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

TheoremdiassdvaN 30401 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 30402* 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 30403 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 30404 A vector (translation) belongs to the 1-dim subspace it generates. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem1 30405 Lemma for dia2dim 30418. 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 30406 Lemma for dia2dim 30418. 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 30407 Lemma for dia2dim 30418. 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 30408 Lemma for dia2dim 30418. 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 30409 Lemma for dia2dim 30418. 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 30410 Lemma for dia2dim 30418. Eliminate auxiliary translations and . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem7 30411 Lemma for dia2dim 30418. Eliminate condition. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem8 30412 Lemma for dia2dim 30418. Eliminate no-longer used auxiliary atoms and . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem9 30413 Lemma for dia2dim 30418. Eliminate , conditions. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem10 30414 Lemma for dia2dim 30418. Convert membership in closed subspace to a lattice ordering. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem11 30415 Lemma for dia2dim 30418. Convert ordering hypothesis on to subspace membership . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem12 30416 Lemma for dia2dim 30418. Obtain subset relation. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem13 30417 Lemma for dia2dim 30418. Eliminate condition. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dim 30418 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 30419 Extend class notation with constructed full vector space H.

Definitiondf-dvech 30420* Define constructed full vector space H. (Contributed by NM, 17-Oct-2013.)
Scalar

Theoremdvhfset 30421* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhset 30422* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhsca 30423 The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)
Scalar

Theoremdvhbase 30424 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 30425* 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 30426* 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 30427 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 30428 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 30429 Vector membership in the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvaddcbv 30430* Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)

Theoremdvhvaddval 30431* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)

Theoremdvhfvadd 30432* 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 30433 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 30434 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 30435* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 30434 and/or dvhfplusr 30425. (Contributed by NM, 26-Sep-2014.)

Theoremdvhvaddcl 30436 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 30437 Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
Scalar

Theoremdvhvaddass 30438 Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Scalar

Theoremdvhvscacbv 30439* Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)

Theoremdvhvscaval 30440* The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)

Theoremdvhfvsca 30441* Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremdvhvsca 30442 Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)

Theoremdvhopvsca 30443 Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvscacl 30444 Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)

Theoremtendoinvcl 30445* 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 30323. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendolinv 30446* Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendorinv 30447* Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremdvhgrp 30448 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 30449 Lemma for dvhlvec 30450. 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 30450 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvhlmod 30451 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvh0g 30452* 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 30453 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 30457 and dihpN 30677. (Contributed by NM, 27-Mar-2015.)

TheoremdvhopclN 30454 Closure of a vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopaddN 30455* Sum of vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopspN 30456* Scalar product of vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopN 30457* 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 30458* Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)

Theoremcdlemm10N 30459* 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 30460 Extend class notation with subspace orthocomplement for partial vector space.

Definitiondf-docaN 30461* 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 30462* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocafvalN 30463* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocavalN 30464* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocaclN 30465 Closure of subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaocN 30466 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 30467 Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdoca3N 30468 Double orthocomplement of partial isomorphism A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdvadiaN 30469 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 30470* 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 30471* 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 30376 for the domain. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

SyntaxcdjaN 30472 Extend class notation with subspace join for partial vector space.

Definitiondf-djaN 30473* Define (closed) subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.)

TheoremdjaffvalN 30474* Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjafvalN 30475* 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 30476 Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjaclN 30477 Closure of subspace join for partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdjajN 30478 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 30479 Extend class notation with isomorphism B.

Definitiondf-dib 30480* 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 30481* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibfval 30482* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibval 30483* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

TheoremdibopelvalN 30484* Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)

Theoremdibval2 30485* Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)

Theoremdibopelval2 30486* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Theoremdibval3N 30487* Value of the partial isomorphism B for a lattice . (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)

Theoremdibelval3 30488* Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)

Theoremdibopelval3 30489* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.)

Theoremdibelval1st 30490 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibelval1st1 30491 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibelval1st2N 30492 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)

Theoremdibelval2nd 30493* Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibn0 30494 The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014.)

Theoremdibfna 30495 Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

Theoremdibdiadm 30496 Domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

TheoremdibfnN 30497* Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdibdmN 30498* Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdibeldmN 30499 Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

Theoremdibord 30500 The isomorphism B for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 24-Feb-2014.)

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