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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdeg1pw 19501 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 19502 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 19503 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 19504 Corollary of deg1mul2 19495: 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 19505 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 )
 
13.1.3  The division algorithm for univariate polynomials
 
Syntaxcmn1 19506 Monic polynomials.
 class Monic1p
 
Syntaxcuc1p 19507 Unitic polynomials.
 class Unic1p
 
Syntaxcq1p 19508 Univariate polynomial quotient.
 class quot1p
 
Syntaxcr1p 19509 Univariate polynomial remainder.
 class rem1p
 
Syntaxcig1p 19510 Univariate polynomial ideal generator.
 class idlGen1p
 
Definitiondf-mon1 19511* 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 19512* 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 19518. (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 19513* Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 19518. We actually use the reversed version for better harmony with our divisibility df-dvdsr 15418. (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 19514* 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 19515* 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.)
 |- 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 19516* 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 19517* Lemma for ply1divalg 19518: 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 19518* 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 19519* Reverse the order of multiplication in ply1divalg 19518 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 19520* 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 19521 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 19522* 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 19523 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 19524 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 19525 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 19526 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 19527 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 19528 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 19529 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 19530 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 19531 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 19532 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 )
 
Theoremdeg1submon1p 19533 The difference of two monic polynomials of the same degree is a polynomial of lesser degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  O  =  (Monic1p `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  .-  =  ( -g `  P )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F  e.  O )   &    |-  ( ph  ->  ( D `  F )  =  X )   &    |-  ( ph  ->  G  e.  O )   &    |-  ( ph  ->  ( D `  G )  =  X )   =>    |-  ( ph  ->  ( D `  ( F 
 .-  G ) )  <  X )
 
Theoremq1pval 19534* Value of the univariate polynomial quotient function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  Q  =  (quot1p `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  .-  =  ( -g `  P )   &    |-  .x.  =  ( .r `  P )   =>    |-  ( ( F  e.  B  /\  G  e.  B )  ->  ( F Q G )  =  ( iota_
 q  e.  B ( D `  ( F 
 .-  ( q  .x.  G ) ) )  < 
 ( D `  G ) ) )
 
Theoremq1peqb 19535 Characterizing property of the polynomial quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  Q  =  (quot1p `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  .-  =  ( -g `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  C  =  (Unic1p `  R )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( ( X  e.  B  /\  ( D `  ( F  .-  ( X 
 .x.  G ) ) )  <  ( D `  G ) )  <->  ( F Q G )  =  X ) )
 
Theoremq1pcl 19536 Closure of the quotient by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  Q  =  (quot1p `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  C  =  (Unic1p `  R )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F Q G )  e.  B )
 
Theoremr1pval 19537 Value of the polynomial remainder function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  E  =  (rem1p `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  Q  =  (quot1p `  R )   &    |-  .x.  =  ( .r `  P )   &    |-  .-  =  ( -g `  P )   =>    |-  ( ( F  e.  B  /\  G  e.  B )  ->  ( F E G )  =  ( F  .-  ( ( F Q G )  .x.  G ) ) )
 
Theoremr1pcl 19538 Closure of remainder following division by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  E  =  (rem1p `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  C  =  (Unic1p `  R )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F E G )  e.  B )
 
Theoremr1pdeglt 19539 The remainder has a degree smaller than the divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  E  =  (rem1p `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  C  =  (Unic1p `  R )   &    |-  D  =  ( deg1  `  R )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( D `  ( F E G ) )  <  ( D `  G ) )
 
Theoremr1pid 19540 Express the original polynomial  F as  F  =  ( q  x.  G )  +  r using the quotient and remainder functions for  q and  r. (Contributed by Mario Carneiro, 5-Jun-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  C  =  (Unic1p `  R )   &    |-  Q  =  (quot1p `  R )   &    |-  E  =  (rem1p `  R )   &    |-  .x.  =  ( .r `  P )   &    |-  .+  =  ( +g  `  P )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  F  =  ( ( ( F Q G )  .x.  G )  .+  ( F E G ) ) )
 
Theoremdvdsq1p 19541 Divisibility in a polynomial ring is witnessed by the quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  .||  =  ( ||r
 `  P )   &    |-  B  =  ( Base `  P )   &    |-  C  =  (Unic1p `  R )   &    |-  .x.  =  ( .r `  P )   &    |-  Q  =  (quot1p `  R )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  <->  F  =  (
 ( F Q G )  .x.  G ) ) )
 
Theoremdvdsr1p 19542 Divisibility in a polynomial ring in terms of the remainder. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  .||  =  ( ||r
 `  P )   &    |-  B  =  ( Base `  P )   &    |-  C  =  (Unic1p `  R )   &    |-  .0.  =  ( 0g `  P )   &    |-  E  =  (rem1p `  R )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  <->  ( F E G )  =  .0.  ) )
 
Theoremply1remlem 19543 A term of the form  x  -  N is linear, monic, and has exactly one zero. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  X  =  (var1 `  R )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   &    |-  G  =  ( X 
 .-  ( A `  N ) )   &    |-  O  =  (eval1 `  R )   &    |-  ( ph  ->  R  e. NzRing )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  N  e.  K )   &    |-  U  =  (Monic1p `  R )   &    |-  D  =  ( deg1  `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ph  ->  ( G  e.  U  /\  ( D `
  G )  =  1  /\  ( `' ( O `  G ) " {  .0.  }
 )  =  { N } ) )
 
Theoremply1rem 19544 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 12716). If a polynomial  F is divided by the linear factor  x  -  A, the remainder is equal to  F ( A ), the evaluation of the polynomial at  A (interpreted as a constant polynomial). (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  X  =  (var1 `  R )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   &    |-  G  =  ( X 
 .-  ( A `  N ) )   &    |-  O  =  (eval1 `  R )   &    |-  ( ph  ->  R  e. NzRing )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  N  e.  K )   &    |-  ( ph  ->  F  e.  B )   &    |-  E  =  (rem1p `  R )   =>    |-  ( ph  ->  ( F E G )  =  ( A `  ( ( O `  F ) `  N ) ) )
 
Theoremfacth1 19545 The factor theorem and its converse. A polynomial  F has a root at  A iff  G  =  x  -  A is a factor of  F. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  X  =  (var1 `  R )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   &    |-  G  =  ( X 
 .-  ( A `  N ) )   &    |-  O  =  (eval1 `  R )   &    |-  ( ph  ->  R  e. NzRing )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  N  e.  K )   &    |-  ( ph  ->  F  e.  B )   &    |-  .0.  =  ( 0g `  R )   &    |-  .||  =  ( ||r
 `  P )   =>    |-  ( ph  ->  ( G  .||  F  <->  ( ( O `
  F ) `  N )  =  .0.  ) )
 
Theoremfta1glem1 19546 Lemma for fta1g 19548. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  O  =  (eval1 `  R )   &    |-  W  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  P )   &    |-  ( ph  ->  R  e. IDomn )   &    |-  ( ph  ->  F  e.  B )   &    |-  K  =  ( Base `  R )   &    |-  X  =  (var1 `  R )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   &    |-  G  =  ( X 
 .-  ( A `  T ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( D `  F )  =  ( N  +  1 ) )   &    |-  ( ph  ->  T  e.  ( `' ( O `  F ) " { W } ) )   =>    |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )
 
Theoremfta1glem2 19547* Lemma for fta1g 19548. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  O  =  (eval1 `  R )   &    |-  W  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  P )   &    |-  ( ph  ->  R  e. IDomn )   &    |-  ( ph  ->  F  e.  B )   &    |-  K  =  ( Base `  R )   &    |-  X  =  (var1 `  R )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   &    |-  G  =  ( X 
 .-  ( A `  T ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( D `  F )  =  ( N  +  1 ) )   &    |-  ( ph  ->  T  e.  ( `' ( O `  F ) " { W } ) )   &    |-  ( ph  ->  A. g  e.  B  ( ( D `
  g )  =  N  ->  ( # `  ( `' ( O `  g
 ) " { W }
 ) )  <_  ( D `  g ) ) )   =>    |-  ( ph  ->  ( # `
  ( `' ( O `  F ) " { W } ) ) 
 <_  ( D `  F ) )
 
Theoremfta1g 19548 The one-sided fundamental theorem of algebra. A polynomial of degree  n has at most  n roots. Unlike the real fundamental theorem fta 20312, which is only true in  CC and other algebraically closed fields, this is true in any integral domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  O  =  (eval1 `  R )   &    |-  W  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  P )   &    |-  ( ph  ->  R  e. IDomn )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  F  =/=  .0.  )   =>    |-  ( ph  ->  ( # `
  ( `' ( O `  F ) " { W } ) ) 
 <_  ( D `  F ) )
 
Theoremfta1blem 19549 Lemma for fta1b 19550. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  O  =  (eval1 `  R )   &    |-  W  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  P )   &    |-  K  =  (
 Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  X  =  (var1 `  R )   &    |- 
 .x.  =  ( .s `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  M  e.  K )   &    |-  ( ph  ->  N  e.  K )   &    |-  ( ph  ->  ( M  .X.  N )  =  W )   &    |-  ( ph  ->  M  =/=  W )   &    |-  ( ph  ->  ( ( M 
 .x.  X )  e.  ( B  \  {  .0.  }
 )  ->  ( # `  ( `' ( O `  ( M  .x.  X ) )
 " { W }
 ) )  <_  ( D `  ( M  .x.  X ) ) ) )   =>    |-  ( ph  ->  N  =  W )
 
Theoremfta1b 19550* The assumption that  R be a domain in fta1g 19548 is necessary. Here we show that the statement is strong enough to prove that  R is a domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  O  =  (eval1 `  R )   &    |-  W  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  P )   =>    |-  ( R  e. IDomn  <->  ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
  f ) " { W } ) ) 
 <_  ( D `  f
 ) ) )
 
Theoremdrnguc1p 19551 Over a division ring, all nonzero polynomials are unitic. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  P )   &    |-  C  =  (Unic1p `  R )   =>    |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/=  .0.  )  ->  F  e.  C )
 
Theoremig1peu 19552* There is a unique monic polynomial of minimal degree in any nonzero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  M  =  (Monic1p `  R )   &    |-  D  =  ( deg1  `  R )   =>    |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
 )  ->  E! g  e.  ( I  i^i  M ) ( D `  g )  =  sup ( ( D "
 ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
 
Theoremig1pval 19553* Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  U  =  (LIdeal `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  M  =  (Monic1p `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  U )  ->  ( G `  I )  =  if ( I  =  {  .0.  } ,  .0.  ,  ( iota_ g  e.  ( I  i^i  M ) ( D `  g )  =  sup ( ( D " ( I 
 \  {  .0.  }
 ) ) ,  RR ,  `'  <  ) ) ) )
 
Theoremig1pval2 19554 Generator of the zero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |- 
 .0.  =  ( 0g `  P )   =>    |-  ( R  e.  Ring  ->  ( G `  {  .0.  } )  =  .0.  )
 
Theoremig1pval3 19555 Characterizing properties of the monic generator of a nonzero ideal of polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  U  =  (LIdeal `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  M  =  (Monic1p `  R )   =>    |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
 )  ->  ( ( G `  I )  e.  I  /\  ( G `
  I )  e.  M  /\  ( D `
  ( G `  I ) )  = 
 sup ( ( D
 " ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
 
Theoremig1pcl 19556 The monic generator of an ideal is always in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |-  U  =  (LIdeal `  P )   =>    |-  ( ( R  e.  DivRing  /\  I  e.  U ) 
 ->  ( G `  I
 )  e.  I )
 
Theoremig1pdvds 19557 The monic generator of an ideal divides all elements of the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  .||  =  ( ||r
 `  P )   =>    |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I
 )  ->  ( G `  I )  .||  X )
 
Theoremig1prsp 19558 Any ideal of polynomials over a division ring is generated by the ideal's canonical generator. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  K  =  (RSpan `  P )   =>    |-  ( ( R  e.  DivRing  /\  I  e.  U ) 
 ->  I  =  ( K `  { ( G `
  I ) }
 ) )
 
Theoremply1lpir 19559 The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  DivRing  ->  P  e. LPIR )
 
Theoremply1pid 19560 The polynomials over a field are a PID. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e. Field  ->  P  e. PID )
 
13.1.4  Elementary properties of complex polynomials
 
Syntaxcply 19561 Extend class notation to include the set of complex polynomials.
 class Poly
 
Syntaxcidp 19562 Extend class notation to include the identity polynomial.
 class  X p
 
Syntaxccoe 19563 Extend class notation to include the coefficient function on polynomials.
 class coeff
 
Syntaxcdgr 19564 Extend class notation to include the degree function on polynomials.
 class deg
 
Definitiondf-ply 19565* Define the set of polynomials on the complexes with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |- Poly  =  ( x  e.  ~P CC  |->  { f  |  E. n  e.  NN0  E. a  e.  ( ( x  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  ( z ^
 k ) ) ) } )
 
Definitiondf-idp 19566 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  X p  =  (  _I  |`  CC )
 
Definitiondf-coe 19567* Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |- coeff  =  ( f  e.  (Poly `  CC )  |->  ( iota_ a  e.  ( CC  ^m  NN0 ) E. n  e. 
 NN0  ( ( a
 " ( ZZ>= `  ( n  +  1 )
 ) )  =  {
 0 }  /\  f  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... n ) ( ( a `  k
 )  x.  ( z ^ k ) ) ) ) ) )
 
Definitiondf-dgr 19568 Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |- deg 
 =  ( f  e.  (Poly `  CC )  |-> 
 sup ( ( `' (coeff `  f ) " ( CC  \  {
 0 } ) ) ,  NN0 ,  <  ) )
 
Theoremplyco0 19569* Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( ( N  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 }  <->  A. k  e.  NN0  ( ( A `  k )  =/=  0  ->  k  <_  N )
 ) )
 
Theoremplyval 19570* Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( S  C_  CC  ->  (Poly `  S )  =  { f  |  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  ( z ^
 k ) ) ) } )
 
Theoremplybss 19571 Reverse closure of the parameter  S of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  S  C_ 
 CC )
 
Theoremelply 19572* Definition of a polynomial with coefficients in  S. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  <->  ( S  C_  CC  /\  E. n  e. 
 NN0  E. a  e.  (
 ( S  u.  {
 0 } )  ^m  NN0 ) F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  ( z ^
 k ) ) ) ) )
 
Theoremelply2 19573* The coefficient function can be assumed to have zeroes outside  0 ... n. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( F  e.  (Poly `  S )  <->  ( S  C_  CC  /\  E. n  e. 
 NN0  E. a  e.  (
 ( S  u.  {
 0 } )  ^m  NN0 ) ( ( a
 " ( ZZ>= `  ( n  +  1 )
 ) )  =  {
 0 }  /\  F  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... n ) ( ( a `  k
 )  x.  ( z ^ k ) ) ) ) ) )
 
Theoremplyun0 19574 The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  (Poly `  ( S  u.  { 0 } )
 )  =  (Poly `  S )
 
Theoremplyf 19575 The polynomial is a function on the complexes. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  F : CC --> CC )
 
Theoremplyss 19576 The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( S  C_  T  /\  T  C_  CC )  ->  (Poly `  S )  C_  (Poly `  T ) )
 
Theoremplyssc 19577 Every polynomial ring is contained in the ring of polynomials over  CC. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (Poly `  S )  C_  (Poly `  CC )
 
Theoremelplyr 19578* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0 --> S ) 
 ->  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) )  e.  (Poly `  S ) )
 
Theoremelplyd 19579* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  k  e.  ( 0
 ... N ) ) 
 ->  A  e.  S )   =>    |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  (
 0 ... N ) ( A  x.  ( z ^ k ) ) )  e.  (Poly `  S ) )
 
Theoremply1termlem 19580* Lemma for ply1term 19581. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  F  =  ( z  e.  CC  |->  ( A  x.  ( z ^ N ) ) )   =>    |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ...
 N ) ( if ( k  =  N ,  A ,  0 )  x.  ( z ^
 k ) ) ) )
 
Theoremply1term 19581* A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  F  =  ( z  e.  CC  |->  ( A  x.  ( z ^ N ) ) )   =>    |-  ( ( S  C_  CC  /\  A  e.  S  /\  N  e.  NN0 )  ->  F  e.  (Poly `  S ) )
 
Theoremplypow 19582* A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( S  C_  CC  /\  1  e.  S  /\  N  e.  NN0 )  ->  ( z  e.  CC  |->  ( z ^ N ) )  e.  (Poly `  S ) )
 
Theoremplyconst 19583 A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( S  C_  CC  /\  A  e.  S )  ->  ( CC  X.  { A } )  e.  (Poly `  S )
 )
 
Theoremne0p 19584 A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( ( A  e.  CC  /\  ( F `  A )  =/=  0
 )  ->  F  =/=  0 p )
 
Theoremply0 19585 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( S  C_  CC  ->  0 p  e.  (Poly `  S ) )
 
Theoremplyid 19586 The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( S  C_  CC  /\  1  e.  S )  ->  X p  e.  (Poly `  S ) )
 
Theoremplyeq0lem 19587* Lemma for plyeq0 19588. If  A is the coefficient function for a nonzero polynomial such that  P ( z )  =  sum_ k  e.  NN0 A ( k )  x.  z ^
k  =  0 for every  z  e.  CC and  A ( M ) is the nonzero leading coefficient, then the function  F ( z )  =  P ( z )  /  z ^ M is a sum of powers of  1  /  z, and so the limit of this function as  z 
~~>  +oo is the constant term,  A ( M ). But  F ( z )  =  0 everywhere, so this limit is also equal to zero so that  A ( M )  =  0, a contradiction. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )
 )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( N  +  1 )
 ) )  =  {
 0 } )   &    |-  ( ph  ->  0 p  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  M  =  sup ( ( `' A " ( S  \  {
 0 } ) ) ,  RR ,  <  )   &    |-  ( ph  ->  ( `' A " ( S  \  { 0 } )
 )  =/=  (/) )   =>    |-  -.  ph
 
Theoremplyeq0 19588* If a polynomial is zero at every point (or even just zero at the positive integers), then all the coefficients must be zero. This is the basis for the method of equating coefficients of equal polynomials, and ensures that df-coe 19567 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )
 )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( N  +  1 )
 ) )  =  {
 0 } )   &    |-  ( ph  ->  0 p  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  A  =  ( NN0  X.  {
 0 } ) )
 
Theoremplypf1 19589 Write the set of complex polynomials in a subring in terms of the abstract polynomial construction. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  R  =  (flds  S )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  (
 Base `  P )   &    |-  E  =  (eval1 ` fld )   =>    |-  ( S  e.  (SubRing ` fld ) 
 ->  (Poly `  S )  =  ( E " A ) )
 
Theoremplyaddlem1 19590* Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  B : NN0 --> CC )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( M  +  1
 ) ) )  =  { 0 } )   &    |-  ( ph  ->  ( B "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... M ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  G  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( B `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  ( F  o F  +  G )  =  ( z  e.  CC  |->  sum_ k  e.  (
 0 ... if ( M 
 <_  N ,  N ,  M ) ) ( ( ( A  o F  +  B ) `  k )  x.  (
 z ^ k ) ) ) )
 
Theoremplymullem1 19591* Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  B : NN0 --> CC )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( M  +  1
 ) ) )  =  { 0 } )   &    |-  ( ph  ->  ( B "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... M ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  G  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( B `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  ( F  o F  x.  G )  =  ( z  e.  CC  |->  sum_ n  e.  (
 0 ... ( M  +  N ) ) (
 sum_ k  e.  (
 0 ... n ) ( ( A `  k
 )  x.  ( B `
  ( n  -  k ) ) )  x.  ( z ^ n ) ) ) )
 
Theoremplyaddlem 19592* Lemma for plyadd 19594. (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )
 )   &    |-  ( ph  ->  B  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( M  +  1
 ) ) )  =  { 0 } )   &    |-  ( ph  ->  ( B "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... M ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  G  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( B `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  ( F  o F  +  G )  e.  (Poly `  S ) )
 
Theoremplymullem 19593* Lemma for plymul 19595. (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )
 )   &    |-  ( ph  ->  B  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( M  +  1
 ) ) )  =  { 0 } )   &    |-  ( ph  ->  ( B "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... M ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  G  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( B `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  x.  y )  e.  S )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e.  (Poly `  S ) )
 
Theoremplyadd 19594* The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   =>    |-  ( ph  ->  ( F  o F  +  G )  e.  (Poly `  S ) )
 
Theoremplymul 19595* The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  x.  y )  e.  S )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e.  (Poly `  S ) )
 
Theoremplysub 19596* The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  x.  y )  e.  S )   &    |-  ( ph  ->  -u 1  e.  S )   =>    |-  ( ph  ->  ( F  o F  -  G )  e.  (Poly `  S ) )
 
Theoremplyaddcl 19597 The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  o F  +  G )  e.  (Poly `  CC ) )
 
Theoremplymulcl 19598 The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  o F  x.  G )  e.  (Poly `  CC ) )
 
Theoremplysubcl 19599 The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  o F  -  G )  e.  (Poly `  CC ) )
 
Theoremcoeval 19600* Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  (coeff `  F )  =  (
 iota_ a  e.  ( CC  ^m  NN0 ) E. n  e.  NN0  ( ( a
 " ( ZZ>= `  ( n  +  1 )
 ) )  =  {
 0 }  /\  F  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... n ) ( ( a `  k
 )  x.  ( z ^ k ) ) ) ) ) )
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