P. 147 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14601-14700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	unirnblps 14601	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 14602	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 14603	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 14604	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 14605	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 14606*	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 14607*	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 14608*	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 14609*	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 14610*	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 14611*	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 14612	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 14613	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 14614	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 14615	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 14616	The equivalence classes under the finite separation equivalence relation are infinity balls. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- .~ = ( `' D " RR )   =>    |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> [ P ] .~ = ( P ( ball ` D ) +oo ) )
Theorem	blssec 14617	A ball centered at P is contained in the set of points finitely separated from P. This is just an application of ssbl 14605 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 14618	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 14619	An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 14616, 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 ) )
9.2.4  Open sets of a metric space
Theorem	mopnrel 14620	The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)
|- Rel MetOpen
Theorem	mopnval 14621	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 14623, the open sets of a metric space form a topology J, whose base set is U. J by mopnuni 14624. (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 14622	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 14623	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 14624	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 14625*	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 14626	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 14627	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 14628*	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 14629	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 14630	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 14631	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 14632	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 14633	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 14634	The topology component of an extended 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 14635	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 14636	An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. *MetSp -> M e. TopSp )
Theorem	msxms 14637	A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. MetSp -> M e. *MetSp )
Theorem	mstps 14638	A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. MetSp -> M e. TopSp )
Theorem	xmsxmet 14639	The distance function, suitably truncated, is an extended metric on X. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- X = ( Base ` M )   &    |- D = ( ( dist ` M ) |` ( X X. X ) )   =>    |- ( M e. *MetSp -> D e. ( *Met ` X ) )
Theorem	msmet 14640	The distance function, suitably truncated, is a metric on X. (Contributed by Mario Carneiro, 12-Nov-2013.)
|- X = ( Base ` M )   &    |- D = ( ( dist ` M ) |` ( X X. X ) )   =>    |- ( M e. MetSp -> D e. ( Met ` X ) )
Theorem	msf 14641	The distance function of a metric space is a function into the real numbers. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- X = ( Base ` M )   &    |- D = ( ( dist ` M ) |` ( X X. X ) )   =>    |- ( M e. MetSp -> D : ( X X. X ) --> RR )
Theorem	xmsxmet2 14642	The distance function, suitably truncated, is an extended metric on X. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( M e. *MetSp -> ( D |` ( X X. X ) ) e. ( *Met ` X ) )
Theorem	msmet2 14643	The distance function, suitably truncated, is a metric on X. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( M e. MetSp -> ( D |` ( X X. X ) ) e. ( Met ` X ) )
Theorem	mscl 14644	Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. MetSp /\ A e. X /\ B e. X ) -> ( A D B ) e. RR )
Theorem	xmscl 14645	Closure of the distance function of an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
Theorem	xmsge0 14646	The distance function in an extended metric space is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
Theorem	xmseq0 14647	The distance between two points in an extended metric space is zero iff the two points are identical. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> A = B ) )
Theorem	xmssym 14648	The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
Theorem	xmstri2 14649	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. *MetSp /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( A D B ) <_ ( ( C D A ) +e ( C D B ) ) )
Theorem	mstri2 14650	Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. MetSp /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( A D B ) <_ ( ( C D A ) + ( C D B ) ) )
Theorem	xmstri 14651	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. *MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) +e ( C D B ) ) )
Theorem	mstri 14652	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) + ( C D B ) ) )
Theorem	xmstri3 14653	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. *MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) +e ( B D C ) ) )
Theorem	mstri3 14654	Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) + ( B D C ) ) )
Theorem	msrtri 14655	Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- X = ( Base ` M )   &    |- D = ( dist ` M )   =>    |- ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( abs ` ( ( A D C ) - ( B D C ) ) ) <_ ( A D B ) )
Theorem	xmspropd 14656	Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( B X. B ) ) )   &    |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )   =>    |- ( ph -> ( K e. *MetSp <-> L e. *MetSp ) )
Theorem	mspropd 14657	Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( B X. B ) ) )   &    |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )   =>    |- ( ph -> ( K e. MetSp <-> L e. MetSp ) )
Theorem	setsmsbasg 14658	The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- ( ph -> X = ( Base ` M ) )   &    |- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )   &    |- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )   &    |- ( ph -> M e. V )   &    |- ( ph -> ( MetOpen ` D ) e. W )   =>    |- ( ph -> X = ( Base ` K ) )
Theorem	setsmsdsg 14659	The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- ( ph -> X = ( Base ` M ) )   &    |- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )   &    |- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )   &    |- ( ph -> M e. V )   &    |- ( ph -> ( MetOpen ` D ) e. W )   =>    |- ( ph -> ( dist ` M ) = ( dist ` K ) )
Theorem	setsmstsetg 14660	The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.)
|- ( ph -> X = ( Base ` M ) )   &    |- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )   &    |- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )   &    |- ( ph -> M e. V )   &    |- ( ph -> ( MetOpen ` D ) e. W )   =>    |- ( ph -> ( MetOpen ` D ) = (TopSet ` K ) )
Theorem	mopni 14661*	An open set of a metric space includes a ball around each of its points. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( ( D e. ( *Met ` X ) /\ A e. J /\ P e. A ) -> E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) )
Theorem	mopni2 14662*	An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( ( D e. ( *Met ` X ) /\ A e. J /\ P e. A ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ A )
Theorem	mopni3 14663*	An open set of a metric space includes an arbitrarily small ball around each of its points. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( ( ( D e. ( *Met ` X ) /\ A e. J /\ P e. A ) /\ R e. RR+ ) -> E. x e. RR+ ( x < R /\ ( P ( ball ` D ) x ) C_ A ) )
Theorem	blssopn 14664	The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> ran ( ball ` D ) C_ J )
Theorem	unimopn 14665	The union of a collection of open sets of a metric space is open. Theorem T2 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
|- J = ( MetOpen ` D )   =>    |- ( ( D e. ( *Met ` X ) /\ A C_ J ) -> U. A e. J )
Theorem	mopnin 14666	The intersection of two open sets of a metric space is open. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
|- J = ( MetOpen ` D )   =>    |- ( ( D e. ( *Met ` X ) /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )
Theorem	mopn0 14667	The empty set is an open set of a metric space. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> (/) e. J )
Theorem	rnblopn 14668	A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)
|- J = ( MetOpen ` D )   =>    |- ( ( D e. ( *Met ` X ) /\ B e. ran ( ball ` D ) ) -> B e. J )
Theorem	blopn 14669	A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` D )   =>    |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) e. J )
Theorem	neibl 14670*	The neighborhoods around a point P of a metric space are those subsets containing a ball around P. Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
|- J = ( MetOpen ` D )   =>    |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( N e. ( ( nei ` J ) ` { P } ) <-> ( N C_ X /\ E. r e. RR+ ( P ( ball ` D ) r ) C_ N ) ) )
Theorem	blnei 14671	A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
|- J = ( MetOpen ` D )   =>    |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR+ ) -> ( P ( ball ` D ) R ) e. ( ( nei ` J ) ` { P } ) )
Theorem	blsscls2 14672*	A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
|- J = ( MetOpen ` D )   &    |- S = { z e. X | ( P D z ) <_ R }   =>    |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) -> S C_ ( P ( ball ` D ) T ) )
Theorem	metss 14673*	Two ways of saying that metric D generates a finer topology than metric C. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( J C_ K <-> A. x e. X A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) ) )
Theorem	metequiv 14674*	Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeff Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( J = K <-> A. x e. X ( A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ A. a e. RR+ E. b e. RR+ ( x ( ball ` C ) b ) C_ ( x ( ball ` D ) a ) ) ) )
Theorem	metequiv2 14675*	If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( A. x e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> J = K ) )
Theorem	metss2lem 14676*	Lemma for metss2 14677. (Contributed by Mario Carneiro, 14-Sep-2015.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   &    |- ( ph -> C e. ( Met ` X ) )   &    |- ( ph -> D e. ( Met ` X ) )   &    |- ( ph -> R e. RR+ )   &    |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x C y ) <_ ( R x. ( x D y ) ) )   =>    |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` D ) ( S / R ) ) C_ ( x ( ball ` C ) S ) )
Theorem	metss2 14677*	If the metric D is "strongly finer" than C (meaning that there is a positive real constant R such that C ( x , y ) <_ R x. D ( x , y )), then D generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   &    |- ( ph -> C e. ( Met ` X ) )   &    |- ( ph -> D e. ( Met ` X ) )   &    |- ( ph -> R e. RR+ )   &    |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x C y ) <_ ( R x. ( x D y ) ) )   =>    |- ( ph -> J C_ K )
Theorem	comet 14678*	The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- ( ph -> D e. ( *Met ` X ) )   &    |- ( ph -> F : ( 0 [,] +oo ) --> RR* )   &    |- ( ( ph /\ x e. ( 0 [,] +oo ) ) -> ( ( F ` x ) = 0 <-> x = 0 ) )   &    |- ( ( ph /\ ( x e. ( 0 [,] +oo ) /\ y e. ( 0 [,] +oo ) ) ) -> ( x <_ y -> ( F ` x ) <_ ( F ` y ) ) )   &    |- ( ( ph /\ ( x e. ( 0 [,] +oo ) /\ y e. ( 0 [,] +oo ) ) ) -> ( F ` ( x +e y ) ) <_ ( ( F ` x ) +e ( F ` y ) ) )   =>    |- ( ph -> ( F o. D ) e. ( *Met ` X ) )
Theorem	bdmetval 14679*	Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )   =>    |- ( ( ( C : ( X X. X ) --> RR* /\ R e. RR* ) /\ ( A e. X /\ B e. X ) ) -> ( A D B ) = inf ( { ( A C B ) , R } , RR* , < ) )
Theorem	bdxmet 14680*	The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )   =>    |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )
Theorem	bdmet 14681*	The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )   =>    |- ( ( C e. ( *Met ` X ) /\ R e. RR+ ) -> D e. ( Met ` X ) )
Theorem	bdbl 14682*	The standard bounded metric corresponding to C generates the same balls as C for radii less than R. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )   =>    |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> ( P ( ball ` D ) S ) = ( P ( ball ` C ) S ) )
Theorem	bdmopn 14683*	The standard bounded metric corresponding to C generates the same topology as C. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )   &    |- J = ( MetOpen ` C )   =>    |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> J = ( MetOpen ` D ) )
Theorem	mopnex 14684*	The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- J = ( MetOpen ` D )   =>    |- ( D e. ( *Met ` X ) -> E. d e. ( Met ` X ) J = ( MetOpen ` d ) )
Theorem	metrest 14685	Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
|- D = ( C |` ( Y X. Y ) )   &    |- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( C e. ( *Met ` X ) /\ Y C_ X ) -> ( J ↾t Y ) = K )
Theorem	xmetxp 14686*	The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)
|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   =>    |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )
Theorem	xmetxpbl 14687*	The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point C with radius R. (Contributed by Jim Kingdon, 22-Oct-2023.)
|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   &    |- ( ph -> R e. RR* )   &    |- ( ph -> C e. ( X X. Y ) )   =>    |- ( ph -> ( C ( ball ` P ) R ) = ( ( ( 1st ` C ) ( ball ` M ) R ) X. ( ( 2nd ` C ) ( ball ` N ) R ) ) )
Theorem	xmettxlem 14688*	Lemma for xmettx 14689. (Contributed by Jim Kingdon, 15-Oct-2023.)
|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   &    |- J = ( MetOpen ` M )   &    |- K = ( MetOpen ` N )   &    |- L = ( MetOpen ` P )   =>    |- ( ph -> L C_ ( J tX K ) )
Theorem	xmettx 14689*	The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.)
|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   &    |- J = ( MetOpen ` M )   &    |- K = ( MetOpen ` N )   &    |- L = ( MetOpen ` P )   =>    |- ( ph -> L = ( J tX K ) )
9.2.5  Continuity in metric spaces
Theorem	metcnp3 14690*	Two ways to express that F is continuous at P for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. RR+ E. z e. RR+ ( F " ( P ( ball ` C ) z ) ) C_ ( ( F ` P ) ( ball ` D ) y ) ) ) )
Theorem	metcnp 14691*	Two ways to say a mapping from metric C to metric D is continuous at point P. (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. RR+ E. z e. RR+ A. w e. X ( ( P C w ) < z -> ( ( F ` P ) D ( F ` w ) ) < y ) ) ) )
Theorem	metcnp2 14692*	Two ways to say a mapping from metric C to metric D is continuous at point P. The distance arguments are swapped compared to metcnp 14691 (and Munkres' metcn 14693) for compatibility with df-lm 14369. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. RR+ E. z e. RR+ A. w e. X ( ( w C P ) < z -> ( ( F ` w ) D ( F ` P ) ) < y ) ) ) )
Theorem	metcn 14693*	Two ways to say a mapping from metric C to metric D is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon" y there is a positive "delta" z such that a distance less than delta in C maps to a distance less than epsilon in D. (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. X A. y e. RR+ E. z e. RR+ A. w e. X ( ( x C w ) < z -> ( ( F ` x ) D ( F ` w ) ) < y ) ) ) )
Theorem	metcnpi 14694*	Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 14691. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. RR+ ) ) -> E. x e. RR+ A. y e. X ( ( P C y ) < x -> ( ( F ` P ) D ( F ` y ) ) < A ) )
Theorem	metcnpi2 14695*	Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 14692. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. RR+ ) ) -> E. x e. RR+ A. y e. X ( ( y C P ) < x -> ( ( F ` y ) D ( F ` P ) ) < A ) )
Theorem	metcnpi3 14696*	Epsilon-delta property of a metric space function continuous at P. A variation of metcnpi2 14695 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. RR+ ) ) -> E. x e. RR+ A. y e. X ( ( y C P ) <_ x -> ( ( F ` y ) D ( F ` P ) ) <_ A ) )
Theorem	txmetcnp 14697*	Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   &    |- L = ( MetOpen ` E )   =>    |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( J tX K ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) ) )
Theorem	txmetcn 14698*	Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   &    |- L = ( MetOpen ` E )   =>    |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) -> ( F e. ( ( J tX K ) Cn L ) <-> ( F : ( X X. Y ) --> Z /\ A. x e. X A. y e. Y A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( x C u ) < w /\ ( y D v ) < w ) -> ( ( x F y ) E ( u F v ) ) < z ) ) ) )
Theorem	metcnpd 14699*	Two ways to say a mapping from metric C to metric D is continuous at point P. (Contributed by Jim Kingdon, 14-Jun-2023.)
|- ( ph -> J = ( MetOpen ` C ) )   &    |- ( ph -> K = ( MetOpen ` D ) )   &    |- ( ph -> C e. ( *Met ` X ) )   &    |- ( ph -> D e. ( *Met ` Y ) )   &    |- ( ph -> P e. X )   =>    |- ( ph -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. RR+ E. z e. RR+ A. w e. X ( ( P C w ) < z -> ( ( F ` P ) D ( F ` w ) ) < y ) ) ) )
9.2.6  Topology on the reals
Theorem	qtopbasss 14700*	The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)
|- S C_ RR*   &    |- ( ( x e. S /\ y e. S ) -> sup ( { x , y } , RR* , < ) e. S )   &    |- ( ( x e. S /\ y e. S ) -> inf ( { x , y } , RR* , < ) e. S )   =>    |- ( (,) " ( S X. S ) ) e. TopBases