P. 152 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15101-15200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	blrn 15101*	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 15102	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 15103	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 15104	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 15105	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR+ ) -> P e. ( P ( ball ` D ) R ) )
Theorem	xblm 15106*	A ball is inhabited iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( E. x x e. ( P ( ball ` D ) R ) <-> 0 < R ) )
Theorem	bln0 15107	A ball is not empty. It is also inhabited, as seen at blcntr 15105. (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 15108	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 15109	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 15110	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 15111	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 15112	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 15113	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 15114	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 15115	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 15116*	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 15117*	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 15118*	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 15119*	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 15120*	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 15121*	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 15122	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 15123	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 15124	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 15125	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 15126	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 15127	A ball centered at P is contained in the set of points finitely separated from P. This is just an application of ssbl 15115 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 15128	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 15129	An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 15126, 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 15130	The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)
|- Rel MetOpen
Theorem	mopnval 15131	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 15133, the open sets of a metric space form a topology J, whose base set is U. J by mopnuni 15134. (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 15132	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 15133	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 15134	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 15135*	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 15136	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 15137	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 15138*	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 15139	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 15140	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 15141	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 15142	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 15143	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 15144	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 15145	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 15146	An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. *MetSp -> M e. TopSp )
Theorem	msxms 15147	A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. MetSp -> M e. *MetSp )
Theorem	mstps 15148	A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. MetSp -> M e. TopSp )
Theorem	xmsxmet 15149	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 15150	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 15151	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 15152	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 15153	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 15154	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 15155	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 15156	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 15157	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 15158	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 15159	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 15160	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 15161	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 15162	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 15163	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 15164	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 15165	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 15166	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 15167	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 15168	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 15169	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 15170	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 15171*	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 15172*	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 15173*	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 15174	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 15175	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 15176	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 15177	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 15178	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 15179	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 15180*	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 15181	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 15182*	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 15183*	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 15184*	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 15185*	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 15186*	Lemma for metss2 15187. (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 15187*	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 15188*	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 15189*	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 15190*	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 15191*	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 15192*	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 15193*	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 15194*	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 15195	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 15196*	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 15197*	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 15198*	Lemma for xmettx 15199. (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 15199*	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 15200*	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 ) ) ) )