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Theorem List for Metamath Proof Explorer - 36001-36100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnsrexpcl 36001 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 36002* 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 36003 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 36004* 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 36005 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 36006 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 36007 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 36008 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 36009* 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 36010* 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 36011 Syntax for module endomorphism algebra.
 class MEndo
 
Definitiondf-mend 36012* 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 36013 Lemma to shorten proofs of algbase 36014 through algvsca 36018. (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 36014 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 36015 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 36016 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 36017 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 36018 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 36019* 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 36020 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   =>    |-  ( M LMHom  M )  =  (
 Base `  A )
 
Theoremmendplusgfval 36021* 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 36022 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 36023* 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 36024 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 36025 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 36026* 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 36027 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 36028 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 36029 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 36030 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 36031 Syntax for subfields (sub-division-rings).
 class SubDRing
 
Definitiondf-sdrg 36032* 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 36033 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 36034* 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 36035* 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 36036 Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e.  Ring  ->  (SubRing `  R )  e.  (ACS `  B ) )
 
Theoremsdrgacs 36037 Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e.  DivRing  ->  (SubDRing `  R )  e.  (ACS `  B ) )
 
Theoremcntzsdrg 36038 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 36039* 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 36040* 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 36041 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 36042* 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 36043 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 36044* 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 36045* 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 36046* 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 36047 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 36048 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 36049 Syntax for the sequence of cyclotomic polynomials.
 class CytP
 
Definitiondf-cytp 36050* 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 36051 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 36052 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 36053 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 36054 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 36055 Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- CytP  Fn  NN
 
Theoremcytpval 36056* 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 36057* 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 36058* 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 36059 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 36060 The class of separable toplogies.
 class TopSep
 
Syntaxctoplnd 36061 The class of Lindelöf toplogies.
 class TopLnd
 
Definitiondf-topsep 36062* 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 36063* 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 36064 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 36065 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 36066 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 36067 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 36068* 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 22796 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 36069* 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 36070 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 36071* 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 36072 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 ) ) ) )
 
Theoremifpan23 36073 Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
 |-  (
 (if- ( ph ,  ps ,  ch )  /\ if- (
 ph ,  th ,  ta ) )  <-> if- ( ph ,  ( ps  /\  th ) ,  ( ch  /\  ta ) ) )
 
Theoremifpdfor2 36074 Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  \/  ps )  <-> if- (
 ph ,  ph ,  ps ) )
 
Theoremifporcor 36075 Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)
 |-  (if- ( ph ,  ph ,  ps )  <-> if- ( ps ,  ps ,  ph ) )
 
Theoremifpdfan2 36076 Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ph  /\  ps )  <-> if- (
 ph ,  ps ,  ph ) )
 
Theoremifpancor 36077 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  ps ,  ph )  <-> if- ( ps ,  ph ,  ps ) )
 
Theoremifpdfor 36078 Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  \/  ps )  <-> if- (
 ph , T.  ,  ps ) )
 
Theoremifpdfan 36079 Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  /\  ps )  <-> if- (
 ph ,  ps , F.  ) )
 
Theoremifpbi2 36080 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ch ,  ph ,  th )  <-> if- ( ch ,  ps ,  th ) ) )
 
Theoremifpbi3 36081 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ch ,  th ,  ph )  <-> if- ( ch ,  th ,  ps ) ) )
 
Theoremifpim1 36082 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( -.  ph , T.  ,  ps ) )
 
Theoremifpnot 36083 Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  ( -.  ph  <-> if- ( ph , F.  , T.  ) )
 
Theoremifpid2 36084 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  ( ph 
 <-> if- ( ph , T.  , F.  ) )
 
Theoremifpim2 36085 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( ps , T.  ,  -.  ph ) )
 
Theoremifpbi23 36086 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ta ,  ph ,  ch )  <-> if- ( ta ,  ps ,  th ) ) )
 
Theoremifpdfbi 36087 Define biimplication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  <->  ps )  <-> if- ( ph ,  ps ,  -.  ps ) )
 
Theoremifpbiidcor 36088 Restatement of biid 239. (Contributed by RP, 25-Apr-2020.)
 |- if- ( ph ,  ph ,  -.  ph )
 
Theoremifpbicor 36089 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  ps ,  -.  ps )  <-> if- ( ps ,  ph ,  -.  ph ) )
 
Theoremifpxorcor 36090 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  -.  ps ,  ps )  <-> if- ( ps ,  -.  ph ,  ph ) )
 
Theoremifpbi1 36091 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th ) ) )
 
Theoremifpnot23 36092 Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
 |-  ( -. if- ( ph ,  ps ,  ch )  <-> if- ( ph ,  -.  ps ,  -.  ch )
 )
 
Theoremifpnotnotb 36093 Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  -.  ps ,  -.  ch )  <->  -. if- ( ph ,  ps ,  ch ) )
 
Theoremifpnorcor 36094 Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  -.  ph ,  -.  ps )  <-> if- ( ps ,  -.  ps ,  -.  ph )
 )
 
Theoremifpnancor 36095 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  -.  ps ,  -.  ph )  <-> if- ( ps ,  -.  ph ,  -.  ps )
 )
 
Theoremifpnot23b 36096 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -. if- ( ph ,  -.  ps ,  ch )  <-> if- ( ph ,  ps ,  -.  ch ) )
 
Theoremifpbiidcor2 36097 Restatement of biid 239. (Contributed by RP, 25-Apr-2020.)
 |-  -. if- (
 ph ,  -.  ph ,  ph )
 
Theoremifpnot23c 36098 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -. if- ( ph ,  ps ,  -.  ch )  <-> if- ( ph ,  -.  ps ,  ch ) )
 
Theoremifpnot23d 36099 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -. if- ( ph ,  -.  ps ,  -.  ch )  <-> if- (
 ph ,  ps ,  ch ) )
 
Theoremifpdfnan 36100 Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  -/\  ps )  <-> if- (
 ph ,  -.  ps , T.  ) )
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