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Theorem List for Metamath Proof Explorer - 30401-30500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvhset 30401* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhsca 30402 The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)
Scalar

Theoremdvhbase 30403 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 30404* 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 30405* 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 30406 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 30407 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 30408 Vector membership in the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvaddcbv 30409* Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)

Theoremdvhvaddval 30410* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)

Theoremdvhfvadd 30411* 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 30412 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 30413 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 30414* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 30413 and/or dvhfplusr 30404. (Contributed by NM, 26-Sep-2014.)

Theoremdvhvaddcl 30415 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 30416 Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
Scalar

Theoremdvhvaddass 30417 Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Scalar

Theoremdvhvscacbv 30418* Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)

Theoremdvhvscaval 30419* The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)

Theoremdvhfvsca 30420* Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremdvhvsca 30421 Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)

Theoremdvhopvsca 30422 Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvscacl 30423 Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)

Theoremtendoinvcl 30424* 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 30302. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendolinv 30425* Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendorinv 30426* Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremdvhgrp 30427 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 30428 Lemma for dvhlvec 30429. 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 30429 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvhlmod 30430 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvh0g 30431* 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 30432 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 30436 and dihpN 30656. (Contributed by NM, 27-Mar-2015.)

TheoremdvhopclN 30433 Closure of a vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopaddN 30434* Sum of vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopspN 30435* Scalar product of vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopN 30436* 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 30437* Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)

Theoremcdlemm10N 30438* 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 30439 Extend class notation with subspace orthocomplement for partial vector space.

Definitiondf-docaN 30440* 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 30441* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocafvalN 30442* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocavalN 30443* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocaclN 30444 Closure of subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaocN 30445 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 30446 Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdoca3N 30447 Double orthocomplement of partial isomorphism A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdvadiaN 30448 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 30449* 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 30450* 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 30355 for the domain. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

SyntaxcdjaN 30451 Extend class notation with subspace join for partial vector space.

Definitiondf-djaN 30452* Define (closed) subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.)

TheoremdjaffvalN 30453* Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjafvalN 30454* 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 30455 Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjaclN 30456 Closure of subspace join for partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdjajN 30457 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 30458 Extend class notation with isomorphism B.

Definitiondf-dib 30459* 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 30460* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibfval 30461* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibval 30462* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

TheoremdibopelvalN 30463* Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)

Theoremdibval2 30464* Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)

Theoremdibopelval2 30465* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Theoremdibval3N 30466* Value of the partial isomorphism B for a lattice . (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)

Theoremdibelval3 30467* Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)

Theoremdibopelval3 30468* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.)

Theoremdibelval1st 30469 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibelval1st1 30470 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibelval1st2N 30471 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)

Theoremdibelval2nd 30472* Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibn0 30473 The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014.)

Theoremdibfna 30474 Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

Theoremdibdiadm 30475 Domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

TheoremdibfnN 30476* Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdibdmN 30477* Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdibeldmN 30478 Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

Theoremdibord 30479 The isomorphism B for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 24-Feb-2014.)

Theoremdib11N 30480 The isomorphism B for a lattice is one-to-one in the region under co-atom . (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)

Theoremdibf11N 30481 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.)

TheoremdibclN 30482 Closure of partial isomorphism B for a lattice . (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdibvalrel 30483 The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremdib0 30484 The value of partial isomorphism B at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 27-Mar-2014.)

Theoremdib1dim 30485* Two expressions for the 1-dimensional subspaces of vector space H. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

TheoremdibglbN 30486* Partial isomorphism B of a lattice glb. (Contributed by NM, 9-Mar-2014.) (New usage is discouraged.)

TheoremdibintclN 30487 The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdib1dim2 30488* Two expressions for a 1-dimensional subspace of vector space H (when is a nonzero vector i.e. non-identity translation). (Contributed by NM, 24-Feb-2014.)

Theoremdibss 30489 The partial isomorphism B maps to a set of vectors in full vector space H. (Contributed by NM, 1-Jan-2014.)

Theoremdiblss 30490 The value of partial isomorphism B is a subspace of partial vector space H. TODO: use dib* specific theorems instead of dia* ones to shorten proof? (Contributed by NM, 11-Feb-2014.)

Theoremdiblsmopel 30491* Membership in subspace sum for partial isomorphism B. (Contributed by NM, 21-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Syntaxcdic 30492 Extend class notation with isomorphism C.

Definitiondf-dic 30493* Isomorphism C has domain of lattice atoms that are not less than or equal to the fiducial co-atom . The value is a one-dimensional subspace generated by the pair consisting of the vector below and the endomorphism ring unit. Definition of phi(q) in [Crawley] p. 121. Note that we use the fixed atom k ) to represent the p in their "Choose an atom p..." on line 21. (Contributed by NM, 15-Dec-2013.)

Theoremdicffval 30494* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.)

Theoremdicfval 30495* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.)

Theoremdicval 30496* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)

Theoremdicopelval 30497* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 15-Feb-2014.)

TheoremdicelvalN 30498* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)

Theoremdicval2 30499* The partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.)

Theoremdicelval3 30500* Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014.)

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