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Theorem List for Metamath Proof Explorer - 35701-35800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdva0g 35701 The zero vector of partial vector space A. (Contributed by NM, 9-Sep-2014.)

Syntaxcdia 35702 Extend class notation with partial isomorphism A.

Definitiondf-disoa 35703* Define partial isomorphism A. (Contributed by NM, 15-Oct-2013.)

Theoremdiaffval 35704* The partial isomorphism A for a lattice . (Contributed by NM, 15-Oct-2013.)

Theoremdiafval 35705* The partial isomorphism A for a lattice . (Contributed by NM, 15-Oct-2013.)

Theoremdiaval 35706* The partial isomorphism A for a lattice . Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)

Theoremdiaelval 35707 Member of the partial isomorphism A for a lattice . (Contributed by NM, 3-Dec-2013.)

Theoremdiafn 35708* Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)

Theoremdiadm 35709* Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)

Theoremdiaeldm 35710 Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)

TheoremdiadmclN 35711 A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiadmleN 35712 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 35713 The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)

Theoremdia0eldmN 35714 The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdia1eldmN 35715 The fiducial hyperplane (the largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdiass 35716 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 35717 A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)

Theoremdiatrl 35718 Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)

TheoremdiaelrnN 35719 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 35720 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 35721 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 35722 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 35723 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 35724 Closure of partial isomorphism A for a lattice . (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)

TheoremdiacnvclN 35725 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdia0 35726 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 35727 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 35728 The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiaglbN 35729* Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

TheoremdiameetN 35730 Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiainN 35731 Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaintclN 35732 The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

TheoremdiasslssN 35733 The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdiassdvaN 35734 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 35735* 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 35736 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 35737 A vector (translation) belongs to the 1-dim subspace it generates. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem1 35738 Lemma for dia2dim 35751. 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 35739 Lemma for dia2dim 35751. 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 35740 Lemma for dia2dim 35751. 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 35741 Lemma for dia2dim 35751. 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 35742 Lemma for dia2dim 35751. 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 35743 Lemma for dia2dim 35751. Eliminate auxiliary translations and . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem7 35744 Lemma for dia2dim 35751. Eliminate condition. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem8 35745 Lemma for dia2dim 35751. Eliminate no-longer used auxiliary atoms and . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem9 35746 Lemma for dia2dim 35751. Eliminate , conditions. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem10 35747 Lemma for dia2dim 35751. Convert membership in closed subspace to a lattice ordering. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem11 35748 Lemma for dia2dim 35751. Convert ordering hypothesis on to subspace membership . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem12 35749 Lemma for dia2dim 35751. Obtain subset relation. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem13 35750 Lemma for dia2dim 35751. Eliminate condition. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dim 35751 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 35752 Extend class notation with constructed full vector space H.

Definitiondf-dvech 35753* Define constructed full vector space H. (Contributed by NM, 17-Oct-2013.)
Scalar

Theoremdvhfset 35754* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhset 35755* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhsca 35756 The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)
Scalar

Theoremdvhbase 35757 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 35758* 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 35759* 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 35760 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 35761 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 35762 Vector membership in the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvaddcbv 35763* Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)

Theoremdvhvaddval 35764* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)

Theoremdvhfvadd 35765* 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 35766 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 35767 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 35768* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 35767 and/or dvhfplusr 35758. (Contributed by NM, 26-Sep-2014.)

Theoremdvhvaddcl 35769 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 35770 Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
Scalar

Theoremdvhvaddass 35771 Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Scalar

Theoremdvhvscacbv 35772* Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)

Theoremdvhvscaval 35773* The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)

Theoremdvhfvsca 35774* Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremdvhvsca 35775 Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)

Theoremdvhopvsca 35776 Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvscacl 35777 Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)

Theoremtendoinvcl 35778* 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 35656. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendolinv 35779* Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendorinv 35780* Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremdvhgrp 35781 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 35782 Lemma for dvhlvec 35783. 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 35783 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvhlmod 35784 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvh0g 35785* 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 35786 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 35790 and dihpN 36010. (Contributed by NM, 27-Mar-2015.)

TheoremdvhopclN 35787 Closure of a vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopaddN 35788* Sum of vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopspN 35789* Scalar product of vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopN 35790* 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 35791* Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)

Theoremcdlemm10N 35792* 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 35793 Extend class notation with subspace orthocomplement for partial vector space.

Definitiondf-docaN 35794* 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 35795* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocafvalN 35796* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocavalN 35797* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocaclN 35798 Closure of subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaocN 35799 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 35800 Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

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