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Theorem List for Metamath Proof Explorer - 29901-30000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtendospdi2 29901* Reverse distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

TheoremtendospcanN 29902* Cancellation law for trace-perserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdvaabl 29903 The constructed partial vector space A for a lattice is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdvalveclem 29904 Lemma for dvalvec 29905. (Contributed by NM, 11-Oct-2013.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvalvec 29905 The constructed partial vector space A for a lattice is a left vector space. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdva0g 29906 The zero vector of partial vector space A. (Contributed by NM, 9-Sep-2014.)

Syntaxcdia 29907 Extend class notation with partial isomorphism A.

Definitiondf-disoa 29908* Define partial isomorphism A. (Contributed by NM, 15-Oct-2013.)

Theoremdiaffval 29909* The partial isomorphism A for a lattice . (Contributed by NM, 15-Oct-2013.)

Theoremdiafval 29910* The partial isomorphism A for a lattice . (Contributed by NM, 15-Oct-2013.)

Theoremdiaval 29911* The partial isomorphism A for a lattice . Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)

Theoremdiaelval 29912 Member of the partial isomorphism A for a lattice . (Contributed by NM, 3-Dec-2013.)

Theoremdiafn 29913* Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)

Theoremdiadm 29914* Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)

Theoremdiaeldm 29915 Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)

TheoremdiadmclN 29916 A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiadmleN 29917 A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdian0 29918 The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)

Theoremdia0eldmN 29919 The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdia1eldmN 29920 The fiducial hyperplane (largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdiass 29921 The value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.)

Theoremdiael 29922 A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)

Theoremdiatrl 29923 Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)

TheoremdiaelrnN 29924 Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)

Theoremdialss 29925 The value of partial isomorphism A is a subspace of partial vector space A. Part of Lemma M of [Crawley] p. 120 line 26. (Contributed by NM, 17-Jan-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremdiaord 29926 The partial isomorphism A for a lattice is order-preserving in the region under co-atom . Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)

Theoremdia11N 29927 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 29928 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 29929 Closure of partial isomorphism A for a lattice . (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)

TheoremdiacnvclN 29930 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdia0 29931 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 29932 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 29933 The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiaglbN 29934* Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

TheoremdiameetN 29935 Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiainN 29936 Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaintclN 29937 The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

TheoremdiasslssN 29938 The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdiassdvaN 29939 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 29940* 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 29941 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 29942 A vector (translation) belongs to the 1-dim subspace it generates. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem1 29943 Lemma for dia2dim 29956. 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 29944 Lemma for dia2dim 29956. 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 29945 Lemma for dia2dim 29956. 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 29946 Lemma for dia2dim 29956. 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 29947 Lemma for dia2dim 29956. 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 29948 Lemma for dia2dim 29956. Eliminate auxiliary translations and . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem7 29949 Lemma for dia2dim 29956. Eliminate condition. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem8 29950 Lemma for dia2dim 29956. Eliminate no-longer used auxiliary atoms and . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem9 29951 Lemma for dia2dim 29956. Eliminate , conditions. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem10 29952 Lemma for dia2dim 29956. Convert membership in closed subspace to a lattice ordering. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem11 29953 Lemma for dia2dim 29956. Convert ordering hypothesis on to subspace membership . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem12 29954 Lemma for dia2dim 29956. Obtain subset relation. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem13 29955 Lemma for dia2dim 29956. Eliminate condition. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dim 29956 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 29957 Extend class notation with constructed full vector space H.

Definitiondf-dvech 29958* Define constructed full vector space H. (Contributed by NM, 17-Oct-2013.)
Scalar

Theoremdvhfset 29959* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhset 29960* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhsca 29961 The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)
Scalar

Theoremdvhbase 29962 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 29963* 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 29964* 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 29965 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 29966 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 29967 Vector membership in the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvaddcbv 29968* Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)

Theoremdvhvaddval 29969* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)

Theoremdvhfvadd 29970* 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 29971 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 29972 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 29973* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 29972 and/or dvhfplusr 29963. (Contributed by NM, 26-Sep-2014.)

Theoremdvhvaddcl 29974 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 29975 Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
Scalar

Theoremdvhvaddass 29976 Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Scalar

Theoremdvhvscacbv 29977* Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)

Theoremdvhvscaval 29978* The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)

Theoremdvhfvsca 29979* Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremdvhvsca 29980 Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)

Theoremdvhopvsca 29981 Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvscacl 29982 Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)

Theoremtendoinvcl 29983* 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 29861. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendolinv 29984* Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendorinv 29985* Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremdvhgrp 29986 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 29987 Lemma for dvhlvec 29988. 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 29988 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvhlmod 29989 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvh0g 29990* 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 29991 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 29995 and dihpN 30215. (Contributed by NM, 27-Mar-2015.)

TheoremdvhopclN 29992 Closure of a vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopaddN 29993* Sum of vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopspN 29994* Scalar product of vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopN 29995* 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 29996* Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)

Theoremcdlemm10N 29997* 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 29998 Extend class notation with subspace orthocomplement for partial vector space.

Definitiondf-docaN 29999* 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 30000* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

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