P. 154 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15301-15400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	txdis 15301	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 ) )
Theorem	txdis1cn 15302*	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 ) )
Theorem	txlm 15303*	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 >. ) )
Theorem	lmcn2 15304*	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 ) )
9.1.9  Continuous function-builders
Theorem	cnmptid 15305*	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 ) )
Theorem	cnmptc 15306*	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 ) )
Theorem	cnmpt11 15307*	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 ) )
Theorem	cnmpt11f 15308*	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 ) )
Theorem	cnmpt1t 15309*	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 ) ) )
Theorem	cnmpt12f 15310*	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 ) )
Theorem	cnmpt12 15311*	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 ) )
Theorem	cnmpt1st 15312*	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 ) )
Theorem	cnmpt2nd 15313*	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 ) )
Theorem	cnmpt2c 15314*	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 ) )
Theorem	cnmpt21 15315*	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 ) )
Theorem	cnmpt21f 15316*	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 ) )
Theorem	cnmpt2t 15317*	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 ) ) )
Theorem	cnmpt22 15318*	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 ) )
Theorem	cnmpt22f 15319*	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 ) )
Theorem	cnmpt1res 15320*	The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
|- K = ( J ↾t 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 ) )
Theorem	cnmpt2res 15321*	The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
|- K = ( J ↾t Y )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> Y C_ X )   &    |- N = ( M ↾t 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 ) )
Theorem	cnmptcom 15322*	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 ) )
Theorem	imasnopn 15323	If a relation graph is open, then an image set of a singleton is also open. Corollary of Proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)
|- X = U. J   =>    |- ( ( ( J e. Top /\ K e. Top ) /\ ( R e. ( J tX K ) /\ A e. X ) ) -> ( R " { A } ) e. K )
9.1.10  Homeomorphisms
Syntax	chmeo 15324	Extend class notation with the class of all homeomorphisms.
class Homeo
Definition	df-hmeo 15325*	Function returning all the homeomorphisms from topology j to topology k. (Contributed by FL, 14-Feb-2007.)
|- Homeo = ( j e. Top , k e. Top |-> { f e. ( j Cn k ) | `' f e. ( k Cn j ) } )
Theorem	hmeofn 15326	The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- Homeo Fn ( Top X. Top )
Theorem	hmeofvalg 15327*	The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ( J e. Top /\ K e. Top ) -> ( J Homeo K ) = { f e. ( J Cn K ) | `' f e. ( K Cn J ) } )
Theorem	ishmeo 15328	The predicate F is a homeomorphism between topology J and topology K. Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( F e. ( J Homeo K ) <-> ( F e. ( J Cn K ) /\ `' F e. ( K Cn J ) ) )
Theorem	hmeocn 15329	A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
|- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
Theorem	hmeocnvcn 15330	The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
|- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
Theorem	hmeocnv 15331	The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( F e. ( J Homeo K ) -> `' F e. ( K Homeo J ) )
Theorem	hmeof1o2 15332	A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Homeo K ) ) -> F : X -1-1-onto-> Y )
Theorem	hmeof1o 15333	A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
|- X = U. J   &    |- Y = U. K   =>    |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> Y )
Theorem	hmeoima 15334	The image of an open set by a homeomorphism is an open set. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ( F e. ( J Homeo K ) /\ A e. J ) -> ( F " A ) e. K )
Theorem	hmeoopn 15335	Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. J <-> ( F " A ) e. K ) )
Theorem	hmeocld 15336	Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. ( Clsd ` J ) <-> ( F " A ) e. ( Clsd ` K ) ) )
Theorem	hmeontr 15337	Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) = ( F " ( ( int ` J ) ` A ) ) )
Theorem	hmeoimaf1o 15338*	The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- G = ( x e. J |-> ( F " x ) )   =>    |- ( F e. ( J Homeo K ) -> G : J -1-1-onto-> K )
Theorem	hmeores 15339	The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Homeo K ) /\ Y C_ X ) -> ( F |` Y ) e. ( ( J ↾t Y ) Homeo ( K ↾t ( F " Y ) ) ) )
Theorem	hmeoco 15340	The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> ( G o. F ) e. ( J Homeo L ) )
Theorem	idhmeo 15341	The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- ( J e. (TopOn ` X ) -> ( _I |` X ) e. ( J Homeo J ) )
Theorem	hmeocnvb 15342	The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( Rel F -> ( `' F e. ( J Homeo K ) <-> F e. ( K Homeo J ) ) )
Theorem	txhmeo 15343*	Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- X = U. J   &    |- Y = U. K   &    |- ( ph -> F e. ( J Homeo L ) )   &    |- ( ph -> G e. ( K Homeo M ) )   =>    |- ( ph -> ( x e. X , y e. Y |-> <. ( F ` x ) , ( G ` y ) >. ) e. ( ( J tX K ) Homeo ( L tX M ) ) )
Theorem	txswaphmeolem 15344*	Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- ( ( y e. Y , x e. X |-> <. x , y >. ) o. ( x e. X , y e. Y |-> <. y , x >. ) ) = ( _I |` ( X X. Y ) )
Theorem	txswaphmeo 15345*	There is a homeomorphism from X X. Y to Y X. X. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( x e. X , y e. Y |-> <. y , x >. ) e. ( ( J tX K ) Homeo ( K tX J ) ) )
9.2  Metric spaces
9.2.1  Pseudometric spaces
Theorem	psmetrel 15346	The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
|- Rel PsMet
Theorem	ispsmet 15347*	Express the predicate " D is a pseudometric". (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( X e. V -> ( D e. (PsMet ` X ) <-> ( D : ( X X. X ) --> RR* /\ A. x e. X ( ( x D x ) = 0 /\ A. y e. X A. z e. X ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) ) ) ) )
Theorem	psmetdmdm 15348	Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( D e. (PsMet ` X ) -> X = dom dom D )
Theorem	psmetf 15349	The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
Theorem	psmetcl 15350	Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
Theorem	psmet0 15351	The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )
Theorem	psmettri2 15352	Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( A D B ) <_ ( ( C D A ) +e ( C D B ) ) )
Theorem	psmetsym 15353	The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
Theorem	psmettri 15354	Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) +e ( C D B ) ) )
Theorem	psmetge0 15355	The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.)
|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
Theorem	psmetxrge0 15356	The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
|- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> ( 0 [,] +oo ) )
Theorem	psmetres2 15357	Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) e. (PsMet ` R ) )
Theorem	psmetlecl 15358	Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X ) /\ ( C e. RR /\ ( A D B ) <_ C ) ) -> ( A D B ) e. RR )
Theorem	distspace 15359	A set X together with a (distance) function D which is a pseudometric is a distance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set X equipped with a distance D, which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.) (Revised by AV, 5-Jul-2022.)
|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( D : ( X X. X ) --> RR* /\ ( A D A ) = 0 ) /\ ( 0 <_ ( A D B ) /\ ( A D B ) = ( B D A ) ) ) )
9.2.2  Basic metric space properties
Syntax	cxms 15360	Extend class notation with the class of extended metric spaces.
class *MetSp
Syntax	cms 15361	Extend class notation with the class of metric spaces.
class MetSp
Syntax	ctms 15362	Extend class notation with the function mapping a metric to the metric space it defines.
class toMetSp
Definition	df-xms 15363	Define the (proper) class of extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
Definition	df-ms 15364	Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)
|- MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
Definition	df-tms 15365	Define the function mapping a metric to the metric space which it defines. (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 ) >. ) )
Theorem	metrel 15366	The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
|- Rel Met
Theorem	xmetrel 15367	The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
|- Rel *Met
Theorem	ismet 15368*	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 ) ) ) ) ) )
Theorem	isxmet 15369*	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 ) ) ) ) ) )
Theorem	ismeti 15370*	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 )
Theorem	isxmetd 15371*	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 ) )
Theorem	isxmet2d 15372*	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 ) )
Theorem	metflem 15373*	Lemma for metf 15375 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 ) ) ) ) )
Theorem	xmetf 15374	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* )
Theorem	metf 15375	Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)
|- ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR )
Theorem	xmetcl 15376	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* )
Theorem	metcl 15377	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 )
Theorem	ismet2 15378	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 ) )
Theorem	metxmet 15379	A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
Theorem	xmetdmdm 15380	Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( D e. ( *Met ` X ) -> X = dom dom D )
Theorem	metdmdm 15381	Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( D e. ( Met ` X ) -> X = dom dom D )
Theorem	xmetunirn 15382	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 ) )
Theorem	xmeteq0 15383	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 ) )
Theorem	meteq0 15384	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 ) )
Theorem	xmettri2 15385	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 ) ) )
Theorem	mettri2 15386	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 ) ) )
Theorem	xmet0 15387	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 )
Theorem	met0 15388	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 )
Theorem	xmetge0 15389	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 ) )
Theorem	metge0 15390	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 ) )
Theorem	xmetlecl 15391	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 )
Theorem	xmetsym 15392	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 ) )
Theorem	xmetpsmet 15393	An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( D e. ( *Met ` X ) -> D e. (PsMet ` X ) )
Theorem	xmettpos 15394	The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( D e. ( *Met ` X ) -> tpos D = D )
Theorem	metsym 15395	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 ) )
Theorem	xmettri 15396	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 ) ) )
Theorem	mettri 15397	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 ) ) )
Theorem	xmettri3 15398	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 ) ) )
Theorem	mettri3 15399	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 ) ) )
Theorem	xmetrtri 15400	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 ) )