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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
21.3.8.3  Continuity - misc additions
 
Theoremhauseqcn 27501 In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.)
 |-  X  =  U. J   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( F  |`  A )  =  ( G  |`  A )
 )   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ph  ->  ( ( cls `  J ) `  A )  =  X )   =>    |-  ( ph  ->  F  =  G )
 
21.3.8.4  Topology of the closed unit
 
Theoremunitsscn 27502 The closed unit is a subset of the set of the complex numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 0 [,] 1 )  C_  CC
 
Theoremelunitrn 27503 The closed unit is a subset of the set of the real numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)
 |-  ( A  e.  ( 0 [,] 1 )  ->  A  e.  RR )
 
Theoremelunitcn 27504 The closed unit is a subset of the set of the complext numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)
 |-  ( A  e.  ( 0 [,] 1 )  ->  A  e.  CC )
 
Theoremelunitge0 27505 An element of the closed unit is positive Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 20-Dec-2016.)
 |-  ( A  e.  ( 0 [,] 1 )  ->  0  <_  A )
 
Theoremunitssxrge0 27506 The closed unit is a subset of the set of the extended nonnegative reals. Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 0 [,] 1 )  C_  ( 0 [,] +oo )
 
Theoremunitdivcld 27507 Necessary conditions for a quotient to be in the closed unit. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.)
 |-  (
 ( A  e.  (
 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  ( A  <_  B  <->  ( A  /  B )  e.  (
 0 [,] 1 ) ) )
 
Theoremiistmd 27508 The closed unit forms a topological monoid. (Contributed by Thierry Arnoux, 25-Mar-2017.)
 |-  I  =  ( (mulGrp ` fld )s  ( 0 [,] 1
 ) )   =>    |-  I  e. TopMnd
 
21.3.8.5  Topology of ` ( RR X. RR ) `
 
Theoremunicls 27509 The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  J  e.  Top   &    |-  X  =  U. J   =>    |-  U. ( Clsd `  J )  =  X
 
Theoremtpr2tp 27510 The usual topology on  ( RR  X.  RR ) is the product topology of the usual topology on  RR. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  ( J  tX  J )  e.  (TopOn `  ( RR  X. 
 RR ) )
 
Theoremtpr2uni 27511 The usual topology on  ( RR  X.  RR ) is the product topology of the usual topology on  RR. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  U. ( J  tX  J )  =  ( RR  X.  RR )
 
Theoremxpinpreima 27512 Rewrite the cartesian product of two sets as the intersection of their preimage by  1st and  2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  ( A  X.  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " B ) )
 
Theoremxpinpreima2 27513 Rewrite the cartesian product of two sets as the intersection of their preimage by  1st and  2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  (
 ( A  C_  E  /\  B  C_  F )  ->  ( A  X.  B )  =  ( ( `' ( 1st  |`  ( E  X.  F ) )
 " A )  i^i  ( `' ( 2nd  |`  ( E  X.  F ) ) " B ) ) )
 
Theoremsqsscirc1 27514 The complex square of side  D is a subset of the complex circle of radius  D. (Contributed by Thierry Arnoux, 25-Sep-2017.)
 |-  (
 ( ( ( X  e.  RR  /\  0  <_  X )  /\  ( Y  e.  RR  /\  0  <_  Y ) )  /\  D  e.  RR+ )  ->  ( ( X  <  ( D  /  2 ) 
 /\  Y  <  ( D  /  2 ) ) 
 ->  ( sqr `  (
 ( X ^ 2
 )  +  ( Y ^ 2 ) ) )  <  D ) )
 
Theoremsqsscirc2 27515 The complex square of side  D is a subset of the complex disc of radius  D. (Contributed by Thierry Arnoux, 25-Sep-2017.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC )  /\  D  e.  RR+ )  ->  ( ( ( abs `  ( Re `  ( B  -  A ) ) )  < 
 ( D  /  2
 )  /\  ( abs `  ( Im `  ( B  -  A ) ) )  <  ( D 
 /  2 ) ) 
 ->  ( abs `  ( B  -  A ) )  <  D ) )
 
Theoremcnre2csqlem 27516* Lemma for cnre2csqima 27517 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( G  |`  ( RR  X.  RR ) )  =  ( H  o.  F )   &    |-  F  Fn  ( RR  X.  RR )   &    |-  G  Fn  _V   &    |-  ( x  e.  ( RR  X. 
 RR )  ->  ( G `  x )  e. 
 RR )   &    |-  ( ( x  e.  ran  F  /\  y  e.  ran  F ) 
 ->  ( H `  ( x  -  y ) )  =  ( ( H `
  x )  -  ( H `  y ) ) )   =>    |-  ( ( X  e.  ( RR  X.  RR )  /\  Y  e.  ( RR 
 X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( `' ( G  |`  ( RR 
 X.  RR ) ) "
 ( ( ( G `
  X )  -  D ) (,) (
 ( G `  X )  +  D )
 ) )  ->  ( abs `  ( H `  ( ( F `  Y )  -  ( F `  X ) ) ) )  <  D ) )
 
Theoremcnre2csqima 27517* Image of a centered square by the canonical bijection from  ( RR  X.  RR ) to  CC. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y ) ) )   =>    |-  ( ( X  e.  ( RR  X.  RR )  /\  Y  e.  ( RR 
 X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( (
 ( ( 1st `  X )  -  D ) (,) ( ( 1st `  X )  +  D )
 )  X.  ( (
 ( 2nd `  X )  -  D ) (,) (
 ( 2nd `  X )  +  D ) ) ) 
 ->  ( ( abs `  ( Re `  ( ( F `
  Y )  -  ( F `  X ) ) ) )  <  D  /\  ( abs `  ( Im `  ( ( F `
  Y )  -  ( F `  X ) ) ) )  <  D ) ) )
 
Theoremtpr2rico 27518* For any point of an open set of the usual topology on  ( RR  X.  RR ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the  (
l ^ +oo ) norm  X. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  G  =  ( u  e.  RR ,  v  e.  RR  |->  ( u  +  ( _i  x.  v ) ) )   &    |-  B  =  ran  ( x  e.  ran  (,)
 ,  y  e.  ran  (,)  |->  ( x  X.  y
 ) )   =>    |-  ( ( A  e.  ( J  tX  J ) 
 /\  X  e.  A )  ->  E. r  e.  B  ( X  e.  r  /\  r  C_  A ) )
 
21.3.8.6  Order topology - misc. additions
 
Theoremcnvordtrestixx 27519* The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  A  C_  RR*   &    |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x [,] y )  C_  A )   =>    |-  ( (ordTop `  <_  )t  A )  =  (ordTop `  ( `'  <_  i^i  ( A  X.  A ) ) )
 
Theoremprsdm 27520 Domain of the relation of a preset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   =>    |-  ( K  e.  Preset  ->  dom  .<_  =  B )
 
Theoremprsrn 27521 Range of the relation of a preset. (Contributed by Thierry Arnoux, 11-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   =>    |-  ( K  e.  Preset  ->  ran  .<_  =  B )
 
Theoremprsss 27522 Relation of a subpreset. (Contributed by Thierry Arnoux, 13-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   =>    |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  (  .<_  i^i  ( A  X.  A ) )  =  ( ( le `  K )  i^i  ( A  X.  A ) ) )
 
Theoremprsssdm 27523 Domain of a subpreset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   =>    |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  dom  (  .<_  i^i  ( A  X.  A ) )  =  A )
 
Theoremordtprsval 27524* Value of the order topology for a preset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   &    |-  E  =  ran  ( x  e.  B  |->  { y  e.  B  |  -.  y  .<_  x }
 )   &    |-  F  =  ran  ( x  e.  B  |->  { y  e.  B  |  -.  x  .<_  y } )   =>    |-  ( K  e.  Preset  ->  (ordTop `  .<_  )  =  ( topGen `  ( fi `  ( { B }  u.  ( E  u.  F ) ) ) ) )
 
Theoremordtprsuni 27525* Value of the order topology. (Contributed by Thierry Arnoux, 13-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   &    |-  E  =  ran  ( x  e.  B  |->  { y  e.  B  |  -.  y  .<_  x }
 )   &    |-  F  =  ran  ( x  e.  B  |->  { y  e.  B  |  -.  x  .<_  y } )   =>    |-  ( K  e.  Preset  ->  B  =  U. ( { B }  u.  ( E  u.  F ) ) )
 
TheoremordtcnvNEW 27526 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.) (Revised by Thierry Arnoux, 13-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   =>    |-  ( K  e.  Preset  ->  (ordTop `  `'  .<_  )  =  (ordTop `  .<_  ) )
 
TheoremordtrestNEW 27527 The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   =>    |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  (ordTop `  (  .<_  i^i  ( A  X.  A ) ) )  C_  ( (ordTop `  .<_  )t  A ) )
 
Theoremordtrest2NEWlem 27528* Lemma for ordtrest2NEW 27529 (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   &    |-  ( ph  ->  K  e. Toset )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  { z  e.  B  |  ( x 
 .<_  z  /\  z  .<_  y ) }  C_  A )   =>    |-  ( ph  ->  A. v  e.  ran  ( z  e.  B  |->  { w  e.  B  |  -.  w  .<_  z }
 ) ( v  i^i 
 A )  e.  (ordTop `  (  .<_  i^i  ( A  X.  A ) ) ) )
 
Theoremordtrest2NEW 27529* An interval-closed set  A in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in  RR, but in other sets like  QQ there are interval-closed sets like  ( pi , +oo )  i^i  QQ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   &    |-  ( ph  ->  K  e. Toset )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  { z  e.  B  |  ( x 
 .<_  z  /\  z  .<_  y ) }  C_  A )   =>    |-  ( ph  ->  (ordTop `  (  .<_  i^i  ( A  X.  A ) ) )  =  ( (ordTop `  .<_  )t  A ) )
 
Theoremordtconlem1 27530* Connectedness in the order topology of a toset. This is the "easy" direction of ordtcon 27531. See also reconnlem1 21061. (Contributed by Thierry Arnoux, 14-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   &    |-  J  =  (ordTop `  .<_  )   =>    |-  ( ( K  e. Toset  /\  A  C_  B )  ->  ( ( Jt  A )  e.  Con  ->  A. x  e.  A  A. y  e.  A  A. r  e.  B  ( ( x 
 .<_  r  /\  r  .<_  y )  ->  r  e.  A ) ) )
 
Theoremordtcon 27531 Connectedness in the order topology of a complete uniform totally ordered space. (Contributed by Thierry Arnoux, 15-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( ( le `  K )  i^i  ( B  X.  B ) )   &    |-  J  =  (ordTop `  .<_  )   =>    |- T.
 
21.3.8.7  Continuity in topological spaces - misc. additions
 
Theoremmndpluscn 27532* A mapping that is both a homeomorphism and a monoid homomorphism preserves the "continuousness" of the operation. (Contributed by Thierry Arnoux, 25-Mar-2017.)
 |-  F  e.  ( J Homeo K )   &    |-  .+ 
 : ( B  X.  B ) --> B   &    |-  .*  : ( C  X.  C ) --> C   &    |-  J  e.  (TopOn `  B )   &    |-  K  e.  (TopOn `  C )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .*  ( F `  y ) ) )   &    |-  .+  e.  ( ( J 
 tX  J )  Cn  J )   =>    |- 
 .*  e.  ( ( K  tX  K )  Cn  K )
 
Theoremmhmhmeotmd 27533 Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
 |-  F  e.  ( S MndHom  T )   &    |-  F  e.  ( ( TopOpen `  S ) Homeo ( TopOpen `  T ) )   &    |-  S  e. TopMnd   &    |-  T  e.  TopSp   =>    |-  T  e. TopMnd
 
Theoremrmulccn 27534* Multiplication by a real constant is a continuous function (Contributed by Thierry Arnoux, 23-May-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( x  e.  RR  |->  ( x  x.  C ) )  e.  ( J  Cn  J ) )
 
Theoremraddcn 27535* Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  y ) )  e.  ( ( J  tX  J )  Cn  J )
 
Theoremxrmulc1cn 27536* The operation multiplying an extended real number by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  J  =  (ordTop `  <_  )   &    |-  F  =  ( x  e.  RR*  |->  ( x xe C ) )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  F  e.  ( J  Cn  J ) )
 
Theoremfmcncfil 27537 The image of a Cauchy filter by a continuous filter map is a Cauchy filter. (Contributed by Thierry Arnoux, 12-Nov-2017.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  ( MetOpen `  E )   =>    |-  (
 ( ( D  e.  ( CMet `  X )  /\  E  e.  ( *Met `  Y )  /\  F  e.  ( J  Cn  K ) )  /\  B  e.  (CauFil `  D ) )  ->  ( ( Y  FilMap  F ) `  B )  e.  (CauFil `  E ) )
 
21.3.8.8  Topology of the extended nonnegative real numbers ordered monoid
 
Theoremxrge0hmph 27538 The extended nonnegative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  II  ~=  ( (ordTop `  <_  )t  ( 0 [,] +oo )
 )
 
Theoremxrge0iifcnv 27539* Define a bijection from  [ 0 ,  1 ] to  [ 0 , +oo ]. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 , +oo ,  -u ( log `  x ) ) )   =>    |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  /\  `' F  =  (
 y  e.  ( 0 [,] +oo )  |->  if (
 y  = +oo , 
 0 ,  ( exp `  -u y ) ) ) )
 
Theoremxrge0iifcv 27540* The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 , +oo ,  -u ( log `  x ) ) )   =>    |-  ( X  e.  (
 0 (,] 1 )  ->  ( F `  X )  =  -u ( log `  X ) )
 
Theoremxrge0iifiso 27541* The defined bijection from the closed unit interval and the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 , +oo ,  -u ( log `  x ) ) )   =>    |-  F  Isom  <  ,  `'  <  ( ( 0 [,] 1 ) ,  (
 0 [,] +oo ) )
 
Theoremxrge0iifhmeo 27542* Expose a homeomorphism from the closed unit interval and the extended nonnegative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 , +oo ,  -u ( log `  x ) ) )   &    |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )   =>    |-  F  e.  ( II Homeo J )
 
Theoremxrge0iifhom 27543* The defined function from the closed unit interval and the extended nonnegative reals is also a monoid homomorphism. (Contributed by Thierry Arnoux, 5-Apr-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 , +oo ,  -u ( log `  x ) ) )   &    |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )   =>    |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1
 ) )  ->  ( F `  ( X  x.  Y ) )  =  ( ( F `  X ) +e
 ( F `  Y ) ) )
 
Theoremxrge0iif1 27544* Condition for the defined function,  -u ( log `  x
) to be a monoid homomorphism. (Contributed by Thierry Arnoux, 20-Jun-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 , +oo ,  -u ( log `  x ) ) )   &    |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )   =>    |-  ( F `  1 )  =  0
 
Theoremxrge0iifmhm 27545* The defined function from the closed unit interval and the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 , +oo ,  -u ( log `  x ) ) )   &    |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )   =>    |-  F  e.  (
 ( (mulGrp ` fld )s  ( 0 [,] 1
 ) ) MndHom  ( RR*ss  ( 0 [,] +oo )
 ) )
 
Theoremxrge0pluscn 27546* The addition operation of the extended nonnegative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 , +oo ,  -u ( log `  x ) ) )   &    |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )   &    |-  .+  =  ( +e  |`  ( ( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) )   =>    |-  .+  e.  (
 ( J  tX  J )  Cn  J )
 
Theoremxrge0mulc1cn 27547* The operation multiplying a nonnegative real numbers by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
 )   &    |-  F  =  ( x  e.  ( 0 [,] +oo )  |->  ( x xe C ) )   &    |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )   =>    |-  ( ph  ->  F  e.  ( J  Cn  J ) )
 
Theoremxrge0tps 27548 The extended nonnegative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.)
 |-  ( RR*ss  ( 0 [,] +oo ) )  e.  TopSp
 
Theoremxrge0topn 27549 The topology of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.)
 |-  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
 )
 
Theoremxrge0haus 27550 The topology of the extended nonnegative real numbers is Hausdorff. (Contributed by Thierry Arnoux, 26-Jul-2017.)
 |-  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )  e. 
 Haus
 
Theoremxrge0tmdOLD 27551 The extended nonnegative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( RR*ss  ( 0 [,] +oo ) )  e. TopMnd
 
Theoremxrge0tmd 27552 The extended nonnegative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof Shortened by Thierry Arnoux, 21-Jun-2017.)
 |-  ( RR*ss  ( 0 [,] +oo ) )  e. TopMnd
 
21.3.8.9  Limits - misc additions
 
Theoremlmlim 27553 Relate a limit in a given topology to a complex number limit, provided that topology agrees with the common topology on  CC on the required subset. (Contributed by Thierry Arnoux, 11-Jul-2017.)
 |-  J  e.  (TopOn `  Y )   &    |-  ( ph  ->  F : NN --> X )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( Jt  X )  =  (
 ( TopOpen ` fld )t  X )   &    |-  X  C_  CC   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  F  ~~>  P ) )
 
Theoremlmlimxrge0 27554 Relate a limit in the nonnegative extended reals to a complex limit, provided the considered function is a real function. (Contributed by Thierry Arnoux, 11-Jul-2017.)
 |-  J  =  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )   &    |-  ( ph  ->  F : NN
 --> X )   &    |-  ( ph  ->  P  e.  X )   &    |-  X  C_  ( 0 [,) +oo )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  F  ~~>  P ) )
 
Theoremrge0scvg 27555 Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 14289 (Contributed by Thierry Arnoux, 28-Jul-2017.)
 |-  (
 ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  sup ( ran  seq 1 (  +  ,  F ) ,  RR ,  <  )  e.  RR )
 
Theoremfsumcvg4 27556 A serie with finite support is a finite sum, and therefore converges. (Contributed by Thierry Arnoux, 6-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
 |-  S  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : S --> CC )   &    |-  ( ph  ->  ( `' F " ( CC  \  {
 0 } ) )  e.  Fin )   =>    |-  ( ph  ->  seq
 M (  +  ,  F )  e.  dom  ~~>  )
 
Theorempnfneige0 27557* A neighborhood of +oo contains an unbounded interval based at a real number. See pnfnei 19482 (Contributed by Thierry Arnoux, 31-Jul-2017.)
 |-  J  =  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )   =>    |-  ( ( A  e.  J  /\ +oo  e.  A )  ->  E. x  e.  RR  ( x (,] +oo )  C_  A )
 
Theoremlmxrge0 27558* Express "sequence  F converges to plus infinity" (i.e. diverges), for a sequence of nonnegative extended real numbers. (Contributed by Thierry Arnoux, 2-Aug-2017.)
 |-  J  =  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )   &    |-  ( ph  ->  F : NN
 --> ( 0 [,] +oo ) )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k
 )  =  A )   =>    |-  ( ph  ->  ( F (
 ~~> t `  J ) +oo 
 <-> 
 A. x  e.  RR  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) x  <  A ) )
 
Theoremlmdvg 27559* If a monotonic sequence of real numbers diverges, it is unbounded. (Contributed by Thierry Arnoux, 4-Aug-2017.)
 |-  ( ph  ->  F : NN --> ( 0 [,) +oo ) )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   &    |-  ( ph  ->  -.  F  e.  dom  ~~>  )   =>    |-  ( ph  ->  A. x  e.  RR  E. j  e. 
 NN  A. k  e.  ( ZZ>=
 `  j ) x  <  ( F `  k ) )
 
Theoremlmdvglim 27560* If a monotonic real number sequence 
F diverges, it converges in the extended real numbers and its limit is plus infinity. (Contributed by Thierry Arnoux, 3-Aug-2017.)
 |-  J  =  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )   &    |-  ( ph  ->  F : NN
 --> ( 0 [,) +oo ) )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   &    |-  ( ph  ->  -.  F  e.  dom  ~~>  )   =>    |-  ( ph  ->  F (
 ~~> t `  J ) +oo )
 
21.3.8.10  Univariate polynomials
 
Theorempl1cn 27561 A univariate polynomial is continuous. (Contributed by Thierry Arnoux, 17-Sep-2018.)
 |-  P  =  (Poly1 `  R )   &    |-  E  =  (eval1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  ( Base `  R )   &    |-  J  =  ( TopOpen `  R )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  R  e.  TopRing )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( E `  F )  e.  ( J  Cn  J ) )
 
21.3.9  Uniform Stuctures and Spaces
 
21.3.9.1  Hausdorff uniform completion
 
Syntaxchcmp 27562 Extend class notation with the Hausdorff uniform completion relation.
 class HCmp
 
Definitiondf-hcmp 27563* Definition of the Hausdorff completion. In this definition, a structure  w is a Hausdorff completion of a uniform structure  u if  w is a complete uniform space, in which  u is dense, and which admits the same uniform structure. Theorem 3 of [BourbakiTop1] p. II.21. states the existence and unicity of such a completion. (Contributed by Thierry Arnoux, 5-Mar-2018.)
 |- HCmp  =  { <. u ,  w >.  |  ( ( u  e. 
 U. ran UnifOn  /\  w  e. CUnifSp )  /\  ( (UnifSt `  w )t  dom  U. u )  =  u  /\  ( ( cls `  ( TopOpen `  w ) ) `  dom  U. u )  =  (
 Base `  w ) ) }
 
21.3.10  Topology and algebraic structures
 
21.3.10.1  The norm on the ring of the integer numbers
 
Theoremzringnm 27564 The norm (function) for a ring of integers is the absolute value function (restricted to the integers). (Contributed by AV, 13-Jun-2019.)
 |-  ( norm ` ring )  =  ( abs  |`  ZZ )
 
Theoremzzsnm 27565 The norm of the ring of the integers (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 13-Jun-2019.)
 |-  ( M  e.  ZZ  ->  ( abs `  M )  =  ( ( norm ` ring ) `  M ) )
 
TheoremzzsnmOLD 27566 The norm of the ring of the integers (Contributed by Thierry Arnoux, 8-Nov-2017.) Obsolete version of zzsnm 27565 as of 13-Jun-2019. (New usage is discouraged.)
 |-  Z  =  (flds  ZZ )   =>    |-  ( M  e.  ZZ  ->  ( abs `  M )  =  ( ( norm `  Z ) `  M ) )
 
21.3.10.2  Topological ` ZZ ` -modules
 
Theoremzlm0 27567 Zero of a  ZZ-module. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   &    |- 
 .0.  =  ( 0g `  G )   =>    |- 
 .0.  =  ( 0g `  W )
 
Theoremzlm1 27568 Unit of a  ZZ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   &    |- 
 .1.  =  ( 1r `  G )   =>    |- 
 .1.  =  ( 1r `  W )
 
Theoremzlmds 27569 Distance in a  ZZ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( G  e.  V  ->  D  =  ( dist `  W ) )
 
Theoremzlmtset 27570 Topology in a  ZZ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   &    |-  J  =  (TopSet `  G )   =>    |-  ( G  e.  V  ->  J  =  (TopSet `  W ) )
 
Theoremzlmnm 27571 Norm of a  ZZ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   &    |-  N  =  ( norm `  G )   =>    |-  ( G  e.  V  ->  N  =  ( norm `  W ) )
 
Theoremzhmnrg 27572 The  ZZ-module built from a normed ring is also a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   =>    |-  ( G  e. NrmRing  ->  W  e. NrmRing )
 
Theoremnmmulg 27573 The norm of a group product, provided the  ZZ-module is normed. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  B  =  ( Base `  R )   &    |-  N  =  ( norm `  R )   &    |-  Z  =  ( ZMod `  R )   &    |- 
 .x.  =  (.g `  R )   =>    |-  ( ( Z  e. NrmMod  /\  M  e.  ZZ  /\  X  e.  B )  ->  ( N `  ( M  .x.  X ) )  =  ( ( abs `  M )  x.  ( N `  X ) ) )
 
Theoremzrhnm 27574 The norm of the image by  ZRHom of an integer in a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  B  =  ( Base `  R )   &    |-  N  =  ( norm `  R )   &    |-  Z  =  ( ZMod `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  ( ( ( Z  e. NrmMod  /\  Z  e. NrmRing  /\  R  e. NzRing )  /\  M  e.  ZZ )  ->  ( N `
  ( L `  M ) )  =  ( abs `  M ) )
 
Theoremcnzh 27575 The  ZZ-module of  CC is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.)
 |-  ( ZMod ` fld )  e. NrmMod
 
Theoremrezh 27576 The  ZZ-module of  RR is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.)
 |-  ( ZMod ` RRfld )  e. NrmMod
 
21.3.10.3  Canonical embedding of the field of the rational numbers into a division ring
 
Syntaxcqqh 27577 Map the rationals into a field.
 class QQHom
 
Definitiondf-qqh 27578* Define the canonical homomorphism from the rationals into any field. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.)
 |- QQHom  =  ( r  e.  _V  |->  ran  ( x  e.  ZZ ,  y  e.  ( `' ( ZRHom `  r
 ) " (Unit `  r
 ) )  |->  <. ( x 
 /  y ) ,  ( ( ( ZRHom `  r ) `  x ) (/r `  r ) ( ( ZRHom `  r
 ) `  y )
 ) >. ) )
 
Theoremqqhval 27579* Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  ./  =  (/r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  ( R  e.  _V  ->  (QQHom `  R )  =  ran  ( x  e. 
 ZZ ,  y  e.  ( `' L "
 (Unit `  R )
 )  |->  <. ( x  /  y ) ,  (
 ( L `  x )  ./  ( L `  y ) ) >. ) )
 
Theoremzrhf1ker 27580 The kernel of the homomorphism from the integers to a ring, if it is injective. (Contributed by Thierry Arnoux, 26-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  L  =  ( ZRHom `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( L : ZZ -1-1-> B  <->  ( `' L " {  .0.  } )  =  { 0 } ) )
 
Theoremzrhchr 27581 The kernel of the homomorphism from the integers to a ring is injective if and only if the ring has characteristic 0 . (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  B  =  ( Base `  R )   &    |-  L  =  ( ZRHom `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( (chr `  R )  =  0  <->  L : ZZ -1-1-> B ) )
 
Theoremzrhker 27582 The kernel of the homomorphism from the integers to a ring with characteristic 0. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  B  =  ( Base `  R )   &    |-  L  =  ( ZRHom `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( (chr `  R )  =  0  <->  ( `' L " {  .0.  } )  =  { 0 } )
 )
 
Theoremzrhunitpreima 27583 The preimage by  ZRHom of the unit of a division ring is  ( ZZ  \  { 0 } ). (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  L  =  ( ZRHom `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( `' L " (Unit `  R ) )  =  ( ZZ  \  {
 0 } ) )
 
Theoremelzrhunit 27584 Condition for the image by  ZRHom to be a unit. (Contributed by Thierry Arnoux, 30-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  L  =  ( ZRHom `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  ->  ( L `  M )  e.  (Unit `  R ) )
 
Theoremelzdif0 27585 Lemma for qqhval2 27587 (Contributed by Thierry Arnoux, 29-Oct-2017.)
 |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  e.  NN  \/  -u M  e.  NN ) )
 
Theoremqqhval2lem 27586 Lemma for qqhval2 27587 (Contributed by Thierry Arnoux, 29-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ  /\  Y  =/=  0 ) ) 
 ->  ( ( L `  (numer `  ( X  /  Y ) ) ) 
 ./  ( L `  (denom `  ( X  /  Y ) ) ) )  =  ( ( L `  X ) 
 ./  ( L `  Y ) ) )
 
Theoremqqhval2 27587* Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 26-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R )  =  ( q  e.  QQ  |->  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) )
 
Theoremqqhvval 27588 Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( (QQHom `  R ) `  Q )  =  ( ( L `  (numer `  Q ) ) 
 ./  ( L `  (denom `  Q ) ) ) )
 
Theoremqqh0 27589 The image of  0 by the QQHom homomorphism is the ring's zero. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (
 (QQHom `  R ) `  0 )  =  ( 0g `  R ) )
 
Theoremqqh1 27590 The image of  1 by the QQHom homomorphism is the ring's unit. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (
 (QQHom `  R ) `  1 )  =  ( 1r `  R ) )
 
Theoremqqhf 27591 QQHom as a function. (Contributed by Thierry Arnoux, 28-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R ) : QQ --> B )
 
Theoremqqhvq 27592 The image of a quotient by the QQHom homomorphism. (Contributed by Thierry Arnoux, 28-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ  /\  Y  =/=  0 ) ) 
 ->  ( (QQHom `  R ) `  ( X  /  Y ) )  =  ( ( L `  X )  ./  ( L `
  Y ) ) )
 
Theoremqqhghm 27593 The QQHom homomorphism is a group homomorphism if the target structure is a division ring. (Contributed by Thierry Arnoux, 9-Nov-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   &    |-  Q  =  (flds  QQ )   =>    |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R )  e.  ( Q  GrpHom  R ) )
 
Theoremqqhrhm 27594 The QQHom homomorphism is a ring homomorphism if the target structure is a field. If the target structure is a division ring, it is a group homomorphism, but not a ring homomorphism, because it does not preserve the ring multiplication operation. (Contributed by Thierry Arnoux, 29-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   &    |-  Q  =  (flds  QQ )   =>    |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  (QQHom `  R )  e.  ( Q RingHom  R )
 )
 
Theoremqqhnm 27595 The norm of the image by QQHom of a rational number in a topological division ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  N  =  ( norm `  R )   &    |-  Z  =  ( ZMod `  R )   =>    |-  ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0
 )  /\  Q  e.  QQ )  ->  ( N `
  ( (QQHom `  R ) `  Q ) )  =  ( abs `  Q ) )
 
Theoremqqhcn 27596 The QQHom homomorphism is a continuous function. (Contributed by Thierry Arnoux, 9-Nov-2017.)
 |-  Q  =  (flds  QQ )   &    |-  J  =  ( TopOpen `  Q )   &    |-  Z  =  ( ZMod `  R )   &    |-  K  =  ( TopOpen `  R )   =>    |-  (
 ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R )  e.  ( J  Cn  K ) )
 
Theoremqqhucn 27597 The QQHom homomorphism is uniformly continuous. (Contributed by Thierry Arnoux, 28-Jan-2018.)
 |-  B  =  ( Base `  R )   &    |-  Q  =  (flds  QQ )   &    |-  U  =  (UnifSt `  Q )   &    |-  V  =  (metUnif `  (
 ( dist `  R )  |`  ( B  X.  B ) ) )   &    |-  Z  =  ( ZMod `  R )   &    |-  ( ph  ->  R  e. NrmRing )   &    |-  ( ph  ->  R  e.  DivRing )   &    |-  ( ph  ->  Z  e. NrmMod )   &    |-  ( ph  ->  (chr `  R )  =  0 )   =>    |-  ( ph  ->  (QQHom `  R )  e.  ( U Cnu
 V ) )
 
21.3.10.4  Canonical embedding of the real numbers into a complete ordered field
 
Syntaxcrrh 27598 Map the real numbers into a complete field.
 class RRHom
 
Syntaxcrrext 27599 Extend class notation with the class of extension fields of  RR.
 class ℝExt
 
Definitiondf-rrh 27600 Define the canonical homomorphism from the real numbers to any complete field, as the extension by continuity of the canonical homomorphism from the rational numbers. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.)
 |- RRHom  =  ( r  e.  _V  |->  ( ( ( topGen `  ran  (,) )CnExt ( TopOpen `  r
 ) ) `  (QQHom `  r ) ) )
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