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Theorem List for Metamath Proof Explorer - 17801-17900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxctdrg 17801 The class of all topological division rings.
 class TopDRing
 
Syntaxctlm 17802 The class of all topological modules.
 class TopMod
 
Syntaxctvc 17803 The class of all topological vector spaces.
 class  TopVec
 
Definitiondf-trg 17804 Define a topological ring, which is a ring such that the addition is a topological group operation and the multiplication is continuous. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  TopRing  =  { r  e.  ( TopGrp  i^i  Ring )  |  (mulGrp `  r )  e. TopMnd }
 
Definitiondf-tdrg 17805 Define a topological division ring (which differs from a topological field only in being potentially noncommutative), which is a division ring and topological ring such that the unit group of the division ring (which is the set of nonzero elements) is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- TopDRing  =  { r  e.  ( TopRing  i^i  DivRing )  |  ( (mulGrp `  r )s  (Unit `  r )
 )  e.  TopGrp }
 
Definitiondf-tlm 17806 Define a topological left module, which is just what its name suggests: instead of a group over a ring with a scalar product connecting them, it is a topological group over a topological ring with a continuous scalar product. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- TopMod  =  { w  e.  (TopMnd  i^i  LMod )  |  ( (Scalar `  w )  e.  TopRing  /\  ( .s f `  w )  e.  ( (
 ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w ) )  Cn  ( TopOpen `  w )
 ) ) }
 
Definitiondf-tvc 17807 Define a topological left vector space, which is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  TopVec  =  { w  e. TopMod  |  (Scalar `  w )  e. TopDRing }
 
Theoremistrg 17808 Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  ( R  e.  TopRing  <->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
 
Theoremtrgtmd 17809 The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  ( R  e.  TopRing  ->  M  e. TopMnd )
 
Theoremistdrg 17810 Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  <->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U )  e.  TopGrp ) )
 
Theoremtdrgunit 17811 The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  ( Ms  U )  e.  TopGrp )
 
Theoremtrgtgp 17812 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
 
Theoremtrgtmd2 17813 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e. TopMnd )
 
Theoremtrgtps 17814 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopSp )
 
Theoremtrgrng 17815 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Ring )
 
Theoremtrggrp 17816 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Grp )
 
Theoremtdrgtrg 17817 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. 
 TopRing )
 
Theoremtdrgdrng 17818 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. 
 DivRing )
 
Theoremtdrgrng 17819 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e.  Ring )
 
Theoremtdrgtmd 17820 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. TopMnd )
 
Theoremtdrgtps 17821 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e.  TopSp )
 
Theoremistdrg2 17822 A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. TopDRing  <->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  ( B  \  {  .0.  } ) )  e.  TopGrp ) )
 
Theoremmulrcn 17823 The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  T  =  ( + f `  (mulGrp `  R ) )   =>    |-  ( R  e.  TopRing  ->  T  e.  ( ( J  tX  J )  Cn  J ) )
 
Theoreminvrcn2 17824 The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to itself. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  I  =  (
 invr `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  I  e.  ( ( Jt  U )  Cn  ( Jt  U ) ) )
 
Theoreminvrcn 17825 The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to the field. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  I  =  (
 invr `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  I  e.  ( ( Jt  U )  Cn  J ) )
 
Theoremcnmpt1mulr 17826* Continuity of ring multiplication; analogue of cnmpt12f 17322 which cannot be used directly because 
.r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  TopRing )   &    |-  ( ph  ->  K  e.  (TopOn `  X ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B ) )  e.  ( K  Cn  J ) )
 
Theoremcnmpt2mulr 17827* Continuity of ring multiplication; analogue of cnmpt22f 17331 which cannot be used directly because 
.r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  TopRing )   &    |-  ( ph  ->  K  e.  (TopOn `  X ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( K  tX  L )  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( K 
 tX  L )  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A 
 .x.  B ) )  e.  ( ( K  tX  L )  Cn  J ) )
 
Theoremdvrcn 17828 The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  ./  =  (/r `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  ./  e.  ( ( J  tX  ( Jt  U ) )  Cn  J ) )
 
Theoremistlm 17829 The predicate " W is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- 
 .x.  =  ( .s f `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( TopOpen `  F )   =>    |-  ( W  e. TopMod  <->  ( ( W  e. TopMnd  /\  W  e.  LMod  /\  F  e.  TopRing )  /\  .x. 
 e.  ( ( K 
 tX  J )  Cn  J ) ) )
 
Theoremvscacn 17830 The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- 
 .x.  =  ( .s f `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( TopOpen `  F )   =>    |-  ( W  e. TopMod  ->  .x.  e.  ( ( K  tX  J )  Cn  J ) )
 
Theoremtlmtmd 17831 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e. TopMnd )
 
Theoremtlmtps 17832 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  TopSp )
 
Theoremtlmlmod 17833 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  LMod )
 
Theoremtlmtrg 17834 The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. TopMod  ->  F  e.  TopRing )
 
Theoremtlmscatps 17835 The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. TopMod  ->  F  e.  TopSp )
 
Theoremistvc 17836 A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  TopVec  <->  ( W  e. TopMod  /\  F  e. TopDRing ) )
 
Theoremtvctdrg 17837 The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  TopVec  ->  F  e. TopDRing )
 
Theoremcnmpt1vsca 17838* Continuity of scalar multiplication; analogue of cnmpt12f 17322 which cannot be used directly because  .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  K  =  ( TopOpen `  F )   &    |-  ( ph  ->  W  e. TopMod )   &    |-  ( ph  ->  L  e.  (TopOn `  X ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B ) )  e.  ( L  Cn  J ) )
 
Theoremcnmpt2vsca 17839* Continuity of scalar multiplication; analogue of cnmpt22f 17331 which cannot be used directly because  .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  K  =  ( TopOpen `  F )   &    |-  ( ph  ->  W  e. TopMod )   &    |-  ( ph  ->  L  e.  (TopOn `  X ) )   &    |-  ( ph  ->  M  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( L  tX  M )  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( L 
 tX  M )  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A 
 .x.  B ) )  e.  ( ( L  tX  M )  Cn  J ) )
 
Theoremtlmtgp 17840 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  TopGrp )
 
Theoremtvctlm 17841 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e. TopMod )
 
Theoremtvclmod 17842 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e.  LMod )
 
Theoremtvclvec 17843 A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e.  LVec )
 
11.3  Metric spaces
 
11.3.1  Basic metric space properties
 
Syntaxcxme 17844 Extend class notation with the class of all extended metric spaces.
 class  * MetSp
 
Syntaxcmt 17845 Extend class notation with the class of all metric spaces.
 class  MetSp
 
Syntaxctmt 17846 Extend class notation with the function mapping a metric to a metric space.
 class toMetSp
 
Definitiondf-xms 17847 Define the (proper) class of all extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- 
 * MetSp  =  { f  e.  TopSp  |  ( TopOpen `  f )  =  ( MetOpen `  ( ( dist `  f
 )  |`  ( ( Base `  f )  X.  ( Base `  f ) ) ) ) }
 
Definitiondf-ms 17848 Define the (proper) class of all metric spaces. (Contributed by NM, 27-Aug-2006.)
 |- 
 MetSp  =  { f  e.  * MetSp  |  (
 ( dist `  f )  |`  ( ( Base `  f
 )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f ) ) }
 
Definitiondf-tms 17849 Define the function mapping a metric to a metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- toMetSp  =  ( d  e.  U. ran  * Met  |->  ( { <. ( Base `  ndx ) , 
 dom  dom  d >. ,  <. (
 dist `  ndx ) ,  d >. } sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >. ) )
 
Theoremismet 17850* Express the predicate " D is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( X  e.  A  ->  ( D  e.  ( Met `  X )  <->  ( D :
 ( X  X.  X )
 --> RR  /\  A. x  e.  X  A. y  e.  X  ( ( ( x D y )  =  0  <->  x  =  y
 )  /\  A. z  e.  X  ( x D y )  <_  (
 ( z D x )  +  ( z D y ) ) ) ) ) )
 
Theoremisxmet 17851* Express the predicate " D is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( X  e.  A  ->  ( D  e.  ( * Met `  X )  <->  ( D : ( X  X.  X ) --> RR*  /\  A. x  e.  X  A. y  e.  X  ( ( ( x D y )  =  0  <->  x  =  y
 )  /\  A. z  e.  X  ( x D y )  <_  (
 ( z D x ) + e ( z D y ) ) ) ) ) )
 
Theoremismeti 17852* Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  X  e.  _V   &    |-  D : ( X  X.  X ) --> RR   &    |-  (
 ( x  e.  X  /\  y  e.  X )  ->  ( ( x D y )  =  0  <->  x  =  y
 ) )   &    |-  ( ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( x D y )  <_  ( ( z D x )  +  (
 z D y ) ) )   =>    |-  D  e.  ( Met `  X )
 
Theoremisxmetd 17853* Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  D : ( X  X.  X ) --> RR* )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( ( x D y )  =  0  <-> 
 x  =  y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x D y )  <_  ( ( z D x ) + e
 ( z D y ) ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  X ) )
 
Theoremisxmet2d 17854* It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample:  D ( x ,  y )  =  if ( x  =  y ,  0 , 
-oo ) satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  D : ( X  X.  X ) --> RR* )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
 0  <_  ( x D y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( ( x D y )  <_ 
 0 
 <->  x  =  y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  /\  ( ( z D x )  e. 
 RR  /\  ( z D y )  e. 
 RR ) )  ->  ( x D y ) 
 <_  ( ( z D x )  +  (
 z D y ) ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  X ) )
 
Theoremmetflem 17855* Lemma for metf 17857 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  ( D : ( X  X.  X ) --> RR  /\  A. x  e.  X  A. y  e.  X  (
 ( ( x D y )  =  0  <-> 
 x  =  y ) 
 /\  A. z  e.  X  ( x D y ) 
 <_  ( ( z D x )  +  (
 z D y ) ) ) ) )
 
Theoremxmetf 17856 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
 
Theoremmetf 17857 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)
 |-  ( D  e.  ( Met `  X )  ->  D : ( X  X.  X ) --> RR )
 
Theoremxmetcl 17858 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A D B )  e.  RR* )
 
Theoremmetcl 17859 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  RR )
 
Theoremismet2 17860 An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  <->  ( D  e.  ( * Met `  X )  /\  D : ( X  X.  X ) --> RR ) )
 
Theoremmetxmet 17861 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  D  e.  ( * Met `  X ) )
 
Theoremxmetdmdm 17862 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  X  =  dom  dom  D )
 
Theoremmetdmdm 17863 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  X  =  dom  dom  D )
 
Theoremxmetunirn 17864 Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( D  e.  U. ran  * Met  <->  D  e.  ( * Met `  dom  dom  D ) )
 
Theoremxmeteq0 17865 The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( ( A D B )  =  0  <->  A  =  B ) )
 
Theoremmeteq0 17866 The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A D B )  =  0  <->  A  =  B ) )
 
Theoremxmettri2 17867 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X ) )  ->  ( A D B )  <_  (
 ( C D A ) + e ( C D B ) ) )
 
Theoremmettri2 17868 Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X )
 )  ->  ( A D B )  <_  (
 ( C D A )  +  ( C D B ) ) )
 
Theoremxmet0 17869 The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X )  ->  ( A D A )  =  0
 )
 
Theoremmet0 17870 The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X ) 
 ->  ( A D A )  =  0 )
 
Theoremxmetge0 17871 The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  0  <_  ( A D B ) )
 
Theoremmetge0 17872 The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  0  <_  ( A D B ) )
 
Theoremxmetlecl 17873 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  RR  /\  ( A D B )  <_  C ) )  ->  ( A D B )  e.  RR )
 
Theoremxmetsym 17874 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A D B )  =  ( B D A ) )
 
Theoremxmettpos 17875 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  -> tpos 
 D  =  D )
 
Theoremmetsym 17876 The distance function of a metric space is symmetric. Definition 14-1.1(c) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( B D A ) )
 
Theoremxmettri 17877 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A D B )  <_  (
 ( A D C ) + e ( C D B ) ) )
 
Theoremmettri 17878 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C )  +  ( C D B ) ) )
 
Theoremxmettri3 17879 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A D B )  <_  (
 ( A D C ) + e ( B D C ) ) )
 
Theoremmettri3 17880 Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C )  +  ( B D C ) ) )
 
Theoremxmetrtri 17881 One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A D C ) + e  - e ( B D C ) )  <_  ( A D B ) )
 
Theoremxmetrtri2 17882 The reverse triangle inequality for the distance function of an extended metric. In order to express the "extended absolute value function", we use the distance function xrsdsval 16377 defined on the extended real structure. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  K  =  ( dist ` 
 RR* s )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D C ) K ( B D C ) )  <_  ( A D B ) )
 
Theoremmetrtri 17883 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( abs `  ( ( A D C )  -  ( B D C ) ) )  <_  ( A D B ) )
 
Theoremxmetgt0 17884 The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A  =/=  B  <->  0  <  ( A D B ) ) )
 
Theoremmetgt0 17885 The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of [Gleason] p. 223 and its converse. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A  =/=  B  <->  0  <  ( A D B ) ) )
 
Theoremmetn0 17886 A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  ( D  =/=  (/)  <->  X  =/=  (/) ) )
 
Theoremxmetres2 17887 Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( * Met `  R ) )
 
Theoremmetreslem 17888 Lemma for metres 17891. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( dom  D  =  ( X  X.  X ) 
 ->  ( D  |`  ( R  X.  R ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )
 
Theoremmetres2 17889 Lemma for metres 17891. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( Met `  R ) )
 
Theoremxmetres 17890 A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( * Met `  ( X  i^i  R ) ) )
 
Theoremmetres 17891 A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( Met `  ( X  i^i  R ) ) )
 
Theorem0met 17892 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  (/)  e.  ( Met `  (/) )
 
Theoremprdsdsf 17893* The product metric is a function into the nonnegative extended reals. In general this means that it is not a metric but rather an *extended* metric (even when all the factors are metrics), but it will be a metric when restricted to regions where it does not take infinite values. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( * Met `  V ) )   =>    |-  ( ph  ->  D : ( B  X.  B ) --> ( 0 [,]  +oo ) )
 
Theoremprdsxmetlem 17894* The product metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( * Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  B ) )
 
Theoremprdsxmet 17895* The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 17894. (Contributed by Mario Carneiro, 26-Sep-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( * Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  B ) )
 
Theoremprdsmet 17896* The product metric is a metric when the index set is finite. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( Met `  B ) )
 
Theoremressprdsds 17897* Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( ph  ->  Y  =  ( S X_s ( x  e.  I  |->  R ) ) )   &    |-  ( ph  ->  H  =  ( T X_s ( x  e.  I  |->  ( Rs  A ) ) ) )   &    |-  B  =  (
 Base `  H )   &    |-  D  =  ( dist `  Y )   &    |-  E  =  ( dist `  H )   &    |-  ( ph  ->  S  e.  U )   &    |-  ( ph  ->  T  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  (
 ( ph  /\  x  e.  I )  ->  R  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  A  e.  Z )   =>    |-  ( ph  ->  E  =  ( D  |`  ( B  X.  B ) ) )
 
Theoremresspwsds 17898 Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( ph  ->  Y  =  ( R  ^s  I )
 )   &    |-  ( ph  ->  H  =  ( ( Rs  A ) 
 ^s  I ) )   &    |-  B  =  ( Base `  H )   &    |-  D  =  ( dist `  Y )   &    |-  E  =  ( dist `  H )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  E  =  ( D  |`  ( B  X.  B ) ) )
 
Theoremimasdsf1olem 17899* Lemma for imasdsf1o 17900. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( * Met `  V )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  W  =  ( RR* ss  ( RR*  \  {  -oo } ) )   &    |-  S  =  { h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  (
 ( F `  ( 1st `  ( h `  1 ) ) )  =  ( F `  X )  /\  ( F `
  ( 2nd `  ( h `  n ) ) )  =  ( F `
  Y )  /\  A. i  e.  ( 1
 ... ( n  -  1 ) ) ( F `  ( 2nd `  ( h `  i
 ) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1
 ) ) ) ) ) }   &    |-  T  =  U_ n  e.  NN  ran  (  g  e.  S  |->  ( RR* s 
 gsumg  ( E  o.  g
 ) ) )   =>    |-  ( ph  ->  ( ( F `  X ) D ( F `  Y ) )  =  ( X E Y ) )
 
Theoremimasdsf1o 17900 The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( * Met `  V )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( ( F `  X ) D ( F `  Y ) )  =  ( X E Y ) )
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