P. 214 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xmetrtri 21301	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	xmetrtri2 21302	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 18947 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 ) )
Theorem	metrtri 21303	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 ) )
Theorem	xmetgt0 21304	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 ) ) )
Theorem	metgt0 21305	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 ) ) )
Theorem	metn0 21306	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 21307	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 21308	Lemma for metres 21311. (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 21309	Lemma for metres 21311. (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 21310	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 21311	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 21312	The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- (/) e. ( Met ` (/) )
Theorem	prdsdsf 21313*	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 ) )
Theorem	prdsxmetlem 21314*	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 ) )
Theorem	prdsxmet 21315*	The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 21314. (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 ) )
Theorem	prdsmet 21316*	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 ) )
Theorem	ressprdsds 21317*	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 |-> ( R ↾s 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 ) ) )
Theorem	resspwsds 21318	Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
|- ( ph -> Y = ( R ^s I ) )   &    |- ( ph -> H = ( ( R ↾s 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 ) ) )
Theorem	imasdsf1olem 21319*	Lemma for imasdsf1o 21320. (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*s ↾s ( 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 ) )
Theorem	imasdsf1o 21320	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 ) )
Theorem	imasf1oxmet 21321	The image of an extended metric is an extended metric. (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 -> D e. ( *Met ` B ) )
Theorem	imasf1omet 21322	The image of a metric is a metric. (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 -> D e. ( Met ` B ) )
Theorem	xpsdsfn 21323	Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- T = ( R X.s S )   &    |- X = ( Base ` R )   &    |- Y = ( Base ` S )   &    |- ( ph -> R e. V )   &    |- ( ph -> S e. W )   &    |- P = ( dist ` T )   =>    |- ( ph -> P Fn ( ( X X. Y ) X. ( X X. Y ) ) )
Theorem	xpsdsfn2 21324	Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- T = ( R X.s S )   &    |- X = ( Base ` R )   &    |- Y = ( Base ` S )   &    |- ( ph -> R e. V )   &    |- ( ph -> S e. W )   &    |- P = ( dist ` T )   =>    |- ( ph -> P Fn ( ( Base ` T ) X. ( Base ` T ) ) )
Theorem	xpsxmetlem 21325*	Lemma for xpsxmet 21326. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- T = ( R X.s S )   &    |- X = ( Base ` R )   &    |- Y = ( Base ` S )   &    |- ( ph -> R e. V )   &    |- ( ph -> S e. W )   &    |- P = ( dist ` T )   &    |- M = ( ( dist ` R ) |` ( X X. X ) )   &    |- N = ( ( dist ` S ) |` ( Y X. Y ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   =>    |- ( ph -> ( dist ` ( (Scalar ` R ) X_s `' ( { R } +c { S } ) ) ) e. ( *Met ` ran ( x e. X , y e. Y |-> `' ( { x } +c { y } ) ) ) )
Theorem	xpsxmet 21326	A product metric of extended metrics is an extended metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- T = ( R X.s S )   &    |- X = ( Base ` R )   &    |- Y = ( Base ` S )   &    |- ( ph -> R e. V )   &    |- ( ph -> S e. W )   &    |- P = ( dist ` T )   &    |- M = ( ( dist ` R ) |` ( X X. X ) )   &    |- N = ( ( dist ` S ) |` ( Y X. Y ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   =>    |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )
Theorem	xpsdsval 21327	Value of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- T = ( R X.s S )   &    |- X = ( Base ` R )   &    |- Y = ( Base ` S )   &    |- ( ph -> R e. V )   &    |- ( ph -> S e. W )   &    |- P = ( dist ` T )   &    |- M = ( ( dist ` R ) |` ( X X. X ) )   &    |- N = ( ( dist ` S ) |` ( Y X. Y ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   &    |- ( ph -> A e. X )   &    |- ( ph -> B e. Y )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. Y )   =>    |- ( ph -> ( <. A , B >. P <. C , D >. ) = sup ( { ( A M C ) , ( B N D ) } , RR* , < ) )
Theorem	xpsmet 21328	The direct product of two metric spaces. Definition 14-1.5 of [Gleason] p. 225. (Contributed by NM, 20-Jun-2007.) (Revised by Mario Carneiro, 20-Aug-2015.)
|- T = ( R X.s S )   &    |- X = ( Base ` R )   &    |- Y = ( Base ` S )   &    |- ( ph -> R e. V )   &    |- ( ph -> S e. W )   &    |- P = ( dist ` T )   &    |- M = ( ( dist ` R ) |` ( X X. X ) )   &    |- N = ( ( dist ` S ) |` ( Y X. Y ) )   &    |- ( ph -> M e. ( Met ` X ) )   &    |- ( ph -> N e. ( Met ` Y ) )   =>    |- ( ph -> P e. ( Met ` ( X X. Y ) ) )
12.4.3  Metric space balls
Theorem	blfvalps 21329*	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 21330*	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	blvalps 21331*	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 21332*	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 21333	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 21334	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 21335	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 21336	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 21337	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 21338	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 21339	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 21340	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 21341	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 21342	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 21343	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 21344	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 21345	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 21346	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 21347	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 21350 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 21348	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 21350 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 21349	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 21350	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 21351	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 21352	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 21353	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 21354*	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 21355*	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 21356	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 21357	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 21358	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 21359	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	xbln0 21360	A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( ( P ( ball ` D ) R ) =/= (/) <-> 0 < R ) )
Theorem	bln0 21361	A ball is not empty. (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 21362	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 21363	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 21364	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 21365	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 21366	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	blin 21367	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 ) if ( R <_ S , R , S ) ) )
Theorem	ssblps 21368	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 21369	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 21370*	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 21371*	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 21372*	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 21373*	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 21374*	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 21375*	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 ) )
Theorem	blbas 21376	The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)
|- ( D e. ( *Met ` X ) -> ran ( ball ` D ) e. TopBases )
Theorem	blres 21377	A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
|- C = ( D |` ( Y X. Y ) )   =>    |- ( ( D e. ( *Met ` X ) /\ P e. ( X i^i Y ) /\ R e. RR* ) -> ( P ( ball ` C ) R ) = ( ( P ( ball ` D ) R ) i^i Y ) )
Theorem	xmeterval 21378	Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- .~ = ( `' D " RR )   =>    |- ( D e. ( *Met ` X ) -> ( A .~ B <-> ( A e. X /\ B e. X /\ ( A D B ) e. RR ) ) )
Theorem	xmeter 21379	The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- .~ = ( `' D " RR )   =>    |- ( D e. ( *Met ` X ) -> .~ Er X )
Theorem	xmetec 21380	The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 7419, infinity balls are either identical or disjoint, quite unlike the usual situation with Euclidean balls which admit many kinds of overlap. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- .~ = ( `' D " RR )   =>    |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> [ P ] .~ = ( P ( ball ` D ) +oo ) )
Theorem	blssec 21381	A ball centered at P is contained in the set of points finitely separated from P. This is just an application of ssbl 21369 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- .~ = ( `' D " RR )   =>    |- ( ( D e. ( *Met ` X ) /\ P e. X /\ S e. RR* ) -> ( P ( ball ` D ) S ) C_ [ P ] .~ )
Theorem	blpnfctr 21382	The infinity ball in an extended metric acts like an ultrametric ball in that every point in the ball is also its center. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( P ( ball ` D ) +oo ) = ( A ( ball ` D ) +oo ) )
Theorem	xmetresbl 21383	An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 21380, this shows that any extended metric space can be "factored" into the disjoint union of proper metric spaces, with points in the same region measured by that region's metric, and points in different regions being distance +oo from each other. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- B = ( P ( ball ` D ) R )   =>    |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) e. ( Met ` B ) )
12.4.4  Open sets of a metric space
Theorem	mopnval 21384	An open set is a subset of a metric space which includes a ball around each of its points. Definition 1.3-2 of [Kreyszig] p. 18. The object ( MetOpen ` D ) is the family of all open sets in the metric space determined by the metric D. By mopntop 21386, the open sets of a metric space form a topology J, whose base set is U. J by mopnuni 21387. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> J = ( topGen ` ran ( ball ` D ) ) )
Theorem	mopntopon 21385	The set of open sets of a metric space X is a topology on X. Remark in [Kreyszig] p. 19. This theorem connects the two concepts and makes available the theorems for topologies for use with metric spaces. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> J e. (TopOn ` X ) )
Theorem	mopntop 21386	The set of open sets of a metric space is a topology. (Contributed by NM, 28-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> J e. Top )
Theorem	mopnuni 21387	The union of all open sets in a metric space is its underlying set. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> X = U. J )
Theorem	elmopn 21388*	The defining property of an open set of a metric space. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> ( A e. J <-> ( A C_ X /\ A. x e. A E. y e. ran ( ball ` D ) ( x e. y /\ y C_ A ) ) ) )
Theorem	mopnfss 21389	The family of open sets of a metric space is a collection of subsets of the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> J C_ ~P X )
Theorem	mopnm 21390	The base set of a metric space is open. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> X e. J )
Theorem	elmopn2 21391*	A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> ( A e. J <-> ( A C_ X /\ A. x e. A E. y e. RR+ ( x ( ball ` D ) y ) C_ A ) ) )
Theorem	mopnss 21392	An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.)
|- J = ( MetOpen ` D )   =>    |- ( ( D e. ( *Met ` X ) /\ A e. J ) -> A C_ X )
Theorem	isxms 21393	Express the predicate " <. X , D >. is an extended metric space" with underlying set X and distance function D. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- J = ( TopOpen ` K )   &    |- X = ( Base ` K )   &    |- D = ( ( dist ` K ) |` ( X X. X ) )   =>    |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )
Theorem	isxms2 21394	Express the predicate " <. X , D >. is an extended metric space" with underlying set X and distance function D. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- J = ( TopOpen ` K )   &    |- X = ( Base ` K )   &    |- D = ( ( dist ` K ) |` ( X X. X ) )   =>    |- ( K e. *MetSp <-> ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) )
Theorem	isms 21395	Express the predicate " <. X , D >. is a metric space" with underlying set X and distance function D. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
|- J = ( TopOpen ` K )   &    |- X = ( Base ` K )   &    |- D = ( ( dist ` K ) |` ( X X. X ) )   =>    |- ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )
Theorem	isms2 21396	Express the predicate " <. X , D >. is a metric space" with underlying set X and distance function D. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
|- J = ( TopOpen ` K )   &    |- X = ( Base ` K )   &    |- D = ( ( dist ` K ) |` ( X X. X ) )   =>    |- ( K e. MetSp <-> ( D e. ( Met ` X ) /\ J = ( MetOpen ` D ) ) )
Theorem	xmstopn 21397	The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- J = ( TopOpen ` K )   &    |- X = ( Base ` K )   &    |- D = ( ( dist ` K ) |` ( X X. X ) )   =>    |- ( K e. *MetSp -> J = ( MetOpen ` D ) )
Theorem	mstopn 21398	The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- J = ( TopOpen ` K )   &    |- X = ( Base ` K )   &    |- D = ( ( dist ` K ) |` ( X X. X ) )   =>    |- ( K e. MetSp -> J = ( MetOpen ` D ) )
Theorem	xmstps 21399	A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. *MetSp -> M e. TopSp )
Theorem	msxms 21400	A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. MetSp -> M e. *MetSp )