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Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiicmp 18801 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)

Theoremiicon 18802 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcncfval 18803* The value of the continuous complex function operation is the set of continuous functions from to . (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)

Theoremelcncf 18804* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)

Theoremelcncf2 18805* Version of elcncf 18804 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)

Theoremcncfrss 18806 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremcncfrss2 18807 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremcncff 18808 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremcncfi 18809* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremelcncf1di 18810* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremelcncf1ii 18811* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremrescncf 18812 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremcncffvrn 18813 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)

Theoremcncfss 18814 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremclimcncf 18815 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremabscncf 18816 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremrecncf 18817 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremimcncf 18818 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremcjcncf 18819 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremmulc1cncf 18820* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremdivccncf 18821* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)

Theoremcncfco 18822 The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremcncfmet 18823 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremcncfcn 18824 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
fld       t        t

Theoremcncfcn1 18825 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
fld

Theoremcncfmptc 18826* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremcncfmptid 18827* The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)

Theoremcncfmpt1f 18828* Composition of continuous functions. analog of cnmpt11f 17631. (Contributed by Mario Carneiro, 3-Sep-2014.)

Theoremcncfmpt2f 18829* Composition of continuous functions. analog of cnmpt12f 17633. (Contributed by Mario Carneiro, 3-Sep-2014.)
fld

Theoremcncfmpt2ss 18830* Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
fld

Theoremaddccncf 18831* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremcdivcncf 18832* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)

Theoremnegcncf 18833* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremnegfcncf 18834* The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)

TheoremabscncfALT 18835 Absolute value is continuous. Alternate proof of abscncf 18816. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (Proof modification is discouraged.)

Theoremcncfcnvcn 18836 Rewrite cmphaushmeo 17767 for functions on the complexes. (Contributed by Mario Carneiro, 19-Feb-2015.)
fld       t

Theoremcnmptre 18837* Lemma for iirevcn 18840 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
fld       t        t

Theoremcnmpt2pc 18838* Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
t        t        t                             TopOn

Theoremiirev 18839 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiirevcn 18840 The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.)

Theoremiihalf1 18841 Map the first half of into . (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiihalf1cn 18842 The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
t

Theoremiihalf2 18843 Map the second half of into . (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiihalf2cn 18844 The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
t

Theoremelii1 18845 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theoremelii2 18846 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theoremiimulcl 18847 The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiimulcn 18848* Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.)

Theoremicoopnst 18849 A half-open interval starting at is open in the closed interval from to . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)

Theoremiocopnst 18850 A half-open interval ending at is open in the closed interval from to . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)

Theoremicchmeo 18851* The natural bijection from to an arbitrary nontrivial closed interval is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
fld              t

Theoremicopnfcnv 18852* Define a bijection from to . (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremicopnfhmeo 18853* The defined bijection from to is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
fld       t t

Theoremiccpnfcnv 18854* Define a bijection from to . (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremiccpnfhmeo 18855 The defined bijection from to is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
ordTop t

Theoremxrhmeo 18856* The bijection from to the extended reals is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
fld       ordTop        t ordTop

Theoremxrhmph 18857 The extended reals are homeomorphic to the interval . (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremxrcmp 18858 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 18722), this means that is a compactification of . (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremxrcon 18859 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremicccvx 18860 A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremoprpiece1res1 18861* Restriction to the first part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Theoremoprpiece1res2 18862* Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Theoremcnrehmeo 18863* The canonical bijection from to described in cnref1o 10553 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.)
fld

Theoremcnheiborlem 18864* Lemma for cnheibor 18865. (Contributed by Mario Carneiro, 14-Sep-2014.)
fld       t

Theoremcnheibor 18865* Heine-Borel theorem for complex numbers. A subset of is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
fld       t

Theoremcnllycmp 18866 The topology on the complexes is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
fld       𝑛Locally

Theoremrellycmp 18867 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally

Theorembndth 18868* The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to .) (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremevth 18869* The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremevth2 18870* The Extreme Value Theorem, minimum version. A continuous function from a nonempty compact topological space to the reals attains its minimum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremlebnumlem1 18871* Lemma for lebnum 18874. The function measures the sum of all of the distances to escape the sets of the cover. Since by assumption it is a cover, there is at least one set which covers a given point, and since it is open, the point is a positive distance from the edge of the set. Thus, the sum is a strictly positive number. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremlebnumlem2 18872* Lemma for lebnum 18874. As a finite sum of point-to-set distance functions, which are continuous by metdscn 18771, the function is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremlebnumlem3 18873* Lemma for lebnum 18874. By the previous lemmas, is continuous and positive on a compact set, so it has a positive minimum . Then setting , since for each we have iff , if for all then summing over yields , in contradiction to the assumption that is the minimum of . (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremlebnum 18874* The Lebesgue number lemma, or Lebesgue covering lemma. If is a compact metric space and is an open cover of , then there exists a positive real number such that every ball of size (and every subset of a ball of size , including every subset of diameter less than ) is a subset of some member of the cover. (Contributed by Mario Carneiro, 14-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Sep-2015.)

Theoremxlebnum 18875* Generalize lebnum 18874 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremlebnumii 18876* Specialize the Lebesgue number lemma lebnum 18874 to the unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)

11.4.11  Path homotopy

Syntaxchtpy 18877 Extend class notation with the class of homotopies between two continuous functions.
Htpy

Syntaxcphtpy 18878 Extend class notation with the class of path homotopies between two continuous functions.

Syntaxcphtpc 18879 Extend class notation with the path homotopy relation.

Definitiondf-htpy 18880* Define the function which takes topological spaces and two continuous functions and returns the class of homotopies from to . (Contributed by Mario Carneiro, 22-Feb-2015.)
Htpy

Definitiondf-phtpy 18881* Define the class of path homotopies between two paths ; these are homotopies (in the sense of df-htpy 18880) which also preserve both endpoints of the paths throughout the homotopy. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.)
Htpy

Theoremishtpy 18882* Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
TopOn                     Htpy

Theoremhtpycn 18883 A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                     Htpy

Theoremhtpyi 18884 A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                     Htpy

Theoremishtpyd 18885* Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                                          Htpy

Theoremhtpycom 18886* Given a homotopy from to , produce a homotopy from to . (Contributed by Mario Carneiro, 23-Feb-2015.)
TopOn                            Htpy        Htpy

Theoremhtpyid 18887* A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.)
TopOn              Htpy

Theoremhtpyco1 18888* Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
TopOn                            Htpy        Htpy

Theoremhtpyco2 18889 Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
Htpy        Htpy

Theoremhtpycc 18890* Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
TopOn                            Htpy        Htpy        Htpy

Theoremisphtpy 18891* Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremphtpyhtpy 18892 A path homotopy is a homotopy. (Contributed by Mario Carneiro, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremphtpycn 18893 A path homotopy is a continuous function. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyi 18894 Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpy01 18895 Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremisphtpyd 18896* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremisphtpy2d 18897* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpycom 18898* Given a homotopy from to , produce a homotopy from to . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyid 18899* A homotopy from a path to itself. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyco2 18900 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)

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