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Theorem List for Metamath Proof Explorer - 17301-17400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtxtopi 17301 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 17302 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 17303 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 17304 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 17305* 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 17306* 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 17307* 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 17308 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 17309 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 17310 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 17311 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 17312* 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 17313 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 17314 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 17315 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 17316 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 17317 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 17318* 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 17319 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 17320 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 17321* 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 17322* 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 17323* 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 17324* 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 17325* 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 17326* 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 17327* Lemma for dfac14 17328. 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 17326 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 17328* Theorem ptcls 17326 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 17329* 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 17330* 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 17331* Lemma for ptcnp 17332. (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 17332* 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 17333* 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 17334* 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 17335* 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 17336 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 17337* 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 17338 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 17339 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 17340 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 17341 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 17342 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 17343 Lemma for txindis 17344. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )
 
Theoremtxindis 17344 The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( { (/) ,  A }  tX  { (/) ,  B } )  =  { (/)
 ,  ( A  X.  B ) }
 
Theoremtxdis1cn 17345* 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 17346* 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 17347* 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 17348 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 17349* 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 17350* 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 17351* Lemma for txcmp 17353. (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 17352* Lemma for txcmp 17353. (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 17353 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 17354 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 17355 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 17356 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 17357 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 17358* 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 17359* 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 17360 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 17361 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 17362 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 17363 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 17364 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 17365 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 17366* 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 17394.) (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 17367* 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 17368 independently of the more general xkococn 17370 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 17368* 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 17369* 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 17370* 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 17371* 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 17372* 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 17373* 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 17374* 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 17375* 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 17376* 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 17377* 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 17378* 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 17379* 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 17380* 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 17381* 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 17382* 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 17383* 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 17384* 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 ) )
 
Theoremcnmpt22f 17385* 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  ->  F  e.  ( ( L 
 tX  M )  Cn  N ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A F B ) )  e.  ( ( J 
 tX  K )  Cn  N ) )
 
Theoremcnmpt1res 17386* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
 |-  K  =  ( Jt  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  Y  |->  A )  e.  ( K  Cn  L ) )
 
Theoremcnmpt2res 17387* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  K  =  ( Jt  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  N  =  ( Mt  W )   &    |-  ( ph  ->  M  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  W 
 C_  Z )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Z  |->  A )  e.  ( ( J  tX  M )  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  Y ,  y  e.  W  |->  A )  e.  ( ( K 
 tX  N )  Cn  L ) )
 
Theoremcnmptcom 17388* The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  ( 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  ->  ( y  e.  Y ,  x  e.  X  |->  A )  e.  ( ( K 
 tX  J )  Cn  L ) )
 
Theoremcnmptkc 17389* The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (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  Cn  ( J  ^ k o  K )
 ) )
 
Theoremcnmptkp 17390* The evaluation of the inner function in a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )   &    |-  ( ph  ->  B  e.  Y )   &    |-  (
 y  =  B  ->  A  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  C )  e.  ( J  Cn  L ) )
 
Theoremcnmptk1 17391* The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )   &    |-  ( ph  ->  ( z  e.  Z  |->  B )  e.  ( L  Cn  M ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  C ) )  e.  ( J  Cn  ( M  ^ k o  K ) ) )
 
Theoremcnmpt1k 17392* The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  M  e.  (TopOn `  W )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  L ) )   &    |-  ( ph  ->  ( y  e.  Y  |->  ( z  e.  Z  |->  B ) )  e.  ( K  Cn  ( M  ^ k o  L ) ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( y  e.  Y  |->  ( x  e.  X  |->  C ) )  e.  ( K  Cn  ( M  ^ k o  J ) ) )
 
Theoremcnmptkk 17393* The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  M  e.  (TopOn `  W )
 )   &    |-  ( ph  ->  L  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )   &    |-  ( ph  ->  ( x  e.  X  |->  ( z  e.  Z  |->  B ) )  e.  ( J  Cn  ( M  ^ k o  L )
 ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  C ) )  e.  ( J  Cn  ( M  ^ k o  K )
 ) )
 
Theoremxkofvcn 17394* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 17366.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. R   &    |-  F  =  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( f `  x ) )   =>    |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  F  e.  ( ( ( S  ^ k o  R )  tX  R )  Cn  S ) )
 
Theoremcnmptk1p 17395* The evaluation of a curried function by a one-arg function is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  K  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  K ) )   &    |-  ( y  =  B  ->  A  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  C )  e.  ( J  Cn  L ) )
 
Theoremcnmptk2 17396* The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  K  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )
 
Theoremxkoinjcn 17397* Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  F  =  ( x  e.  X  |->  ( y  e.  Y  |->  <. y ,  x >. ) )   =>    |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  ->  F  e.  ( R  Cn  ( ( S  tX  R )  ^ k o  S ) ) )
 
Theoremcnmpt2k 17398* The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-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  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )
 
Theoremtxcon 17399 The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( ( R  e.  Con  /\  S  e.  Con )  ->  ( R  tX  S )  e.  Con )
 
11.1.18  Quotient maps and quotient topology
 
Syntaxckq 17400 Extend class notation with the Kolmogorov quotient function.
 class KQ
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