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Theorem List for Metamath Proof Explorer - 35701-35800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmncply 35701 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  P  e.  (Poly `  S )
 )
 
Theoremmnccoe 35702 A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  (
 (coeff `  P ) `  (deg `  P )
 )  =  1 )
 
Theoremmncn0 35703 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  P  =/=  0p )
 
21.23.45  Degree and minimal polynomial of algebraic numbers
 
Syntaxcdgraa 35704 Extend class notation to include the degree function for algebraic numbers.
 class degAA
 
Syntaxcmpaa 35705 Extend class notation to include the minimal polynomial for an algebraic number.
 class minPolyAA
 
Definitiondf-dgraa 35706* Define the degree of an algebraic number as the smallest degree of any nonzero polynomial which has said number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |- degAA  =  ( x  e.  AA  |->  sup ( { d  e.  NN  |  E. p  e.  (
 (Poly `  QQ )  \  { 0p }
 ) ( (deg `  p )  =  d  /\  ( p `  x )  =  0 ) } ,  RR ,  `'  <  ) )
 
Definitiondf-mpaa 35707* Define the minimal polynomial of an algebraic number as the unique monic polynomial which achieves the minimum of degAA. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |- minPolyAA  =  ( x  e.  AA  |->  (
 iota_ p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  x )  /\  ( p `  x )  =  0  /\  ( (coeff `  p ) `  (degAA `  x ) )  =  1 ) ) )
 
Theoremdgraaval 35708* Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (degAA `  A )  =  sup ( { d  e.  NN  |  E. p  e.  (
 (Poly `  QQ )  \  { 0p }
 ) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  ) )
 
Theoremdgraalem 35709* Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (degAA `  A )  e. 
 NN  /\  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg `  p )  =  (degAA `  A )  /\  ( p `  A )  =  0 ) ) )
 
Theoremdgraacl 35710 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
 
Theoremdgraaf 35711 Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |- degAA : AA --> NN
 
Theoremdgraaub 35712 Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  (
 ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0p ) 
 /\  ( A  e.  CC  /\  ( P `  A )  =  0
 ) )  ->  (degAA `  A )  <_  (deg `  P ) )
 
Theoremdgraa0p 35713 A rational polynomial of degree less than an algebraic number cannot be zero at that number unless it is the zero polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  (
 ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  ->  ( ( P `  A )  =  0  <->  P  =  0p ) )
 
Theoremmpaaeu 35714* An algebraic number has exactly one monic polynomial of the least degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  E! p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  ( p `
  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 ) )
 
Theoremmpaaval 35715* Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (minPolyAA `  A )  =  (
 iota_ p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 ) ) )
 
Theoremmpaalem 35716 Properties of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (minPolyAA `  A )  e.  (Poly `  QQ )  /\  ( (deg `  (minPolyAA `  A ) )  =  (degAA `  A )  /\  ( (minPolyAA `  A ) `  A )  =  0  /\  ( (coeff `  (minPolyAA `  A ) ) `  (degAA `  A ) )  =  1 ) ) )
 
Theoremmpaacl 35717 Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (minPolyAA `  A )  e.  (Poly `  QQ ) )
 
Theoremmpaadgr 35718 Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (deg `  (minPolyAA `  A ) )  =  (degAA `  A ) )
 
Theoremmpaaroot 35719 The minimal polynomial of an algebraic number has the number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (minPolyAA `  A ) `  A )  =  0
 )
 
Theoremmpaamn 35720 Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (coeff `  (minPolyAA `  A ) ) `  (degAA `  A ) )  =  1 )
 
21.23.46  Algebraic integers I
 
Syntaxcitgo 35721 Extend class notation with the integral-over predicate.
 class IntgOver
 
Syntaxcza 35722 Extend class notation with the class of algebraic integers.
 class
 
Definitiondf-itgo 35723* A complex number is said to be integral over a subset if it is the root of a monic polynomial with coefficients from the subset. This definition is typically not used for fields but it works there, see aaitgo 35726. This definition could work for subsets of an arbitrary ring with a more general definition of polynomials. TODO: use  Monic (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |- IntgOver  =  ( s  e.  ~P CC  |->  { x  e.  CC  |  E. p  e.  (Poly `  s ) ( ( p `  x )  =  0  /\  (
 (coeff `  p ) `  (deg `  p )
 )  =  1 ) } )
 
Definitiondf-za 35724 Define an algebraic integer as a complex number which is the root of a monic integer polynomial. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  =  (IntgOver `  ZZ )
 
Theoremitgoval 35725* Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  ( S  C_  CC  ->  (IntgOver `  S )  =  { x  e.  CC  |  E. p  e.  (Poly `  S ) ( ( p `
  x )  =  0  /\  ( (coeff `  p ) `  (deg `  p ) )  =  1 ) } )
 
Theoremaaitgo 35726 The standard algebraic numbers  AA are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  AA  =  (IntgOver `  QQ )
 
Theoremitgoss 35727 An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  (
 ( S  C_  T  /\  T  C_  CC )  ->  (IntgOver `  S )  C_  (IntgOver `  T )
 )
 
Theoremitgocn 35728 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  (IntgOver `  S )  C_  CC
 
Theoremcnsrexpcl 35729 Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  S  e.  (SubRing ` fld ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  NN0 )   =>    |-  ( ph  ->  ( X ^ Y )  e.  S )
 
Theoremfsumcnsrcl 35730* Finite sums are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  S  e.  (SubRing ` fld ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  S )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  S )
 
Theoremcnsrplycl 35731 Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  S  e.  (SubRing ` fld ) )   &    |-  ( ph  ->  P  e.  (Poly `  C ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  C  C_  S )   =>    |-  ( ph  ->  ( P `  X )  e.  S )
 
Theoremrgspnval 35732* Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  N  =  (RingSpan `  R ) )   &    |-  ( ph  ->  U  =  ( N `  A ) )   =>    |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R )  |  A  C_  t }
 )
 
Theoremrgspncl 35733 The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  N  =  (RingSpan `  R ) )   &    |-  ( ph  ->  U  =  ( N `  A ) )   =>    |-  ( ph  ->  U  e.  (SubRing `  R ) )
 
Theoremrgspnssid 35734 The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  N  =  (RingSpan `  R ) )   &    |-  ( ph  ->  U  =  ( N `  A ) )   =>    |-  ( ph  ->  A 
 C_  U )
 
Theoremrgspnmin 35735 The ring-span is contained in all subspaces which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  N  =  (RingSpan `  R ) )   &    |-  ( ph  ->  U  =  ( N `  A ) )   &    |-  ( ph  ->  S  e.  (SubRing `  R ) )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ph  ->  U  C_  S )
 
Theoremrgspnid 35736 The span of a subring is itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  A  e.  (SubRing `  R ) )   &    |-  ( ph  ->  S  =  ( (RingSpan `  R ) `  A ) )   =>    |-  ( ph  ->  S  =  A )
 
Theoremrngunsnply 35737* Adjoining one element to a ring results in a set of polynomial evaluations. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  B  e.  (SubRing ` fld ) )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  S  =  ( (RingSpan ` fld ) `  ( B  u.  { X }
 ) ) )   =>    |-  ( ph  ->  ( V  e.  S  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
 
Theoremflcidc 35738* Finite linear combinations with an indicator function. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( ph  ->  F  =  ( j  e.  S  |->  if ( j  =  K ,  1 ,  0 ) ) )   &    |-  ( ph  ->  S  e.  Fin )   &    |-  ( ph  ->  K  e.  S )   &    |-  ( ( ph  /\  i  e.  S ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ i  e.  S  ( ( F `  i )  x.  B )  =  [_ K  /  i ]_ B )
 
21.23.47  Endomorphism algebra
 
Syntaxcmend 35739 Syntax for module endomorphism algebra.
 class MEndo
 
Definitiondf-mend 35740* Define the endomorphism algebra of a module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |- MEndo  =  ( m  e.  _V  |->  [_ ( m LMHom  m )  /  b ]_ ( { <. (
 Base `  ndx ) ,  b >. ,  <. ( +g  ` 
 ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  oF (
 +g  `  m )
 y ) ) >. , 
 <. ( .r `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s
 `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m )
 ) ,  y  e.  b  |->  ( ( (
 Base `  m )  X.  { x } )  oF ( .s `  m ) y ) ) >. } ) )
 
Theoremalgstr 35741 Lemma to shorten proofs of algbase 35742 through algvsca 35746. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  A Struct  <. 1 ,  6
 >.
 
Theoremalgbase 35742 The base set of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  ( B  e.  V  ->  B  =  ( Base `  A ) )
 
Theoremalgaddg 35743 The additive operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .+  e.  V  ->  .+  =  ( +g  `  A ) )
 
Theoremalgmulr 35744 The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .X.  e.  V  -> 
 .X.  =  ( .r `  A ) )
 
Theoremalgsca 35745 The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  ( S  e.  V  ->  S  =  (Scalar `  A ) )
 
Theoremalgvsca 35746 The scalar product operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .x.  e.  V  ->  .x.  =  ( .s
 `  A ) )
 
Theoremmendval 35747* Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  B  =  ( M LMHom  M )   &    |-  .+  =  ( x  e.  B ,  y  e.  B  |->  ( x  oF ( +g  `  M ) y ) )   &    |-  .X. 
 =  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y ) )   &    |-  S  =  (Scalar `  M )   &    |-  .x.  =  ( x  e.  ( Base `  S ) ,  y  e.  B  |->  ( ( ( Base `  M )  X.  { x }
 )  oF ( .s `  M ) y ) )   =>    |-  ( M  e.  X  ->  (MEndo `  M )  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } ) )
 
Theoremmendbas 35748 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   =>    |-  ( M LMHom  M )  =  (
 Base `  A )
 
Theoremmendplusgfval 35749* Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( +g  `  A )  =  ( x  e.  B ,  y  e.  B  |->  ( x  oF  .+  y ) )
 
Theoremmendplusg 35750 A specific addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   &    |-  .+  =  ( +g  `  M )   &    |-  .+b  =  ( +g  `  A )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+b  Y )  =  ( X  oF  .+  Y ) )
 
Theoremmendmulrfval 35751* Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   =>    |-  ( .r `  A )  =  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y
 ) )
 
Theoremmendmulr 35752 A specific multiplication in the module endormoprhism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .r `  A )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  =  ( X  o.  Y ) )
 
Theoremmendsca 35753 The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  S  =  (Scalar `  M )   =>    |-  S  =  (Scalar `  A )
 
Theoremmendvscafval 35754* Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  .x.  =  ( .s `  M )   &    |-  B  =  ( Base `  A )   &    |-  S  =  (Scalar `  M )   &    |-  K  =  (
 Base `  S )   &    |-  E  =  ( Base `  M )   =>    |-  ( .s `  A )  =  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y )
 )
 
Theoremmendvsca 35755 A specific scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  .x.  =  ( .s `  M )   &    |-  B  =  ( Base `  A )   &    |-  S  =  (Scalar `  M )   &    |-  K  =  (
 Base `  S )   &    |-  E  =  ( Base `  M )   &    |-  .xb  =  ( .s `  A )   =>    |-  ( ( X  e.  K  /\  Y  e.  B )  ->  ( X  .xb  Y )  =  ( ( E  X.  { X } )  oF  .x.  Y ) )
 
Theoremmendring 35756 The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  (MEndo `  M )   =>    |-  ( M  e.  LMod  ->  A  e.  Ring )
 
Theoremmendlmod 35757 The module endomorphism algebra is a left module. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  S  =  (Scalar `  M )   =>    |-  (
 ( M  e.  LMod  /\  S  e.  CRing )  ->  A  e.  LMod )
 
Theoremmendassa 35758 The module endomorphism algebra is an algebra. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  S  =  (Scalar `  M )   =>    |-  (
 ( M  e.  LMod  /\  S  e.  CRing )  ->  A  e. AssAlg )
 
21.23.48  Subfields
 
Syntaxcsdrg 35759 Syntax for subfields (sub-division-rings).
 class SubDRing
 
Definitiondf-sdrg 35760* A sub-division-ring is a subset of a division ring's set which is a division ring under the induced operation. If the overring is commutative this is a field; no special consideration is made of the fields in the center of a skew field. (Contributed by Stefan O'Rear, 3-Oct-2015.)
 |- SubDRing  =  ( w  e.  DivRing  |->  { s  e.  (SubRing `  w )  |  ( ws  s )  e.  DivRing }
 )
 
Theoremissdrg 35761 Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
 |-  ( S  e.  (SubDRing `  R ) 
 <->  ( R  e.  DivRing  /\  S  e.  (SubRing `  R )  /\  ( Rs  S )  e.  DivRing ) )
 
Theoremissdrg2 35762* Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  I  =  ( invr `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( S  e.  (SubDRing `  R )  <->  ( R  e.  DivRing  /\  S  e.  (SubRing `  R )  /\  A. x  e.  ( S  \  {  .0.  } ) ( I `
  x )  e.  S ) )
 
Theoremacsfn1p 35763* Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  A. b  e.  Y  E  e.  X )  ->  { a  e.  ~P X  |  A. b  e.  ( a  i^i  Y ) E  e.  a }  e.  (ACS `  X ) )
 
Theoremsubrgacs 35764 Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e.  Ring  ->  (SubRing `  R )  e.  (ACS `  B ) )
 
Theoremsdrgacs 35765 Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e.  DivRing  ->  (SubDRing `  R )  e.  (ACS `  B ) )
 
Theoremcntzsdrg 35766 Centralizers in division rings/fields are subfields. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  B  =  ( Base `  R )   &    |-  M  =  (mulGrp `  R )   &    |-  Z  =  (Cntz `  M )   =>    |-  (
 ( R  e.  DivRing  /\  S  C_  B )  ->  ( Z `  S )  e.  (SubDRing `  R )
 )
 
21.23.49  Cyclic groups and order
 
Theoremidomrootle 35767* No element of an integral domain can have more than  N  N-th roots. (Contributed by Stefan O'Rear, 11-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  .^  =  (.g `  (mulGrp `  R )
 )   =>    |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( # `  { y  e.  B  |  ( N 
 .^  y )  =  X } )  <_  N )
 
Theoremidomodle 35768* Limit on the number of  N-th roots of unity in an integral domain. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  G  =  ( (mulGrp `  R )s  (Unit `  R ) )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  ( # `  { x  e.  B  |  ( O `
  x )  ||  N } )  <_  N )
 
Theoremfiuneneq 35769 Two finite sets of equal size have a union of the same size iff they were equal. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  (
 ( A  ~~  B  /\  A  e.  Fin )  ->  ( ( A  u.  B )  ~~  A  <->  A  =  B ) )
 
Theoremidomsubgmo 35770* The units of an integral domain have at most one subgroup of any single finite cardinality. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Revised by NM, 17-Jun-2017.)
 |-  G  =  ( (mulGrp `  R )s  (Unit `  R ) )   =>    |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  E* y  e.  (SubGrp `  G ) ( # `  y )  =  N )
 
Theoremproot1mul 35771 Any primitive  N-th root of unity is a multiple of any other. (Contributed by Stefan O'Rear, 2-Nov-2015.)
 |-  G  =  ( (mulGrp `  R )s  (Unit `  R ) )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G )
 )   =>    |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N } )  /\  Y  e.  ( `' O " { N } ) ) ) 
 ->  X  e.  ( K `
  { Y }
 ) )
 
Theoremhashgcdlem 35772* A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  A  =  { y  e.  (
 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }   &    |-  B  =  { z  e.  (
 0..^ M )  |  ( z  gcd  M )  =  N }   &    |-  F  =  ( x  e.  A  |->  ( x  x.  N ) )   =>    |-  ( ( M  e.  NN  /\  N  e.  NN  /\  N  ||  M )  ->  F : A -1-1-onto-> B )
 
Theoremhashgcdeq 35773* Number of initial positive integers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  (
 ( M  e.  NN  /\  N  e.  NN )  ->  ( # `  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }
 )  =  if ( N  ||  M ,  ( phi `  ( M  /  N ) ) ,  0 ) )
 
Theoremphisum 35774* The divisor sum identity of the totient function. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  ( N  e.  NN  ->  sum_
 d  e.  { x  e.  NN  |  x  ||  N }  ( phi `  d )  =  N )
 
Theoremproot1hash 35775 If an integral domain has a primitive  N-th root of unity, it has exactly  ( phi `  N ) of them. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  G  =  ( (mulGrp `  R )s  (Unit `  R ) )   &    |-  O  =  ( od `  G )   =>    |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N }
 ) )  ->  ( # `
  ( `' O " { N } )
 )  =  ( phi `  N ) )
 
Theoremproot1ex 35776 The complex field has primitive  N-th roots of unity for all  N. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  G  =  ( (mulGrp ` fld )s  ( CC  \  {
 0 } ) )   &    |-  O  =  ( od `  G )   =>    |-  ( N  e.  NN  ->  ( -u 1  ^c 
 ( 2  /  N ) )  e.  ( `' O " { N } ) )
 
21.23.50  Cyclotomic polynomials
 
Syntaxccytp 35777 Syntax for the sequence of cyclotomic polynomials.
 class CytP
 
Definitiondf-cytp 35778* The Nth cyclotomic polynomial is the polynomial which has as its zeros precisely the primitive Nth roots of unity. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- CytP  =  ( n  e.  NN  |->  ( (mulGrp `  (Poly1 ` fld ) )  gsumg  ( r  e.  ( `' ( od `  (
 (mulGrp ` fld )s  ( CC  \  {
 0 } ) ) ) " { n } )  |->  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
 ) ( (algSc `  (Poly1 ` fld ) ) `  r
 ) ) ) ) )
 
Theoremisdomn3 35779 Nonzero elements form a multiplicative submonoid of any domain. (Contributed by Stefan O'Rear, 11-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  (mulGrp `  R )   =>    |-  ( R  e. Domn  <->  ( R  e.  Ring  /\  ( B  \  {  .0.  } )  e.  (SubMnd `  U ) ) )
 
Theoremmon1pid 35780 Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  .1.  =  ( 1r `  P )   &    |-  M  =  (Monic1p `  R )   &    |-  D  =  ( deg1  `  R )   =>    |-  ( R  e. NzRing  ->  (  .1.  e.  M  /\  ( D `  .1.  )  =  0 ) )
 
Theoremmon1psubm 35781 Monic polynomials are a multiplicative submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  M  =  (Monic1p `
  R )   &    |-  U  =  (mulGrp `  P )   =>    |-  ( R  e. NzRing  ->  M  e.  (SubMnd `  U ) )
 
Theoremdeg1mhm 35782 Homomorphic property of the polynomial degree. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (Poly1 `  R )   &    |-  .0.  =  ( 0g `  P )   &    |-  Y  =  ( (mulGrp `  P )s  ( B  \  {  .0.  } ) )   &    |-  N  =  (flds  NN0 )   =>    |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } )
 )  e.  ( Y MndHom  N ) )
 
Theoremcytpfn 35783 Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- CytP  Fn  NN
 
Theoremcytpval 35784* Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  T  =  ( (mulGrp ` fld )s  ( CC  \  {
 0 } ) )   &    |-  O  =  ( od `  T )   &    |-  P  =  (Poly1 ` fld )   &    |-  X  =  (var1 ` fld )   &    |-  Q  =  (mulGrp `  P )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   =>    |-  ( N  e.  NN  ->  (CytP `  N )  =  ( Q  gsumg  ( r  e.  ( `' O " { N } )  |->  ( X 
 .-  ( A `  r ) ) ) ) )
 
21.23.51  Miscellaneous topology
 
Theoremfgraphopab 35785* Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( F : A --> B  ->  F  =  { <. a ,  b >.  |  (
 ( a  e.  A  /\  b  e.  B )  /\  ( F `  a )  =  b
 ) } )
 
Theoremfgraphxp 35786* Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( F : A --> B  ->  F  =  { x  e.  ( A  X.  B )  |  ( F `  ( 1st `  x ) )  =  ( 2nd `  x ) }
 )
 
Theoremhausgraph 35787 The graph of a continuous function into a Hausdorff space is closed. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  (
 ( K  e.  Haus  /\  F  e.  ( J  Cn  K ) ) 
 ->  F  e.  ( Clsd `  ( J  tX  K ) ) )
 
Syntaxctopsep 35788 The class of separable toplogies.
 class TopSep
 
Syntaxctoplnd 35789 The class of Lindelöf toplogies.
 class TopLnd
 
Definitiondf-topsep 35790* A topology is separable iff it has a countable dense subset. (Contributed by Stefan O'Rear, 8-Jan-2015.)
 |- TopSep  =  {
 j  e.  Top  |  E. x  e.  ~P  U. j ( x  ~<_  om 
 /\  ( ( cls `  j ) `  x )  =  U. j ) }
 
Definitiondf-toplnd 35791* A topology is Lindelöf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.)
 |- TopLnd  =  { x  e.  Top  |  A. y  e.  ~P  x ( U. x  =  U. y  ->  E. z  e.  ~P  x ( z  ~<_  om 
 /\  U. x  =  U. z ) ) }
 
21.24  Mathbox for Jon Pennant
 
Theoremioounsn 35792 The closure of the upper end of an open real interval. (Contributed by Jon Pennant, 8-Jun-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  (
 ( A (,) B )  u.  { B }
 )  =  ( A (,] B ) )
 
Theoremiocunico 35793 Split an open interval into two pieces at point B, Co-author TA. (Contributed by Jon Pennant, 8-Jun-2019.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  ( ( A (,] B )  u.  ( B [,) C ) )  =  ( A (,) C ) )
 
Theoremiocinico 35794 The intersection of two sets that meet at a point is that point. (Contributed by Jon Pennant, 12-Jun-2019.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  ( ( A (,] B )  i^i  ( B [,) C ) )  =  { B }
 )
 
Theoremiocmbl 35795 An open-below, closed-above real interval is measurable. (Contributed by Jon Pennant, 12-Jun-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  e.  dom  vol )
 
Theoremcnioobibld 35796* A bounded, continuous function on an open bounded interval is integrable. The function must be bounded. For a counterexample, consider  F  =  ( x  e.  ( 0 (,) 1 )  |->  ( 1  /  x ) ). See cniccibl 22675 for closed bounded intervals. (Contributed by Jon Pennant, 31-May-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  dom  F ( abs `  ( F `  y ) )  <_  x )   =>    |-  ( ph  ->  F  e.  L^1 )
 
Theoremitgpowd 35797* The integral of a monomial on a closed bounded interval of the real line. Co-authors TA and MC. (Contributed by Jon Pennant, 31-May-2019.) (Revised by Thierry Arnoux, 14-Jun-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  S. ( A [,] B ) ( x ^ N )  _d x  =  ( ( ( B ^
 ( N  +  1 ) )  -  ( A ^ ( N  +  1 ) ) ) 
 /  ( N  +  1 ) ) )
 
Theoremarearect 35798 The area of a rectangle whose sides are parallel to the coordinate axes in  ( RR  X.  RR ) is its width multiplied by its height. (Contributed by Jon Pennant, 19-Mar-2019.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  D  e.  RR   &    |-  A  <_  B   &    |-  C  <_  D   &    |-  S  =  ( ( A [,] B )  X.  ( C [,] D ) )   =>    |-  (area `  S )  =  ( ( B  -  A )  x.  ( D  -  C ) )
 
Theoremareaquad 35799* The area of a quadrilateral with two sides which are parallel to the y-axis in  ( RR  X.  RR ) is its width multiplied by the average height of its higher edge minus the average height of its lower edge. Co-author TA. (Contributed by Jon Pennant, 31-May-2019.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  D  e.  RR   &    |-  E  e.  RR   &    |-  F  e.  RR   &    |-  A  <  B   &    |-  C  <_  E   &    |-  D  <_  F   &    |-  U  =  ( C  +  ( ( ( x  -  A )  /  ( B  -  A ) )  x.  ( D  -  C ) ) )   &    |-  V  =  ( E  +  (
 ( ( x  -  A )  /  ( B  -  A ) )  x.  ( F  -  E ) ) )   &    |-  S  =  { <. x ,  y >.  |  ( x  e.  ( A [,] B )  /\  y  e.  ( U [,] V ) ) }   =>    |-  (area `  S )  =  ( (
 ( ( F  +  E )  /  2
 )  -  ( ( D  +  C ) 
 /  2 ) )  x.  ( B  -  A ) )
 
21.25  Mathbox for Richard Penner
 
21.25.1  Short Studies
 
21.25.1.1  Additional work on conditional logical operator
 
Theoremifpan123g 35800 Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  ta )  /\ if- ( ps ,  th ,  et ) )  <->  ( ( ( -.  ph  \/  ch )  /\  ( ph  \/  ta ) )  /\  ( ( -.  ps  \/  th )  /\  ( ps  \/  et ) ) ) )
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