P. 151 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15001-15100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	uptx 15001*	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 ) ) )
Theorem	txcn 15002	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 ) ) ) )
Theorem	txrest 15003	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 ) ) = ( ( R ↾t A ) tX ( S ↾t B ) ) )
Theorem	txdis 15004	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 15005*	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 15006*	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 15007*	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 15008*	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 15009*	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 15010*	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 15011*	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 15012*	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 15013*	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 15014*	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 15015*	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 15016*	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 15017*	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 15018*	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 15019*	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 15020*	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 15021*	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 15022*	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 15023*	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 15024*	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 15025*	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 15026	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 15027	Extend class notation with the class of all homeomorphisms.
class Homeo
Definition	df-hmeo 15028*	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 15029	The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- Homeo Fn ( Top X. Top )
Theorem	hmeofvalg 15030*	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 15031	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 15032	A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
|- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
Theorem	hmeocnvcn 15033	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 15034	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 15035	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 15036	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 15037	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 15038	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 15039	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 15040	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 15041*	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 15042	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 15043	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 15044	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 15045	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 15046*	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 15047*	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 15048*	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 15049	The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
|- Rel PsMet
Theorem	ispsmet 15050*	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 15051	Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( D e. (PsMet ` X ) -> X = dom dom D )
Theorem	psmetf 15052	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 15053	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 15054	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 15055	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 15056	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 15057	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 15058	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 15059	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 15060	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 15061	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 15062	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 15063	Extend class notation with the class of extended metric spaces.
class *MetSp
Syntax	cms 15064	Extend class notation with the class of metric spaces.
class MetSp
Syntax	ctms 15065	Extend class notation with the function mapping a metric to the metric space it defines.
class toMetSp
Definition	df-xms 15066	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 15067	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 15068	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 15069	The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
|- Rel Met
Theorem	xmetrel 15070	The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
|- Rel *Met
Theorem	ismet 15071*	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 15072*	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 15073*	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 15074*	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 15075*	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 15076*	Lemma for metf 15078 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 15077	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 15078	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 15079	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 15080	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 15081	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 15082	A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
Theorem	xmetdmdm 15083	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 15084	Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( D e. ( Met ` X ) -> X = dom dom D )
Theorem	xmetunirn 15085	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 15086	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 15087	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 15088	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 15089	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 15090	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 15091	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 15092	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 15093	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 15094	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 15095	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 15096	An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( D e. ( *Met ` X ) -> D e. (PsMet ` X ) )
Theorem	xmettpos 15097	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 15098	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 15099	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 15100	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 ) ) )