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Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelptr2 17101* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  W  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  S  e.  ( F `  k
 ) )   &    |-  ( ( ph  /\  k  e.  ( A 
 \  W ) ) 
 ->  S  =  U. ( F `  k ) )   =>    |-  ( ph  ->  X_ k  e.  A  S  e.  B )
 
Theoremptbasid 17102* The base set of the product topology is a basic open set. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  X_ k  e.  A  U. ( F `
  k )  e.  B )
 
Theoremptuni2 17103* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  X_ k  e.  A  U. ( F `
  k )  = 
 U. B )
 
Theoremptbasin 17104* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X  i^i  Y )  e.  B )
 
Theoremptbasin2 17105* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  ( fi
 `  B )  =  B )
 
Theoremptbas 17106* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  B  e.  TopBases )
 
Theoremptpjpre2 17107* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   &    |-  X  =  X_ n  e.  A  U. ( F `  n )   =>    |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `
  I ) ) )  ->  ( `' ( w  e.  X  |->  ( w `  I ) ) " U )  e.  B )
 
Theoremptbasfi 17108* The basis for the product topology can also be written as the set of finite intersections of "cylinder sets", the preimages of projections into one factor from open sets in the factor. (We have to add  X itself to the list because if  A is empty we get  ( fi `  (/) )  =  (/) while  B  =  { (/) }.) (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   &    |-  X  =  X_ n  e.  A  U. ( F `  n )   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  B  =  ( fi `  ( { X }  u.  ran  (  k  e.  A ,  u  e.  ( F `  k )  |->  ( `' ( w  e.  X  |->  ( w `  k ) ) " u ) ) ) ) )
 
Theorempttop 17109 The product topology is a topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  ( ( A  e.  V  /\  F : A --> Top )  ->  ( Xt_ `  F )  e.  Top )
 
Theoremptopn 17110* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Top )   &    |-  ( ph  ->  W  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  S  e.  ( F `  k
 ) )   &    |-  ( ( ph  /\  k  e.  ( A 
 \  W ) ) 
 ->  S  =  U. ( F `  k ) )   =>    |-  ( ph  ->  X_ k  e.  A  S  e.  ( Xt_ `  F ) )
 
Theoremptopn2 17111* A sub-basic open set in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Top )   &    |-  ( ph  ->  O  e.  ( F `  Y ) )   =>    |-  ( ph  ->  X_ k  e.  A  if ( k  =  Y ,  O ,  U. ( F `  k ) )  e.  ( Xt_ `  F ) )
 
Theoremxkotf 17112* Functionality of function  T. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  X  =  U. R   &    |-  K  =  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }   &    |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k
 )  C_  v }
 )   =>    |-  T : ( K  X.  S ) --> ~P ( R  Cn  S )
 
Theoremxkobval 17113* Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. R   &    |-  K  =  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }   &    |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k
 )  C_  v }
 )   =>    |- 
 ran  T  =  {
 s  |  E. k  e.  ~P  X E. v  e.  S  ( ( Rt  k )  e.  Comp  /\  s  =  { f  e.  ( R  Cn  S )  |  ( f " k
 )  C_  v }
 ) }
 
Theoremxkoval 17114* Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  X  =  U. R   &    |-  K  =  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }   &    |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k
 )  C_  v }
 )   =>    |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ k o  R )  =  (
 topGen `  ( fi `  ran  T ) ) )
 
Theoremxkotop 17115 The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ k o  R )  e.  Top )
 
Theoremxkoopn 17116* A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  X  =  U. R   &    |-  ( ph  ->  R  e.  Top )   &    |-  ( ph  ->  S  e.  Top )   &    |-  ( ph  ->  A 
 C_  X )   &    |-  ( ph  ->  ( Rt  A )  e.  Comp )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  e.  ( S  ^ k o  R ) )
 
Theoremtxtopi 17117 The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)
 |-  R  e.  Top   &    |-  S  e.  Top   =>    |-  ( R  tX  S )  e.  Top
 
Theoremtxtopon 17118 The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  ->  ( R 
 tX  S )  e.  (TopOn `  ( X  X.  Y ) ) )
 
Theoremtxuni 17119 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  X  =  U. R   &    |-  Y  =  U. S   =>    |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( X  X.  Y )  =  U. ( R 
 tX  S ) )
 
Theoremtxunii 17120 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)
 |-  R  e.  Top   &    |-  S  e.  Top   &    |-  X  =  U. R   &    |-  Y  =  U. S   =>    |-  ( X  X.  Y )  =  U. ( R 
 tX  S )
 
Theoremptuni 17121* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  J  =  ( Xt_ `  F )   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  X_ x  e.  A  U. ( F `
  x )  = 
 U. J )
 
Theoremptunimpt 17122* Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  J  =  ( Xt_ `  ( x  e.  A  |->  K ) )   =>    |-  ( ( A  e.  V  /\  A. x  e.  A  K  e.  Top )  ->  X_ x  e.  A  U. K  =  U. J )
 
Theorempttopon 17123* The base set for the product topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  J  =  ( Xt_ `  ( x  e.  A  |->  K ) )   =>    |-  ( ( A  e.  V  /\  A. x  e.  A  K  e.  (TopOn `  B )
 )  ->  J  e.  (TopOn `  X_ x  e.  A  B ) )
 
Theorempttoponconst 17124 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  J  =  ( Xt_ `  ( A  X.  { R } ) )   =>    |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X )
 )  ->  J  e.  (TopOn `  ( X  ^m  A ) ) )
 
Theoremptuniconst 17125 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  J  =  ( Xt_ `  ( A  X.  { R } ) )   &    |-  X  =  U. R   =>    |-  ( ( A  e.  V  /\  R  e.  Top )  ->  ( X  ^m  A )  =  U. J )
 
Theoremxkouni 17126 The base set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  J  =  ( S 
 ^ k o  R )   =>    |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( R  Cn  S )  =  U. J )
 
Theoremxkotopon 17127 The base set of the compact-open topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  J  =  ( S 
 ^ k o  R )   =>    |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  J  e.  (TopOn `  ( R  Cn  S ) ) )
 
Theoremptval2 17128* The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015.)
 |-  J  =  ( Xt_ `  F )   &    |-  X  =  U. J   &    |-  G  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  J  =  ( topGen `  ( fi `  ( { X }  u.  ran  G ) ) ) )
 
Theoremtxopn 17129 The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( ( R  e.  V  /\  S  e.  W )  /\  ( A  e.  R  /\  B  e.  S )
 )  ->  ( A  X.  B )  e.  ( R  tX  S ) )
 
Theoremtxcld 17130 The product of two closed sets is closed in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  ( ( A  e.  ( Clsd `  R )  /\  B  e.  ( Clsd `  S ) )  ->  ( A  X.  B )  e.  ( Clsd `  ( R  tX  S ) ) )
 
Theoremtxcls 17131 Closure of a rectangle in the product topology. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  /\  ( A  C_  X  /\  B  C_  Y ) ) 
 ->  ( ( cls `  ( R  tX  S ) ) `
  ( A  X.  B ) )  =  ( ( ( cls `  R ) `  A )  X.  ( ( cls `  S ) `  B ) ) )
 
Theoremtxss12 17132 Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( ( B  e.  V  /\  D  e.  W )  /\  ( A  C_  B  /\  C  C_  D ) )  ->  ( A  tX  C ) 
 C_  ( B  tX  D ) )
 
Theoremtxbasval 17133 It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( ( topGen `  R )  tX  ( topGen `  S ) )  =  ( R  tX  S ) )
 
Theoremtxcnpi 17134* Continuity of a two-argument function at a point. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  F  e.  ( ( ( J 
 tX  K )  CnP  L ) `  <. A ,  B >. ) )   &    |-  ( ph  ->  U  e.  L )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  ( A F B )  e.  U )   =>    |-  ( ph  ->  E. u  e.  J  E. v  e.  K  ( A  e.  u  /\  B  e.  v  /\  ( u  X.  v
 )  C_  ( `' F " U ) ) )
 
Theoremtx1cn 17135 Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  ->  ( 1st  |`  ( X  X.  Y ) )  e.  (
 ( R  tX  S )  Cn  R ) )
 
Theoremtx2cn 17136 Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  ->  ( 2nd  |`  ( X  X.  Y ) )  e.  (
 ( R  tX  S )  Cn  S ) )
 
Theoremptpjcn 17137* Continuity of a projection map into a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Feb-2015.)
 |-  Y  =  U. J   &    |-  J  =  ( Xt_ `  F )   =>    |-  ( ( A  e.  V  /\  F : A --> Top  /\  I  e.  A )  ->  ( x  e.  Y  |->  ( x `  I ) )  e.  ( J  Cn  ( F `  I ) ) )
 
Theoremptpjopn 17138* The projection map is an open map. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  Y  =  U. J   &    |-  J  =  ( Xt_ `  F )   =>    |-  ( ( ( A  e.  V  /\  F : A --> Top  /\  I  e.  A )  /\  U  e.  J )  ->  (
 ( x  e.  Y  |->  ( x `  I ) ) " U )  e.  ( F `  I ) )
 
Theoremptcld 17139* A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Top )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  ( F `  k ) ) )   =>    |-  ( ph  ->  X_ k  e.  A  C  e.  ( Clsd `  ( Xt_ `  F ) ) )
 
Theoremptcldmpt 17140* A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  J  e.  Top )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  J )
 )   =>    |-  ( ph  ->  X_ k  e.  A  C  e.  ( Clsd `  ( Xt_ `  (
 k  e.  A  |->  J ) ) ) )
 
Theoremptclsg 17141* The closure of a box in the product topology is the box formed from the closures of the factors. The proof uses the axiom of choice; the last hypothesis is the choice assumption. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  J  =  ( Xt_ `  ( k  e.  A  |->  R ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  R  e.  (TopOn `  X )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  S  C_  X )   &    |-  ( ph  ->  U_ k  e.  A  S  e. AC  A )   =>    |-  ( ph  ->  (
 ( cls `  J ) `  X_ k  e.  A  S )  =  X_ k  e.  A  ( ( cls `  R ) `  S ) )
 
Theoremptcls 17142* The closure of a box in the product topology is the box formed from the closures of the factors. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( Xt_ `  ( k  e.  A  |->  R ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  R  e.  (TopOn `  X )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  S  C_  X )   =>    |-  ( ph  ->  (
 ( cls `  J ) `  X_ k  e.  A  S )  =  X_ k  e.  A  ( ( cls `  R ) `  S ) )
 
Theoremdfac14lem 17143* Lemma for dfac14 17144. By equipping  S  u.  { P } for some  P  e/  S with the particular point topology, we can show that  P is in the closure of  S; hence the sequence  P ( x ) is in the product of the closures and we can utilize this instance of ptcls 17142 to extract an element of the closure of  X_ k  e.  I S. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  ( ph  ->  I  e.  V )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  S  e.  W )   &    |-  ( ( ph  /\  x  e.  I )  ->  S  =/= 
 (/) )   &    |-  P  =  ~P U. S   &    |-  R  =  {
 y  e.  ~P ( S  u.  { P }
 )  |  ( P  e.  y  ->  y  =  ( S  u.  { P } ) ) }   &    |-  J  =  ( Xt_ `  ( x  e.  I  |->  R ) )   &    |-  ( ph  ->  ( ( cls `  J ) `  X_ x  e.  I  S )  =  X_ x  e.  I  ( ( cls `  R ) `  S ) )   =>    |-  ( ph  ->  X_ x  e.  I  S  =/= 
 (/) )
 
Theoremdfac14 17144* Theorem ptcls 17142 is an equivalent of the axiom of choice. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  (CHOICE  <->  A. f ( f : dom  f --> Top  ->  A. s  e.  X_  k  e.  dom  f ~P U. ( f `  k
 ) ( ( cls `  ( Xt_ `  f
 ) ) `  X_ k  e.  dom  f ( s `
  k ) )  =  X_ k  e.  dom  f ( ( cls `  ( f `  k
 ) ) `  (
 s `  k )
 ) ) )
 
Theoremxkoccn 17145* The "constant function" function which maps  x  e.  Y to the constant function  z  e.  X  |->  x is a continuous function from  X into the space of continuous functions from  Y to  X. This can also be understood as the currying of the first projection function. (The currying of the second projection function is  x  e.  Y  |->  ( z  e.  X  |->  z ), which we already know is continuous because it is a constant function.) (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  ->  ( x  e.  Y  |->  ( X  X.  { x }
 ) )  e.  ( S  Cn  ( S  ^ k o  R )
 ) )
 
Theoremtxcnp 17146* If two functions are continuous at 
D, then the ordered pair of them is continuous at  D into the product topology. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( ( J  CnP  K ) `
  D ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( ( J  CnP  L ) `  D ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  <. A ,  B >. )  e.  (
 ( J  CnP  ( K  tX  L ) ) `
  D ) )
 
Theoremptcnplem 17147* Lemma for ptcnp 17148. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  K  =  ( Xt_ `  F )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> Top )   &    |-  ( ph  ->  D  e.  X )   &    |-  (
 ( ph  /\  k  e.  I )  ->  ( x  e.  X  |->  A )  e.  ( ( J 
 CnP  ( F `  k ) ) `  D ) )   &    |-  F/ k ps   &    |-  ( ( ph  /\ 
 ps )  ->  G  Fn  I )   &    |-  ( ( (
 ph  /\  ps )  /\  k  e.  I
 )  ->  ( G `  k )  e.  ( F `  k ) )   &    |-  ( ( ph  /\  ps )  ->  W  e.  Fin )   &    |-  ( ( ( ph  /\ 
 ps )  /\  k  e.  ( I  \  W ) )  ->  ( G `
  k )  = 
 U. ( F `  k ) )   &    |-  (
 ( ph  /\  ps )  ->  ( ( x  e.  X  |->  ( k  e.  I  |->  A ) ) `
  D )  e.  X_ k  e.  I  ( G `  k ) )   =>    |-  ( ( ph  /\  ps )  ->  E. z  e.  J  ( D  e.  z  /\  ( ( x  e.  X  |->  ( k  e.  I  |->  A ) )
 " z )  C_  X_ k  e.  I  ( G `  k ) ) )
 
Theoremptcnp 17148* If every projection of a function is continuous at  D, then the function itself is continuous at  D into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  K  =  ( Xt_ `  F )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> Top )   &    |-  ( ph  ->  D  e.  X )   &    |-  (
 ( ph  /\  k  e.  I )  ->  ( x  e.  X  |->  A )  e.  ( ( J 
 CnP  ( F `  k ) ) `  D ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( k  e.  I  |->  A ) )  e.  (
 ( J  CnP  K ) `  D ) )
 
Theoremupxp 17149* Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  P  =  ( 1st  |`  ( B  X.  C ) )   &    |-  Q  =  ( 2nd  |`  ( B  X.  C ) )   =>    |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  E! h ( h : A
 --> ( B  X.  C )  /\  F  =  ( P  o.  h ) 
 /\  G  =  ( Q  o.  h ) ) )
 
Theoremtxcnmpt 17150* A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  W  =  U. U   &    |-  H  =  ( x  e.  W  |->  <.
 ( F `  x ) ,  ( G `  x ) >. )   =>    |-  ( ( F  e.  ( U  Cn  R )  /\  G  e.  ( U  Cn  S ) )  ->  H  e.  ( U  Cn  ( R  tX  S ) ) )
 
Theoremuptx 17151* Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
 |-  T  =  ( R 
 tX  S )   &    |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  Z  =  ( X  X.  Y )   &    |-  P  =  ( 1st  |`  Z )   &    |-  Q  =  ( 2nd  |`  Z )   =>    |-  ( ( F  e.  ( U  Cn  R ) 
 /\  G  e.  ( U  Cn  S ) ) 
 ->  E! h  e.  ( U  Cn  T ) ( F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )
 
Theoremtxcn 17152 A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  Z  =  ( X  X.  Y )   &    |-  W  =  U. U   &    |-  P  =  ( 1st  |`  Z )   &    |-  Q  =  ( 2nd  |`  Z )   =>    |-  ( ( R  e.  Top  /\  S  e.  Top  /\  F : W --> Z ) 
 ->  ( F  e.  ( U  Cn  ( R  tX  S ) )  <->  ( ( P  o.  F )  e.  ( U  Cn  R )  /\  ( Q  o.  F )  e.  ( U  Cn  S ) ) ) )
 
Theoremptcn 17153* If every projection of a function is continuous, then the function itself is continuous into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  K  =  ( Xt_ `  F )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> Top )   &    |-  ( ( ph  /\  k  e.  I ) 
 ->  ( x  e.  X  |->  A )  e.  ( J  Cn  ( F `  k ) ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( k  e.  I  |->  A ) )  e.  ( J  Cn  K ) )
 
Theoremprdstopn 17154 Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  O  =  (
 TopOpen `  Y )   =>    |-  ( ph  ->  O  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
 
Theoremprdstps 17155 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R : I --> TopSp )   =>    |-  ( ph  ->  Y  e.  TopSp )
 
Theorempwstps 17156 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  TopSp  /\  I  e.  V )  ->  Y  e.  TopSp )
 
Theoremtxrest 17157 The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( ( R  e.  V  /\  S  e.  W )  /\  ( A  e.  X  /\  B  e.  Y )
 )  ->  ( ( R  tX  S )t  ( A  X.  B ) )  =  ( ( Rt  A )  tX  ( St  B ) ) )
 
Theoremtxdis 17158 The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  tX 
 ~P B )  =  ~P ( A  X.  B ) )
 
Theoremtxindislem 17159 Lemma for txindis 17160. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )
 
Theoremtxindis 17160 The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( { (/) ,  A }  tX  { (/) ,  B } )  =  { (/)
 ,  ( A  X.  B ) }
 
Theoremtxdis1cn 17161* A function is jointly continuous on a discrete left topology iff it is continuous as a function of its right argument, for each fixed left value. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  J  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  K  e.  Top )   &    |-  ( ph  ->  F  Fn  ( X  X.  Y ) )   &    |-  ( ( ph  /\  x  e.  X )  ->  (
 y  e.  Y  |->  ( x F y ) )  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  F  e.  ( ( ~P X  tX  J )  Cn  K ) )
 
Theoremtxlly 17162* If the property  A is preserved under topological products, then so is the property of being locally  A. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  ( ( j  e.  A  /\  k  e.  A )  ->  (
 j  tX  k )  e.  A )   =>    |-  ( ( R  e. Locally  A 
 /\  S  e. Locally  A ) 
 ->  ( R  tX  S )  e. Locally  A )
 
Theoremtxnlly 17163* If the property  A is preserved under topological products, then so is the property of being n-locally  A. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  ( ( j  e.  A  /\  k  e.  A )  ->  (
 j  tX  k )  e.  A )   =>    |-  ( ( R  e. 𝑛Locally  A  /\  S  e. 𝑛Locally  A )  ->  ( R  tX  S )  e. 𝑛Locally  A )
 
Theorempthaus 17164 The product of a collection of Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( A  e.  V  /\  F : A --> Haus )  ->  ( Xt_ `  F )  e.  Haus )
 
Theoremptrescn 17165* Restriction is a continuous function on product topologies. (Contributed by Mario Carneiro, 7-Feb-2015.)
 |-  X  =  U. J   &    |-  J  =  ( Xt_ `  F )   &    |-  K  =  ( Xt_ `  ( F  |`  B ) )   =>    |-  ( ( A  e.  V  /\  F : A --> Top  /\  B  C_  A )  ->  ( x  e.  X  |->  ( x  |`  B ) )  e.  ( J  Cn  K ) )
 
Theoremtxtube 17166* The "tube lemma". If  X is compact and there is an open set  U containing the line  X  X.  { A }, then there is a "tube"  X  X.  u for some neighborhood  u of  A which is entirely contained within  U. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  ( ph  ->  R  e.  Comp )   &    |-  ( ph  ->  S  e.  Top )   &    |-  ( ph  ->  U  e.  ( R  tX  S ) )   &    |-  ( ph  ->  ( X  X.  { A } )  C_  U )   &    |-  ( ph  ->  A  e.  Y )   =>    |-  ( ph  ->  E. u  e.  S  ( A  e.  u  /\  ( X  X.  u )  C_  U ) )
 
Theoremtxcmplem1 17167* Lemma for txcmp 17169. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  ( ph  ->  R  e.  Comp )   &    |-  ( ph  ->  S  e.  Comp )   &    |-  ( ph  ->  W 
 C_  ( R  tX  S ) )   &    |-  ( ph  ->  ( X  X.  Y )  =  U. W )   &    |-  ( ph  ->  A  e.  Y )   =>    |-  ( ph  ->  E. u  e.  S  ( A  e.  u  /\  E. v  e.  ( ~P W  i^i  Fin )
 ( X  X.  u )  C_  U. v ) )
 
Theoremtxcmplem2 17168* Lemma for txcmp 17169. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  ( ph  ->  R  e.  Comp )   &    |-  ( ph  ->  S  e.  Comp )   &    |-  ( ph  ->  W 
 C_  ( R  tX  S ) )   &    |-  ( ph  ->  ( X  X.  Y )  =  U. W )   =>    |-  ( ph  ->  E. v  e.  ( ~P W  i^i  Fin ) ( X  X.  Y )  =  U. v )
 
Theoremtxcmp 17169 The topological product of two compact spaces is compact. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened 21-Mar-2015.)
 |-  ( ( R  e.  Comp  /\  S  e.  Comp )  ->  ( R  tX  S )  e.  Comp )
 
Theoremtxcmpb 17170 The topological product of two nonempty topologies is compact iff the component topologies are both compact. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  X  =  U. R   &    |-  Y  =  U. S   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  ->  (
 ( R  tX  S )  e.  Comp  <->  ( R  e.  Comp  /\  S  e.  Comp )
 ) )
 
Theoremhausdiag 17171 A topology is Hausdorff iff the diagonal set is closed in the topology's product with itself. EDITORIAL: very clumsy proof, can probably be shortened substantially. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Haus  <->  ( J  e.  Top  /\  (  _I  |`  X )  e.  ( Clsd `  ( J  tX  J ) ) ) )
 
Theoremhauseqlcld 17172 In a Hausdorff topology, the equalizer of two continuous functions is closed (thus, two continuous functions which agree on a dense set agree everywhere). (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  dom  (  F  i^i  G )  e.  ( Clsd `  J )
 )
 
Theoremtxhaus 17173 The topological product of two Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ( R  e.  Haus  /\  S  e.  Haus )  ->  ( R  tX  S )  e.  Haus )
 
Theoremtxlm 17174* Two sequences converge iff the sequence of their ordered pairs converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  F : Z --> X )   &    |-  ( ph  ->  G : Z --> Y )   &    |-  H  =  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )   =>    |-  ( ph  ->  ( ( F ( ~~> t `  J ) R  /\  G ( ~~> t `  K ) S )  <->  H ( ~~> t `  ( J  tX  K ) ) <. R ,  S >. ) )
 
Theoremlmcn2 17175* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 15-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  F : Z --> X )   &    |-  ( ph  ->  G : Z --> Y )   &    |-  ( ph  ->  F ( ~~> t `  J ) R )   &    |-  ( ph  ->  G ( ~~> t `  K ) S )   &    |-  ( ph  ->  O  e.  ( ( J 
 tX  K )  Cn  N ) )   &    |-  H  =  ( n  e.  Z  |->  ( ( F `  n ) O ( G `  n ) ) )   =>    |-  ( ph  ->  H (
 ~~> t `  N ) ( R O S ) )
 
Theoremtx1stc 17176 The topological product of two first-countable spaces is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( R  e.  1stc  /\  S  e.  1stc )  ->  ( R  tX  S )  e.  1stc )
 
Theoremtx2ndc 17177 The topological product of two second-countable spaces is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( R  e.  2ndc  /\  S  e.  2ndc )  ->  ( R  tX  S )  e.  2ndc )
 
Theoremtxkgen 17178 The topological product of a locally compact space and a compactly generated Hausdorff space is compactly generated. (The condition on  S can also be replaced with either "comactly generated weak Hausdorff (CGWH)" or "compact Hausdorff-ly generated (CHG)", where WH means that all images of compact Hausdorff spaces are closed and CHG means that a set is open iff it is open in all compact Hausdorff spaces.) (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  ( ran 𝑘Gen  i^i  Haus ) )  ->  ( R  tX  S )  e. 
 ran 𝑘Gen )
 
Theoremxkohaus 17179 If the codomain space is Hausdorff, then the compact-open topology of continuous functions is also Hausdorff. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( R  e.  Top  /\  S  e.  Haus )  ->  ( S  ^ k o  R )  e.  Haus )
 
Theoremxkoptsub 17180 The compact-open topology is finer than the product topology restricted to continuous functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  X  =  U. R   &    |-  J  =  ( Xt_ `  ( X  X.  { S }
 ) )   =>    |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( Jt  ( R  Cn  S ) )  C_  ( S 
 ^ k o  R ) )
 
Theoremxkopt 17181 The compact-open topology on a discrete set coincides with the product topology where all the factors are the same. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  ( ( R  e.  Top  /\  A  e.  V ) 
 ->  ( R  ^ k o  ~P A )  =  ( Xt_ `  ( A  X.  { R }
 ) ) )
 
Theoremxkopjcn 17182* Continuity of a projection map from the space of continuous functions. (This theorem can be strengthened, to joint continuity in both  f and  A as a function on  ( S  ^ k o  R ) 
tX  R, but not without stronger assumptions on  R; see xkofvcn 17210.) (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. R   =>    |-  (
 ( R  e.  Top  /\  S  e.  Top  /\  A  e.  X )  ->  ( f  e.  ( R  Cn  S )  |->  ( f `  A ) )  e.  ( ( S  ^ k o  R )  Cn  S ) )
 
Theoremxkoco1cn 17183* If  F is a continuous function, then  g  |->  g  o.  F is a continuous function on function spaces. (The reason we prove this and xkoco2cn 17184 independently of the more general xkococn 17186 is because that requires some inconvenient extra assumptions on  S.) (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( ph  ->  T  e.  Top )   &    |-  ( ph  ->  F  e.  ( R  Cn  S ) )   =>    |-  ( ph  ->  ( g  e.  ( S  Cn  T )  |->  ( g  o.  F ) )  e.  ( ( T  ^ k o  S )  Cn  ( T  ^ k o  R ) ) )
 
Theoremxkoco2cn 17184* If  F is a continuous function, then  g  |->  F  o.  g is a continuous function on function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ph  ->  R  e.  Top )   &    |-  ( ph  ->  F  e.  ( S  Cn  T ) )   =>    |-  ( ph  ->  ( g  e.  ( R  Cn  S )  |->  ( F  o.  g ) )  e.  ( ( S  ^ k o  R )  Cn  ( T  ^ k o  R ) ) )
 
Theoremxkococnlem 17185* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  F  =  ( f  e.  ( S  Cn  T ) ,  g  e.  ( R  Cn  S )  |->  ( f  o.  g ) )   &    |-  ( ph  ->  S  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  K  C_  U. R )   &    |-  ( ph  ->  ( Rt  K )  e.  Comp )   &    |-  ( ph  ->  V  e.  T )   &    |-  ( ph  ->  A  e.  ( S  Cn  T ) )   &    |-  ( ph  ->  B  e.  ( R  Cn  S ) )   &    |-  ( ph  ->  ( ( A  o.  B ) " K )  C_  V )   =>    |-  ( ph  ->  E. z  e.  ( ( T  ^ k o  S )  tX  ( S  ^ k o  R ) ) (
 <. A ,  B >.  e.  z  /\  z  C_  ( `' F " { h  e.  ( R  Cn  T )  |  ( h " K )  C_  V } ) ) )
 
Theoremxkococn 17186* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  F  =  ( f  e.  ( S  Cn  T ) ,  g  e.  ( R  Cn  S )  |->  ( f  o.  g ) )   =>    |-  ( ( R  e.  Top  /\  S  e. 𝑛Locally  Comp  /\  T  e.  Top )  ->  F  e.  ( ( ( T  ^ k o  S )  tX  ( S  ^ k o  R ) )  Cn  ( T  ^ k o  R ) ) )
 
11.1.17  Continuous function-builders
 
Theoremcnmptid 17187* The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   =>    |-  ( ph  ->  ( x  e.  X  |->  x )  e.  ( J  Cn  J ) )
 
Theoremcnmptc 17188* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  P  e.  Y )   =>    |-  ( ph  ->  ( x  e.  X  |->  P )  e.  ( J  Cn  K ) )
 
Theoremcnmpt11 17189* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( y  e.  Y  |->  B )  e.  ( K  Cn  L ) )   &    |-  ( y  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  C )  e.  ( J  Cn  L ) )
 
Theoremcnmpt11f 17190* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  F  e.  ( K  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( F `  A ) )  e.  ( J  Cn  L ) )
 
Theoremcnmpt1t 17191* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  <. A ,  B >. )  e.  ( J  Cn  ( K  tX  L ) ) )
 
Theoremcnmpt12f 17192* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )   &    |-  ( ph  ->  F  e.  ( ( K 
 tX  L )  Cn  M ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( J  Cn  M ) )
 
Theoremcnmpt12 17193* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  ( y  e.  Y ,  z  e.  Z  |->  C )  e.  ( ( K 
 tX  L )  Cn  M ) )   &    |-  (
 ( y  =  A  /\  z  =  B )  ->  C  =  D )   =>    |-  ( ph  ->  ( x  e.  X  |->  D )  e.  ( J  Cn  M ) )
 
Theoremcnmpt1st 17194* The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  x )  e.  ( ( J 
 tX  K )  Cn  J ) )
 
Theoremcnmpt2nd 17195* The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  y )  e.  ( ( J 
 tX  K )  Cn  K ) )
 
Theoremcnmpt2c 17196* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  P  e.  Z )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  P )  e.  ( ( J 
 tX  K )  Cn  L ) )
 
Theoremcnmpt21 17197* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  ( z  e.  Z  |->  B )  e.  ( L  Cn  M ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  C )  e.  ( ( J 
 tX  K )  Cn  M ) )
 
Theoremcnmpt21f 17198* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  F  e.  ( L  Cn  M ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( F `  A ) )  e.  ( ( J  tX  K )  Cn  M ) )
 
Theoremcnmpt2t 17199* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  <. A ,  B >. )  e.  (
 ( J  tX  K )  Cn  ( L  tX  M ) ) )
 
Theoremcnmpt22 17200* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  M  e.  (TopOn `  W ) )   &    |-  ( ph  ->  ( z  e.  Z ,  w  e.  W  |->  C )  e.  ( ( L 
 tX  M )  Cn  N ) )   &    |-  (
 ( z  =  A  /\  w  =  B )  ->  C  =  D )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  D )  e.  ( ( J 
 tX  K )  Cn  N ) )
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