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Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremdiarnN 35801* 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 35802* 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 35707 for the domain. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

SyntaxcdjaN 35803 Extend class notation with subspace join for partial vector space.

Definitiondf-djaN 35804* Define (closed) subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.)

TheoremdjaffvalN 35805* Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjafvalN 35806* 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 35807 Subspace join for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdjaclN 35808 Closure of subspace join for partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdjajN 35809 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 35810 Extend class notation with isomorphism B.

Definitiondf-dib 35811* 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 35812* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibfval 35813* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

Theoremdibval 35814* The partial isomorphism B for a lattice . (Contributed by NM, 8-Dec-2013.)

TheoremdibopelvalN 35815* Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)

Theoremdibval2 35816* Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)

Theoremdibopelval2 35817* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Theoremdibval3N 35818* Value of the partial isomorphism B for a lattice . (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)

Theoremdibelval3 35819* Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)

Theoremdibopelval3 35820* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.)

Theoremdibelval1st 35821 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibelval1st1 35822 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibelval1st2N 35823 Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)

Theoremdibelval2nd 35824* Membership in value of the partial isomorphism B for a lattice . (Contributed by NM, 13-Feb-2014.)

Theoremdibn0 35825 The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014.)

Theoremdibfna 35826 Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

Theoremdibdiadm 35827 Domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

TheoremdibfnN 35828* Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdibdmN 35829* Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdibeldmN 35830 Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

Theoremdibord 35831 The isomorphism B for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 24-Feb-2014.)

Theoremdib11N 35832 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 35833 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 35834 Closure of partial isomorphism B for a lattice . (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdibvalrel 35835 The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremdib0 35836 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 35837* 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 35838* Partial isomorphism B of a lattice glb. (Contributed by NM, 9-Mar-2014.) (New usage is discouraged.)

TheoremdibintclN 35839 The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdib1dim2 35840* 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 35841 The partial isomorphism B maps to a set of vectors in full vector space H. (Contributed by NM, 1-Jan-2014.)

Theoremdiblss 35842 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 35843* Membership in subspace sum for partial isomorphism B. (Contributed by NM, 21-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Syntaxcdic 35844 Extend class notation with isomorphism C.

Definitiondf-dic 35845* 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 35846* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.)

Theoremdicfval 35847* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.)

Theoremdicval 35848* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)

Theoremdicopelval 35849* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 15-Feb-2014.)

TheoremdicelvalN 35850* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)

Theoremdicval2 35851* The partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.)

Theoremdicelval3 35852* Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014.)

Theoremdicopelval2 35853* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.)

Theoremdicelval2N 35854* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)

TheoremdicfnN 35855* Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdicdmN 35856* Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdicvalrelN 35857 The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdicssdvh 35858 The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014.)

Theoremdicelval1sta 35859* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.)

Theoremdicelval1stN 35860 Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.) (New usage is discouraged.)

Theoremdicelval2nd 35861 Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.)

Theoremdicvaddcl 35862 Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)

Theoremdicvscacl 35863 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 35864 The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)

Theoremdiclss 35865 The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)

Theoremdiclspsn 35866* 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 35867* Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)

Theoremcdlemn2a 35868* Part of proof of Lemma N of [Crawley] p. 121. (Contributed by NM, 24-Feb-2014.)

Theoremcdlemn3 35869* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.)

Theoremcdlemn4 35870* 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 35871* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.)

Theoremcdlemn5pre 35872* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)

Theoremcdlemn5 35873 Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)

Theoremcdlemn6 35874* Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn7 35875* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn8 35876* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn9 35877* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn10 35878 Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11a 35879* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11b 35880* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11c 35881* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11pre 35882* Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 35879, cdlemn11b 35880, cdlemn11c 35881, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11 35883 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn 35884 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)

Theoremdihordlem6 35885* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7 35886* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7b 35887* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihjustlem 35888 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 35889 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.)

Theoremdihord1 35890 Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change to using lhpmcvr3 34696, here and all theorems below. (Contributed by NM, 2-Mar-2014.)

Theoremdihord2a 35891 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2b 35892 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2cN 35893* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)

Theoremdihord11b 35894* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord10 35895* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord11c 35896* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2pre 35897* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2pre2 35898* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.)

Theoremdihord2 35899 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. Todo: do we need and ? (Contributed by NM, 4-Mar-2014.)

Syntaxcdih 35900 Extend class notation with isomorphism H.

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