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	blbas 15101	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 15102	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 15103	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 15104	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 15105	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 15106	A ball centered at P is contained in the set of points finitely separated from P. This is just an application of ssbl 15094 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 15107	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 15108	An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 15105, 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 15109	The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)
|- Rel MetOpen
Theorem	mopnval 15110	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 15112, the open sets of a metric space form a topology J, whose base set is U. J by mopnuni 15113. (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 15111	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 15112	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 15113	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 15114*	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 15115	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 15116	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 15117*	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 15118	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 15119	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 15120	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 15121	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 15122	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 15123	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 15124	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 15125	An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. *MetSp -> M e. TopSp )
Theorem	msxms 15126	A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. MetSp -> M e. *MetSp )
Theorem	mstps 15127	A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( M e. MetSp -> M e. TopSp )
Theorem	xmsxmet 15128	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 15129	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 15130	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 15131	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 15132	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 15133	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 15134	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 15135	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 15136	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 15137	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 15138	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 15139	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 15140	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 15141	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 15142	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 15143	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 15144	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 15145	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 15146	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 15147	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 15148	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 15149	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 15150*	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 15151*	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 15152*	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 15153	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 15154	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 15155	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 15156	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 15157	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 15158	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 15159*	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 15160	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 15161*	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 15162*	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 15163*	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 15164*	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 15165*	Lemma for metss2 15166. (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 15166*	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 15167*	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 15168*	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 15169*	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 15170*	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 15171*	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 15172*	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 15173*	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 15174	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 15175*	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 15176*	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 15177*	Lemma for xmettx 15178. (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 15178*	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 15179*	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 15180*	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 15181*	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 15180 (and Munkres' metcn 15182) for compatibility with df-lm 14858. 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 15182*	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 15183*	Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 15180. (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 15184*	Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 15181. (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 15185*	Epsilon-delta property of a metric space function continuous at P. A variation of metcnpi2 15184 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 15186*	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 15187*	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 15188*	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 15189*	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
Theorem	qtopbas 15190	The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
|- ( (,) " ( QQ X. QQ ) ) e. TopBases
Theorem	retopbas 15191	A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)
|- ran (,) e. TopBases
Theorem	retop 15192	The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
|- ( topGen ` ran (,) ) e. Top
Theorem	uniretop 15193	The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
|- RR = U. ( topGen ` ran (,) )
Theorem	retopon 15194	The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- ( topGen ` ran (,) ) e. (TopOn ` RR )
Theorem	retps 15195	The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
|- K = { <. ( Base ` ndx ) , RR >. , <. (TopSet ` ndx ) , ( topGen ` ran (,) ) >. }   =>    |- K e. TopSp
Theorem	iooretopg 15196	Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) e. ( topGen ` ran (,) ) )
Theorem	cnmetdval 15197	Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- D = ( abs o. - )   =>    |- ( ( A e. CC /\ B e. CC ) -> ( A D B ) = ( abs ` ( A - B ) ) )
Theorem	cnmet 15198	The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
|- ( abs o. - ) e. ( Met ` CC )
Theorem	cnxmet 15199	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- ( abs o. - ) e. ( *Met ` CC )
Theorem	cntoptopon 15200	The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- J e. (TopOn ` CC )