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	psmetres2 15001	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 15002	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 15003	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 15004	Extend class notation with the class of extended metric spaces.
class *MetSp
Syntax	cms 15005	Extend class notation with the class of metric spaces.
class MetSp
Syntax	ctms 15006	Extend class notation with the function mapping a metric to the metric space it defines.
class toMetSp
Definition	df-xms 15007	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 15008	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 15009	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 15010	The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
|- Rel Met
Theorem	xmetrel 15011	The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
|- Rel *Met
Theorem	ismet 15012*	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 15013*	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 15014*	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 15015*	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 15016*	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 15017*	Lemma for metf 15019 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 15018	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 15019	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 15020	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 15021	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 15022	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 15023	A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
Theorem	xmetdmdm 15024	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 15025	Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( D e. ( Met ` X ) -> X = dom dom D )
Theorem	xmetunirn 15026	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 15027	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 15028	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 15029	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 15030	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 15031	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 15032	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 15033	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 15034	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 15035	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 15036	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 15037	An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( D e. ( *Met ` X ) -> D e. (PsMet ` X ) )
Theorem	xmettpos 15038	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 15039	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 15040	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 15041	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 15042	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 15043	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 15044	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 ) )
Theorem	metrtri 15045	Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 21-Apr-2023.)
|- ( ( D e. ( Met ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( abs ` ( ( A D C ) - ( B D C ) ) ) <_ ( A D B ) )
Theorem	metn0 15046	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 =/= (/) ) )
Theorem	xmetres2 15047	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 ) )
Theorem	metreslem 15048	Lemma for metres 15051. (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 ) ) ) )
Theorem	metres2 15049	Lemma for metres 15051. (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 ) )
Theorem	xmetres 15050	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 ) ) )
Theorem	metres 15051	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 ) ) )
Theorem	0met 15052	The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- (/) e. ( Met ` (/) )
9.2.3  Metric space balls
Theorem	blfvalps 15053*	The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.)
|- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )
Theorem	blfval 15054*	The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Proof shortened by Thierry Arnoux, 11-Feb-2018.)
|- ( D e. ( *Met ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )
Theorem	blex 15055	A ball is a set. Also see blfn 14509 in case you just know D is a set, not D e. ( *Met ` X ). (Contributed by Jim Kingdon, 4-May-2023.)
|- ( D e. ( *Met ` X ) -> ( ball ` D ) e. _V )
Theorem	blvalps 15056*	The ball around a point P is the set of all points whose distance from P is less than the ball's radius R. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) = { x e. X | ( P D x ) < R } )
Theorem	blval 15057*	The ball around a point P is the set of all points whose distance from P is less than the ball's radius R. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) = { x e. X | ( P D x ) < R } )
Theorem	elblps 15058	Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A e. X /\ ( P D A ) < R ) ) )
Theorem	elbl 15059	Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A e. X /\ ( P D A ) < R ) ) )
Theorem	elbl2ps 15060	Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( ( D e. (PsMet ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( P D A ) < R ) )
Theorem	elbl2 15061	Membership in a ball. (Contributed by NM, 9-Mar-2007.)
|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( P D A ) < R ) )
Theorem	elbl3ps 15062	Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
|- ( ( ( D e. (PsMet ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A D P ) < R ) )
Theorem	elbl3 15063	Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A D P ) < R ) )
Theorem	blcomps 15064	Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( ( D e. (PsMet ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> P e. ( A ( ball ` D ) R ) ) )
Theorem	blcom 15065	Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)
|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> P e. ( A ( ball ` D ) R ) ) )
Theorem	xblpnfps 15066	The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( A e. ( P ( ball ` D ) +oo ) <-> ( A e. X /\ ( P D A ) e. RR ) ) )
Theorem	xblpnf 15067	The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( A e. ( P ( ball ` D ) +oo ) <-> ( A e. X /\ ( P D A ) e. RR ) ) )
Theorem	blpnf 15068	The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ( D e. ( Met ` X ) /\ P e. X ) -> ( P ( ball ` D ) +oo ) = X )
Theorem	bldisj 15069	Two balls are disjoint if the center-to-center distance is more than the sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.)
|- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ Q e. X ) /\ ( R e. RR* /\ S e. RR* /\ ( R +e S ) <_ ( P D Q ) ) ) -> ( ( P ( ball ` D ) R ) i^i ( Q ( ball ` D ) S ) ) = (/) )
Theorem	blgt0 15070	A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ A e. ( P ( ball ` D ) R ) ) -> 0 < R )
Theorem	bl2in 15071	Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( ( ( D e. ( Met ` X ) /\ P e. X /\ Q e. X ) /\ ( R e. RR /\ R <_ ( ( P D Q ) / 2 ) ) ) -> ( ( P ( ball ` D ) R ) i^i ( Q ( ball ` D ) R ) ) = (/) )
Theorem	xblss2ps 15072	One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 15075 for extended metrics, we have to assume the balls are a finite distance apart, or else P will not even be in the infinity ball around Q. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ph -> D e. (PsMet ` X ) )   &    |- ( ph -> P e. X )   &    |- ( ph -> Q e. X )   &    |- ( ph -> R e. RR* )   &    |- ( ph -> S e. RR* )   &    |- ( ph -> ( P D Q ) e. RR )   &    |- ( ph -> ( P D Q ) <_ ( S +e -e R ) )   =>    |- ( ph -> ( P ( ball ` D ) R ) C_ ( Q ( ball ` D ) S ) )
Theorem	xblss2 15073	One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 15075 for extended metrics, we have to assume the balls are a finite distance apart, or else P will not even be in the infinity ball around Q. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ph -> D e. ( *Met ` X ) )   &    |- ( ph -> P e. X )   &    |- ( ph -> Q e. X )   &    |- ( ph -> R e. RR* )   &    |- ( ph -> S e. RR* )   &    |- ( ph -> ( P D Q ) e. RR )   &    |- ( ph -> ( P D Q ) <_ ( S +e -e R ) )   =>    |- ( ph -> ( P ( ball ` D ) R ) C_ ( Q ( ball ` D ) S ) )
Theorem	blss2ps 15074	One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ Q e. X ) /\ ( R e. RR /\ S e. RR /\ ( P D Q ) <_ ( S - R ) ) ) -> ( P ( ball ` D ) R ) C_ ( Q ( ball ` D ) S ) )
Theorem	blss2 15075	One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ Q e. X ) /\ ( R e. RR /\ S e. RR /\ ( P D Q ) <_ ( S - R ) ) ) -> ( P ( ball ` D ) R ) C_ ( Q ( ball ` D ) S ) )
Theorem	blhalf 15076	A ball of radius R / 2 is contained in a ball of radius R centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.)
|- ( ( ( M e. ( *Met ` X ) /\ Y e. X ) /\ ( R e. RR /\ Z e. ( Y ( ball ` M ) ( R / 2 ) ) ) ) -> ( Y ( ball ` M ) ( R / 2 ) ) C_ ( Z ( ball ` M ) R ) )
Theorem	blfps 15077	Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( D e. (PsMet ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
Theorem	blf 15078	Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
Theorem	blrnps 15079*	Membership in the range of the ball function. Note that ran ( ball ` D ) is the collection of all balls for metric D. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( D e. (PsMet ` X ) -> ( A e. ran ( ball ` D ) <-> E. x e. X E. r e. RR* A = ( x ( ball ` D ) r ) ) )
Theorem	blrn 15080*	Membership in the range of the ball function. Note that ran ( ball ` D ) is the collection of all balls for metric D. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( D e. ( *Met ` X ) -> ( A e. ran ( ball ` D ) <-> E. x e. X E. r e. RR* A = ( x ( ball ` D ) r ) ) )
Theorem	xblcntrps 15081	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. ( P ( ball ` D ) R ) )
Theorem	xblcntr 15082	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. ( P ( ball ` D ) R ) )
Theorem	blcntrps 15083	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> P e. ( P ( ball ` D ) R ) )
Theorem	blcntr 15084	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR+ ) -> P e. ( P ( ball ` D ) R ) )
Theorem	xblm 15085*	A ball is inhabited iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( E. x x e. ( P ( ball ` D ) R ) <-> 0 < R ) )
Theorem	bln0 15086	A ball is not empty. It is also inhabited, as seen at blcntr 15084. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR+ ) -> ( P ( ball ` D ) R ) =/= (/) )
Theorem	blelrnps 15087	A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) e. ran ( ball ` D ) )
Theorem	blelrn 15088	A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) e. ran ( ball ` D ) )
Theorem	blssm 15089	A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ X )
Theorem	unirnblps 15090	The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) = X )
Theorem	unirnbl 15091	The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( D e. ( *Met ` X ) -> U. ran ( ball ` D ) = X )
Theorem	blininf 15092	The intersection of two balls with the same center is the smaller of them. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( ( P ( ball ` D ) R ) i^i ( P ( ball ` D ) S ) ) = ( P ( ball ` D )inf ( { R , S } , RR* , < ) ) )
Theorem	ssblps 15093	The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) /\ R <_ S ) -> ( P ( ball ` D ) R ) C_ ( P ( ball ` D ) S ) )
Theorem	ssbl 15094	The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
|- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) /\ R <_ S ) -> ( P ( ball ` D ) R ) C_ ( P ( ball ` D ) S ) )
Theorem	blssps 15095*	Any point P in a ball B can be centered in another ball that is a subset of B. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ B e. ran ( ball ` D ) /\ P e. B ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ B )
Theorem	blss 15096*	Any point P in a ball B can be centered in another ball that is a subset of B. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ B e. ran ( ball ` D ) /\ P e. B ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ B )
Theorem	blssexps 15097*	Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
Theorem	blssex 15098*	Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
Theorem	ssblex 15099*	A nested ball exists whose radius is less than any desired amount. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> E. x e. RR+ ( x < R /\ ( P ( ball ` D ) x ) C_ ( P ( ball ` D ) S ) ) )
Theorem	blin2 15100*	Given any two balls and a point in their intersection, there is a ball contained in the intersection with the given center point. (Contributed by Mario Carneiro, 12-Nov-2013.)
|- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) )