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Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-phtpy 18301* Define the class of path homotopies between two paths ; these are homotopies (in the sense of df-htpy 18300) 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 18302* 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 18303 A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                     Htpy

Theoremhtpyi 18304 A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                     Htpy

Theoremishtpyd 18305* Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                                          Htpy

Theoremhtpycom 18306* Given a homotopy from to , produce a homotopy from to . (Contributed by Mario Carneiro, 23-Feb-2015.)
TopOn                            Htpy        Htpy

Theoremhtpyid 18307* A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.)
TopOn              Htpy

Theoremhtpyco1 18308* Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
TopOn                            Htpy        Htpy

Theoremhtpyco2 18309 Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
Htpy        Htpy

Theoremhtpycc 18310* Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
TopOn                            Htpy        Htpy        Htpy

Theoremisphtpy 18311* 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 18312 A path homotopy is a homotopy. (Contributed by Mario Carneiro, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremphtpycn 18313 A path homotopy is a continuous function. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyi 18314 Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpy01 18315 Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremisphtpyd 18316* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremisphtpy2d 18317* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpycom 18318* 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 18319* A homotopy from a path to itself. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyco2 18320 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theoremphtpycc 18321* Concatenate two path homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Definitiondf-phtpc 18322* 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 18323 The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)

Theoremisphtpc 18324 The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremphtpcer 18325 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 18326 Path homotopic paths have the same endpoints. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremreparphti 18327* Lemma for reparpht 18328. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theoremreparpht 18328 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 18329 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 6-Jul-2015.)

11.3.11  The fundamental group

Syntaxcpco 18330 Extend class notation with the concatenation operation for paths in a topological space.

Syntaxcomi 18331 Extend class notation with the loop space.

Syntaxcomn 18332 Extend class notation with the higher loop spaces.

Syntaxcpi1 18333 Extend class notation with the fundamental group.

Syntaxcpin 18334 Extend class notation with the higher homotopy groups.

Definitiondf-pco 18335* Define the concatenation of two paths in a topological space . For simplicity of definition, we define it on all paths, not just those whose endpoints line up. Definition of [Hatcher] p. 26. Hatcher denotes path concatenation with a square dot; other authors, such as Munkres, use a star. (Contributed by Jeff Madsen, 15-Jun-2010.)

Definitiondf-om1 18336* Define the loop space of a topological space, with a magma structure on it given by concatenation of loops. This structure is not a group, but the operation is compatible with homotopy, which allows the homotopy groups to be defined based on this operation. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopSet

Definitiondf-omn 18337* Define the n-th iterated loop space of a topological space. Unlike this is actually a pointed topological space, which is to say a tuple of a topological space (a member of , not ) and a point in the space. Higher loop spaces select the constant loop at the point from the lower loop space for the distinguished point. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopSet

Definitiondf-pi1 18338* Define the fundamental group, whose operation is given by concatenation of homotopy classes of loops. Definition of [Hatcher] p. 26. (Contributed by Mario Carneiro, 11-Feb-2015.)
s

Definitiondf-pin 18339* Define the n-th homotopy group, which is formed by taking the -th loop space and forming the quotient under the relation of path homotopy equivalence in the base space of the -th loop space, which is the -th loop space. For , since this is not well-defined we replace this relation with the path-connectedness relation, so that the -th homotopy group is the set of path components of . (Since the -th loop space does not have a group operation, neither does the -th homotopy group, but the rest are genuine groups.) (Contributed by Mario Carneiro, 11-Feb-2015.)
s

Theorempcofval 18340* The value of the path concatenation function on a topological space. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theorempcoval 18341* The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theorempcovalg 18342 Evaluate the concatenation of two paths. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theorempcoval1 18343 Evaluate the concatenation of two paths on the first half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theorempco0 18344 The starting point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theorempco1 18345 The ending point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theorempcoval2 18346 Evaluate the concatenation of two paths on the second half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Theorempcocn 18347 The concatenation of two paths is a path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Theoremcopco 18348 The composition of a concatenation of paths with a continuous function. (Contributed by Mario Carneiro, 9-Jul-2015.)

Theorempcohtpylem 18349* Lemma for pcohtpy 18350. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorempcohtpy 18350 Homotopy invariance of path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorempcoptcl 18351 A constant function is a path from to itself. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 19-Mar-2015.)
TopOn

Theorempcopt 18352 Concatenation with a point does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theorempcopt2 18353 Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcoass 18354* Order of concatenation does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)

Theorempcorevcl 18355* Closure for a reversed path. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcorevlem 18356* Lemma for pcorev 18357. Prove continuity of the homotopy function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)

Theorempcorev 18357* Concatenation with the reverse path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theorempcorev2 18358* Concatenation with the reverse path. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcophtb 18359* The path homotopy equivalence relation on two paths with the same start and end point can be written in terms of the loop formed by concatenating with the inverse of . Thus all the homotopy information in is available if we restrict our attention to closed loops, as in the definition of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theoremom1val 18360* The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn              TopSet

Theoremom1bas 18361* The base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremom1elbas 18362 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremom1addcl 18363 Closure of the group operation of the loop space. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 5-Sep-2015.)
TopOn

Theoremom1plusg 18364 The group operation (which isn't much more than a magma) of the loop space. (Contributed by Mario Carneiro, 11-Feb-2015.)
TopOn

Theoremom1tset 18365 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn              TopSet

Theoremom1opn 18366 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn                            t

Theorempi1val 18367 The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn                     s

Theorempi1bas 18368 The base set of the fundamental group of a topological space at a given base point. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1blem 18369 Lemma for pi1buni 18370. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1buni 18370 Another way to write the loop space base in terms of the base of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1bas2 18371 The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1eluni 18372 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1bas3 18373 The base set of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1cpbl 18374 The group operation, loop concatenation, is compatible with homotopy equivalence. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremelpi1 18375* The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremelpi1i 18376 The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1addf 18377 The group operation of is a binary operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1addval 18378 The concatenation of two path-homotopy classes in the fundamental group. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1grplem 18379 Lemma for pi1grp 18380. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1grp 18380 The fundamental group is a group. Proposition 1.3 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1id 18381 The identity element of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1inv 18382* An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1xfrf 18383* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfrval 18384* The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfr 18385* Given a path and its inverse between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.)
TopOn

Theorempi1xfrcnvlem 18386* Given a path between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfrcnv 18387* Given a path between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfrgim 18388* The mapping between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.)
TopOn                     GrpIso

Theorempi1cof 18389* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1coval 18390* The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 10-Aug-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1coghm 18391* The mapping between fundamental groups is a group homomorphism. (Contributed by Mario Carneiro, 10-Aug-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
TopOn

11.3.12  Complex left modules

Syntaxcclm 18392 Complex module.
CMod

Definitiondf-clm 18393* Define a complex module, which is just a left module over a subring of , which allows us to use conventional addition, multiplication, etc. in the left module theorems. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod Scalar flds SubRingfld

Theoremisclm 18394 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod flds SubRingfld

Theoremclmsca 18395 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod flds

Theoremclmsubrg 18396 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod SubRingfld

Theoremclmlmod 18397 A complex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmgrp 18398 A complex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmabl 18399 A complex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmrng 18400 The scalar ring of a complex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

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