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Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreldmevls 19401 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |- 
 Rel  dom evalSub
 
Theoremmpfrcl 19402 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 19403* 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 19404* 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 19405 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 19406 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 19407* 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 19408 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 19409 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 19410* 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 19411* 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 19412 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 19413 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 19414 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 19415 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 19416 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 19417 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 19418 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 19419 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 19420 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 19421 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 19422 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 19423* 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 19424 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 19425 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 19426 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 19427 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 19428* 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 19429 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 19430 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 19431 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 19432 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 19433 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 19434* 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 19435* 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 19436 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 19437 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 19438* 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 19439 Multivariate polynomial degree.
 class mDeg
 
Syntaxcdg1 19440 Univariate polynomial degree.
 class deg1
 
Definitiondf-mdeg 19441* 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 19573. (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 19442 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |- deg1  =  ( r  e.  _V  |->  ( 1o mDeg  r )
 )
 
Theoremreldmmdeg 19443 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |- 
 Rel  dom mDeg
 
Theoremtdeglem1 19444* 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 19445* 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 19446* 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 19447 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 19448* 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 19449* 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 19450* 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 19451* 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 19452* 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 19453 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 19454 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 19455 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 19456 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 19457 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 19458 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 19459 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 19460 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 19461 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 19462 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 19463 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 19464* Lemma for mdegmulle2 19465. (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 19465 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 19466 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 19467 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 19468 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 19469 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 19470* 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 19471 Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  ( deg1  `  R )  =  ( deg1  `  (  _I  `  R ) )
 
Theoremdeg1propd 19472* 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 19473 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 19474 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 19475 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 19476 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 19477 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 19478 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 19479 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 19480 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 19481* 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 19482 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 19483 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 19484 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 19485 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 19486 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 19487 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 19488 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 19489 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 19490 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 19491 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 19492 The degree of the a 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  =  ( inv g `  Y )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( D `  ( N `
  F ) )  =  ( D `  F ) )
 
Theoremdeg1suble 19493 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 19494 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 19495 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 19496 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 19497 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 19498 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 19499 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 19500 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 ) ) )
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