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Theorem List for Metamath Proof Explorer - 35001-35100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdia11N 35001 The partial isomorphism A for a lattice is one-to-one in the region under co-atom . Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)

Theoremdiaf11N 35002 The partial isomorphism A for a lattice is a one-to-one function. Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)

TheoremdiaclN 35003 Closure of partial isomorphism A for a lattice . (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)

TheoremdiacnvclN 35004 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdia0 35005 The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)

Theoremdia1N 35006 The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)

Theoremdia1elN 35007 The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiaglbN 35008* Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

TheoremdiameetN 35009 Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiainN 35010 Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaintclN 35011 The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

TheoremdiasslssN 35012 The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdiassdvaN 35013 The partial isomorphism A maps to a set of vectors in partial vector space A. (Contributed by NM, 1-Jan-2014.) (New usage is discouraged.)

Theoremdia1dim 35014* Two expressions for the 1-dimensional subspaces of partial vector space A (when is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdia1dim2 35015 Two expressions for a 1-dimensional subspace of partial vector space A (when is a nonzero vector i.e. non-identity translation). (Contributed by NM, 15-Jan-2014.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdia1dimid 35016 A vector (translation) belongs to the 1-dim subspace it generates. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem1 35017 Lemma for dia2dim 35030. Show properties of the auxiliary atom . Part of proof of Lemma M in [Crawley] p. 121 line 3. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem2 35018 Lemma for dia2dim 35030. Define a translation whose trace is atom . Part of proof of Lemma M in [Crawley] p. 121 line 4. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem3 35019 Lemma for dia2dim 35030. Define a translation whose trace is atom . Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem4 35020 Lemma for dia2dim 35030. Show that the composition (sum) of translations (vectors) and equals . Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem5 35021 Lemma for dia2dim 35030. The sum of vectors and belongs to the sum of the subspaces generated by them. Thus, belongs to the subspace sum. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem6 35022 Lemma for dia2dim 35030. Eliminate auxiliary translations and . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem7 35023 Lemma for dia2dim 35030. Eliminate condition. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem8 35024 Lemma for dia2dim 35030. Eliminate no-longer used auxiliary atoms and . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem9 35025 Lemma for dia2dim 35030. Eliminate , conditions. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem10 35026 Lemma for dia2dim 35030. Convert membership in closed subspace to a lattice ordering. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem11 35027 Lemma for dia2dim 35030. Convert ordering hypothesis on to subspace membership . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem12 35028 Lemma for dia2dim 35030. Obtain subset relation. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem13 35029 Lemma for dia2dim 35030. Eliminate condition. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dim 35030 A two-dimensional subspace of partial vector space A is closed, or equivalently, the isomorphism of a join of two atoms is a subset of the subspace sum of the isomorphisms of each atom (and thus they are equal, as shown later for the full vector space H). (Contributed by NM, 9-Sep-2014.)

Syntaxcdvh 35031 Extend class notation with constructed full vector space H.

Definitiondf-dvech 35032* Define constructed full vector space H. (Contributed by NM, 17-Oct-2013.)
Scalar

Theoremdvhfset 35033* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhset 35034* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhsca 35035 The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)
Scalar

Theoremdvhbase 35036 The ring base set of the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhfplusr 35037* Ring addition operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhfmulr 35038* Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhmulr 35039 Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhvbase 35040 The vectors (vector base set) of the constructed full vector space H are all translations (for a fiducial co-atom ). (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdvhelvbasei 35041 Vector membership in the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvaddcbv 35042* Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)

Theoremdvhvaddval 35043* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)

Theoremdvhfvadd 35044* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremdvhvadd 35045 The vector sum operation for the constructed full vector space H. (Contributed by NM, 11-Feb-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremdvhopvadd 35046 The vector sum operation for the constructed full vector space H. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.)
Scalar

Theoremdvhopvadd2 35047* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 35046 and/or dvhfplusr 35037. (Contributed by NM, 26-Sep-2014.)

Theoremdvhvaddcl 35048 Closure of the vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

TheoremdvhvaddcomN 35049 Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
Scalar

Theoremdvhvaddass 35050 Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Scalar

Theoremdvhvscacbv 35051* Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)

Theoremdvhvscaval 35052* The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)

Theoremdvhfvsca 35053* Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremdvhvsca 35054 Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)

Theoremdvhopvsca 35055 Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvscacl 35056 Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)

Theoremtendoinvcl 35057* Closure of multiplicative inverse for endomorphism. We use the scalar inverse of the vector space since it is much simpler than the direct inverse of cdleml8 34935. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendolinv 35058* Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendorinv 35059* Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremdvhgrp 35060 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar

Theoremdvhlveclem 35061 Lemma for dvhlvec 35062. TODO: proof substituting inner part first shorter/longer than substituting outer part first? TODO: break up into smaller lemmas? TODO: does method shorten proof? (Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
Scalar

Theoremdvhlvec 35062 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvhlmod 35063 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvh0g 35064* The zero vector of vector space H has the zero translation as its first member and the zero trace-preserving endomorphism as the second. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremdvheveccl 35065 Properties of a unit vector that we will use later as a convenient reference vector. This vector is called "e" in the remark after Lemma M of [Crawley] p. 121. line 17. See also dvhopN 35069 and dihpN 35289. (Contributed by NM, 27-Mar-2015.)

TheoremdvhopclN 35066 Closure of a vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopaddN 35067* Sum of vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopspN 35068* Scalar product of vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopN 35069* Decompose a vector expressed as an ordered pair into the sum of two components, the first from the translation group vector base of and the other from the one-dimensional vector subspace . Part of Lemma M of [Crawley] p. 121, line 18. We represent their e, sigma, f by , , . We swapped the order of vector sum (their juxtaposition i.e. composition) to show first. Note that and are the zero and one of the division ring , and is the zero of the translation group. is the scalar product. (Contributed by NM, 21-Nov-2013.) (New usage is discouraged.)

Theoremdvhopellsm 35070* Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)

Theoremcdlemm10N 35071* The image of the map is the entire one-dimensional subspace . Remark after Lemma M of [Crawley] p. 121 line 23. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)

SyntaxcocaN 35072 Extend class notation with subspace orthocomplement for partial vector space.

Definitiondf-docaN 35073* Define subspace orthocomplement for partial vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 6-Dec-2013.)

TheoremdocaffvalN 35074* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocafvalN 35075* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocavalN 35076* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocaclN 35077 Closure of subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaocN 35078 Value of partial isomorphism A at lattice orthocomplement (using a Sasaki projection to get orthocomplement relative to the fiducial co-atom ). (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdoca2N 35079 Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdoca3N 35080 Double orthocomplement of partial isomorphism A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdvadiaN 35081 Any closed subspace is a member of the range of partial isomorphism A, showing the isomorphism maps onto the set of closed subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdiarnN 35082* Partial isomorphism A maps onto the set of all closed subspaces of partial vector space A. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

Theoremdiaf1oN 35083* The partial isomorphism A for a lattice is a one-to-one, onto function. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. See diadm 34988 for the domain. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

SyntaxcdjaN 35084 Extend class notation with subspace join for partial vector space.

Definitiondf-djaN 35085* Define (closed) subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.)

TheoremdjaffvalN 35086* Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjafvalN 35087* Subspace join for partial vector space. TODO: take out hypothesis .i, no longer used. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjavalN 35088 Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjaclN 35089 Closure of subspace join for partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdjajN 35090 Transfer lattice join to partial vector space closed subspace join. Part of Lemma M of [Crawley] p. 120 line 29, with closed subspace join rather than subspace sum. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Syntaxcdib 35091 Extend class notation with isomorphism B.

Definitiondf-dib 35092* Isomorphism B is isomorphism A extended with an extra dimension set to the zero vector component i.e. the zero endormorphism. Its domain is lattice elements less than or equal to the fiducial co-atom . (Contributed by NM, 8-Dec-2013.)

Theoremdibffval 35093* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibfval 35094* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibval 35095* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

TheoremdibopelvalN 35096* Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)

Theoremdibval2 35097* Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)

Theoremdibopelval2 35098* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Theoremdibval3N 35099* Value of the partial isomorphism B for a lattice . (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)

Theoremdibelval3 35100* Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)

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