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Theorem List for Metamath Proof Explorer - 25601-25700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrdr 25601 Two ways of expressing the remainder when  A is divided by 
B. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  -  ( A  mod  B ) )  /  B )  =  ( |_ `  ( A  /  B ) ) )
 
TheoremeluzaddOLD 25602 Membership in a later set of upper integers. (Moved to eluzadd 10135 in main set.mm and may be deleted by mathbox owner, JM. --NM 2-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( N  e.  ( ZZ>=
 `  M )  /\  K  e.  ZZ )  ->  ( N  +  K )  e.  ( ZZ>= `  ( M  +  K ) ) )
 
TheoremeluzsubOLD 25603 Membership in an earlier set of upper integers. (Moved to eluzsub 10136 in main set.mm and may be deleted by mathbox owner, JM. --NM 2-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  K ) ) )  ->  ( N  -  K )  e.  ( ZZ>= `  M ) )
 
Theoremuzm1OLD 25604 Choices for an element of an upper interval of integers. (Moved to uzm1 10137 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-May-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( N  =  M  \/  ( N  -  1 )  e.  ( ZZ>= `  M )
 ) )
 
Theoremuzp1OLD 25605 Choices for an element of an upper interval of integers. (Moved to uzp1 10140 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-May-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
 
TheoremfzfiOLD 25606 A finite interval of integers is finite. (Moved to fzfi 10912 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( M
 ... N )  e. 
 Fin )
 
Theoremfzfi2OLD 25607 Variant of fzfi 10912 with hypothesis weakened. (Moved to fzfi 10912 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( N  e.  A  ->  ( M ... N )  e.  Fin )
 
Theoremfz10OLD 25608 There are no integers between 1 and 0. (Moved to fz10 10692 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 1 ... 0 )  =  (/)
 
Theoremfzmul 25609 Membership of a product in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  NN )  ->  ( J  e.  ( M ... N )  ->  ( K  x.  J )  e.  ( ( K  x.  M ) ... ( K  x.  N ) ) ) )
 
Theoremfzadd2 25610 Membership of a sum in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  (
 ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( O  e.  ZZ  /\  P  e.  ZZ ) )  ->  ( ( J  e.  ( M
 ... N )  /\  K  e.  ( O ... P ) )  ->  ( J  +  K )  e.  ( ( M  +  O ) ... ( N  +  P ) ) ) )
 
TheoremfzsplitOLD 25611 Split a finite interval of integers into two parts. (Moved to fzsplit 10694 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( N  e.  A  /\  K  e.  ( M
 ... N ) ) 
 ->  ( M ... N )  =  ( ( M ... K )  u.  ( ( K  +  1 ) ... N ) ) )
 
TheoremfzdisjOLD 25612 Condition for two finite intervals of integers to be disjoint. (Moved to fzdisj 10695 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( K  e.  A  /\  K  <  M ) 
 ->  ( ( J ... K )  i^i  ( M
 ... N ) )  =  (/) )
 
Theoremfzp1elp1OLD 25613 Add one to an element of a finite set of integers. (Moved to fzp1elp1 10717 in main set.mm and may be deleted by mathbox owner, JM. --NM 28-Feb-2014.) (Contributed by Jeff Madsen, 6-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( N  e.  A  /\  K  e.  ( M
 ... N ) ) 
 ->  ( K  +  1 )  e.  ( M
 ... ( N  +  1 ) ) )
 
TheoremabszOLD 25614 A real number is an integer iff its absolute value is an integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to absz 11673 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 25615  A  mod  B is zero iff  A is evenly divisible by  B. (Moved to mod0 10856 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 25616  A is divisible by  B iff its negative is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to negmod0 10857 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 25617  A is divisible by  B iff its absolute value is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to absmod0 11665 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 )
 )
 
16.13.3  Sequences and sums
 
Theoremsdclem2 25618* Lemma for sdc 25620. (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 25619* Lemma for sdc 25620. (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 25620* 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 25621* 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 25622* Variant of fdc 25621 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 25623* 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 25624* 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 25625* 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 25626* 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 25627* 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 25628* 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 25629* 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 ) ) ) )
 
16.13.4  Topology
 
TheoremunopnOLD 25630 The union of two open sets is open. (Moved to unopn 16481 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 25631 The intersection of two closed sets is closed. (Moved to incld 16612 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 25632 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 25633 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 25634 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 ) )
 
16.13.5  Metric spaces
 
Theoremmetf1o 25635* 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 25636 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 25637 Two ways of expressing a subspace of  RR. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to resubmet 18140 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 25638 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 17792 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 25639* 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 25640* 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 25641* 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 25642 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 25643* 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 )
 )
 
16.13.6  Continuous maps and homeomorphisms
 
Theoremconstcncf 25644* A constant function is a continuous function on  CC. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved into main set.mm as cncfmptc 18247 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 25645* 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 25646 The identity function is a continuous function on  CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 18248 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 25647* 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 25648* 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 25649* 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 25650 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 25651* 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 )
 
16.13.7  Product topologies
 
TheoremtxtopiOLD 25652 The product of two topologies is a topology. (Moved to txtopi 17117 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 25653 The underlying set of the product of two topologies. (Moved to txunii 17120 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 25654 The product of two open sets is open in the product topology. (Moved to txopn 17129 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 25655 The product of two closed sets is closed in the product topology. (Moved to txcld 17130 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 ) )
 
16.13.8  Boundedness
 
Syntaxctotbnd 25656 Extend class notation with the class of totally bounded metric spaces.
 class  TotBnd
 
Syntaxcbnd 25657 Extend class notation with the class of bounded metric spaces.
 class  Bnd
 
Definitiondf-totbnd 25658* 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 25659* 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 25660* 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 25661* 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 25662 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 25663 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 25664* 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 25665* 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 25666* 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 25667 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 25668* 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 25669* 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 25670* 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 25671 A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( M  e.  ( Bnd `  X )  ->  M  e.  ( Met `  X ) )
 
Theoremisbndx 25672* 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 25673* 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 25674* 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 25675* 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 25676 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 25677 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 25678* 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 25679 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 25659 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 25680* 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 25681 Lemma for equivbnd2 25682 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 25682* 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 25683* 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 25684* 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 25685* 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 25686 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 25687 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 ) ) )
 
16.13.9  Isometries
 
Syntaxcismty 25688 Extend class notation with the class of metric space isometries.
 class  Ismty
 
Definitiondf-ismty 25689* 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 25690* 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 25691* 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 25692 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 25693 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 25694 Lemma for ismtyhmeo 25695. (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 25695 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 25696 Lemma for ismtybnd 25697. (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 25697 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 25698 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 ) )
 
16.13.10  Heine-Borel Theorem
 
Theoremheibor1lem 25699 Lemma for heibor1 25700. 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 25700 One half of heibor 25711, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 18575 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 ) ) )
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