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Theorem List for Metamath Proof Explorer - 26501-26600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
18.15.8  Boundedness
 
Syntaxctotbnd 26501 Extend class notation with the class of totally bounded metric spaces.
 class  TotBnd
 
Syntaxcbnd 26502 Extend class notation with the class of bounded metric spaces.
 class  Bnd
 
Definitiondf-totbnd 26503* Define the class of totally bounded metrics. A metric space is totally bounded iff it can be covered by a finite number of balls of any given radius. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  TotBnd  =  ( x  e.  _V  |->  { m  e.  ( Met `  x )  |  A. d  e.  RR+  E. v  e.  Fin  ( U. v  =  x  /\  A. b  e.  v  E. y  e.  x  b  =  ( y ( ball `  m ) d ) ) } )
 
Theoremistotbnd 26504* The predicate "is a totally bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ( TotBnd `  X )  <->  ( M  e.  ( Met `  X )  /\  A. d  e.  RR+  E. v  e.  Fin  ( U. v  =  X  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M )
 d ) ) ) )
 
Theoremistotbnd2 26505* The predicate "is a totally bounded metric space." (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ( Met `  X )  ->  ( M  e.  ( TotBnd `  X )  <->  A. d  e.  RR+  E. v  e.  Fin  ( U. v  =  X  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M )
 d ) ) ) )
 
Theoremistotbnd3 26506* A metric space is totally bounded iff there is a finite ε-net for every positive ε. This differs from the definition in providing a finite set of ball centers rather than a finite set of balls. (Contributed by Mario Carneiro, 12-Sep-2015.)
 |-  ( M  e.  ( TotBnd `  X )  <->  ( M  e.  ( Met `  X )  /\  A. d  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) d )  =  X ) )
 
Theoremtotbndmet 26507 The predicate "totally bounded" implies  M is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ( TotBnd `  X )  ->  M  e.  ( Met `  X )
 )
 
Theorem0totbnd 26508 The metric (there is only one) on the empty set is totally bounded. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  X )  <->  M  e.  ( Met `  X ) ) )
 
Theoremsstotbnd2 26509* Condition for a subset of a metric space to be totally bounded. (Contributed by Mario Carneiro, 12-Sep-2015.)
 |-  N  =  ( M  |`  ( Y  X.  Y ) )   =>    |-  ( ( M  e.  ( Met `  X )  /\  Y  C_  X )  ->  ( N  e.  ( TotBnd `
  Y )  <->  A. d  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) Y  C_  U_ x  e.  v  ( x ( ball `  M ) d ) ) )
 
Theoremsstotbnd 26510* Condition for a subset of a metric space to be totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  N  =  ( M  |`  ( Y  X.  Y ) )   =>    |-  ( ( M  e.  ( Met `  X )  /\  Y  C_  X )  ->  ( N  e.  ( TotBnd `
  Y )  <->  A. d  e.  RR+  E. v  e.  Fin  ( Y  C_  U. v  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M ) d ) ) ) )
 
Theoremsstotbnd3 26511* Use a net that is not necessarily finite, but for which only finitely many balls meet the subset. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  N  =  ( M  |`  ( Y  X.  Y ) )   =>    |-  ( ( M  e.  ( Met `  X )  /\  Y  C_  X )  ->  ( N  e.  ( TotBnd `
  Y )  <->  A. d  e.  RR+  E. v  e.  ~P  X ( Y  C_  U_ x  e.  v  ( x ( ball `  M )
 d )  /\  { x  e.  v  |  ( ( x (
 ball `  M ) d )  i^i  Y )  =/=  (/) }  e.  Fin ) ) )
 
Theoremtotbndss 26512 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 26513* 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 26514* 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 26515* 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 26516 A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( M  e.  ( Bnd `  X )  ->  M  e.  ( Met `  X ) )
 
Theoremisbndx 26517* 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 26518* 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 26519* 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 26520* 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 26521 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 26522 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 26523* 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 26524 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 26504 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 26525* 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 26526 Lemma for equivbnd2 26527 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 26527* 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 26528* 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 26529* 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 26530* 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 26531 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 26532 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 26533 Extend class notation with the class of metric space isometries.
 class  Ismty
 
Definitiondf-ismty 26534* 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 26535* 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 26536* 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 26537 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 26538 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 26539 Lemma for ismtyhmeo 26540. (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 26540 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 26541 Lemma for ismtybnd 26542. (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 26542 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 26543 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 26544 Lemma for heibor1 26545. 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 26545 One half of heibor 26556, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 18745 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 26546* Lemma for heibor 26556. 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 26547* Lemma for heibor 26556. 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 26548* Lemma for heibor 26556. Using countable choice ax-cc 8063, 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 26546 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 8063 via iunctb 8198), and so we can apply ax-cc 8063 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 26549* Lemma for heibor 26556. Using the function  T constructed in heiborlem3 26548, 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 26550* Lemma for heibor 26556. The function  M is a set of point-and-radius pairs suitable for application to caubl 18735. (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 26551* Lemma for heibor 26556. 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 26552* Lemma for heibor 26556. 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 26553* Lemma for heibor 26556. 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 26554* Lemma for heibor 26556. Discharge the hypotheses of heiborlem8 26553 by applying caubl 18735 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 26555* Lemma for heibor 26556. 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 26546, 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 26556 Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 26545 and heiborlem1 26546 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 26557* Lemma for bfp 26559. 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 26558* Lemma for bfp 26559. Using the point found in bfplem1 26557, 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 26559* 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 26560 Extend class notation with the n-dimensional Euclidean space.
 class  Rn
 
Definitiondf-rrn 26561* 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 26562* 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 26563* 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 26564 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 26565 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 26566* Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 26565 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 26567* Lemma for rrncms 26568. (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 26568 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 26569 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 26570 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 26571 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 26572 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 26573* 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 26574 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 26575 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 26576 A closed interval in  RR is compact. Alternate proof of icccmp 18332 using the Heine-Borel theorem heibor 26556. (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 26577 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 26578* Lemma for exidres 26579 and exidresid 26580. (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 26579 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 26580 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 26581 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 26582 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 26583 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 26584 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 26585 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 26586 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 26587 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 26588 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 26589 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 26590 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 26591 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 26592 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 26593 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 26594 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 26595 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 26596 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 26597* In a unitary ring a left invertible element is not a zero divisor. Generalization of zerdivemp1 25447 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 26598 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 26599 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 26600* 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 ) ) )
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