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Theorem List for Metamath Proof Explorer - 18501-18600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcpi1 18501 Extend class notation with the fundamental group.

Syntaxcpin 18502 Extend class notation with the higher homotopy groups.

Definitiondf-pco 18503* 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 18504* 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 18505* 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 18506* 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 18507* 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 18508* 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 18509* The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theorempcovalg 18510 Evaluate the concatenation of two paths. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theorempcoval1 18511 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 18512 The starting point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theorempco1 18513 The ending point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theorempcoval2 18514 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 18515 The concatenation of two paths is a path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Theoremcopco 18516 The composition of a concatenation of paths with a continuous function. (Contributed by Mario Carneiro, 9-Jul-2015.)

Theorempcohtpylem 18517* Lemma for pcohtpy 18518. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorempcohtpy 18518 Homotopy invariance of path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorempcoptcl 18519 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 18520 Concatenation with a point does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theorempcopt2 18521 Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcoass 18522* Order of concatenation does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)

Theorempcorevcl 18523* Closure for a reversed path. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcorevlem 18524* Lemma for pcorev 18525. Prove continuity of the homotopy function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)

Theorempcorev 18525* Concatenation with the reverse path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theorempcorev2 18526* Concatenation with the reverse path. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcophtb 18527* 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 18528* The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn              TopSet

Theoremom1bas 18529* The base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremom1elbas 18530 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremom1addcl 18531 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 18532 The group operation (which isn't much more than a magma) of the loop space. (Contributed by Mario Carneiro, 11-Feb-2015.)
TopOn

Theoremom1tset 18533 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn              TopSet

Theoremom1opn 18534 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn                            t

Theorempi1val 18535 The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn                     s

Theorempi1bas 18536 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 18537 Lemma for pi1buni 18538. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1buni 18538 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 18539 The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1eluni 18540 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1bas3 18541 The base set of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1cpbl 18542 The group operation, loop concatenation, is compatible with homotopy equivalence. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremelpi1 18543* The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremelpi1i 18544 The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1addf 18545 The group operation of is a binary operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1addval 18546 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 18547 Lemma for pi1grp 18548. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1grp 18548 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 18549 The identity element of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1inv 18550* An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1xfrf 18551* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfrval 18552* 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 18553* 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 18554* 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 18555* 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 18556* The mapping between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.)
TopOn                     GrpIso

Theorempi1cof 18557* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1coval 18558* 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 18559* The mapping between fundamental groups is a group homomorphism. (Contributed by Mario Carneiro, 10-Aug-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
TopOn

11.4  Complex metric vector spaces

11.4.1  Complex left modules

Syntaxcclm 18560 Complex module.
CMod

Definitiondf-clm 18561* 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 18562 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 18563 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod flds

Theoremclmsubrg 18564 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod SubRingfld

Theoremclmlmod 18565 A complex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmgrp 18566 A complex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmabl 18567 A complex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmrng 18568 The scalar ring of a complex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmfgrp 18569 The scalar ring of a complex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclm0 18570 The zero of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclm1 18571 The identity of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmadd 18572 The addition of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmmul 18573 The multiplication of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmcj 18574 The conjugation of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremisclmi 18575 Reverse direction of isclm 18562. (Contributed by Mario Carneiro, 30-Oct-2015.)
Scalar       flds SubRingfld CMod

Theoremclmzss 18576 The scalar ring of a complex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsscn 18577 The scalar ring of a complex module is a subset of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsub 18578 Subtraction in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmneg 18579 Negation in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmabs 18580 Norm in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmacl 18581 Closure of ring addition for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmmcl 18582 Closure of ring multiplication for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsubcl 18583 Closure of ring subtraction for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremlmhmclm 18584 The domain of a linear operator is a complex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
LMHom CMod CMod

Theoremclmvsass 18585 Associative law for scalar product. (lmodvsass 15654 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremclmvsdir 18586 Distributive law for scalar product. (lmodvsdir 15652 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                            CMod

Theoremclmvs1 18587 Scalar product with ring unit. (lmodvs1 15658 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclm0vs 18588 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 15663 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremclmvneg1 18589 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 15667 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmvsneg 18590 Multiplication of a vector by a negated scalar. (lmodvsneg 15669 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                            CMod

Theoremclmmulg 18591 The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
.g              CMod

Theoremclmsubdir 18592 Scalar multiplication distributive law for subtraction. (lmodsubdir 15683 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremzlmclm 18593 The -module operation turns an arbitrary abelian group into a complex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
Mod       CMod

Theoremclmzlmvsca 18594 The scalar product of a complex module matches the scalar product of the derived -module, which implies, together with zlmbas 16472 and zlmplusg 16473, that any module over is structure-equivalent to the canonical -module Mod. (Contributed by Mario Carneiro, 30-Oct-2015.)
Mod              CMod

Theoremnmoleub2lem 18595* Lemma for nmoleub2a 18598 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2lem3 18596* Lemma for nmoleub2a 18598 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2lem2 18597* Lemma for nmoleub2a 18598 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2a 18598* The operator norm is the supremum of the value of a linear operator in the closed unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2b 18599* The operator norm is the supremum of the value of a linear operator in the open unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub3 18600* The operator norm is the supremum of the value of a linear operator on the closed unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

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