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Theorem List for Metamath Proof Explorer - 34801-34900   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremdibclN 34801 Closure of partial isomorphism B for a lattice . (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdibvalrel 34802 The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremdib0 34803 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 34804* 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 34805* Partial isomorphism B of a lattice glb. (Contributed by NM, 9-Mar-2014.) (New usage is discouraged.)

TheoremdibintclN 34806 The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdib1dim2 34807* 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 34808 The partial isomorphism B maps to a set of vectors in full vector space H. (Contributed by NM, 1-Jan-2014.)

Theoremdiblss 34809 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 34810* Membership in subspace sum for partial isomorphism B. (Contributed by NM, 21-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Syntaxcdic 34811 Extend class notation with isomorphism C.

Definitiondf-dic 34812* 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 34813* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.)

Theoremdicfval 34814* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.)

Theoremdicval 34815* The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)

Theoremdicopelval 34816* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 15-Feb-2014.)

TheoremdicelvalN 34817* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)

Theoremdicval2 34818* The partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.)

Theoremdicelval3 34819* Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014.)

Theoremdicopelval2 34820* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.)

Theoremdicelval2N 34821* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)

TheoremdicfnN 34822* Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdicdmN 34823* Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

TheoremdicvalrelN 34824 The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Theoremdicssdvh 34825 The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014.)

Theoremdicelval1sta 34826* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.)

Theoremdicelval1stN 34827 Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.) (New usage is discouraged.)

Theoremdicelval2nd 34828 Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.)

Theoremdicvaddcl 34829 Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)

Theoremdicvscacl 34830 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 34831 The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)

Theoremdiclss 34832 The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)

Theoremdiclspsn 34833* 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 34834* Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)

Theoremcdlemn2a 34835* Part of proof of Lemma N of [Crawley] p. 121. (Contributed by NM, 24-Feb-2014.)

Theoremcdlemn3 34836* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.)

Theoremcdlemn4 34837* 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 34838* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.)

Theoremcdlemn5pre 34839* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)

Theoremcdlemn5 34840 Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)

Theoremcdlemn6 34841* Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn7 34842* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn8 34843* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn9 34844* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn10 34845 Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11a 34846* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11b 34847* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11c 34848* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11pre 34849* Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 34846, cdlemn11b 34847, cdlemn11c 34848, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11 34850 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn 34851 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)

Theoremdihordlem6 34852* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7 34853* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7b 34854* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihjustlem 34855 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 34856 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 34857 Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change to using lhpmcvr3 33661, here and all theorems below. (Contributed by NM, 2-Mar-2014.)

Theoremdihord2a 34858 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2b 34859 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2cN 34860* 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 34861* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord10 34862* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord11c 34863* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2pre 34864* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2pre2 34865* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.)

Theoremdihord2 34866 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 34867 Extend class notation with isomorphism H.

Definitiondf-dih 34868* Define isomorphism H. (Contributed by NM, 28-Jan-2014.)

Theoremdihffval 34869* The isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)

Theoremdihfval 34870* Isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)

Theoremdihval 34871* Value of isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-2014.)

Theoremdihvalc 34872* Value of isomorphism H for a lattice when . (Contributed by NM, 4-Mar-2014.)

Theoremdihlsscpre 34873 Closure of isomorphism H for a lattice when . (Contributed by NM, 6-Mar-2014.)

Theoremdihvalcqpre 34874 Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 6-Mar-2014.)

Theoremdihvalcq 34875 Value of isomorphism H for a lattice when , given auxiliary atom . TODO: Use dihvalcq2 34886 (with lhpmcvr3 33661 for simplification) that changes and to and make this obsolete. Do to other theorems as well. (Contributed by NM, 6-Mar-2014.)

Theoremdihvalb 34876 Value of isomorphism H for a lattice when . (Contributed by NM, 4-Mar-2014.)

TheoremdihopelvalbN 34877* Ordered pair member of the partial isomorphism H for argument under . (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihvalcqat 34878 Value of isomorphism H for a lattice at an atom not under . (Contributed by NM, 27-Mar-2014.)

Theoremdih1dimb 34879* Two expressions for a 1-dimensional subspace of vector space H (when is a nonzero vector i.e. non-identity translation). (Contributed by NM, 27-Apr-2014.)

Theoremdih1dimb2 34880* Isomorphism H at an atom under . (Contributed by NM, 27-Apr-2014.)

Theoremdih1dimc 34881* Isomorphism H at an atom not under . (Contributed by NM, 27-Apr-2014.)

Theoremdib2dim 34882 Extend dia2dim 34716 to partial isomorphism B. (Contributed by NM, 22-Sep-2014.)

Theoremdih2dimb 34883 Extend dib2dim 34882 to isomorphism H. (Contributed by NM, 22-Sep-2014.)

Theoremdih2dimbALTN 34884 Extend dia2dim 34716 to isomorphism H. (This version combines dib2dim 34882 and dih2dimb 34883 for shorter overall proof, but may be less easy to understand. TODO: decide which to use.) (Contributed by NM, 22-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdihopelvalcqat 34885* Ordered pair member of the partial isomorphism H for atom argument not under . TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014.)

Theoremdihvalcq2 34886 Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 26-Sep-2014.)

Theoremdihopelvalcpre 34887* Member of value of isomorphism H for a lattice when , given auxiliary atom . TODO: refactor to be shorter and more understandable; add lemmas? (Contributed by NM, 13-Mar-2014.)

Theoremdihopelvalc 34888* Member of value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 13-Mar-2014.)

Theoremdihlss 34889 The value of isomorphism H is a subspace. (Contributed by NM, 6-Mar-2014.)

Theoremdihss 34890 The value of isomorphism H is a set of vectors. (Contributed by NM, 14-Mar-2014.)

Theoremdihssxp 34891 An isomorphism H value is included in the vector space (expressed as ). (Contributed by NM, 26-Sep-2014.)

Theoremdihopcl 34892 Closure of an ordered pair (vector) member of a value of isomorphism H. (Contributed by NM, 26-Sep-2014.)

TheoremxihopellsmN 34893* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.)

Theoremdihopellsm 34894* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.)

Theoremdihord6apre 34895* Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord3 34896 The isomorphism H for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 6-Mar-2014.)

Theoremdihord4 34897 The isomorphism H for a lattice is order-preserving in the region not under co-atom . TODO: reformat q e. A /\ -. q .<_ W to eliminate adant*. (Contributed by NM, 6-Mar-2014.)