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Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvhmulr 31201 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 31202 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 31203 Vector membership in the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvaddcbv 31204* Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)

Theoremdvhvaddval 31205* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)

Theoremdvhfvadd 31206* 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 31207 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 31208 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 31209* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 31208 and/or dvhfplusr 31199. (Contributed by NM, 26-Sep-2014.)

Theoremdvhvaddcl 31210 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 31211 Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
Scalar

Theoremdvhvaddass 31212 Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Scalar

Theoremdvhvscacbv 31213* Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)

Theoremdvhvscaval 31214* The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)

Theoremdvhfvsca 31215* Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremdvhvsca 31216 Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)

Theoremdvhopvsca 31217 Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)

Theoremdvhvscacl 31218 Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)

Theoremtendoinvcl 31219* 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 31097. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendolinv 31220* Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremtendorinv 31221* Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar

Theoremdvhgrp 31222 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 31223 Lemma for dvhlvec 31224. 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 31224 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvhlmod 31225 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)

Theoremdvh0g 31226* 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 31227 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 31231 and dihpN 31451. (Contributed by NM, 27-Mar-2015.)

TheoremdvhopclN 31228 Closure of a vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopaddN 31229* Sum of vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopspN 31230* Scalar product of vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

TheoremdvhopN 31231* 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 31232* Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)

Theoremcdlemm10N 31233* 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 31234 Extend class notation with subspace orthocomplement for partial vector space.

Definitiondf-docaN 31235* 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 31236* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocafvalN 31237* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocavalN 31238* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocaclN 31239 Closure of subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaocN 31240 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 31241 Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdoca3N 31242 Double orthocomplement of partial isomorphism A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdvadiaN 31243 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 31244* 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 31245* 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 31150 for the domain. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

SyntaxcdjaN 31246 Extend class notation with subspace join for partial vector space.

Definitiondf-djaN 31247* Define (closed) subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.)

TheoremdjaffvalN 31248* Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjafvalN 31249* 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 31250 Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjaclN 31251 Closure of subspace join for partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdjajN 31252 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 31253 Extend class notation with isomorphism B.

Definitiondf-dib 31254* 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 31255* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibfval 31256* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibval 31257* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

TheoremdibopelvalN 31258* Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)

Theoremdibval2 31259* Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)

Theoremdibopelval2 31260* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Theoremdibval3N 31261* Value of the partial isomorphism B for a lattice . (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)

Theoremdibelval3 31262* Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)

Theoremdibopelval3 31263* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.)

Theoremdibelval1st 31264 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibelval1st1 31265 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibelval1st2N 31266 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)

Theoremdibelval2nd 31267* Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibn0 31268 The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014.)

Theoremdibfna 31269 Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

Theoremdibdiadm 31270 Domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

TheoremdibfnN 31271* Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdibdmN 31272* Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdibeldmN 31273 Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

Theoremdibord 31274 The isomorphism B for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 24-Feb-2014.)

Theoremdib11N 31275 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 31276 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 31277 Closure of partial isomorphism B for a lattice . (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdibvalrel 31278 The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremdib0 31279 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 31280* 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 31281* Partial isomorphism B of a lattice glb. (Contributed by NM, 9-Mar-2014.) (New usage is discouraged.)

TheoremdibintclN 31282 The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdib1dim2 31283* 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 31284 The partial isomorphism B maps to a set of vectors in full vector space H. (Contributed by NM, 1-Jan-2014.)

Theoremdiblss 31285 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 31286* Membership in subspace sum for partial isomorphism B. (Contributed by NM, 21-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Syntaxcdic 31287 Extend class notation with isomorphism C.

Definitiondf-dic 31288* 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 31289* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.)

Theoremdicfval 31290* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.)

Theoremdicval 31291* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)

Theoremdicopelval 31292* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 15-Feb-2014.)

TheoremdicelvalN 31293* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)

Theoremdicval2 31294* The partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.)

Theoremdicelval3 31295* Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014.)

Theoremdicopelval2 31296* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.)

Theoremdicelval2N 31297* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)

TheoremdicfnN 31298* Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdicdmN 31299* Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdicvalrelN 31300 The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

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