ) )   &    |-  ( x  =  g  ->  ( ps  <->  et ) )   &    |-  ( x  =  ( f  o F  .+  g )  ->  ( ps 
 <->  ze ) )   &    |-  ( x  =  ( f  o F  .x.  g ) 
 ->  ( ps  <->  si ) )   &    |-  ( x  =  A  ->  ( ps  <->  rh ) )   &    |-  (
 ( ph  /\  f  e.  R )  ->  ch )   &    |-  (
 ( ph  /\  f  e.  I )  ->  th )   &    |-  ( ph  ->  A  e.  Q )   =>    |-  ( ph  ->  rh )
 
Theorempf1const 19445 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 )
 
Theorempf1id 19446 The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  Q  =  ran  (eval1 `  R )   =>    |-  ( R  e.  CRing  ->  (  _I  |`  B )  e.  Q )
 
Theorempf1subrg 19447 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  Q  =  ran  (eval1 `  R )   =>    |-  ( R  e.  CRing  ->  Q  e.  (SubRing `  ( R  ^s  B ) ) )
 
Theorempf1rcl 19448 Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   =>    |-  ( X  e.  Q  ->  R  e.  CRing )
 
Theorempf1f 19449 Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( F  e.  Q  ->  F : B --> B )
 
Theoremmpfpf1 19450* Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   &    |-  B  =  (
 Base `  R )   &    |-  E  =  ran  ( 1o eval  R )   =>    |-  ( F  e.  E  ->  ( F  o.  (
 y  e.  B  |->  ( 1o  X.  { y } ) ) )  e.  Q )
 
Theorempf1mpf 19451* Convert a univariate polynomial function to multivariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   &    |-  B  =  (
 Base `  R )   &    |-  E  =  ran  ( 1o eval  R )   =>    |-  ( F  e.  Q  ->  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
  (/) ) ) )  e.  E )
 
Theorempf1addcl 19452 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( F  e.  Q  /\  G  e.  Q )  ->  ( F  o F  .+  G )  e.  Q )
 
Theorempf1mulcl 19453 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( F  e.  Q  /\  G  e.  Q )  ->  ( F  o F  .x.  G )  e.  Q )
 
Theorempf1ind 19454* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  Q  =  ran  (eval1 `  R )   &    |-  ( ( ph  /\  (
 ( f  e.  Q  /\  ta )  /\  (
 g  e.  Q  /\  et ) ) )  ->  ze )   &    |-  ( ( ph  /\  ( ( f  e.  Q  /\  ta )  /\  ( g  e.  Q  /\  et ) ) ) 
 ->  si )   &    |-  ( x  =  ( B  X.  {
 f } )  ->  ( ps  <->  ch ) )   &    |-  ( x  =  (  _I  |`  B )  ->  ( ps 
 <-> 
 th ) )   &    |-  ( x  =  f  ->  ( ps  <->  ta ) )   &    |-  ( x  =  g  ->  ( ps  <->  et ) )   &    |-  ( x  =  ( f  o F  .+  g ) 
 ->  ( ps  <->  ze ) )   &    |-  ( x  =  ( f  o F  .x.  g ) 
 ->  ( ps  <->  si ) )   &    |-  ( x  =  A  ->  ( ps  <->  rh ) )   &    |-  (
 ( ph  /\  f  e.  B )  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  A  e.  Q )   =>    |-  ( ph  ->  rh )
 
13.1.2  Polynomial degrees
 
Syntaxcmdg 19455 Multivariate polynomial degree.
 class mDeg
 
Syntaxcdg1 19456 Univariate polynomial degree.
 class deg1
 
Definitiondf-mdeg 19457* 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 19589. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |- mDeg  =  ( i  e.  _V ,  r  e.  _V  |->  ( f  e.  ( Base `  ( i mPoly  r
 ) )  |->  sup ( ran  ( h  e.  ( `' f " ( _V  \  { ( 0g `  r ) } )
 )  |->  (fld 
 gsumg  h ) ) , 
 RR* ,  <  ) ) )
 
Definitiondf-deg1 19458 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |- deg1  =  ( r  e.  _V  |->  ( 1o mDeg  r )
 )
 
Theoremreldmmdeg 19459 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |- 
 Rel  dom mDeg
 
Theoremtdeglem1 19460* Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( I  e.  V  ->  H : A --> NN0 )
 
Theoremtdeglem3 19461* Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-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  /\  Y  e.  A )  ->  ( H `  ( X  o F  +  Y ) )  =  ( ( H `  X )  +  ( H `  Y ) ) )
 
Theoremtdeglem4 19462* 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 19463 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 19464* Value of the multivariate degree function. (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 ) )   =>    |-  D  =  ( f  e.  B  |->  sup (
 ( H " ( `' f " ( _V  \  {  .0.  } )
 ) ) ,  RR* ,  <  ) )
 
Theoremmdegval 19465* Value of the multivariate degree function at some particular polynomial. (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  ->  ( D `  F )  =  sup ( ( H " ( `' F " ( _V  \  {  .0.  } )
 ) ) ,  RR* ,  <  ) )
 
Theoremmdegleb 19466* 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 19467* 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.)
 |-  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 19468* 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 19469 Closure of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  ( D `  F )  e.  RR* )
 
Theoremmdegxrf 19470 Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   =>    |-  D : B --> RR*
 
Theoremmdegcl 19471 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 19472 Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  .0.  =  ( 0g `  P )   =>    |-  ( ( I  e.  V  /\  R  e.  Ring
 )  ->  ( D ` 
 .0.  )  =  -oo )
 
Theoremmdegnn0cl 19473 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 19474 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 19475 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 19476 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 19477 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 19478 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.)
 |-  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 19479 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 19480* Lemma for mdegmulle2 19481. (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 19481 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 19482 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 19483 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 19484 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 19485 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 19486* 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  ->  ( I mDeg  R )  =  ( I mDeg 
 S ) )
 
Theoremdeg1fvi 19487 Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  ( deg1  `  R )  =  ( deg1  `  (  _I  `  R ) )
 
Theoremdeg1propd 19488* 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 19489 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 19490 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 19491 Degree image of a set of polynomials whcih 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 19492 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 19493 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 19494 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 19495 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 19496 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 19497* 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 19498 Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  (coe1 `  F )   =>    |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( `' A " ( _V  \  {  .0.  } )
 ) ,  RR* ,  <  ) )
 
Theoremdeg1lt 19499 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 19500 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 ) )
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