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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembfp 25901* Banach fixed point theorem, also known as contraction mapping theorem. A contraction on a complete metric space has a unique fixed point. We show existence in the lemmas, and uniqueness here - if  F has two fixed points, then the distance between them is less than  K times itself, a contradiction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
 |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  X  =/=  (/) )   &    |-  ( ph  ->  K  e.  RR+ )   &    |-  ( ph  ->  K  <  1 )   &    |-  ( ph  ->  F : X --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( ( F `  x ) D ( F `  y ) )  <_  ( K  x.  ( x D y ) ) )   =>    |-  ( ph  ->  E! z  e.  X  ( F `  z )  =  z )
 
16.14.12  Euclidean space
 
Syntaxcrrn 25902 Extend class notation with the n-dimensional Euclidean space.
 class  Rn
 
Definitiondf-rrn 25903* Define n-dimensional Euclidean space as a metric space with the standard Euclidean norm given by the quadratic mean. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  Rn  =  ( i  e.  Fin  |->  ( x  e.  ( RR  ^m  i ) ,  y  e.  ( RR 
 ^m  i )  |->  ( sqr `  sum_ k  e.  i  ( ( ( x `  k )  -  ( y `  k ) ) ^
 2 ) ) ) )
 
Theoremrrnval 25904* The n-dimensional Euclidean space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   =>    |-  ( I  e.  Fin  ->  ( Rn `  I )  =  ( x  e.  X ,  y  e.  X  |->  ( sqr `  sum_ k  e.  I  ( (
 ( x `  k
 )  -  ( y `
  k ) ) ^ 2 ) ) ) )
 
Theoremrrnmval 25905* The value of the Euclidean metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   =>    |-  ( ( I  e. 
 Fin  /\  F  e.  X  /\  G  e.  X ) 
 ->  ( F ( Rn `  I ) G )  =  ( sqr `  sum_ k  e.  I  ( (
 ( F `  k
 )  -  ( G `
  k ) ) ^ 2 ) ) )
 
Theoremrrnmet 25906 Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
 |-  X  =  ( RR  ^m  I
 )   =>    |-  ( I  e.  Fin  ->  ( Rn `  I )  e.  ( Met `  X ) )
 
Theoremrrndstprj1 25907 The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   &    |-  M  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( ( I  e.  Fin  /\  A  e.  I )  /\  ( F  e.  X  /\  G  e.  X ) )  ->  ( ( F `  A ) M ( G `  A ) )  <_  ( F ( Rn `  I ) G ) )
 
Theoremrrndstprj2 25908* Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 25907 can be used to show that the supremum norm and Euclidean norm are equivalent. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   &    |-  M  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( ( I  e.  ( Fin  \  { (/)
 } )  /\  F  e.  X  /\  G  e.  X )  /\  ( R  e.  RR+  /\  A. n  e.  I  ( ( F `  n ) M ( G `  n ) )  <  R ) )  ->  ( F ( Rn `  I ) G )  <  ( R  x.  ( sqr `  ( # `
  I ) ) ) )
 
Theoremrrncmslem 25909* Lemma for rrncms 25910. (Contributed by Jeff Madsen, 6-Jun-2014.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   &    |-  M  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   &    |-  J  =  (
 MetOpen `  ( Rn `  I
 ) )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ph  ->  F  e.  ( Cau `  ( Rn `  I
 ) ) )   &    |-  ( ph  ->  F : NN --> X )   &    |-  P  =  ( m  e.  I  |->  (  ~~>  `  ( t  e.  NN  |->  ( ( F `  t ) `  m ) ) ) )   =>    |-  ( ph  ->  F  e.  dom  ( ~~> t `  J ) )
 
Theoremrrncms 25910 Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   =>    |-  ( I  e.  Fin  ->  ( Rn `  I )  e.  ( CMet `  X ) )
 
Theoremrepwsmet 25911 The supremum metric on  RR ^ I is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
 |-  Y  =  ( (flds  RR )  ^s  I )   &    |-  D  =  (
 dist `  Y )   &    |-  X  =  ( RR  ^m  I
 )   =>    |-  ( I  e.  Fin  ->  D  e.  ( Met `  X ) )
 
Theoremrrnequiv 25912 The supremum metric on  RR ^ I is equivalent to the  Rn metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
 |-  Y  =  ( (flds  RR )  ^s  I )   &    |-  D  =  (
 dist `  Y )   &    |-  X  =  ( RR  ^m  I
 )   &    |-  ( ph  ->  I  e.  Fin )   =>    |-  ( ( ph  /\  ( F  e.  X  /\  G  e.  X )
 )  ->  ( ( F D G )  <_  ( F ( Rn `  I
 ) G )  /\  ( F ( Rn `  I
 ) G )  <_  ( ( sqr `  ( # `
  I ) )  x.  ( F D G ) ) ) )
 
Theoremrrntotbnd 25913 A set in Euclidean space is totally bounded iff its is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   &    |-  M  =  ( ( Rn `  I )  |`  ( Y  X.  Y ) )   =>    |-  ( I  e.  Fin  ->  ( M  e.  ( TotBnd `
  Y )  <->  M  e.  ( Bnd `  Y ) ) )
 
Theoremrrnheibor 25914 Heine-Borel theorem for Euclidean space. A subset of Euclidean space is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   &    |-  M  =  ( ( Rn `  I )  |`  ( Y  X.  Y ) )   &    |-  T  =  (
 MetOpen `  M )   &    |-  U  =  ( MetOpen `  ( Rn `  I ) )   =>    |-  ( ( I  e.  Fin  /\  Y  C_  X )  ->  ( T  e.  Comp  <->  ( Y  e.  ( Clsd `  U )  /\  M  e.  ( Bnd `  Y ) ) ) )
 
16.14.13  Intervals (continued)
 
Theoremismrer1 25915* An isometry between  RR and  RR ^ 1. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  R  =  ( ( abs  o.  -  )  |`  ( RR  X. 
 RR ) )   &    |-  F  =  ( x  e.  RR  |->  ( { A }  X.  { x } ) )   =>    |-  ( A  e.  V  ->  F  e.  ( R 
 Ismty  ( Rn `  { A } ) ) )
 
Theoremreheibor 25916 Heine-Borel theorem for real numbers. A subset of  RR is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  M  =  ( ( abs  o.  -  )  |`  ( Y  X.  Y ) )   &    |-  T  =  ( MetOpen `  M )   &    |-  U  =  ( topGen `  ran  (,) )   =>    |-  ( Y  C_  RR  ->  ( T  e.  Comp  <->  ( Y  e.  ( Clsd `  U )  /\  M  e.  ( Bnd `  Y ) ) ) )
 
Theoremiccbnd 25917 A closed interval in  RR is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Sep-2015.)
 |-  J  =  ( A [,] B )   &    |-  M  =  ( ( abs  o.  -  )  |`  ( J  X.  J ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  M  e.  ( Bnd `  J ) )
 
TheoremicccmpALT 25918 A closed interval in  RR is compact. Alternate proof of icccmp 18278 using the Heine-Borel theorem heibor 25898. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Aug-2014.)
 |-  J  =  ( A [,] B )   &    |-  M  =  ( ( abs  o.  -  )  |`  ( J  X.  J ) )   &    |-  T  =  (
 MetOpen `  M )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  T  e.  Comp )
 
16.14.14  Groups and related structures
 
Theoremexidcl 25919 Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  X  =  ran  G   =>    |-  ( ( G  e.  ( Magma  i^i  ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
 
Theoremexidreslem 25920* Lemma for exidres 25921 and exidresid 25922. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  H  =  ( G  |`  ( Y  X.  Y ) )   =>    |-  ( ( G  e.  ( Magma  i^i  ExId  ) 
 /\  Y  C_  X  /\  U  e.  Y ) 
 ->  ( U  e.  dom  dom 
 H  /\  A. x  e. 
 dom  dom  H ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
 
Theoremexidres 25921 The restriction of a binary operation with identity to a subset containing the identity has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  H  =  ( G  |`  ( Y  X.  Y ) )   =>    |-  ( ( G  e.  ( Magma  i^i  ExId  ) 
 /\  Y  C_  X  /\  U  e.  Y ) 
 ->  H  e.  ExId  )
 
Theoremexidresid 25922 The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  H  =  ( G  |`  ( Y  X.  Y ) )   =>    |-  ( ( ( G  e.  ( Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma ) 
 ->  (GId `  H )  =  U )
 
Theoremablo4pnp 25923 A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.)
 |-  X  =  ran  G   &    |-  D  =  ( 
 /g  `  G )   =>    |-  (
 ( G  e.  AbelOp  /\  ( ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( A G B ) D ( C G F ) )  =  ( ( A D C ) G ( B D F ) ) )
 
Theoremgrpoeqdivid 25924 Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  D  =  ( 
 /g  `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A  =  B  <->  ( A D B )  =  U ) )
 
Theoremghomf 25925 Mapping property of a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.)
 |-  X  =  ran  G   &    |-  W  =  ran  H   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  F : X
 --> W )
 
Theoremghomco 25926 The composition of two group homomorphisms is a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  (
 ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  K  e.  GrpOp )  /\  ( S  e.  ( G GrpOpHom  H )  /\  T  e.  ( H GrpOpHom  K ) ) )  ->  ( T  o.  S )  e.  ( G GrpOpHom  K ) )
 
Theoremghomdiv 25927 Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  X  =  ran  G   &    |-  D  =  ( 
 /g  `  G )   &    |-  C  =  (  /g  `  H )   =>    |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A D B ) )  =  (
 ( F `  A ) C ( F `  B ) ) )
 
Theoremgrpokerinj 25928 A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  X  =  ran  G   &    |-  W  =  (GId `  G )   &    |-  Y  =  ran  H   &    |-  U  =  (GId `  H )   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  ( F : X -1-1-> Y  <->  ( `' F " { U } )  =  { W } )
 )
 
16.14.15  Rings
 
Theoremrngonegcl 25929 A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( N `  A )  e.  X )
 
Theoremrngoaddneg1 25930 Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( A G ( N `  A ) )  =  Z )
 
Theoremrngoaddneg2 25931 Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( ( N `  A ) G A )  =  Z )
 
Theoremrngosub 25932 Subtraction in a ring, in terms of addition and negation. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )
 
Theoremrngonegmn1l 25933 Negation in a ring is the same as left multiplication by  -u 1. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( N `  A )  =  ( ( N `  U ) H A ) )
 
Theoremrngonegmn1r 25934 Negation in a ring is the same as right multiplication by  -u 1. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( N `  A )  =  ( A H ( N `  U ) ) )
 
Theoremrngoneglmul 25935 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( N `
  A ) H B ) )
 
Theoremrngonegrmul 25936 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( A H ( N `  B ) ) )
 
Theoremrngosubdi 25937 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A H ( B D C ) )  =  ( ( A H B ) D ( A H C ) ) )
 
Theoremrngosubdir 25938 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B ) H C )  =  ( ( A H C ) D ( B H C ) ) )
 
Theoremzerdivemp1x 25939* In a unitary ring a left invertible element is not a zero divisor. Generalization of zerdivemp1 24789 by Frederic Line. (Contributed by Jeff Madsen, 18-Apr-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) )
 
Theoremisdrngo1 25940 The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   =>    |-  ( R  e.  DivRingOps  <->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
 ) ) )  e. 
 GrpOp ) )
 
Theoremdivrngcl 25941 The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  DivRingOps  /\  A  e.  ( X 
 \  { Z }
 )  /\  B  e.  ( X  \  { Z } ) )  ->  ( A H B )  e.  ( X  \  { Z } ) )
 
Theoremisdrngo2 25942* A division ring is a ring in which  1  =/=  0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  DivRingOps  <->  ( R  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z }
 ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  U ) ) )
 
Theoremisdrngo3 25943* A division ring is a ring in which  1  =/=  0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  DivRingOps  <->  ( R  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z }
 ) E. y  e.  X  ( y H x )  =  U ) ) )
 
16.14.16  Ring homomorphisms
 
Syntaxcrnghom 25944 Extend class notation with the class of ring homomorphisms.
 class  RngHom
 
Syntaxcrngiso 25945 Extend class notation with the class of ring isomorphisms.
 class  RngIso
 
Syntaxcrisc 25946 Extend class notation with the ring isomorphism relation.
 class  ~=r
 
Definitiondf-rngohom 25947* Define the function which gives the set of ring homomorphisms between two given rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  RngHom  =  ( r  e.  RingOps ,  s  e. 
 RingOps 
 |->  { f  e.  ( ran  ( 1st `  s
 )  ^m  ran  ( 1st `  r ) )  |  ( ( f `  (GId `  ( 2nd `  r
 ) ) )  =  (GId `  ( 2nd `  s ) )  /\  A. x  e.  ran  ( 1st `  r ) A. y  e.  ran  ( 1st `  r ) ( ( f `  ( x ( 1st `  r
 ) y ) )  =  ( ( f `
  x ) ( 1st `  s )
 ( f `  y
 ) )  /\  (
 f `  ( x ( 2nd `  r )
 y ) )  =  ( ( f `  x ) ( 2nd `  s ) ( f `
  y ) ) ) ) } )
 
Theoremrngohomval 25948* The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   &    |-  J  =  ( 1st `  S )   &    |-  K  =  ( 2nd `  S )   &    |-  Y  =  ran  J   &    |-  V  =  (GId `  K )   =>    |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngHom  S )  =  { f  e.  ( Y  ^m  X )  |  ( (
 f `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( f `
  ( x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `
  ( x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) ) }
 )
 
Theoremisrngohom 25949* The predicate "is a ring homomorphism from  R to  S." (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   &    |-  J  =  ( 1st `  S )   &    |-  K  =  ( 2nd `  S )   &    |-  Y  =  ran  J   &    |-  V  =  (GId `  K )   =>    |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : X
 --> Y  /\  ( F `
  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `
  ( x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `
  ( x H y ) )  =  ( ( F `  x ) K ( F `  y ) ) ) ) ) )
 
Theoremrngohomf 25950 A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : X
 --> Y )
 
Theoremrngohomcl 25951 Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  A  e.  X )  ->  ( F `
  A )  e.  Y )
 
Theoremrngohom1 25952 A ring homomorphism preserves  1. (Contributed by Jeff Madsen, 24-Jun-2011.)
 |-  H  =  ( 2nd `  R )   &    |-  U  =  (GId `  H )   &    |-  K  =  ( 2nd `  S )   &    |-  V  =  (GId `  K )   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  U )  =  V )
 
Theoremrngohomadd 25953 Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   =>    |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R 
 RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A G B ) )  =  (
 ( F `  A ) J ( F `  B ) ) )
 
Theoremrngohommul 25954 Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  H  =  ( 2nd `  R )   &    |-  K  =  ( 2nd `  S )   =>    |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R 
 RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A H B ) )  =  (
 ( F `  A ) K ( F `  B ) ) )
 
Theoremrngogrphom 25955 A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  J  =  ( 1st `  S )   =>    |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F  e.  ( G GrpOpHom  J ) )
 
Theoremrngohom0 25956 A ring homomorphism preserves  0. (Contributed by Jeff Madsen, 2-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  Z  =  (GId `  G )   &    |-  J  =  ( 1st `  S )   &    |-  W  =  (GId `  J )   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  Z )  =  W )
 
Theoremrngohomsub 25957 Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  H  =  (  /g  `  G )   &    |-  J  =  ( 1st `  S )   &    |-  K  =  ( 
 /g  `  J )   =>    |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A H B ) )  =  ( ( F `
  A ) K ( F `  B ) ) )
 
Theoremrngohomco 25958 The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e. 
 RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngHom  T ) )
 
Theoremrngokerinj 25959 A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  W  =  (GId `  G )   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   &    |-  Z  =  (GId `  J )   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : X -1-1-> Y  <->  ( `' F " { Z } )  =  { W } )
 )
 
Definitiondf-rngoiso 25960* Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  RngIso  =  ( r  e.  RingOps ,  s  e. 
 RingOps 
 |->  { f  e.  (
 r  RngHom  s )  |  f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s ) }
 )
 
Theoremrngoisoval 25961* The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngIso  S )  =  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y } )
 
Theoremisrngoiso 25962 The predicate "is a ring isomorphism between  R and 
S." (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  <->  ( F  e.  ( R  RngHom  S ) 
 /\  F : X -1-1-onto-> Y ) ) )
 
Theoremrngoiso1o 25963 A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F : X
 -1-1-onto-> Y )
 
Theoremrngoisohom 25964 A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
 
Theoremrngoisocnv 25965 The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  `' F  e.  ( S  RngIso  R ) )
 
Theoremrngoisoco 25966 The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e. 
 RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngIso  T ) )
 
Definitiondf-risc 25967* Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  ~=r  =  { <. r ,  s >.  |  ( ( r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  ( r  RngIso  s ) ) }
 
Theoremisriscg 25968* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  A  /\  S  e.  B ) 
 ->  ( R  ~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) ) )
 
Theoremisrisc 25969* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  R  e.  _V   &    |-  S  e.  _V   =>    |-  ( R  ~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) )
 
Theoremrisc 25970* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  ~=r  S  <->  E. f  f  e.  ( R  RngIso  S ) ) )
 
Theoremrisci 25971 Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  R  ~=r  S )
 
Theoremriscer 25972 Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ~=r  Er  dom  ~=r
 
16.14.17  Commutative rings
 
Syntaxccring 25973 Extend class notation with the class of commutative rings.
 class CRingOps
 
Definitiondf-crngo 25974 Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |- CRingOps  =  (
 RingOps  i^i  Com2 )
 
Theoremiscrngo 25975 The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  ( R  e. CRingOps  <->  ( R  e.  RingOps  /\  R  e.  Com2 )
 )
 
Theoremiscrngo2 25976* The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e. CRingOps  <->  ( R  e.  RingOps  /\ 
 A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) )
 
Theoremiscringd 25977* Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  ( ph  ->  G  e.  AbelOp )   &    |-  ( ph  ->  X  =  ran  G )   &    |-  ( ph  ->  H : ( X  X.  X ) --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )
 )  ->  ( ( x H y ) H z )  =  ( x H ( y H z ) ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) ) )   &    |-  ( ph  ->  U  e.  X )   &    |-  (
 ( ph  /\  y  e.  X )  ->  (
 y H U )  =  y )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x H y )  =  ( y H x ) )   =>    |-  ( ph  ->  <. G ,  H >.  e. CRingOps )
 
Theoremcrngorngo 25978 A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e. CRingOps  ->  R  e.  RingOps )
 
Theoremcrngocom 25979 The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  =  ( B H A ) )
 
Theoremcrngm23 25980 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A H B ) H C )  =  ( ( A H C ) H B ) )
 
Theoremcrngm4 25981 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A H B ) H ( C H D ) )  =  ( ( A H C ) H ( B H D ) ) )
 
Theoremfldcrng 25982 A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  ( K  e.  Fld  ->  K  e. CRingOps )
 
Theoremisfld2 25983 The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( K  e.  Fld  <->  ( K  e.  DivRingOps  /\  K  e. CRingOps ) )
 
Theoremcrngohomfo 25984 The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e. CRingOps )
 
16.14.18  Ideals
 
Syntaxcidl 25985 Extend class notation with the class of ideals.
 class  Idl
 
Syntaxcpridl 25986 Extend class notation with the class of prime ideals.
 class  PrIdl
 
Syntaxcmaxidl 25987 Extend class notation with the class of maximal ideals.
 class  MaxIdl
 
Definitiondf-idl 25988* Define the class of (two-sided) ideals of a ring  R. A subset of  R is an ideal if it contains  0, is closed under addition, and is closed under multiplication on either side by any element of  R. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  Idl  =  ( r  e.  RingOps  |->  { i  e.  ~P ran  ( 1st `  r )  |  ( (GId `  ( 1st `  r ) )  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x ( 1st `  r
 ) y )  e.  i  /\  A. z  e.  ran  ( 1st `  r
 ) ( ( z ( 2nd `  r
 ) x )  e.  i  /\  ( x ( 2nd `  r
 ) z )  e.  i ) ) ) } )
 
Definitiondf-pridl 25989* Define the class of prime ideals of a ring  R. A proper ideal  I of  R is prime if whenever  A B  C_  I for ideals  A and  B, either  A  C_  I or  B  C_  I. The more familiar definition using elements rather than ideals is equivalent provided  R is commutative; see ispridl2 26016 and ispridlc 26048. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  PrIdl  =  ( r  e.  RingOps  |->  { i  e.  ( Idl `  r
 )  |  ( i  =/=  ran  ( 1st `  r )  /\  A. a  e.  ( Idl `  r ) A. b  e.  ( Idl `  r
 ) ( A. x  e.  a  A. y  e.  b  ( x ( 2nd `  r )
 y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
 
Definitiondf-maxidl 25990* Define the class of maximal ideals of a ring  R. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  MaxIdl  =  ( r  e.  RingOps  |->  { i  e.  ( Idl `  r
 )  |  ( i  =/=  ran  ( 1st `  r )  /\  A. j  e.  ( Idl `  r ) ( i 
 C_  j  ->  (
 j  =  i  \/  j  =  ran  ( 1st `  r ) ) ) ) } )
 
Theoremidlval 25991* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  RingOps  ->  ( Idl `  R )  =  { i  e.  ~P X  |  ( Z  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x G y )  e.  i  /\  A. z  e.  X  ( ( z H x )  e.  i  /\  ( x H z )  e.  i ) ) ) } )
 
Theoremisidl 25992* The predicate "is an ideal of the ring  R." (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  RingOps  ->  ( I  e.  ( Idl `  R )  <->  ( I  C_  X  /\  Z  e.  I  /\  A. x  e.  I  ( A. y  e.  I  ( x G y )  e.  I  /\  A. z  e.  X  (
 ( z H x )  e.  I  /\  ( x H z )  e.  I ) ) ) ) )
 
Theoremisidlc 25993* The predicate "is an ideal of the commutative ring  R." (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e. CRingOps  ->  ( I  e.  ( Idl `  R )  <->  ( I  C_  X  /\  Z  e.  I  /\  A. x  e.  I  ( A. y  e.  I  ( x G y )  e.  I  /\  A. z  e.  X  (
 z H x )  e.  I ) ) ) )
 
Theoremidlss 25994 An ideal of  R is a subset of  R. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  ->  I  C_  X )
 
Theoremidlcl 25995 An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  A  e.  X )
 
Theoremidl0cl 25996 An ideal contains  0. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  ->  Z  e.  I )
 
Theoremidladdcl 25997 An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   =>    |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R )
 )  /\  ( A  e.  I  /\  B  e.  I ) )  ->  ( A G B )  e.  I )
 
Theoremidllmulcl 25998 An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R )
 )  /\  ( A  e.  I  /\  B  e.  X ) )  ->  ( B H A )  e.  I )
 
Theoremidlrmulcl 25999 An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R )
 )  /\  ( A  e.  I  /\  B  e.  X ) )  ->  ( A H B )  e.  I )
 
Theoremidlnegcl 26000 An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R )
 )  /\  A  e.  I )  ->  ( N `
  A )  e.  I )
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