P. 209 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-xms 20801	Define the (proper) class of all extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
Definition	df-ms 20802	Define the (proper) class of all metric spaces. (Contributed by NM, 27-Aug-2006.)
|- MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
Definition	df-tms 20803	Define the function mapping a metric to a metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- toMetSp = ( d e. U. ran *Met |-> ( { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } sSet <. (TopSet ` ndx ) , ( MetOpen ` d ) >. ) )
Theorem	ismet 20804*	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 20805*	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 20806*	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 20807*	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 20808*	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 20809*	Lemma for metf 20811 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 20810	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 20811	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 20812	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 20813	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 20814	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 20815	A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
Theorem	xmetdmdm 20816	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 20817	Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( D e. ( Met ` X ) -> X = dom dom D )
Theorem	xmetunirn 20818	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 20819	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 20820	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 20821	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 20822	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 20823	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 20824	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 20825	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 20826	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 20827	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 20828	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 20829	An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( D e. ( *Met ` X ) -> D e. (PsMet ` X ) )
Theorem	xmettpos 20830	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 20831	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 20832	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 20833	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 20834	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 20835	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 20836	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 20837	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 18441 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 20838	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 20839	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 20840	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 20841	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 20842	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 20843	Lemma for metres 20846. (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 20844	Lemma for metres 20846. (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 20845	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 20846	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 20847	The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- (/) e. ( Met ` (/) )
Theorem	prdsdsf 20848*	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 20849*	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 20850*	The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 20849. (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 20851*	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 20852*	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 20853	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 20854*	Lemma for imasdsf1o 20855. (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 20855	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 20856	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 20857	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 20858	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 20859	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 20860*	Lemma for xpsxmet 20861. (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 20861	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 20862	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 20863	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 20864*	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 20865*	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 20866*	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 20867*	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 20868	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 20869	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 20870	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 20871	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 20872	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 20873	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 20874	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 20875	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 20876	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 20877	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 20878	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 20879	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 20880	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 20881	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 20882	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 20885 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 20883	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 20885 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 20884	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 20885	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 20886	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 20887	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 20888	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 20889*	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 20890*	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 20891	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 20892	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 20893	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 20894	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 20895	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 20896	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 20897	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 20898	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 20899	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 20900	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 )