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

Theorempcopt2 19001 Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcoass 19002* Order of concatenation does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)

Theorempcorevcl 19003* Closure for a reversed path. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcorevlem 19004* Lemma for pcorev 19005. Prove continuity of the homotopy function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)

Theorempcorev 19005* Concatenation with the reverse path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theorempcorev2 19006* Concatenation with the reverse path. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcophtb 19007* 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 19008* The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn              TopSet

Theoremom1bas 19009* The base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremom1elbas 19010 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremom1addcl 19011 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 19012 The group operation (which isn't much more than a magma) of the loop space. (Contributed by Mario Carneiro, 11-Feb-2015.)
TopOn

Theoremom1tset 19013 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn              TopSet

Theoremom1opn 19014 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn                            t

Theorempi1val 19015 The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn                     s

Theorempi1bas 19016 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 19017 Lemma for pi1buni 19018. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1buni 19018 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 19019 The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1eluni 19020 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1bas3 19021 The base set of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1cpbl 19022 The group operation, loop concatenation, is compatible with homotopy equivalence. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremelpi1 19023* The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremelpi1i 19024 The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1addf 19025 The group operation of is a binary operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1addval 19026 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 19027 Lemma for pi1grp 19028. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1grp 19028 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 19029 The identity element of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1inv 19030* An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1xfrf 19031* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfrval 19032* 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 19033* 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 19034* 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 19035* 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 19036* The mapping between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.)
TopOn                     GrpIso

Theorempi1cof 19037* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1coval 19038* 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 19039* 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 19040 Complex module.
CMod

Definitiondf-clm 19041* 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 19042 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 19043 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod flds

Theoremclmsubrg 19044 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod SubRingfld

Theoremclmlmod 19045 A complex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmgrp 19046 A complex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmabl 19047 A complex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmrng 19048 The scalar ring of a complex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmfgrp 19049 The scalar ring of a complex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclm0 19050 The zero of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclm1 19051 The identity of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmadd 19052 The addition of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmmul 19053 The multiplication of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmcj 19054 The conjugation of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremisclmi 19055 Reverse direction of isclm 19042. (Contributed by Mario Carneiro, 30-Oct-2015.)
Scalar       flds SubRingfld CMod

Theoremclmzss 19056 The scalar ring of a complex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsscn 19057 The scalar ring of a complex module is a subset of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsub 19058 Subtraction in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmneg 19059 Negation in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmabs 19060 Norm in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmacl 19061 Closure of ring addition for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmmcl 19062 Closure of ring multiplication for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsubcl 19063 Closure of ring subtraction for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremlmhmclm 19064 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 19065 Associative law for scalar product. (lmodvsass 15930 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremclmvsdir 19066 Distributive law for scalar product. (lmodvsdir 15929 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                            CMod

Theoremclmvs1 19067 Scalar product with ring unit. (lmodvs1 15933 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclm0vs 19068 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 15938 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremclmvneg1 19069 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 15942 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmvsneg 19070 Multiplication of a vector by a negated scalar. (lmodvsneg 15943 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                            CMod

Theoremclmmulg 19071 The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
.g              CMod

Theoremclmsubdir 19072 Scalar multiplication distributive law for subtraction. (lmodsubdir 15957 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremzlmclm 19073 The -module operation turns an arbitrary abelian group into a complex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
Mod       CMod

Theoremclmzlmvsca 19074 The scalar product of a complex module matches the scalar product of the derived -module, which implies, together with zlmbas 16754 and zlmplusg 16755, that any module over is structure-equivalent to the canonical -module Mod. (Contributed by Mario Carneiro, 30-Oct-2015.)
Mod              CMod

Theoremnmoleub2lem 19075* Lemma for nmoleub2a 19078 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2lem3 19076* Lemma for nmoleub2a 19078 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2lem2 19077* Lemma for nmoleub2a 19078 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2a 19078* 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 19079* 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 19080* 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

Theoremnmhmcn 19081 A linear operator over a normed complex module is bounded iff it is continuous. (Contributed by Mario Carneiro, 22-Oct-2015.)
Scalar              NrmMod CMod NrmMod CMod NMHom LMHom

11.5.2  Complex pre-Hilbert space

Syntaxccph 19082 Extend class notation with a complex pre-Hilbert space.

Syntaxctch 19083 Function to put a norm on a Hilbert space.
toCHil

Definitiondf-cph 19084* Define a complex pre-Hilbert space. By restricting the scalar field to a quadratically closed subfield of , we have enough structure to define a norm, with the associated connection to a metric and topology. (Contributed by Mario Carneiro, 8-Oct-2015.)
NrmMod Scalar flds

Definitiondf-tch 19085* Define a function to augment a (pre-)Hilbert space with a norm. No extra parameters are needed, but some conditions must be satisfied to ensure that this in fact creates a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
toCHil toNrmGrp

Theoremiscph 19086* A complex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the complexes, with a norm defined (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar              NrmMod flds

Theoremcphphl 19087 A complex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphnlm 19088 A complex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
NrmMod

Theoremcphngp 19089 A complex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
NrmGrp

Theoremcphlmod 19090 A complex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphlvec 19091 A complex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphnvc 19092 A complex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
NrmVec

Theoremcphsubrglem 19093 Lemma for cphsubrg 19096. (Contributed by Mario Carneiro, 9-Oct-2015.)
flds               flds SubRingfld

Theoremcphreccllem 19094 Lemma for cphreccl 19097. (Contributed by Mario Carneiro, 8-Oct-2015.)
flds

Theoremcphsca 19095 A complex pre-Hilbert space is a vector space over a subfield of . (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar              flds

Theoremcphsubrg 19096 The scalar field of a complex pre-Hilbert space is a subring of . (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar              SubRingfld

Theoremcphreccl 19097 The scalar field of a complex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

Theoremcphdivcl 19098 The scalar field of a complex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.)
Scalar

Theoremcphcjcl 19099 The scalar field of a complex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.)
Scalar

Theoremcphsqrcl 19100 The scalar field of a complex pre-Hilbert space is closed under square roots of positive reals (i.e. it is quadratically closed relative to ). (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

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