HomeHome Metamath Proof Explorer
Theorem List (p. 269 of 313)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21423)
  Hilbert Space Explorer  Hilbert Space Explorer
(21424-22946)
  Users' Mathboxes  Users' Mathboxes
(22947-31284)
 

Theorem List for Metamath Proof Explorer - 26801-26900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmatplusg 26801 The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( +g  `  G )  =  ( +g  `  A ) )
 
Theoremmatsca 26802 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  (Scalar `  G )  =  (Scalar `  A )
 )
 
Theoremmatvsca 26803 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( .s `  G )  =  ( .s `  A ) )
 
Theoremmat0 26804 The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( 0g `  G )  =  ( 0g `  A ) )
 
Theoremmatinvg 26805 The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( inv g `  G )  =  ( inv g `  A ) )
 
Theoremmatsca2 26806 The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  R  =  (Scalar `  A ) )
 
Theoremmatbas2 26807 The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  K  =  ( Base `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( K  ^m  ( N  X.  N ) )  =  ( Base `  A ) )
 
Theoremmatbas2i 26808 A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  K  =  ( Base `  R )   &    |-  B  =  (
 Base `  A )   =>    |-  ( M  e.  B  ->  M  e.  ( K  ^m  ( N  X.  N ) ) )
 
Theoremmatplusg2 26809 Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  .+b  =  ( +g  `  A )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+b  Y )  =  ( X  o F  .+  Y ) )
 
Theoremmatvsca2 26810 Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  K  =  (
 Base `  R )   &    |-  .x.  =  ( .s `  A )   &    |-  .X. 
 =  ( .r `  R )   &    |-  C  =  ( N  X.  N )   =>    |-  ( ( X  e.  K  /\  Y  e.  B )  ->  ( X  .x.  Y )  =  ( ( C  X.  { X } )  o F  .X.  Y ) )
 
Theoremmatlmod 26811 The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
 
Theoremmatrng 26812 Existence of the matrix ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
 
Theoremmatassa 26813 Existence of the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e. AssAlg )
 
Theoremmat1 26814* Value of an identity matrix. (Contributed by Stefan O'Rear, 7-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  ( 1r `  A )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  .1.  ,  .0.  ) ) )
 
16.16.58  The determinant
 
Syntaxcmdat 26815 Syntax for the matrix determinant function.
 class maDet
 
Syntaxcmadu 26816 Syntax for the matrix adjugate function.
 class maAdju
 
Definitiondf-mdet 26817* Determinant of a square matrix... (Contributed by Stefan O'Rear, 9-Sep-2015.)
 |- maDet  =  ( n  e.  _V ,  r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) )  |->  ( r  gsumg  ( p  e.  ( Base `  ( SymGrp `  n ) )  |->  ( ( ( ZRHom `  r
 ) `  ( (pmSgn `  n ) `  p ) ) ( .r
 `  r ) ( (mulGrp `  r )  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
 
Definitiondf-madu 26818* Define the adjunct (matrix of cofactors) of a square matrix. This definition gives the standard cofactors, however the internal minors are not the standard minors. (Contributed by Stefan O'Rear, 7-Sep-2015.)
 |- maAdju  =  ( n  e.  _V ,  r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) )  |->  ( i  e.  n ,  j  e.  n  |->  ( if ( i  =  j ,  ( 1r
 `  r ) ,  ( ( inv g `  r ) `  ( 1r `  r ) ) ) ( .r `  r ) ( ( ( n  \  {
 i } ) maDet  r
 ) `  ( k  e.  ( n  \  {
 i } ) ,  l  e.  ( n 
 \  { i }
 )  |->  ( if (
 k  =  j ,  i ,  k ) m l ) ) ) ) ) ) )
 
Theoremmdetfval 26819* First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.)
 |-  D  =  ( N maDet  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  ( Base `  ( SymGrp `  N ) )   &    |-  Y  =  ( ZRHom `  R )   &    |-  S  =  (pmSgn `  N )   &    |-  .x.  =  ( .r `  R )   &    |-  U  =  (mulGrp `  R )   =>    |-  D  =  ( m  e.  B  |->  ( R 
 gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
 
Theoremmdetleib 26820* Full substitution of our determinant definition (also known as Leibniz' Formula). (Contributed by Stefan O'Rear, 3-Oct-2015.)
 |-  D  =  ( N maDet  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  ( Base `  ( SymGrp `  N ) )   &    |-  Y  =  ( ZRHom `  R )   &    |-  S  =  (pmSgn `  N )   &    |-  .x.  =  ( .r `  R )   &    |-  U  =  (mulGrp `  R )   =>    |-  ( M  e.  B  ->  ( D `  M )  =  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
  p ) ) 
 .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) ) )
 
16.16.59  Endomorphism algebra
 
Syntaxcmend 26821 Syntax for module endomorphism algebra.
 class MEndo
 
Definitiondf-mend 26822* 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  o F (
 +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 } )  o F ( .s `  m ) y ) ) >. } ) )
 
Theoremmendval 26823* 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  o F ( +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 }
 )  o F ( .s `  M ) y ) )   =>    |-  ( M  e.  X  ->  (MEndo `  M )  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } ) )
 
Theoremmendbas 26824 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   =>    |-  ( M LMHom  M )  =  (
 Base `  A )
 
Theoremmendplusgfval 26825* 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  o F  .+  y ) )
 
Theoremmendplusg 26826 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  o F  .+  Y ) )
 
Theoremmendmulrfval 26827* 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 26828 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 26829 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 26830* 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 } )  o F  .x.  y )
 )
 
Theoremmendvsca 26831 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 } )  o F  .x.  Y ) )
 
Theoremmendrng 26832 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 26833 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 26834 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 )
 
16.16.60  Subfields
 
Syntaxcsdrg 26835 Syntax for subfields (sub-division-rings).
 class SubDRing
 
Definitiondf-sdrg 26836* 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 26837 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 26838* 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 26839* 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 26840 Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e.  Ring  ->  (SubRing `  R )  e.  (ACS `  B ) )
 
Theoremsdrgacs 26841 Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e.  DivRing  ->  (SubDRing `  R )  e.  (ACS `  B ) )
 
Theoremcntzsdrg 26842 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 )
 )
 
16.16.61  Cyclic groups and order
 
Theoremidomrootle 26843* 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 26844* 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 26845 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 26846* The units of an integral domain have at most one subgroup of any single finite cardinality. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  G  =  ( (mulGrp `  R )s  (Unit `  R ) )   =>    |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  E* y ( y  e.  (SubGrp `  G )  /\  ( # `  y
 )  =  N ) )
 
Theoremproot1mul 26847 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 26848* 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 26849* Number of initial natural numbers 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 26850* 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 26851 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 26852 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 } ) )
 
16.16.62  Cyclotomic polynomials
 
Syntaxccytp 26853 Syntax for the sequence of cyclotomic polynomials.
 class CytP
 
Definitiondf-cytp 26854* 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 26855 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 26856 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 26857 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 26858 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 26859 Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- CytP  Fn  NN
 
Theoremcytpval 26860* 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 ) ) ) ) )
 
16.16.63  Miscellaneous topology
 
Theoremfgraphopab 26861* Express a function as a subset of the cross 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 26862* Express a function as a subset of the cross product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( F : A --> B  ->  F  =  { x  e.  ( A  X.  B )  |  ( F `  ( 1st `  x ) )  =  ( 2nd `  x ) }
 )
 
Theoremhausgraph 26863 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 26864 The class of separable toplogies.
 class TopSep
 
Syntaxctoplnd 26865 The class of Lindelöf toplogies.
 class TopLnd
 
Definitiondf-topsep 26866* 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 26867* 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 ) ) }
 
16.17  Mathbox for Steve Rodriguez
 
16.17.1  Miscellanea
 
Theoremiso0 26868 The empty set is an  R ,  S isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
 |-  (/)  Isom  R ,  S  ( (/) ,  (/) )
 
Theoremssrecnpr 26869  RR is a subset of both  RR and  CC. (Contributed by Steve Rodriguez, 22-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  RR  C_  S )
 
Theoremseff 26870 Let set  S be the reals or complexes. Then the exponential function restricted to  S is a mapping from  S to  S. (Contributed by Steve Rodriguez, 6-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   =>    |-  ( ph  ->  ( exp  |`  S ) : S --> S )
 
Theoremsblpnf 26871 The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 17881. (Contributed by Steve Rodriguez, 8-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  D  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )   =>    |-  ( ( ph  /\  P  e.  S ) 
 ->  ( P ( ball `  D )  +oo )  =  S )
 
16.17.2  Function operations
 
Theoremcaofcan 26872* Transfer a cancellation law like mulcan 9338 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> T )   &    |-  ( ph  ->  G : A --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y )  =  ( x R z )  <->  y  =  z
 ) )   =>    |-  ( ph  ->  (
 ( F  o F R G )  =  ( F  o F R H )  <->  G  =  H ) )
 
Theoremofsubid 26873 Function analog of subid 9000. (Contributed by Steve Rodriguez, 5-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC )  ->  ( F  o F  -  F )  =  ( A  X.  { 0 } ) )
 
Theoremofmul12 26874 Function analog of mul12 8911. (Contributed by Steve Rodriguez, 13-Nov-2015.)
 |-  (
 ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A --> CC  /\  H : A --> CC )
 )  ->  ( F  o F  x.  ( G  o F  x.  H ) )  =  ( G  o F  x.  ( F  o F  x.  H ) ) )
 
Theoremofdivrec 26875 Function analog of divrec 9373, a division analog of ofnegsub 9677. (Contributed by Steve Rodriguez, 3-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC  \  { 0 } )
 )  ->  ( F  o F  x.  (
 ( A  X.  {
 1 } )  o F  /  G ) )  =  ( F  o F  /  G ) )
 
Theoremofdivcan4 26876 Function analog of divcan4 9382. (Contributed by Steve Rodriguez, 4-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC  \  { 0 } )
 )  ->  ( ( F  o F  x.  G )  o F  /  G )  =  F )
 
Theoremofdivdiv2 26877 Function analog of divdiv2 9405. (Contributed by Steve Rodriguez, 23-Nov-2015.)
 |-  (
 ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A --> ( CC  \  { 0 } )  /\  H : A --> ( CC  \  { 0 } )
 ) )  ->  ( F  o F  /  ( G  o F  /  H ) )  =  (
 ( F  o F  x.  H )  o F  /  G ) )
 
16.17.3  Calculus
 
Theoremlhe4.4ex1a 26878 Example of the Fundamental Theorem of Calculus, part two (ftc2 19318):  S. ( 1 (,) 2 ) ( ( x ^ 2 )  -  3 )  _d x  =  -u ( 2  /  3
). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 19318 as simply the "Fundamental Theorem of Calculus", then ftc1 19316 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)
 |-  S. ( 1 (,) 2
 ) ( ( x ^ 2 )  -  3 )  _d x  =  -u ( 2  / 
 3 )
 
Theoremdvsconst 26879 Derivative of a constant function on the reals or complexes. The function may return a complex  A even if  S is  RR. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  (
 ( S  e.  { RR ,  CC }  /\  A  e.  CC )  ->  ( S  _D  ( S  X.  { A }
 ) )  =  ( S  X.  { 0 } ) )
 
Theoremdvsid 26880 Derivative of the identity function on the reals or complexes. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  (  _I  |`  S ) )  =  ( S  X.  { 1 } ) )
 
Theoremdvsef 26881 Derivative of the exponential function on the reals or complexes. (Contributed by Steve Rodriguez, 12-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( exp  |`  S ) )  =  ( exp  |`  S ) )
 
Theoremexpgrowthi 26882* Exponential growth and decay model. See expgrowth 26884 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  K  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  Y  =  ( t  e.  S  |->  ( C  x.  ( exp `  ( K  x.  t
 ) ) ) )   =>    |-  ( ph  ->  ( S  _D  Y )  =  ( ( S  X.  { K } )  o F  x.  Y ) )
 
Theoremdvconstbi 26883* The derivative of a function on  S is zero iff it is a constant function. Roughly a biconditional  S analog of dvconst 19193 and dveq0 19274. Corresponds to integration formula " S. 0  _d x  =  C " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  Y : S --> CC )   &    |-  ( ph  ->  dom  (  S  _D  Y )  =  S )   =>    |-  ( ph  ->  ( ( S  _D  Y )  =  ( S  X.  {
 0 } )  <->  E. c  e.  CC  Y  =  ( S  X.  { c } )
 ) )
 
Theoremexpgrowth 26884* Exponential growth and decay model. The derivative of a function y of variable t equals a constant k times y itself, iff y equals some constant C times the exponential of kt. This theorem and expgrowthi 26882 illustrate one of the simplest and most crucial classes of differential equations, equations that relate functions to their derivatives.

Section 6.3 of [Strang] p. 242 calls y' = ky "the most important differential equation in applied mathematics". In the field of population ecology it is known as the Malthusian growth model or exponential law, and C, k, and t correspond to initial population size, growth rate, and time respectively (https://en.wikipedia.org/wiki/Malthusian_growth_model); and in finance, the model appears in a similar role in continuous compounding with C as the initial amount of money. In exponential decay models, k is often expressed as the negative of a positive constant λ.

Here y' is given as  ( S  _D  Y
), C as  c, and ky as  ( ( S  X.  { K }
)  o F  x.  Y ).  ( S  X.  { K } ) is the constant function that maps any real or complex input to k and  o F  x. is multiplication as a function operation.

The leftward direction of the biconditional is as given in http://www.saylor.org/site/wp-content/uploads/2011/06/MA221-2.1.1.pdf pp. 1-2, which also notes the reverse direction ("While we will not prove this here, it turns out that these are the only functions that satisfy this equation."). The rightward direction is Theorem 5.1 of [LarsonHostetlerEdwards] p. 375 (which notes " C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0."); its proof here closely follows the proof of y' = y in https://proofwiki.org/wiki/Exponential_Growth_Equation/Special_Case.

Statements for this and expgrowthi 26882 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  K  e.  CC )   &    |-  ( ph  ->  Y : S --> CC )   &    |-  ( ph  ->  dom  (  S  _D  Y )  =  S )   =>    |-  ( ph  ->  (
 ( S  _D  Y )  =  ( ( S  X.  { K }
 )  o F  x.  Y )  <->  E. c  e.  CC  Y  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t ) ) ) ) ) )
 
16.18  Mathbox for Andrew Salmon
 
16.18.1  Principia Mathematica * 10
 
Theorempm10.12 26885* Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x ( ph  \/  ps )  ->  ( ph  \/  A. x ps )
 )
 
Theorempm10.14 26886 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  (
 ( A. x ph  /\  A. x ps )  ->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps )
 )
 
Theorempm10.251 26887 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x  -.  ph  ->  -. 
 A. x ph )
 
Theorempm10.252 26888 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( -.  E. x ph  <->  A. x  -.  ph )
 
Theorempm10.253 26889 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( -.  A. x ph  <->  E. x  -.  ph )
 
Theoremalbitr 26890 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ( ph  <->  ps )  /\  A. x ( ps  <->  ch ) )  ->  A. x ( ph  <->  ch ) )
 
Theorempm10.42 26891 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  (
 ( E. x ph  \/  E. x ps )  <->  E. x ( ph  \/  ps ) )
 
Theorempm10.52 26892* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x ph  ->  ( A. x ( ph  ->  ps )  <->  ps ) )
 
Theorempm10.53 26893 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x ph  ->  A. x ( ph  ->  ps ) )
 
Theorempm10.541 26894* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ch  \/  ps )
 ) 
 <->  ( ch  \/  A. x ( ph  ->  ps ) ) )
 
Theorempm10.542 26895* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ch  ->  ps )
 ) 
 <->  ( ch  ->  A. x ( ph  ->  ps )
 ) )
 
Theorempm10.55 26896 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( E. x (
 ph  /\  ps )  /\  A. x ( ph  ->  ps ) )  <->  ( E. x ph 
 /\  A. x ( ph  ->  ps ) ) )
 
Theorempm10.56 26897 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ( ph  ->  ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x ( ps  /\  ch )
 )
 
Theorempm10.57 26898 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ps  \/  ch )
 )  ->  ( A. x ( ph  ->  ps )  \/  E. x ( ph  /\  ch )
 ) )
 
16.18.2  Principia Mathematica * 11
 
Theorem2alanimi 26899 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( A. x A. y ph  /\  A. x A. y ps )  ->  A. x A. y ch )
 
Theorem2al2imi 26900 Removes two universal qunatifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x A. y ph  ->  ( A. x A. y ps  ->  A. x A. y ch ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31284
  Copyright terms: Public domain < Previous  Next >