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Theorem List for Metamath Proof Explorer - 31601-31700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcdlemm10N 31601* 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 31602 Extend class notation with subspace orthocomplement for partial vector space.

Definitiondf-docaN 31603* 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 31604* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocafvalN 31605* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocavalN 31606* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdocaclN 31607 Closure of subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaocN 31608 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 31609 Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdoca3N 31610 Double orthocomplement of partial isomorphism A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdvadiaN 31611 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 31612* 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 31613* 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 31518 for the domain. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

SyntaxcdjaN 31614 Extend class notation with subspace join for partial vector space.

Definitiondf-djaN 31615* Define (closed) subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.)

TheoremdjaffvalN 31616* Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjafvalN 31617* 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 31618 Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjaclN 31619 Closure of subspace join for partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdjajN 31620 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 31621 Extend class notation with isomorphism B.

Definitiondf-dib 31622* 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 31623* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibfval 31624* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibval 31625* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

TheoremdibopelvalN 31626* Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)

Theoremdibval2 31627* Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)

Theoremdibopelval2 31628* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Theoremdibval3N 31629* Value of the partial isomorphism B for a lattice . (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)

Theoremdibelval3 31630* Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)

Theoremdibopelval3 31631* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.)

Theoremdibelval1st 31632 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibelval1st1 31633 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibelval1st2N 31634 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)

Theoremdibelval2nd 31635* Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibn0 31636 The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014.)

Theoremdibfna 31637 Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

Theoremdibdiadm 31638 Domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

TheoremdibfnN 31639* Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdibdmN 31640* Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdibeldmN 31641 Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

Theoremdibord 31642 The isomorphism B for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 24-Feb-2014.)

Theoremdib11N 31643 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 31644 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 31645 Closure of partial isomorphism B for a lattice . (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdibvalrel 31646 The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremdib0 31647 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 31648* 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 31649* Partial isomorphism B of a lattice glb. (Contributed by NM, 9-Mar-2014.) (New usage is discouraged.)

TheoremdibintclN 31650 The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdib1dim2 31651* 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 31652 The partial isomorphism B maps to a set of vectors in full vector space H. (Contributed by NM, 1-Jan-2014.)

Theoremdiblss 31653 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 31654* Membership in subspace sum for partial isomorphism B. (Contributed by NM, 21-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Syntaxcdic 31655 Extend class notation with isomorphism C.

Definitiondf-dic 31656* 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 31657* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.)

Theoremdicfval 31658* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.)

Theoremdicval 31659* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)

Theoremdicopelval 31660* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 15-Feb-2014.)

TheoremdicelvalN 31661* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)

Theoremdicval2 31662* The partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.)

Theoremdicelval3 31663* Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014.)

Theoremdicopelval2 31664* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.)

Theoremdicelval2N 31665* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)

TheoremdicfnN 31666* Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdicdmN 31667* Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdicvalrelN 31668 The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdicssdvh 31669 The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014.)

Theoremdicelval1sta 31670* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.)

Theoremdicelval1stN 31671 Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.) (New usage is discouraged.)

Theoremdicelval2nd 31672 Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.)

Theoremdicvaddcl 31673 Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)

Theoremdicvscacl 31674 Membership in value of the partial isomorphism C is closed under scalar product. (Contributed by NM, 16-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremdicn0 31675 The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)

Theoremdiclss 31676 The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)

Theoremdiclspsn 31677* The value of isomorphism C is spanned by vector . Part of proof of Lemma N of [Crawley] p. 121 line 29. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremcdlemn2 31678* Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)

Theoremcdlemn2a 31679* Part of proof of Lemma N of [Crawley] p. 121. (Contributed by NM, 24-Feb-2014.)

Theoremcdlemn3 31680* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.)

Theoremcdlemn4 31681* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremcdlemn4a 31682* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.)

Theoremcdlemn5pre 31683* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)

Theoremcdlemn5 31684 Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)

Theoremcdlemn6 31685* Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn7 31686* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn8 31687* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn9 31688* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn10 31689 Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11a 31690* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11b 31691* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11c 31692* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11pre 31693* Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 31690, cdlemn11b 31691, cdlemn11c 31692, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11 31694 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn 31695 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)

Theoremdihordlem6 31696* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7 31697* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7b 31698* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihjustlem 31699 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)

Theoremdihjust 31700 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)

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