mmtheorems79 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7801-7900 - Page 79 of 108

Type	Label	Description
Statement
Theorem	ismsi 7801	Properties that determine a metric space.
|- D e. V   &   |- D:(X X. X)-->RR   &   |- ((x e. X /\ y e. X) -> ((xDy) = 0 <-> x = y))   &   |- ((x e. X /\ y e. X /\ z e. X) -> (xDy) <_ ((zDx) + (zDy)))   &   |- M = <.X, D>.   =>   |- M e. MetSp
Theorem	ismeti 7802	Properties that determine a metric.
|- X e. V   &   |- D:(X X. X)-->RR   &   |- ((x e. X /\ y e. X) -> ((xDy) = 0 <-> x = y))   &   |- ((x e. X /\ y e. X /\ z e. X) -> (xDy) <_ ((zDx) + (zDy)))   =>   |- D e. Met
Theorem	msflem 7803	Lemma for msf 7804 and others.
Theorem	msf 7804	Mapping of the distance function of a metric space.
|- X = (1st` M)   &   |- D = (2nd` M)   =>   |- (M e. MetSp -> D:(X X. X)-->RR)
Theorem	mscl 7805	Closure of the distance function of a metric space.
|- X = (1st` M)   &   |- D = (2nd` M)   =>   |- ((M e. MetSp /\ A e. X /\ B e. X) -> (ADB) e. RR)
Theorem	metflem 7806	Lemma for metf 7807 and others.
Theorem	metf 7807	Mapping of the distance function of a metric space.
|- X = dom dom D   =>   |- (D e. Met -> D:(X X. X)-->RR)
Theorem	metdmdm 7808	The base set of a metric space in terms of its distance function.
|- X = dom dom D   =>   |- (D e. Met -> X = dom dom D)
Theorem	metssba 7809	The base set of a metric subspace.
|- X = dom dom D   =>   |- (D e. Met -> (X i^i Y) = dom dom ( D |` (Y X. Y)))
Theorem	metssba2 7810	The base set of a metric subspace.
|- X = dom dom D   =>   |- ((D e. Met /\ Y (_ X) -> Y = dom dom ( D |` (Y X. Y)))
Theorem	metcl 7811	Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X /\ B e. X) -> (ADB) e. RR)
Theorem	meteq0 7812	The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X /\ B e. X) -> ((ADB) = 0 <-> A = B))
Theorem	mettri2 7813	Triangle inequality for the distance function of a metric space.
|- X = dom dom D   =>   |- ((D e. Met /\ (C e. X /\ A e. X /\ B e. X)) -> (ADB) <_ ((CDA) + (CDB)))
Theorem	mettri4 7814	Triangle inequality for the distance function of a metric space.
|- X = dom dom D   =>   |- (((D e. Met /\ A e. X) /\ (B e. X /\ C e. X)) -> (ADB) <_ ((CDA) + (CDB)))
Theorem	met0 7815	The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X) -> (ADA) = 0)
Theorem	metsym 7816	The distance function of a metric space is symmetric. Definition 14-1.1(c) of [Gleason] p. 223.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X /\ B e. X) -> (ADB) = (BDA))
Theorem	mettri 7817	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223.
|- X = dom dom D   =>   |- ((D e. Met /\ (A e. X /\ B e. X /\ C e. X)) -> (ADB) <_ ((ADC) + (CDB)))
Theorem	mettri3 7818	Triangle inequality for the distance function of a metric space.
|- X = dom dom D   =>   |- ((D e. Met /\ (A e. X /\ B e. X /\ C e. X)) -> (ADB) <_ ((ADC) + (BDC)))
Theorem	metge0 7819	The distance function of a metric space is nonnegative.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X /\ B e. X) -> 0 <_ (ADB))
Theorem	metgt0 7820	The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of [Gleason] p. 223 and its converse.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X /\ B e. X) -> (A =/= B <-> 0 < (ADB)))
Theorem	metne0 7821	A metric space is nonempty iff its base set is nonempty.
|- X = dom dom D   =>   |- (D e. Met -> (D =/= (/) <-> X =/= (/)))
Theorem	metreslem 7822	Lemma for metres 7823. (Contributed by FL, 10-Nov-2006.)
Theorem	metres 7823	A restriction of a metric is a metric.
|- (D e. Met -> (D |` (R X. R)) e. Met)
Theorem	metss 7824	If two metrics are in a subset relationship, so are their base sets.
|- X = dom dom C   &   |- Y = dom dom D   =>   |- (C (_ D -> X (_ Y)
Theorem	0met 7825	The empty metric.
|- (/) e. Met
Theorem	metxplem1 7826	Lemma for metxp 7834.
Theorem	metxplem2 7827	Lemma for metxp 7834.
Theorem	metxplem3 7828	Lemma for metxp 7834.
Theorem	metxpdval 7829	Value of the distance function of the direct product of two metric spaces. Based on Definition 14-1.5 of [Gleason] p. 225.
|- X = dom dom B   &   |- Y = dom dom C   &   |- B e. Met   &   |- C e. Met   &   |- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}   &   |- F = (1st` R)   &   |- G = (2nd` R)   &   |- H = (1st` S)   &   |- J = (2nd` S)   =>   |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (RDS) = if((GCJ) < (FBH), (FBH), (GCJ)))
Theorem	metxptval 7830	One case of the value of the distance function of the direct product of two metric spaces. Based on Definition 14-1.5 of [Gleason] p. 225.
|- X = dom dom B   &   |- Y = dom dom C   &   |- B e. Met   &   |- C e. Met   &   |- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}   &   |- F = (1st` R)   &   |- G = (2nd` R)   &   |- H = (1st` S)   &   |- J = (2nd` S)   =>   |- (((R e. (X X. Y) /\ S e. (X X. Y)) /\ (GCJ) <_ (FBH)) -> (RDS) = (FBH))
Theorem	metxpfval 7831	One case of the value of the distance function of the direct product of two metric spaces. Based on Definition 14-1.5 of [Gleason] p. 225.
|- X = dom dom B   &   |- Y = dom dom C   &   |- B e. Met   &   |- C e. Met   &   |- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}   &   |- F = (1st` R)   &   |- G = (2nd` R)   &   |- H = (1st` S)   &   |- J = (2nd` S)   =>   |- (((R e. (X X. Y) /\ S e. (X X. Y)) /\ (FBH) <_ (GCJ)) -> (RDS) = (GCJ))
Theorem	metxpcl 7832	Closure of the distance function of the direct product of two metric spaces. Based on Definition 14-1.5 of [Gleason] p. 225.
|- X = dom dom B   &   |- Y = dom dom C   &   |- B e. Met   &   |- C e. Met   &   |- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}   =>   |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (RDS) e. RR)
Theorem	metxplem4 7833	Lemma for metxp 7834. Triangle inequality. Warning: The HTML proof page is 0.6 megabyte in size.
Theorem	metxp 7834	The direct product of two metric spaces. Definition 14-1.5 of [Gleason] p. 225.
|- X = dom dom B   &   |- Y = dom dom C   &   |- B e. Met   &   |- C e. Met   &   |- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}   =>   |- D e. Met
Metric space balls
Theorem	blfval 7835	The value of the ball function.
|- X = dom dom D   =>   |- (D e. Met -> ( ball ` D) = {<.<.x, y>., z>. | ((x e. X /\ y e. RR) /\ (0 < y /\ z = {w e. X | (xDw) < y}))})
Theorem	blfval2 7836	Alternate value of the ball function that simplifies its use with operation theorems.
|- X = dom dom D   =>   |- (D e. Met -> ( ball ` D) = {<.<.x, y>., z>. | ((x e. X /\ y e. {v e. RR