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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrescncf 18401 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 18402 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)

Theoremcncfss 18403 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremclimcncf 18404 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremabscncf 18405 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremrecncf 18406 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremimcncf 18407 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremcjcncf 18408 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremmulc1cncf 18409* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremdivccncf 18410* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)

Theoremcncfco 18411 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 18412 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremcncfcn 18413 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
fld       t        t

Theoremcncfcn1 18414 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
fld

Theoremcncfmptc 18415* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremcncfmptid 18416* The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)

Theoremcncfmpt1f 18417* Composition of continuous functions. analog of cnmpt11f 17358. (Contributed by Mario Carneiro, 3-Sep-2014.)

Theoremcncfmpt2f 18418* Composition of continuous functions. analog of cnmpt12f 17360. (Contributed by Mario Carneiro, 3-Sep-2014.)
fld

Theoremcncfmpt2ss 18419* Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
fld

Theoremcdivcncf 18420* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)

Theoremnegcncf 18421* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremnegfcncf 18422* The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)

TheoremabscncfALT 18423 Absolute value is continuous. Alternate proof of abscncf 18405. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (Proof modification is discouraged.)

Theoremcncfcnvcn 18424 Rewrite cmphaushmeo 17491 for functions on the complexes. (Contributed by Mario Carneiro, 19-Feb-2015.)
fld       t

Theoremcnmptre 18425* Lemma for iirevcn 18428 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
fld       t        t

Theoremcnmpt2pc 18426* 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 18427 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiirevcn 18428 The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.)

Theoremiihalf1 18429 Map the first half of into . (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiihalf1cn 18430 The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
t

Theoremiihalf2 18431 Map the second half of into . (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiihalf2cn 18432 The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
t

Theoremelii1 18433 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theoremelii2 18434 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theoremiimulcl 18435 The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiimulcn 18436* Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.)

Theoremicoopnst 18437 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 18438 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 18439* The natural bijection from to an arbitrary nontrivial closed interval is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
fld              t

Theoremicopnfcnv 18440* Define a bijection from to . (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremicopnfhmeo 18441* The defined bijection from to is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
fld       t t

Theoremiccpnfcnv 18442* Define a bijection from to . (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremiccpnfhmeo 18443 The defined bijection from to is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
ordTop t

Theoremxrhmeo 18444* 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 18445 The extended reals are homeomorphic to the interval . (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremxrcmp 18446 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 18312), this means that is a compactification of . (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremxrcon 18447 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremicccvx 18448 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 18449* 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 18450* 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 18451* The canonical bijection from to described in cnref1o 10349 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 18452* Lemma for cnheibor 18453. (Contributed by Mario Carneiro, 14-Sep-2014.)
fld       t

Theoremcnheibor 18453* 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 18454 The topology on the complexes is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
fld       𝑛Locally

Theoremrellycmp 18455 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally

Theorembndth 18456* 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 18457* 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 18458* 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 18459* Lemma for lebnum 18462. 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 18460* Lemma for lebnum 18462. As a finite sum of point-to-set distance functions, which are continuous by metdscn 18360, the function is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremlebnumlem3 18461* Lemma for lebnum 18462. 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 18462* 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 18463* Generalize lebnum 18462 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremlebnumii 18464* Specialize the Lebesgue number lemma lebnum 18462 to the unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)

11.3.10  Path homotopy

Syntaxchtpy 18465 Extend class notation with the class of homotopies between two continuous functions.
Htpy

Syntaxcphtpy 18466 Extend class notation with the class of path homotopies between two continuous functions.

Syntaxcphtpc 18467 Extend class notation with the path homotopy relation.

Definitiondf-htpy 18468* 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 18469* Define the class of path homotopies between two paths ; these are homotopies (in the sense of df-htpy 18468) 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 18470* 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 18471 A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                     Htpy

Theoremhtpyi 18472 A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                     Htpy

Theoremishtpyd 18473* Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                                          Htpy

Theoremhtpycom 18474* Given a homotopy from to , produce a homotopy from to . (Contributed by Mario Carneiro, 23-Feb-2015.)
TopOn                            Htpy        Htpy

Theoremhtpyid 18475* A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.)
TopOn              Htpy

Theoremhtpyco1 18476* Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
TopOn                            Htpy        Htpy

Theoremhtpyco2 18477 Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
Htpy        Htpy

Theoremhtpycc 18478* Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
TopOn                            Htpy        Htpy        Htpy

Theoremisphtpy 18479* 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 18480 A path homotopy is a homotopy. (Contributed by Mario Carneiro, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremphtpycn 18481 A path homotopy is a continuous function. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyi 18482 Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpy01 18483 Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremisphtpyd 18484* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremisphtpy2d 18485* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpycom 18486* 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 18487* A homotopy from a path to itself. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyco2 18488 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theoremphtpycc 18489* Concatenate two path homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Definitiondf-phtpc 18490* Define the function which takes a topology and returns the path homotopy relation on that topology. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theoremphtpcrel 18491 The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)

Theoremisphtpc 18492 The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremphtpcer 18493 Path homotopy is an equivalence relation. Proposition 1.2 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 6-Jul-2015.)

Theoremphtpc01 18494 Path homotopic paths have the same endpoints. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremreparphti 18495* Lemma for reparpht 18496. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theoremreparpht 18496 Reparametrization lemma. The reparametrization of a path by any continuous map with and is path-homotopic to the original path. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Feb-2015.)

Theoremphtpcco2 18497 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 6-Jul-2015.)

11.3.11  The fundamental group

Syntaxcpco 18498 Extend class notation with the concatenation operation for paths in a topological space.

Syntaxcomi 18499 Extend class notation with the loop space.

Syntaxcomn 18500 Extend class notation with the higher loop spaces.

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