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Theorem List for Metamath Proof Explorer - 23301-23400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiistmd 23301 The closed unit monoid is a topological monoid. (Contributed by Thierry Arnoux, 25-Mar-2017.)
 |-  I  =  ( (mulGrp ` fld )s  ( 0 [,] 1
 ) )   =>    |-  I  e. TopMnd
 
18.3.20  Topology of ` ( RR X. RR ) `
 
Theoremtpr2tp 23302 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 23303 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 )
 
Theoremclduni 23304 For any topology, the union of the closed sets is the base set. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  ( J  e.  Top  ->  U. ( Clsd `  J )  = 
 U. J )
 
Theoremxpinpreima 23305 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 23306 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 23307 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 23308 The complex square of side  D is a subset of the complex circle 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 23309* Lemma for cnre2csqima 23310 (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 23310* 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 23311* For any point of an open set of the usual topology on  ( RR  X.  RR ) there is a closed below opened above square which contains that point and is entirely in the open set. This is square is actually similar to a ball by the  ( l ^  +oo ) norm, closed below, centered on  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 ) )
 
18.3.21  Order topology - misc. additions
 
Theoremcnvordtrestixx 23312* 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 ) ) )
 
18.3.22  Continuity in topological spaces - misc. additions
 
Theoremressplusf 23313 The group operation function  + f of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.)
 |-  B  =  ( Base `  G )   &    |-  H  =  ( Gs  A )   &    |-  .+^  =  ( +g  `  G )   &    |-  .+^  Fn  ( B  X.  B )   &    |-  A  C_  B   =>    |-  ( + f `  H )  =  (  .+^  |`  ( A  X.  A ) )
 
Theoremmndpluscn 23314* 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 23315 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 23316* 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 23317* 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 23318* The operation multiplying an extended real number by a non-negative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  J  =  (ordTop `  <_  )   &    |-  F  =  ( x  e.  RR*  |->  ( x x e C ) )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  F  e.  ( J  Cn  J ) )
 
18.3.23  Extended reals Structure - misc additions
 
Theoremxaddeq0 23319 Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A + e B )  =  0  <->  A  =  - e B ) )
 
Theoremxrs0 23320 The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 10585 and df-xrs 13419), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.)
 |-  0  =  ( 0g `  RR* s )
 
Theoremxrsinvgval 23321 The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 10585 and df-xrs 13419), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.)
 |-  ( B  e.  RR*  ->  (
 ( inv g `  RR* s ) `
  B )  =  - e B )
 
Theoremxrsmulgzz 23322 The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  RR* )  ->  ( A (.g `  RR* s ) B )  =  ( A x e B ) )
 
Theoremressmulgnn 23323 Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 12-Jun-2017.)
 |-  H  =  ( Gs  A )   &    |-  A  C_  ( Base `  G )   &    |-  .*  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( N  e.  NN  /\  X  e.  A )  ->  ( N (.g `  H ) X )  =  ( N  .*  X ) )
 
Theoremressmulgnn0 23324 Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 14-Jun-2017.)
 |-  H  =  ( Gs  A )   &    |-  A  C_  ( Base `  G )   &    |-  .*  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   &    |-  ( 0g `  G )  =  ( 0g `  H )   =>    |-  ( ( N  e.  NN0  /\  X  e.  A )  ->  ( N (.g `  H ) X )  =  ( N  .*  X ) )
 
18.3.24  The extended non-negative real numbers monoid
 
Theoremxrge0base 23325 The base of the extended non-negative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  (
 0 [,]  +oo )  =  ( Base `  ( RR* ss  ( 0 [,]  +oo ) ) )
 
Theoremxrge00 23326 The zero of the extended non-negative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  0  =  ( 0g `  ( RR* ss  ( 0 [,]  +oo ) ) )
 
Theoremxrge0plusg 23327 The additive law of the extended non-negative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.)
 |-  + e  =  ( +g  `  ( RR* ss  ( 0 [,]  +oo ) ) )
 
Theoremxrge0mulgnn0 23328 The group multiple function in the extended non-negative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
 |-  (
 ( A  e.  NN0  /\  B  e.  ( 0 [,]  +oo ) )  ->  ( A (.g `  ( RR* ss  ( 0 [,]  +oo ) ) ) B )  =  ( A x e B ) )
 
Theoremxrge0hmph 23329 The extended non-negative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  II  ~=  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
 )
 
Theoremxrge0iifcnv 23330* 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 23331* 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 23332* The defined bijection from the closed unit interval and the extended non-negative 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 23333* Expose a homeomorphism from the closed unit interval and the extended non-negative 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 23334* The defined function from the closed unit interval and the extended non-negative 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 23335* 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 23336* The defined function from the closed unit interval and the extended non-negative 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 23337* The addition operation of the extended non-negative 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 23338* The operation multiplying a non-negative real numbers by a non-negative constant is continuous. (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
 )   &    |-  F  =  ( x  e.  ( 0 [,]  +oo )  |->  ( x x e C ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  F  e.  ( J  Cn  J ) )
 
Theoremxrge0tps 23339 The extended non-negative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e.  TopSp
 
Theoremxrge0topn 23340 The topology of the extended non-negative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.)
 |-  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
 )
 
Theoremxrge0haus 23341 The topology of the extended non-negative real numbers is Hausdorf. (Contributed by Thierry Arnoux, 26-Jul-2017.)
 |-  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )  e. 
 Haus
 
Theoremxrge0tmdALT 23342 The extended non-negative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e. TopMnd
 
Theoremxrge0tmd 23343 The extended non-negative 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
 
Theoremxrge0addass 23344 Associativity of extended non-negative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)
 |-  (
 ( A  e.  (
 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  C  e.  ( 0 [,]  +oo ) )  ->  ( ( A + e B ) + e C )  =  ( A + e ( B + e C ) ) )
 
Theoremxrge0neqmnf 23345 An extended non-negative real cannot be minus infinity. (Contributed by Thierry Arnoux, 9-Jun-2017.)
 |-  ( A  e.  ( 0 [,]  +oo )  ->  A  =/=  -oo )
 
Theoremxrge0addgt0 23346 The sum of nonnegative and positive numbers is positive. See addgtge0 9278 (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  (
 ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
 0 [,]  +oo ) ) 
 /\  0  <  A )  ->  0  <  ( A + e B ) )
 
Theoremxrge0adddir 23347 Distributivity of extended non-negative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)
 |-  (
 ( A  e.  (
 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  C  e.  ( 0 [,]  +oo ) )  ->  ( ( A + e B ) x e C )  =  ( ( A x e C ) + e ( B x e C ) ) )
 
Theoremxrge0npcan 23348 Extended non-negative real version of npcan 9076. (Contributed by Thierry Arnoux, 9-Jun-2017.)
 |-  (
 ( A  e.  (
 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  B  <_  A )  ->  (
 ( A + e  - e B ) + e B )  =  A )
 
Theoremfsumrp0cl 23349* Closure of a finite sum of positive integers. (Contributed by Thierry Arnoux, 25-Jun-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  (
 0 [,)  +oo ) )
 
18.3.25  Countable Sets
 
Theoremnnct 23350  NN is countable (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  NN  ~<_  om
 
Theoremctex 23351 A countable set is a set (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  A  e.  _V )
 
Theoremssct 23352 The subset of a countable set is countable (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
 
Theoremxpct 23353 The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  (
 ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( A  X.  B )  ~<_  om )
 
Theoremsnct 23354 A singleton is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)
 |-  ( A  e.  V  ->  { A }  ~<_  om )
 
Theoremprct 23355 An unordered pair is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  { A ,  B } 
 ~<_  om )
 
Theoremfnct 23356 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  (
 ( F  Fn  A  /\  A  ~<_  om )  ->  F  ~<_  om )
 
Theoremdmct 23357 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  dom  A  ~<_  om )
 
Theoremcnvct 23358 If a set is countable, its converse is as well. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  `' A  ~<_  om )
 
Theoremrnct 23359 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  ran  A  ~<_  om )
 
Theoremmptct 23360* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  ( x  e.  A  |->  B )  ~<_  om )
 
Theoremmpt2cti 23361* An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.)
 |-  A. x  e.  A  A. y  e.  B  C  e.  V   =>    |-  (
 ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( x  e.  A ,  y  e.  B  |->  C )  ~<_ 
 om )
 
Theoremabrexct 23362* An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  { y  |  E. x  e.  A  y  =  B }  ~<_  om )
 
Theoremmptctf 23363 A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  ~<_  om  ->  ( x  e.  A  |->  B )  ~<_  om )
 
Theoremabrexctf 23364* An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  ~<_  om  ->  { y  |  E. x  e.  A  y  =  B } 
 ~<_  om )
 
18.3.26  Disjointness - misc additions
 
Theoremcbvdisjf 23365* Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y  ->  B  =  C )   =>    |-  (Disj  x  e.  A B  <-> Disj  y  e.  A C )
 
Theoremdisjss1f 23366 A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  (Disj  x  e.  B C  -> Disj  x  e.  A C ) )
 
Theoremdisjdifprg 23367* A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 -> Disj 
 x  e.  { ( B  \  A ) ,  A } x )
 
Theoremdisjdifprg2 23368* A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( A  e.  V  -> Disj  x  e.  { ( A  \  B ) ,  ( A  i^i  B ) } x )
 
Theoremdisji2f 23369* Property of a disjoint collection: if  B ( x )  =  C and  B ( Y )  =  D, and  x  =/=  Y, then  B and  C are disjoint. (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/_ x C   &    |-  ( x  =  Y  ->  B  =  C )   =>    |-  ( (Disj  x  e.  A B  /\  ( x  e.  A  /\  Y  e.  A )  /\  x  =/= 
 Y )  ->  ( B  i^i  C )  =  (/) )
 
Theoremdisjif 23370* Property of a disjoint collection: if  B ( x ) and 
B ( Y )  =  D have a common element  Z, then  x  =  Y. (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/_ x C   &    |-  ( x  =  Y  ->  B  =  C )   =>    |-  ( (Disj  x  e.  A B  /\  ( x  e.  A  /\  Y  e.  A )  /\  ( Z  e.  B  /\  Z  e.  C ) )  ->  x  =  Y )
 
Theoremdisjorf 23371* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ i A   &    |-  F/_ j A   &    |-  ( i  =  j  ->  B  =  C )   =>    |-  (Disj  i  e.  A B 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
 
Theoremdisjorsf 23372* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  (Disj  x  e.  A B 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
 
Theoremdisjif2 23373* Property of a disjoint collection: if  B ( x ) and 
B ( Y )  =  D have a common element  Z, then  x  =  Y. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x C   &    |-  ( x  =  Y  ->  B  =  C )   =>    |-  ( (Disj  x  e.  A B  /\  ( x  e.  A  /\  Y  e.  A )  /\  ( Z  e.  B  /\  Z  e.  C )
 )  ->  x  =  Y )
 
Theoremdisjabrex 23374* Rewriting a disjoint collection into a partition of its image set (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  (Disj  x  e.  A B  -> Disj  y  e.  { z  |  E. x  e.  A  z  =  B } y )
 
Theoremdisjabrexf 23375* Rewriting a disjoint collection into a partition of its image set (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
 |-  F/_ x A   =>    |-  (Disj  x  e.  A B  -> Disj  y  e.  { z  |  E. x  e.  A  z  =  B }
 y )
 
Theoremdisjpreima 23376* A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
 |-  (
 ( Fun  F  /\ Disj  x  e.  A B )  -> Disj  x  e.  A ( `' F " B ) )
 
Theoremdisjin 23377 If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  (Disj  x  e.  B C  -> Disj  x  e.  B ( C  i^i  A ) )
 
Theoremiundisjfi 23378* Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 18921 (Contributed by Thierry Arnoux, 15-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   =>    |-  U_ n  e.  ( 1..^ N ) A  =  U_ n  e.  ( 1..^ N ) ( A 
 \  U_ k  e.  (
 1..^ n ) B )
 
Theoremiundisj2fi 23379* A disjoint union is disjoint, finite version. Cf. iundisj2 18922 (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   =>    |- Disj  n  e.  ( 1..^ N ) ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremiundisjf 23380* Rewrite a countable union as a disjoint union. Cf. iundisj 18921 (Contributed by Thierry Arnoux, 31-Dec-2016.)
 |-  F/_ k A   &    |-  F/_ n B   &    |-  ( n  =  k  ->  A  =  B )   =>    |-  U_ n  e.  NN  A  =  U_ n  e. 
 NN  ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremiundisj2f 23381* A disjoint union is disjoint. Cf. iundisj2 18922 (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/_ k A   &    |-  F/_ n B   &    |-  ( n  =  k  ->  A  =  B )   =>    |- Disj  n  e.  NN ( A 
 \  U_ k  e.  (
 1..^ n ) B )
 
Theoremiundisjcnt 23382* Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ M ) ) )   =>    |-  ( ph  ->  U_ n  e.  N  A  =  U_ n  e.  N  ( A  \  U_ k  e.  ( 1..^ n ) B ) )
 
Theoremiundisj2cnt 23383* A countable disjoint union is disjoint. Cf. iundisj2 18922 (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ M ) ) )   =>    |-  ( ph  -> Disj  n  e.  N ( A  \  U_ k  e.  ( 1..^ n ) B ) )
 
Theoremdisjdsct 23384* A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 5326) (Contributed by Thierry Arnoux, 28-Feb-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( V  \  { (/)
 } ) )   &    |-  ( ph  -> Disj  x  e.  A B )   =>    |-  ( ph  ->  Fun  `' ( x  e.  A  |->  B ) )
 
Theoremdisjrdx 23385* Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)
 |-  ( ph  ->  F : A -1-1-onto-> C )   &    |-  ( ( ph  /\  y  =  ( F `  x ) )  ->  D  =  B )   =>    |-  ( ph  ->  (Disj  x  e.  A B  <-> Disj  y  e.  C D ) )
 
18.3.27  Limits - misc additions
 
Theoremlmlim 23386 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 23387 Relate a limit in the non-negative 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 23388 Implication of convergence for a non-negative series. This could be used to shorten prmreclem6 12984 (Contributed by Thierry Arnoux, 28-Jul-2017.)
 |-  (
 ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  e.  RR )
 
Theorempnfneige0 23389* A neighborhood of  +oo contains an unbounded interval based at a real number. See pnfnei 16966 (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 23390* Express "sequence  F converges to plus infinity" (i.e. diverges), for a sequence of non-negative 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 23391* 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 23392* 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 )
 
18.3.28  Finitely supported group sums - misc additions
 
Theoremgsumsn2 23393* Group sum of a singleton. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  B  =  ( Base `  G )   &    |-  G  e.  Mnd   &    |-  ( ( ph  /\  k  =  M )  ->  A  =  C )   &    |-  ( ph  ->  M  e.  V )   &    |-  ( ph  ->  C  e.  B )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  { M }  |->  A ) )  =  C )
 
Theoremgsumpropd2lem 23394* Lemma for gsumpropd2 23395 (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H ) )   &    |-  (
 ( ph  /\  ( s  e.  ( Base `  G )  /\  t  e.  ( Base `  G ) ) )  ->  ( s
 ( +g  `  G ) t )  e.  ( Base `  G ) )   &    |-  ( ( ph  /\  (
 s  e.  ( Base `  G )  /\  t  e.  ( Base `  G )
 ) )  ->  (
 s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )   &    |-  ( ph  ->  Fun 
 F )   &    |-  ( ph  ->  ran 
 F  C_  ( Base `  G ) )   &    |-  A  =  ( `' F "
 ( _V  \  {
 s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G ) t )  =  t  /\  ( t ( +g  `  G ) s )  =  t ) } )
 )   &    |-  B  =  ( `' F " ( _V  \  { s  e.  ( Base `  H )  | 
 A. t  e.  ( Base `  H ) ( ( s ( +g  `  H ) t )  =  t  /\  (
 t ( +g  `  H ) s )  =  t ) } )
 )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumpropd2 23395* A stronger version of gsumpropd 14469, working for magma, where only the closure of the addition operation on a common base is required. (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H ) )   &    |-  (
 ( ph  /\  ( s  e.  ( Base `  G )  /\  t  e.  ( Base `  G ) ) )  ->  ( s
 ( +g  `  G ) t )  e.  ( Base `  G ) )   &    |-  ( ( ph  /\  (
 s  e.  ( Base `  G )  /\  t  e.  ( Base `  G )
 ) )  ->  (
 s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )   &    |-  ( ph  ->  Fun 
 F )   &    |-  ( ph  ->  ran 
 F  C_  ( Base `  G ) )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumconstf 23396* Sum of a constant series (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  F/_ k X   &    |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  A  e.  Fin  /\  X  e.  B )  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  =  ( ( # `  A )  .x.  X ) )
 
Theoremxrge0tsmsd 23397* Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 30-Jan-2017.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  S  =  sup ( ran  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s ) ) ) ,  RR* ,  <  ) )   =>    |-  ( ph  ->  ( G tsums  F )  =  { S } )
 
Theoremxrge0tsmsbi 23398 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   =>    |-  ( ph  ->  ( C  e.  ( G tsums  F )  <->  C  =  U. ( G tsums  F ) ) )
 
Theoremxrge0tsmseq 23399 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  C  e.  ( G tsums  F ) )   =>    |-  ( ph  ->  C  =  U. ( G tsums  F ) )
 
Theoremhashunif 23400* The cardinality of a disjoint finite union of finite sets. Cf. hashuni 12299 (Contributed by Thierry Arnoux, 17-Feb-2017.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  Fin )   &    |-  ( ph  -> Disj  x  e.  A x )   =>    |-  ( ph  ->  ( # `
  U. A )  = 
 sum_ x  e.  A  ( # `  x ) )
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