mmtheorems78 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7701-7800 - Page 78 of 107

Type	Label	Description
Statement
Limit points
Syntax	clp 7701	Extend class notation with the limit point function for topologies.
class limPt
Definition	df-lp 7702	Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 7704.
|- limPt = {<.z, w>. | (z e. Top /\ w = {<.x, y>. | (x (_ U.z /\ y = {v | v e. ((cls` J)` (x \ {v}))})})}
Theorem	lpfval 7703	The limit point function on the subsets of a topology's base set.
|- X = U.J   =>   |- (J e. Top -> (limPt` J) = {<.x, y>. | (x (_ X /\ y = {v | v e. ((cls` J)` (x \ {v}))})})
Theorem	lpval 7704	The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((limPt` J)` S) = {x | x e. ((cls` J)` (S \ {x}))})
Theorem	islp 7705	The predicate "P is a limit point of S."
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (P e. ((limPt` J)` S) <-> P e. ((cls` J)` (S \ {P}))))
Theorem	lpsscls 7706	The limits points of a subset are included in the subset's closure.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((limPt` J)` S) (_ ((cls` J)` S))
Theorem	lpss 7707	The limits points of a subset are included in the base set.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((limPt` J)` S) (_ X)
Theorem	islp2 7708	The predicate "P is a limit point of S," in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X /\ P e. X) -> (P e. ((limPt` J)` S) <-> A.n e. ((nei` J)` {P})(n i^i (S \ {P})) =/= (/)))
Theorem	clslp 7709	The closure of a subset of a topological space is the subset together with its limit points. Theorem 6.6 of [Munkres] p. 97.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) = (S u. ((limPt` J)` S)))
Theorem	islpi 7710	A point belonging to a set's closure but not the set itself is a limit point.
|- X = U.J   =>   |- (((J e. Top /\ S (_ X) /\ (P e. ((cls` J)` S) /\ -. P e. S)) -> P e. ((limPt` J)` S))
Theorem	cldlp 7711	A subset of a topological space is closed iff it contains all its limit points. Corollary 6.7 of [Munkres] p. 97.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (S e. (Clsd` J) <-> ((limPt` J)` S) (_ S))
Theorem	qdensere 7712	QQ is dense in the standard topology on RR.
|- ((cls` (topGen` ran (,)))` QQ) = RR
Continuity
Syntax	ccn 7713	Extend class notation with the set of continuous functions between topologies.
class Cn
Syntax	ccnp 7714	Extend class notation with the set of functions between topologies continuous at a point.
class CnP
Definition	df-cn 7715	Define a function on two topologies whose value is the set of continuous mappings from the first topology to the second. Based on definition of continuous function in [Munkres] p. 102. See iscn 7719 for the predicate form.
|- Cn = {<.<.j, k>., z>. | ((j e. Top /\ k e. Top) /\ z = {f e. (U.k ^m U.j) | A.y e. k (`'f"y) e. j})}
Definition	df-cnp 7716	Define a function on two topologies whose value is the set of continuous mappings at a specified point in the first topology. Based on Theorem 7.2(g) of [Munkres] p. 107.
|- CnP = {<.<.j, k>., z>. | ((j e. Top /\ k e. Top) /\ z = {<.x, y>. | (x e. U.j /\ y = {f e. (U.k ^m U.j) | A.w e. k ((f` x) e. w -> E.v e. j (x e. v /\ (f"v) (_ w))})})}
Theorem	cnfval 7717	The set of all continuous functions from topology J to topology K.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top) -> (J Cn K) = {f e. (Y ^m X) | A.y e. K (`'f"y) e. J})
Theorem	cnpfval 7718	The function mapping the points in a topology J to the set of all functions from J to topology K continuous at that point.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top) -> (J CnP K) = {<.x, y>. | (x e. X /\ y = {f e. (Y ^m X) | A.w e. K ((f` x) e. w -> E.v e. J (x e. v /\ (f"v) (_ w))})})
Theorem	iscn 7719	The predicate "F is a continuous function from topology J to topology K." Definition of continuous function in [Munkres] p. 102.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.y e. K (`'F"y) e. J)))
Theorem	cnpval 7720	The set of all functions from topology J to topology K that are continuous at a point P.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ P e. X) -> ((J CnP K)` P) = {f e. (Y ^m X) | A.y e. K ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y))})
Theorem	iscnp 7721	The predicate "F is a continuous function from topology J to topology K at point P." Based on Theorem 7.2(g) of [Munkres] p. 107.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ P e. X) -> (F e. ((J CnP K)` P) <-> (F:X-->Y /\ A.y e. K ((F` P) e. y -> E.x e. J (P e. x /\ (F"x) (_ y)))))
Theorem	iscnp2 7722	The predicate "F is a continuous function from topology J to topology K at point P."
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ P e. X) -> (F e. ((J CnP K)` P) <-> (F:X-->Y /\ A.y e. K ((F` P) e. y -> E.x e. J (P e. x /\ x (_ (`'F"y))))))
Theorem	cnf 7723	A continuous function is a mapping. (Contributed by FL, 15-Dec-2006.)
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> F:X-->Y)
Theorem	cnpf 7724	A continuous function at point P is a mapping. (Contributed by FL, 31-Dec-2006.)
|- X = U.J   &   |- Y = U.K   =>   |- (((J e. Top /\ K e. Top /\ P e. X) /\ F e. ((J CnP K)` P)) -> F:X-->Y)
Theorem	cnpcl 7725	The value of a continuous function from J to K at point P belongs to the underlying set of topology K. (Contributed by FL, 31-Dec-2006.)
|- X = U.J   &   |- Y = U.K   =>   |- (((J e. Top /\ K e. Top /\ P e. X) /\ (F e. ((J CnP K)` P) /\ A e. X)) -> (F` A) e. Y)
Theorem	cnpimaex 7726	Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)
|- X = U.J   =>   |- (((J e. Top /\ K e. Top /\ P e. X) /\ (F e. ((J CnP K)` P) /\ A e. K /\ (F` P) e. A)) -> E.x e. J (P e. x /\ (F"x) (_ A))
Theorem	idcn 7727	A restricted identity function is a continuous function. (Contributed by FL, 31-Dec-2006.)
|- X = U.J   =>   |- (J e. Top -> (I |` X) e. (J Cn J))
Theorem	cnima 7728	An open subset of the codomain of a continuous function has an open pre-image. (Contributed by FL, 15-Dec-2006.)
|- (((J e. Top /\ K e. Top /\ F e. (J Cn K)) /\ A e. K) -> (`'F"A) e. J)
Theorem	cnco 7729	The composition of two continuous functions is a continuous function. (Contributed by FL, 15-Dec-2006.)
|- (((J e. Top /\ K e. Top /\ L e. Top) /\ (F e. (J Cn K) /\ G e. (K Cn L))) -> (G o. F) e. (J Cn L))
Theorem	cnpco 7730	The composition of two continuous functions at point P is a continuous function at point P. Bourbaki TG I.9. (Contributed by FL, 31-Dec-2006.) Warning: The HTML proof page is 0.5MB in size.
|- X = U.J   =>   |- (((J e. Top /\ L e. Top /\ P e. X) /\ (K e. Top /\ F e. ((J CnP K)` P) /\ G e. ((K CnP L)` (F` P)))) -> (G o. F) e. ((J CnP L)` P))
Theorem	iscncl 7731	A definition of a continuous function using closed sets. Bourbaki TG I.9 Th. 1 (d). (Contributed by FL, 30-Jan-2007.)
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J))))
Theorem	cnclima 7732	A closed subset of the codomain of a continuous function has a closed pre-image.
|- (((J e. Top /\ K e. Top /\ F e. (J Cn K)) /\ A e. (Clsd` K)) -> (`'F"A) e. (Clsd` J))
Theorem	cnsscnp 7733	The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.)
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ P e. X) -> (J Cn K) (_ ((J CnP K)` P))
Theorem	cncnpi 7734	A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.)
|- X = U.J   &   |- Y = U.K   =>   |- (((J e. Top /\ K e. Top) /\ (F e. (J Cn K) /\ A e. X)) -> F e. ((J CnP K)` A))
Theorem	cncnplem1 7735	Lemma for cncnp2 7740.
Theorem	cncnplem2 7736	Lemma for cncnp2 7740.
Theorem	cncnplem3 7737	Lemma for cncnp2 7740.
Theorem	cncnplem4 7738	Lemma for cncnp2 7740.
Theorem	cncnp 7739	A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ F:X-->Y) -> (F e. (J Cn K) <-> A.x e. X F e. (