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Theorem List for Metamath Proof Explorer - 25801-25900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremabszOLD 25801 A real number is an integer iff its absolute value is an integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to absz 11747 in main set.mm and may be deleted by mathbox owner, JM. --NM 10-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  ( A  e.  ZZ  <->  ( abs `  A )  e.  ZZ )
 )
 
Theoremmod0OLD 25802  A  mod  B is zero iff  A is evenly divisible by  B. (Moved to mod0 10930 in main set.mm and may be deleted by mathbox owner, JM. --NM 19-Apr-2014.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  mod  B )  =  0  <->  ( A  /  B )  e.  ZZ ) )
 
Theoremnegmod0OLD 25803  A is divisible by  B iff its negative is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to negmod0 10931 in main set.mm and may be deleted by mathbox owner, JM. --NM 10-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  mod  B )  =  0  <->  ( -u A  mod  B )  =  0 ) )
 
Theoremabsmod0OLD 25804  A is divisible by  B iff its absolute value is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to absmod0 11739 in main set.mm and may be deleted by mathbox owner, JM. --NM 10-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  mod  B )  =  0  <->  ( ( abs `  A )  mod  B )  =  0 )
 )
 
16.14.3  Sequences and sums
 
Theoremsdclem2 25805* Lemma for sdc 25807. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  (
 g  =  ( f  |`  ( M ... n ) )  ->  ( ps  <->  ch ) )   &    |-  ( n  =  M  ->  ( ps  <->  ta ) )   &    |-  ( n  =  k  ->  ( ps  <->  th ) )   &    |-  ( ( g  =  h  /\  n  =  ( k  +  1 ) )  ->  ( ps 
 <-> 
 si ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. g ( g : { M } --> A  /\  ta ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( ( g : ( M ... k
 ) --> A  /\  th )  ->  E. h ( h : ( M ... ( k  +  1
 ) ) --> A  /\  g  =  ( h  |`  ( M ... k
 ) )  /\  si ) ) )   &    |-  J  =  { g  |  E. n  e.  Z  (
 g : ( M
 ... n ) --> A  /\  ps ) }   &    |-  F  =  ( w  e.  Z ,  x  e.  J  |->  { h  |  E. k  e.  Z  ( h : ( M
 ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k
 ) )  /\  si ) } )   &    |-  F/ k ph   &    |-  ( ph  ->  G : Z --> J )   &    |-  ( ph  ->  ( G `  M ) : ( M ... M ) --> A )   &    |-  (
 ( ph  /\  w  e.  Z )  ->  ( G `  ( w  +  1 ) )  e.  ( w F ( G `  w ) ) )   =>    |-  ( ph  ->  E. f
 ( f : Z --> A  /\  A. n  e.  Z  ch ) )
 
Theoremsdclem1 25806* Lemma for sdc 25807. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  (
 g  =  ( f  |`  ( M ... n ) )  ->  ( ps  <->  ch ) )   &    |-  ( n  =  M  ->  ( ps  <->  ta ) )   &    |-  ( n  =  k  ->  ( ps  <->  th ) )   &    |-  ( ( g  =  h  /\  n  =  ( k  +  1 ) )  ->  ( ps 
 <-> 
 si ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. g ( g : { M } --> A  /\  ta ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( ( g : ( M ... k
 ) --> A  /\  th )  ->  E. h ( h : ( M ... ( k  +  1
 ) ) --> A  /\  g  =  ( h  |`  ( M ... k
 ) )  /\  si ) ) )   &    |-  J  =  { g  |  E. n  e.  Z  (
 g : ( M
 ... n ) --> A  /\  ps ) }   &    |-  F  =  ( w  e.  Z ,  x  e.  J  |->  { h  |  E. k  e.  Z  ( h : ( M
 ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k
 ) )  /\  si ) } )   =>    |-  ( ph  ->  E. f
 ( f : Z --> A  /\  A. n  e.  Z  ch ) )
 
Theoremsdc 25807* Strong dependent choice. Suppose we may choose an element of  A such that property  ps holds, and suppose that if we have already chosen the first  k elements (represented here by a function from  1 ... k to  A), we may choose another element so that all  k  +  1 elements taken together have property  ps. Then there exists an infinite sequence of elements of  A such that the first  n terms of this sequence satisfy  ps for all  n. This theorem allows us to construct infinite seqeunces where each term depends on all the previous terms in the sequence. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  (
 g  =  ( f  |`  ( M ... n ) )  ->  ( ps  <->  ch ) )   &    |-  ( n  =  M  ->  ( ps  <->  ta ) )   &    |-  ( n  =  k  ->  ( ps  <->  th ) )   &    |-  ( ( g  =  h  /\  n  =  ( k  +  1 ) )  ->  ( ps 
 <-> 
 si ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. g ( g : { M } --> A  /\  ta ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( ( g : ( M ... k
 ) --> A  /\  th )  ->  E. h ( h : ( M ... ( k  +  1
 ) ) --> A  /\  g  =  ( h  |`  ( M ... k
 ) )  /\  si ) ) )   =>    |-  ( ph  ->  E. f ( f : Z --> A  /\  A. n  e.  Z  ch ) )
 
Theoremfdc 25808* Finite version of dependent choice. Construct a function whose value depends on the previous function value, except at a final point at which no new value can be chosen. The final hypothesis ensures that the process will terminate. The proof does not use the Axiom of Choice. (Contributed by Jeff Madsen, 18-Jun-2010.)
 |-  A  e.  _V   &    |-  M  e.  ZZ   &    |-  Z  =  ( ZZ>= `  M )   &    |-  N  =  ( M  +  1 )   &    |-  ( a  =  ( f `  (
 k  -  1 ) )  ->  ( ph  <->  ps ) )   &    |-  ( b  =  ( f `  k
 )  ->  ( ps  <->  ch ) )   &    |-  ( a  =  ( f `  n )  ->  ( th  <->  ta ) )   &    |-  ( et  ->  C  e.  A )   &    |-  ( et  ->  R  Fr  A )   &    |-  ( ( et 
 /\  a  e.  A )  ->  ( th  \/  E. b  e.  A  ph ) )   &    |-  ( ( ( et  /\  ph )  /\  ( a  e.  A  /\  b  e.  A ) )  ->  b R a )   =>    |-  ( et  ->  E. n  e.  Z  E. f ( f : ( M
 ... n ) --> A  /\  ( ( f `  M )  =  C  /\  ta )  /\  A. k  e.  ( N ... n ) ch )
 )
 
Theoremfdc1 25809* Variant of fdc 25808 with no specified base value. (Contributed by Jeff Madsen, 18-Jun-2010.)
 |-  A  e.  _V   &    |-  M  e.  ZZ   &    |-  Z  =  ( ZZ>= `  M )   &    |-  N  =  ( M  +  1 )   &    |-  ( a  =  ( f `  M )  ->  ( ze  <->  si ) )   &    |-  (
 a  =  ( f `
  ( k  -  1 ) )  ->  ( ph  <->  ps ) )   &    |-  (
 b  =  ( f `
  k )  ->  ( ps  <->  ch ) )   &    |-  (
 a  =  ( f `
  n )  ->  ( th  <->  ta ) )   &    |-  ( et  ->  E. a  e.  A  ze )   &    |-  ( et  ->  R  Fr  A )   &    |-  (
 ( et  /\  a  e.  A )  ->  ( th  \/  E. b  e.  A  ph ) )   &    |-  ( ( ( et 
 /\  ph )  /\  (
 a  e.  A  /\  b  e.  A )
 )  ->  b R a )   =>    |-  ( et  ->  E. n  e.  Z  E. f ( f : ( M
 ... n ) --> A  /\  ( si  /\  ta )  /\  A. k  e.  ( N ... n ) ch ) )
 
Theoremseqpo 25810* Two ways to say that a sequence respects a partial order. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( R  Po  A  /\  F : NN --> A ) 
 ->  ( A. s  e. 
 NN  ( F `  s ) R ( F `  ( s  +  1 ) )  <->  A. m  e.  NN  A. n  e.  ( ZZ>= `  ( m  +  1
 ) ) ( F `
  m ) R ( F `  n ) ) )
 
Theoremincsequz 25811* An increasing sequence of natural numbers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( F : NN --> NN  /\  A. m  e. 
 NN  ( F `  m )  <  ( F `
  ( m  +  1 ) )  /\  A  e.  NN )  ->  E. n  e.  NN  ( F `  n )  e.  ( ZZ>= `  A ) )
 
Theoremincsequz2 25812* An increasing sequence of natural numbers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( F : NN --> NN  /\  A. m  e. 
 NN  ( F `  m )  <  ( F `
  ( m  +  1 ) )  /\  A  e.  NN )  ->  E. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( F `
  k )  e.  ( ZZ>= `  A )
 )
 
Theoremnnubfi 25813* A bounded above set of natural numbers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)
 |-  (
 ( A  C_  NN  /\  B  e.  NN )  ->  { x  e.  A  |  x  <  B }  e.  Fin )
 
Theoremnninfnub 25814* An infinite set of natural numbers is unbounded above. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)
 |-  (
 ( A  C_  NN  /\ 
 -.  A  e.  Fin  /\  B  e.  NN )  ->  { x  e.  A  |  B  <  x }  =/= 
 (/) )
 
Theoremcsbrn 25815* Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  RR )   =>    |-  ( ph  ->  ( sum_ k  e.  A  ( B  x.  C ) ^ 2
 )  <_  ( sum_ k  e.  A  ( B ^ 2 )  x. 
 sum_ k  e.  A  ( C ^ 2 ) ) )
 
Theoremtrirn 25816* Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  RR )   =>    |-  ( ph  ->  ( sqr `  sum_ k  e.  A  ( ( B  +  C ) ^
 2 ) )  <_  ( ( sqr `  sum_ k  e.  A  ( B ^
 2 ) )  +  ( sqr `  sum_ k  e.  A  ( C ^
 2 ) ) ) )
 
16.14.4  Topology
 
TheoremunopnOLD 25817 The union of two open sets is open. (Moved to unopn 16597 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  u.  B )  e.  J )
 
TheoremincldOLD 25818 The intersection of two closed sets is closed. (Moved to incld 16728 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( J  e.  Top  /\  A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J )
 )  ->  ( A  i^i  B )  e.  ( Clsd `  J ) )
 
Theoremsubspopn 25819 An open set is open in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  (
 ( ( J  e.  Top  /\  A  e.  V ) 
 /\  ( B  e.  J  /\  B  C_  A ) )  ->  B  e.  ( Jt  A ) )
 
Theoremneificl 25820 Neighborhoods are closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Nov-2013.)
 |-  (
 ( ( J  e.  Top  /\  N  C_  ( ( nei `  J ) `  S ) )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) ) 
 ->  |^| N  e.  (
 ( nei `  J ) `  S ) )
 
Theoremlpss2 25821 Limit points of a subset are limit points of the larger set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  ( ( limPt `  J ) `  B )  C_  ( ( limPt `  J ) `  A ) )
 
16.14.5  Metric spaces
 
Theoremmetf1o 25822* Use a bijection with a metric space to construct a metric on a set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  N  =  ( x  e.  Y ,  y  e.  Y  |->  ( ( F `  x ) M ( F `  y ) ) )   =>    |-  ( ( Y  e.  A  /\  M  e.  ( Met `  X )  /\  F : Y -1-1-onto-> X )  ->  N  e.  ( Met `  Y ) )
 
Theoremblssp 25823 A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.)
 |-  N  =  ( M  |`  ( S  X.  S ) )   =>    |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X )  /\  ( Y  e.  S  /\  R  e.  RR+ ) )  ->  ( Y ( ball `  N ) R )  =  (
 ( Y ( ball `  M ) R )  i^i  S ) )
 
TheoremstiooOLD 25824 Two ways of expressing a subspace of  RR. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to resubmet 18256 in main set.mm and may be deleted by mathbox owner, SF. --MC 23-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  RR  ->  (
 ( topGen `  ran  (,) )t  A )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( A  X.  A ) ) ) )
 
TheoremblhalfOLD 25825 A ball of radius  R  /  2 is contained in a ball of radius  R centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to blhalf 17908 in main set.mm and may be deleted by mathbox owner, JM. --NM 21-May-2014.) (Revised by Mario Carneiro, 14-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( M  e.  ( Met `  X )  /\  Y  e.  X ) 
 /\  ( R  e.  RR+  /\  Z  e.  ( Y ( ball `  M )
 ( R  /  2
 ) ) ) ) 
 ->  ( Y ( ball `  M ) ( R 
 /  2 ) ) 
 C_  ( Z (
 ball `  M ) R ) )
 
Theoremmettrifi 25826* Generalized triangle inequality for arbitrary finite sums. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e.  X )   =>    |-  ( ph  ->  (
 ( F `  M ) D ( F `  N ) )  <_  sum_ k  e.  ( M
 ... ( N  -  1 ) ) ( ( F `  k
 ) D ( F `
  ( k  +  1 ) ) ) )
 
Theoremlmclim2 25827* A sequence in a metric space converges to a point iff the distance between the point and the elements of the sequence converges to 0. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
 |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  F : NN
 --> X )   &    |-  J  =  (
 MetOpen `  D )   &    |-  G  =  ( x  e.  NN  |->  ( ( F `  x ) D Y ) )   &    |-  ( ph  ->  Y  e.  X )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) Y  <->  G  ~~>  0 ) )
 
Theoremgeomcau 25828* If the distance between consecutive points in a sequence is bounded by a geometric sequence, then the sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
 |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  F : NN
 --> X )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  B  <  1 )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( ( F `  k ) D ( F `  ( k  +  1 ) ) )  <_  ( A  x.  ( B ^ k
 ) ) )   =>    |-  ( ph  ->  F  e.  ( Cau `  D ) )
 
Theoremcaures 25829 The restriction of a Cauchy sequence to a set of upper integers is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  F  e.  ( X  ^pm  CC ) )   =>    |-  ( ph  ->  ( F  e.  ( Cau `  D )  <->  ( F  |`  Z )  e.  ( Cau `  D ) ) )
 
Theoremcaushft 25830* A shifted Cauchy sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  W  =  (
 ZZ>= `  ( M  +  N ) )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( G `  (
 k  +  N ) ) )   &    |-  ( ph  ->  F  e.  ( Cau `  D ) )   &    |-  ( ph  ->  G : W --> X )   =>    |-  ( ph  ->  G  e.  ( Cau `  D )
 )
 
16.14.6  Continuous maps and homeomorphisms
 
Theoremconstcncf 25831* A constant function is a continuous function on  CC. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved into main set.mm as cncfmptc 18363 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  A )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremaddccncf 25832* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  ( x  +  A ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremidcncf 25833 The identity function is a continuous function on  CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 18364 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  x )   =>    |-  F  e.  ( CC
 -cn-> CC )
 
Theoremsub1cncf 25834* Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  ( x  -  A ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremsub2cncf 25835* Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  ( A  -  x ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremcnres2 25836* The restriction of a continuous function to a subset is continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A 
 C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K )  /\  A. x  e.  A  ( F `  x )  e.  B ) )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  B ) ) )
 
Theoremcnresima 25837 A continuous function is continuous onto its image. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Cn  K ) )  ->  F  e.  ( J  Cn  ( Kt  ran  F ) ) )
 
Theoremcncfres 25838* A continuous function on complex numbers restricted to a subset. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  A  C_ 
 CC   &    |-  B  C_  CC   &    |-  F  =  ( x  e.  CC  |->  C )   &    |-  G  =  ( x  e.  A  |->  C )   &    |-  ( x  e.  A  ->  C  e.  B )   &    |-  F  e.  ( CC -cn-> CC )   &    |-  J  =  (
 MetOpen `  ( ( abs 
 o.  -  )  |`  ( A  X.  A ) ) )   &    |-  K  =  (
 MetOpen `  ( ( abs 
 o.  -  )  |`  ( B  X.  B ) ) )   =>    |-  G  e.  ( J  Cn  K )
 
16.14.7  Product topologies
 
TheoremtxtopiOLD 25839 The product of two topologies is a topology. (Moved to txtopi 17233 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  R  e.  Top   &    |-  S  e.  Top   =>    |-  ( R  tX  S )  e.  Top
 
TheoremtxuniiOLD 25840 The underlying set of the product of two topologies. (Moved to txunii 17236 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  R  e.  Top   &    |-  S  e.  Top   &    |-  X  =  U. R   &    |-  Y  =  U. S   =>    |-  ( X  X.  Y )  =  U. ( R 
 tX  S )
 
TheoremtxopnOLD 25841 The product of two open sets is open in the product topology. (Moved to txopn 17245 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  T  =  ( R  tX  S )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( A  e.  R  /\  B  e.  S ) )  ->  ( A  X.  B )  e.  T )
 
TheoremtxcldOLD 25842 The product of two closed sets is closed in the product topology. (Moved to txcld 17246 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  T  =  ( R  tX  S )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( A  e.  ( Clsd `  R )  /\  B  e.  ( Clsd `  S ) ) )  ->  ( A  X.  B )  e.  ( Clsd `  T ) )
 
16.14.8  Boundedness
 
Syntaxctotbnd 25843 Extend class notation with the class of totally bounded metric spaces.
 class  TotBnd
 
Syntaxcbnd 25844 Extend class notation with the class of bounded metric spaces.
 class  Bnd
 
Definitiondf-totbnd 25845* 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 25846* 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 25847* 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 25848* 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 25849 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 25850 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 25851* 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 25852* 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 25853* 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 25854 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 25855* 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 25856* 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 25857* 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 25858 A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( M  e.  ( Bnd `  X )  ->  M  e.  ( Met `  X ) )
 
Theoremisbndx 25859* 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 25860* 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 25861* 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 25862* 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 25863 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 25864 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 25865* 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 25866 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 25846 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 25867* 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 25868 Lemma for equivbnd2 25869 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 25869* 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 25870* 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 25871* 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 25872* 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 25873 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 25874 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 ) ) )
 
16.14.9  Isometries
 
Syntaxcismty 25875 Extend class notation with the class of metric space isometries.
 class  Ismty
 
Definitiondf-ismty 25876* 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 25877* 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 25878* 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 25879 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 25880 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 25881 Lemma for ismtyhmeo 25882. (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 25882 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 25883 Lemma for ismtybnd 25884. (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 25884 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 25885 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 ) )
 
16.14.10  Heine-Borel Theorem
 
Theoremheibor1lem 25886 Lemma for heibor1 25887. 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 25887 One half of heibor 25898, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 18691 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 25888* Lemma for heibor 25898. 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 25889* Lemma for heibor 25898. 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 25890* Lemma for heibor 25898. Using countable choice ax-cc 8015, 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 25888 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 8015 via iunctb 8150), and so we can apply ax-cc 8015 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 25891* Lemma for heibor 25898. Using the function  T constructed in heiborlem3 25890, 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 25892* Lemma for heibor 25898. The function  M is a set of point-and-radius pairs suitable for application to caubl 18681. (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 25893* Lemma for heibor 25898. 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 25894* Lemma for heibor 25898. 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 25895* Lemma for heibor 25898. 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 25896* Lemma for heibor 25898. Discharge the hypotheses of heiborlem8 25895 by applying caubl 18681 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 25897* Lemma for heibor 25898. 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 25888, 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 25898 Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 25887 and heiborlem1 25888 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 ) ) )
 
16.14.11  Banach Fixed Point Theorem
 
Theorembfplem1 25899* Lemma for bfp 25901. 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 25900* Lemma for bfp 25901. Using the point found in bfplem1 25899, 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
 )
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