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Theorem List for Metamath Proof Explorer - 19001-19100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisphtpyd 19001* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremisphtpy2d 19002* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpycom 19003* 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 19004* A homotopy from a path to itself. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyco2 19005 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theoremphtpycc 19006* Concatenate two path homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Definitiondf-phtpc 19007* 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 19008 The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)

Theoremisphtpc 19009 The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremphtpcer 19010 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 19011 Path homotopic paths have the same endpoints. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremreparphti 19012* Lemma for reparpht 19013. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theoremreparpht 19013 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 19014 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 6-Jul-2015.)

11.4.13  The fundamental group

Syntaxcpco 19015 Extend class notation with the concatenation operation for paths in a topological space.

Syntaxcomi 19016 Extend class notation with the loop space.

Syntaxcomn 19017 Extend class notation with the higher loop spaces.

Syntaxcpi1 19018 Extend class notation with the fundamental group.

Syntaxcpin 19019 Extend class notation with the higher homotopy groups.

Definitiondf-pco 19020* 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 19021* 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 19022* 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 19023* 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 19024* 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 19025* 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 19026* The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theorempcovalg 19027 Evaluate the concatenation of two paths. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theorempcoval1 19028 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 19029 The starting point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theorempco1 19030 The ending point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theorempcoval2 19031 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 19032 The concatenation of two paths is a path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Theoremcopco 19033 The composition of a concatenation of paths with a continuous function. (Contributed by Mario Carneiro, 9-Jul-2015.)

Theorempcohtpylem 19034* Lemma for pcohtpy 19035. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorempcohtpy 19035 Homotopy invariance of path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorempcoptcl 19036 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 19037 Concatenation with a point does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theorempcopt2 19038 Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcoass 19039* Order of concatenation does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)

Theorempcorevcl 19040* Closure for a reversed path. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcorevlem 19041* Lemma for pcorev 19042. Prove continuity of the homotopy function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)

Theorempcorev 19042* Concatenation with the reverse path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theorempcorev2 19043* Concatenation with the reverse path. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcophtb 19044* 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 19045* The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn              TopSet

Theoremom1bas 19046* The base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremom1elbas 19047 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremom1addcl 19048 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 19049 The group operation (which isn't much more than a magma) of the loop space. (Contributed by Mario Carneiro, 11-Feb-2015.)
TopOn

Theoremom1tset 19050 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn              TopSet

Theoremom1opn 19051 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn                            t

Theorempi1val 19052 The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn                     s

Theorempi1bas 19053 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 19054 Lemma for pi1buni 19055. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1buni 19055 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 19056 The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1eluni 19057 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1bas3 19058 The base set of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1cpbl 19059 The group operation, loop concatenation, is compatible with homotopy equivalence. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremelpi1 19060* The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremelpi1i 19061 The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1addf 19062 The group operation of is a binary operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1addval 19063 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 19064 Lemma for pi1grp 19065. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1grp 19065 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 19066 The identity element of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1inv 19067* An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1xfrf 19068* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfrval 19069* 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 19070* 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 19071* 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 19072* 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 19073* The mapping between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.)
TopOn                     GrpIso

Theorempi1cof 19074* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1coval 19075* 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 19076* The mapping between fundamental groups is a group homomorphism. (Contributed by Mario Carneiro, 10-Aug-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
TopOn

11.5  Complex metric vector spaces

11.5.1  Complex left modules

Syntaxcclm 19077 Complex module.
CMod

Definitiondf-clm 19078* 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 19079 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 19080 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod flds

Theoremclmsubrg 19081 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod SubRingfld

Theoremclmlmod 19082 A complex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmgrp 19083 A complex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmabl 19084 A complex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmrng 19085 The scalar ring of a complex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmfgrp 19086 The scalar ring of a complex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclm0 19087 The zero of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclm1 19088 The identity of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmadd 19089 The addition of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmmul 19090 The multiplication of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmcj 19091 The conjugation of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremisclmi 19092 Reverse direction of isclm 19079. (Contributed by Mario Carneiro, 30-Oct-2015.)
Scalar       flds SubRingfld CMod

Theoremclmzss 19093 The scalar ring of a complex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsscn 19094 The scalar ring of a complex module is a subset of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsub 19095 Subtraction in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmneg 19096 Negation in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmabs 19097 Norm in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmacl 19098 Closure of ring addition for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmmcl 19099 Closure of ring multiplication for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsubcl 19100 Closure of ring subtraction for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

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