HomeHome Metamath Proof Explorer
Theorem List (p. 266 of 322)
< 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
(1-21498)
  Hilbert Space Explorer  Hilbert Space Explorer
(21499-23021)
  Users' Mathboxes  Users' Mathboxes
(23022-32154)
 

Theorem List for Metamath Proof Explorer - 26501-26600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtotbndss 26501 A subset of a totally bounded metric space is totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( M  e.  ( TotBnd `
  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( TotBnd `  S ) )
 
Theoremequivtotbnd 26502* If the metric  M is "strongly finer" than  N (meaning that there is a positive real constant 
R such that  N ( x ,  y )  <_  R  x.  M (
x ,  y )), then total boundedness of  M implies total boundedness of 
N. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then one is totally bounded iff the other is.) (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  ( ph  ->  M  e.  ( TotBnd `
  X ) )   &    |-  ( ph  ->  N  e.  ( Met `  X )
 )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x N y )  <_  ( R  x.  ( x M y ) ) )   =>    |-  ( ph  ->  N  e.  ( TotBnd `  X )
 )
 
Definitiondf-bnd 26503* Define the class of bounded metrics. A metric space is bounded iff it can be covered by a single ball. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  Bnd  =  ( x  e.  _V  |->  { m  e.  ( Met `  x )  |  A. y  e.  x  E. r  e.  RR+  x  =  ( y ( ball `  m ) r ) } )
 
Theoremisbnd 26504* The predicate "is a bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  ( M  e.  ( Bnd `  X )  <->  ( M  e.  ( Met `  X )  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
 
Theorembndmet 26505 A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( M  e.  ( Bnd `  X )  ->  M  e.  ( Met `  X ) )
 
Theoremisbndx 26506* A "bounded extended metric" (meaning that it satisfies the same condition as a bounded metric, but with "metric" replaced with "extended metric") is a metric and thus is bounded in the conventional sense. (Contributed by Mario Carneiro, 12-Sep-2015.)
 |-  ( M  e.  ( Bnd `  X )  <->  ( M  e.  ( * Met `  X )  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M )
 r ) ) )
 
Theoremisbnd2 26507* The predicate "is a bounded metric space". Uses a single point instead of an arbitrary point in the space. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( M  e.  ( Bnd `  X )  /\  X  =/=  (/) )  <->  ( M  e.  ( * Met `  X )  /\  E. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M )
 r ) ) )
 
Theoremisbnd3 26508* A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |-  ( M  e.  ( Bnd `  X )  <->  ( M  e.  ( Met `  X )  /\  E. x  e.  RR  M : ( X  X.  X ) --> ( 0 [,] x ) ) )
 
Theoremisbnd3b 26509* A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  ( M  e.  ( Bnd `  X )  <->  ( M  e.  ( Met `  X )  /\  E. x  e.  RR  A. y  e.  X  A. z  e.  X  (
 y M z ) 
 <_  x ) )
 
Theorembndss 26510 A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( M  e.  ( Bnd `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( Bnd `  S ) )
 
Theoremblbnd 26511 A ball is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 15-Jan-2014.)
 |-  (
 ( M  e.  ( * Met `  X )  /\  Y  e.  X  /\  R  e.  RR )  ->  ( M  |`  ( ( Y ( ball `  M ) R )  X.  ( Y ( ball `  M ) R ) ) )  e.  ( Bnd `  ( Y ( ball `  M ) R ) ) )
 
Theoremssbnd 26512* A subset of a metric space is bounded iff it is contained in a ball around  P, for any  P in the larger space. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  N  =  ( M  |`  ( Y  X.  Y ) )   =>    |-  ( ( M  e.  ( Met `  X )  /\  P  e.  X ) 
 ->  ( N  e.  ( Bnd `  Y )  <->  E. d  e.  RR  Y  C_  ( P (
 ball `  M ) d ) ) )
 
Theoremtotbndbnd 26513 A totally bounded metric space is bounded. This theorem fails for extended metrics - a bounded extended metric is a metric, but there are totally bounded extended metrics that are not metrics (if we were to weaken istotbnd 26493 to only require that  M be an extended metric). A counterexample is the discrete extended metric (assigning distinct points distance  +oo) on a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  ( M  e.  ( TotBnd `  X )  ->  M  e.  ( Bnd `  X )
 )
 
Theoremequivbnd 26514* If the metric  M is "strongly finer" than  N (meaning that there is a positive real constant 
R such that  N ( x ,  y )  <_  R  x.  M (
x ,  y )), then boundedness of  M implies boundedness of  N. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then one is bounded iff the other is.) (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  ( ph  ->  M  e.  ( Bnd `  X ) )   &    |-  ( ph  ->  N  e.  ( Met `  X )
 )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x N y )  <_  ( R  x.  ( x M y ) ) )   =>    |-  ( ph  ->  N  e.  ( Bnd `  X ) )
 
Theorembnd2lem 26515 Lemma for equivbnd2 26516 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.)
 |-  D  =  ( M  |`  ( Y  X.  Y ) )   =>    |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y ) )  ->  Y  C_  X )
 
Theoremequivbnd2 26516* If balls are totally bounded in the metric  M, then balls are totally bounded in the equivalent metric  N. (Contributed by Mario Carneiro, 15-Sep-2015.)
 |-  ( ph  ->  M  e.  ( Met `  X ) )   &    |-  ( ph  ->  N  e.  ( Met `  X )
 )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  S  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x N y )  <_  ( R  x.  ( x M y ) ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x M y )  <_  ( S  x.  ( x N y ) ) )   &    |-  C  =  ( M  |`  ( Y  X.  Y ) )   &    |-  D  =  ( N  |`  ( Y  X.  Y ) )   &    |-  ( ph  ->  ( C  e.  ( TotBnd `  Y )  <->  C  e.  ( Bnd `  Y ) ) )   =>    |-  ( ph  ->  ( D  e.  ( TotBnd `  Y )  <->  D  e.  ( Bnd `  Y ) ) )
 
Theoremprdsbnd 26517* The product metric over finite index set is bounded if all the factors are bounded. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |-  Y  =  ( S X_s R )   &    |-  B  =  (
 Base `  Y )   &    |-  V  =  ( Base `  ( R `  x ) )   &    |-  E  =  ( ( dist `  ( R `  x ) )  |`  ( V  X.  V ) )   &    |-  D  =  (
 dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ph  ->  R  Fn  I )   &    |-  (
 ( ph  /\  x  e.  I )  ->  E  e.  ( Bnd `  V ) )   =>    |-  ( ph  ->  D  e.  ( Bnd `  B ) )
 
Theoremprdstotbnd 26518* The product metric over finite index set is totally bounded if all the factors are totally bounded. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  Y  =  ( S X_s R )   &    |-  B  =  (
 Base `  Y )   &    |-  V  =  ( Base `  ( R `  x ) )   &    |-  E  =  ( ( dist `  ( R `  x ) )  |`  ( V  X.  V ) )   &    |-  D  =  (
 dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ph  ->  R  Fn  I )   &    |-  (
 ( ph  /\  x  e.  I )  ->  E  e.  ( TotBnd `  V )
 )   =>    |-  ( ph  ->  D  e.  ( TotBnd `  B )
 )
 
Theoremprdsbnd2 26519* If balls are totally bounded in each factor, then balls are bounded in a metric product. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  Y  =  ( S X_s R )   &    |-  B  =  (
 Base `  Y )   &    |-  V  =  ( Base `  ( R `  x ) )   &    |-  E  =  ( ( dist `  ( R `  x ) )  |`  ( V  X.  V ) )   &    |-  D  =  (
 dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ph  ->  R  Fn  I )   &    |-  C  =  ( D  |`  ( A  X.  A ) )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( Met `  V ) )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  ( ( E  |`  ( y  X.  y ) )  e.  ( TotBnd `  y
 ) 
 <->  ( E  |`  ( y  X.  y ) )  e.  ( Bnd `  y
 ) ) )   =>    |-  ( ph  ->  ( C  e.  ( TotBnd `  A )  <->  C  e.  ( Bnd `  A ) ) )
 
Theoremcntotbnd 26520 A subset of the complexes is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( X  X.  X ) )   =>    |-  ( D  e.  ( TotBnd `  X )  <->  D  e.  ( Bnd `  X ) )
 
Theoremcnpwstotbnd 26521 A subset of  A ^ I, where  A 
C_  CC, is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  Y  =  ( (flds  A )  ^s  I )   &    |-  D  =  ( ( dist `  Y )  |`  ( X  X.  X ) )   =>    |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  ( D  e.  ( TotBnd `  X )  <->  D  e.  ( Bnd `  X ) ) )
 
18.15.9  Isometries
 
Syntaxcismty 26522 Extend class notation with the class of metric space isometries.
 class  Ismty
 
Definitiondf-ismty 26523* Define a function which takes two metric spaces and returns the set of isometries between the spaces. An isometry is a bijection which preserves distance. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  Ismty  =  ( m  e.  U. ran  * Met ,  n  e. 
 U. ran  * Met  |->  { f  |  ( f : dom  dom  m -1-1-onto-> dom  dom 
 n  /\  A. x  e. 
 dom  dom  m A. y  e.  dom  dom  m ( x m y )  =  ( ( f `  x ) n ( f `  y ) ) ) } )
 
Theoremismtyval 26524* The set of isometries between two metric spaces. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y ) ) 
 ->  ( M  Ismty  N )  =  { f  |  ( f : X -1-1-onto-> Y  /\  A. x  e.  X  A. y  e.  X  ( x M y )  =  ( ( f `
  x ) N ( f `  y
 ) ) ) }
 )
 
Theoremisismty 26525* The condition "is an isometry". (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y ) ) 
 ->  ( F  e.  ( M  Ismty  N )  <->  ( F : X
 -1-1-onto-> Y  /\  A. x  e.  X  A. y  e.  X  ( x M y )  =  ( ( F `  x ) N ( F `  y ) ) ) ) )
 
Theoremismtycnv 26526 The inverse of an isometry is an isometry. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y ) ) 
 ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
 
Theoremismtyima 26527 The image of a ball under an isometry is another ball. (Contributed by Jeff Madsen, 31-Jan-2014.)
 |-  (
 ( ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y )  /\  F  e.  ( M 
 Ismty  N ) )  /\  ( P  e.  X  /\  R  e.  RR* )
 )  ->  ( F " ( P ( ball `  M ) R ) )  =  ( ( F `  P ) ( ball `  N ) R ) )
 
Theoremismtyhmeolem 26528 Lemma for ismtyhmeo 26529. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  J  =  ( MetOpen `  M )   &    |-  K  =  ( MetOpen `  N )   &    |-  ( ph  ->  M  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  N  e.  ( * Met `  Y ) )   &    |-  ( ph  ->  F  e.  ( M  Ismty  N ) )   =>    |-  ( ph  ->  F  e.  ( J  Cn  K ) )
 
Theoremismtyhmeo 26529 An isometry is a homeomorphism on the induced topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  J  =  ( MetOpen `  M )   &    |-  K  =  ( MetOpen `  N )   =>    |-  (
 ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y ) ) 
 ->  ( M  Ismty  N ) 
 C_  ( J  Homeo  K ) )
 
Theoremismtybndlem 26530 Lemma for ismtybnd 26531. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 19-Jan-2014.)
 |-  (
 ( N  e.  ( * Met `  Y )  /\  F  e.  ( M 
 Ismty  N ) )  ->  ( M  e.  ( Bnd `  X )  ->  N  e.  ( Bnd `  Y ) ) )
 
Theoremismtybnd 26531 Isometries preserve boundedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 19-Jan-2014.)
 |-  (
 ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y )  /\  F  e.  ( M  Ismty  N ) )  ->  ( M  e.  ( Bnd `  X )  <->  N  e.  ( Bnd `  Y ) ) )
 
Theoremismtyres 26532 A restriction of an isometry is an isometry. The condition  A  C_  X is not necessary but makes the proof easier. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  B  =  ( F " A )   &    |-  S  =  ( M  |`  ( A  X.  A ) )   &    |-  T  =  ( N  |`  ( B  X.  B ) )   =>    |-  ( ( ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y ) ) 
 /\  ( F  e.  ( M  Ismty  N ) 
 /\  A  C_  X ) )  ->  ( F  |`  A )  e.  ( S  Ismty  T ) )
 
18.15.10  Heine-Borel Theorem
 
Theoremheibor1lem 26533 Lemma for heibor1 26534. A compact metric space is complete. This proof works by considering the collection  cls ( F " ( ZZ>=
`  n ) ) for each  n  e.  NN, which has the finite intersection property because any finite intersection of upper integer sets is another upper integer set, so any finite intersection of the image closures will contain  ( F "
( ZZ>= `  m )
) for some  m. Thus by compactness the intersection contains a point  y, which must then be the convergent point of  F. (Contributed by Jeff Madsen, 17-Jan-2014.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp
 )   &    |-  ( ph  ->  F  e.  ( Cau `  D ) )   &    |-  ( ph  ->  F : NN --> X )   =>    |-  ( ph  ->  F  e.  dom  ( ~~> t `  J ) )
 
Theoremheibor1 26534 One half of heibor 26545, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 18743 and total boundedness here, which follows trivially from the fact that the set of all  r-balls is an open cover of  X, so finitely many cover  X. (Contributed by Jeff Madsen, 16-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   =>    |-  (
 ( D  e.  ( Met `  X )  /\  J  e.  Comp )  ->  ( D  e.  ( CMet `  X )  /\  D  e.  ( TotBnd `  X ) ) )
 
Theoremheiborlem1 26535* Lemma for heibor 26545. We work with a fixed open cover  U throughout. The set  K is the set of all subsets of  X that admit no finite subcover of  U. (We wish to prove that  K is empty.) If a set  C has no finite subcover, then any finite cover of  C must contain a set that also has no finite subcover. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  B  e.  _V   =>    |-  ( ( A  e.  Fin  /\  C  C_  U_ x  e.  A  B  /\  C  e.  K )  ->  E. x  e.  A  B  e.  K )
 
Theoremheiborlem2 26536* Lemma for heibor 26545. Substitutions for the set  G. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  A  e.  _V   &    |-  C  e.  _V   =>    |-  ( A G C  <->  ( C  e.  NN0  /\  A  e.  ( F `  C )  /\  ( A B C )  e.  K ) )
 
Theoremheiborlem3 26537* Lemma for heibor 26545. Using countable choice ax-cc 8061, we have fixed in advance a collection of finite  2 ^ -u n nets  ( F `  n ) for  X (note that an  r-net is a set of points in  X whose  r -balls cover  X). The set  G is the subset of these points whose corresponding balls have no finite subcover (i.e. in the set  K). If the theorem was false, then  X would be in  K, and so some ball at each level would also be in  K. But we can say more than this; given a ball 
( y B n ) on level  n, since level  n  +  1 covers the space and thus also  (
y B n ), using heiborlem1 26535 there is a ball on the next level whose intersection with  ( y B n ) also has no finite subcover. Now since the set 
G is a countable union of finite sets, it is countable (which needs ax-cc 8061 via iunctb 8196), and so we can apply ax-cc 8061 to  G directly to get a function from  G to itself, which points from each ball in  K to a ball on the next level in  K, and such that the intersection between these balls is also in  K. (Contributed by Jeff Madsen, 18-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   =>    |-  ( ph  ->  E. g A. x  e.  G  ( ( g `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( g `  x ) B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )
 
Theoremheiborlem4 26538* Lemma for heibor 26545. Using the function  T constructed in heiborlem3 26537, construct an infinite path in  G. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   =>    |-  ( ( ph  /\  A  e.  NN0 )  ->  ( S `  A ) G A )
 
Theoremheiborlem5 26539* Lemma for heibor 26545. The function  M is a set of point-and-radius pairs suitable for application to caubl 18733. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   &    |-  M  =  ( n  e.  NN  |->  <. ( S `  n ) ,  (
 3  /  ( 2 ^ n ) ) >. )   =>    |-  ( ph  ->  M : NN
 --> ( X  X.  RR+ )
 )
 
Theoremheiborlem6 26540* Lemma for heibor 26545. Since the sequence of balls connected by the function  T ensures that each ball nontrivially intersects with the next (since the empty set has a finite subcover, the intersection of any two successive balls in the sequence is nonempty), and each ball is half the size of the previous one, the distance between the centers is at most  3  /  2 times the size of the larger, and so if we expand each ball by a factor of  3 we get a nested sequence of balls. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   &    |-  M  =  ( n  e.  NN  |->  <. ( S `  n ) ,  (
 3  /  ( 2 ^ n ) ) >. )   =>    |-  ( ph  ->  A. k  e. 
 NN  ( ( ball `  D ) `  ( M `  ( k  +  1 ) ) ) 
 C_  ( ( ball `  D ) `  ( M `  k ) ) )
 
Theoremheiborlem7 26541* Lemma for heibor 26545. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   &    |-  M  =  ( n  e.  NN  |->  <. ( S `  n ) ,  (
 3  /  ( 2 ^ n ) ) >. )   =>    |-  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k ) )  <  r
 
Theoremheiborlem8 26542* Lemma for heibor 26545. The previous lemmas establish that the sequence  M is Cauchy, so using completeness we now consider the convergent point 
Y. By assumption,  U is an open cover, so  Y is an element of some  Z  e.  U, and some ball centered at  Y is contained in  Z. But the sequence contains arbitrarily small balls close to  Y, so some element  ball ( M `  n ) of the sequence is contained in  Z. And finally we arrive at a contradiction, because  { Z } is a finite subcover of  U that covers  ball ( M `  n ), yet  ball ( M `  n )  e.  K. For convenience, we write this contradiction as 
ph  ->  ps where  ph is all the accumulated hypotheses and  ps is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   &    |-  M  =  ( n  e.  NN  |->  <. ( S `  n ) ,  (
 3  /  ( 2 ^ n ) ) >. )   &    |-  ( ph  ->  U  C_  J )   &    |-  Y  e.  _V   &    |-  ( ph  ->  Y  e.  Z )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  ( 1st  o.  M ) ( ~~> t `  J ) Y )   =>    |-  ( ph  ->  ps )
 
Theoremheiborlem9 26543* Lemma for heibor 26545. Discharge the hypotheses of heiborlem8 26542 by applying caubl 18733 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   &    |-  M  =  ( n  e.  NN  |->  <. ( S `  n ) ,  (
 3  /  ( 2 ^ n ) ) >. )   &    |-  ( ph  ->  U  C_  J )   &    |-  ( ph  ->  U. U  =  X )   =>    |-  ( ph  ->  ps )
 
Theoremheiborlem10 26544* Lemma for heibor 26545. The last remaining piece of the proof is to find an element  C such that  C G 0, i.e. 
C is an element of  ( F ` 
0 ) that has no finite subcover, which is true by heiborlem1 26535, since  ( F `  0 ) is a finite cover of  X, which has no finite subcover. Thus the rest of the proof follows to a contradiction, and thus there must be a finite subcover of  U that covers  X, i.e.  X is compact. (Contributed by Jeff Madsen, 22-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   =>    |-  ( ( ph  /\  ( U  C_  J  /\  U. J  =  U. U ) )  ->  E. v  e.  ( ~P U  i^i  Fin ) U. J  =  U. v )
 
Theoremheibor 26545 Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 26534 and heiborlem1 26535 for a description of the proof. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   =>    |-  (
 ( D  e.  ( Met `  X )  /\  J  e.  Comp )  <->  ( D  e.  ( CMet `  X )  /\  D  e.  ( TotBnd `  X ) ) )
 
18.15.11  Banach Fixed Point Theorem
 
Theorembfplem1 26546* Lemma for bfp 26548. The sequence  G, which simply starts from any point in the space and iterates  F, satisfies the property that the distance from  G ( n ) to  G ( n  + 
1 ) decreases by at least  K after each step. Thus the total distance from any  G ( i ) to  G ( j ) is bounded by a geometric series, and the sequence is Cauchy. Therefore, it converges to a point  ( ( ~~> t `  J
) `  G ) since the space is complete. (Contributed by Jeff Madsen, 17-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 ) ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  A  e.  X )   &    |-  G  =  seq  1
 ( ( F  o.  1st ) ,  ( NN 
 X.  { A } )
 )   =>    |-  ( ph  ->  G (
 ~~> t `  J ) ( ( ~~> t `  J ) `  G ) )
 
Theorembfplem2 26547* Lemma for bfp 26548. Using the point found in bfplem1 26546, we show that this convergent point is a fixed point of  F. Since for any positive  x, the sequence  G is in  B ( x  /  2 ,  P ) for all  k  e.  (
ZZ>= `  j ) (where  P  =  ( ( ~~> t `  J ) `  G
)), we have  D ( G ( j  +  1 ) ,  F ( P ) )  <_  D ( G ( j ) ,  P
)  <  x  / 
2 and  D ( G ( j  +  1 ) ,  P )  <  x  /  2, so  F ( P ) is in every neighborhood of  P and  P is a fixed point of  F. (Contributed by Jeff Madsen, 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 ) ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  A  e.  X )   &    |-  G  =  seq  1
 ( ( F  o.  1st ) ,  ( NN 
 X.  { A } )
 )   =>    |-  ( ph  ->  E. z  e.  X  ( F `  z )  =  z
 )
 
Theorembfp 26548* 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 )
 
18.15.12  Euclidean space
 
Syntaxcrrn 26549 Extend class notation with the n-dimensional Euclidean space.
 class  Rn
 
Definitiondf-rrn 26550* 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 26551* 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 26552* 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 26553 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 26554 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 26555* Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 26554 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 26556* Lemma for rrncms 26557. (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 26557 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 26558 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 26559 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 26560 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 26561 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 ) ) ) )
 
18.15.13  Intervals (continued)
 
Theoremismrer1 26562* 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 26563 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 26564 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 26565 A closed interval in  RR is compact. Alternate proof of icccmp 18330 using the Heine-Borel theorem heibor 26545. (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 )
 
18.15.14  Groups and related structures
 
Theoremexidcl 26566 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 26567* Lemma for exidres 26568 and exidresid 26569. (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 26568 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 26569 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 26570 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 26571 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 26572 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 26573 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 26574 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 26575 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 } )
 )
 
18.15.15  Rings
 
Theoremrngonegcl 26576 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 26577 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 26578 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 26579 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 26580 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 26581 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 26582 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 26583 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 26584 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 26585 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 26586* In a unitary ring a left invertible element is not a zero divisor. Generalization of zerdivemp1 25436 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 26587 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 26588 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 26589* 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 26590* 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 ) ) )
 
18.15.16  Ring homomorphisms
 
Syntaxcrnghom 26591 Extend class notation with the class of ring homomorphisms.
 class  RngHom
 
Syntaxcrngiso 26592 Extend class notation with the class of ring isomorphisms.
 class  RngIso
 
Syntaxcrisc 26593 Extend class notation with the ring isomorphism relation.
 class  ~=r
 
Definitiondf-rngohom 26594* 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 26595* 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 26596* 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 26597 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 26598 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 26599 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 26600 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 ) ) )
    < 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 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32154
  Copyright terms: Public domain < Previous  Next >