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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dibclN 34801 | Closure of partial isomorphism B for a lattice . (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.) |
Theorem | dibvalrel 34802 | The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.) |
Theorem | dib0 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.) |
Theorem | dib1dim 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.) |
Theorem | dibglbN 34805* | Partial isomorphism B of a lattice glb. (Contributed by NM, 9-Mar-2014.) (New usage is discouraged.) |
Theorem | dibintclN 34806 | The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.) |
Theorem | dib1dim2 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.) |
Theorem | dibss 34808 | The partial isomorphism B maps to a set of vectors in full vector space H. (Contributed by NM, 1-Jan-2014.) |
Theorem | diblss 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.) |
Theorem | diblsmopel 34810* | Membership in subspace sum for partial isomorphism B. (Contributed by NM, 21-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.) |
Syntax | cdic 34811 | Extend class notation with isomorphism C. |
Definition | df-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.) |
Theorem | dicffval 34813* | The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.) |
Theorem | dicfval 34814* | The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.) |
Theorem | dicval 34815* | The partial isomorphism C for a lattice . (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.) |
Theorem | dicopelval 34816* | Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 15-Feb-2014.) |
Theorem | dicelvalN 34817* | Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.) |
Theorem | dicval2 34818* | The partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.) |
Theorem | dicelval3 34819* | Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014.) |
Theorem | dicopelval2 34820* | Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 20-Feb-2014.) |
Theorem | dicelval2N 34821* | Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.) |
Theorem | dicfnN 34822* | Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.) |
Theorem | dicdmN 34823* | Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.) |
Theorem | dicvalrelN 34824 | The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.) |
Theorem | dicssdvh 34825 | The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014.) |
Theorem | dicelval1sta 34826* | Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.) |
Theorem | dicelval1stN 34827 | Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.) (New usage is discouraged.) |
Theorem | dicelval2nd 34828 | Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.) |
Theorem | dicvaddcl 34829 | Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.) |
Theorem | dicvscacl 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.) |
Theorem | dicn0 34831 | The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.) |
Theorem | diclss 34832 | The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.) |
Theorem | diclspsn 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.) |
Theorem | cdlemn2 34834* | Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.) |
Theorem | cdlemn2a 34835* | Part of proof of Lemma N of [Crawley] p. 121. (Contributed by NM, 24-Feb-2014.) |
Theorem | cdlemn3 34836* | Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.) |
Theorem | cdlemn4 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.) |
Theorem | cdlemn4a 34838* | Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.) |
Theorem | cdlemn5pre 34839* | Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.) |
Theorem | cdlemn5 34840 | Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.) |
Theorem | cdlemn6 34841* | Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.) |
Theorem | cdlemn7 34842* | Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.) |
Theorem | cdlemn8 34843* | Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.) |
Theorem | cdlemn9 34844* | Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.) |
Theorem | cdlemn10 34845 | Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.) |
Theorem | cdlemn11a 34846* | Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.) |
Theorem | cdlemn11b 34847* | Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.) |
Theorem | cdlemn11c 34848* | Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.) |
Theorem | cdlemn11pre 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.) |
Theorem | cdlemn11 34850 | Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.) |
Theorem | cdlemn 34851 | Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.) |
Theorem | dihordlem6 34852* | Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.) |
Theorem | dihordlem7 34853* | Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.) |
Theorem | dihordlem7b 34854* | Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.) |
Theorem | dihjustlem 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.) |
Theorem | dihjust 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.) |
Theorem | dihord1 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.) |
Theorem | dihord2a 34858 | Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.) |
Theorem | dihord2b 34859 | Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.) |
Theorem | dihord2cN 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.) |
Theorem | dihord11b 34861* | Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.) |
Theorem | dihord10 34862* | Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.) |
Theorem | dihord11c 34863* | Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.) |
Theorem | dihord2pre 34864* | Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.) |
Theorem | dihord2pre2 34865* | Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.) |
Theorem | dihord2 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.) |
Syntax | cdih 34867 | Extend class notation with isomorphism H. |
Definition | df-dih 34868* | Define isomorphism H. (Contributed by NM, 28-Jan-2014.) |
Theorem | dihffval 34869* | The isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.) |
Theorem | dihfval 34870* | Isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.) |
Theorem | dihval 34871* | Value of isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-2014.) |
Theorem | dihvalc 34872* | Value of isomorphism H for a lattice when . (Contributed by NM, 4-Mar-2014.) |
Theorem | dihlsscpre 34873 | Closure of isomorphism H for a lattice when . (Contributed by NM, 6-Mar-2014.) |
Theorem | dihvalcqpre 34874 | Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 6-Mar-2014.) |
Theorem | dihvalcq 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.) |
Theorem | dihvalb 34876 | Value of isomorphism H for a lattice when . (Contributed by NM, 4-Mar-2014.) |
Theorem | dihopelvalbN 34877* | Ordered pair member of the partial isomorphism H for argument under . (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.) |
Theorem | dihvalcqat 34878 | Value of isomorphism H for a lattice at an atom not under . (Contributed by NM, 27-Mar-2014.) |
Theorem | dih1dimb 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.) |
Theorem | dih1dimb2 34880* | Isomorphism H at an atom under . (Contributed by NM, 27-Apr-2014.) |
Theorem | dih1dimc 34881* | Isomorphism H at an atom not under . (Contributed by NM, 27-Apr-2014.) |
Theorem | dib2dim 34882 | Extend dia2dim 34716 to partial isomorphism B. (Contributed by NM, 22-Sep-2014.) |
Theorem | dih2dimb 34883 | Extend dib2dim 34882 to isomorphism H. (Contributed by NM, 22-Sep-2014.) |
Theorem | dih2dimbALTN 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.) |
Theorem | dihopelvalcqat 34885* | Ordered pair member of the partial isomorphism H for atom argument not under . TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014.) |
Theorem | dihvalcq2 34886 | Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 26-Sep-2014.) |
Theorem | dihopelvalcpre 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.) |
Theorem | dihopelvalc 34888* | Member of value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 13-Mar-2014.) |
Theorem | dihlss 34889 | The value of isomorphism H is a subspace. (Contributed by NM, 6-Mar-2014.) |
Theorem | dihss 34890 | The value of isomorphism H is a set of vectors. (Contributed by NM, 14-Mar-2014.) |
Theorem | dihssxp 34891 | An isomorphism H value is included in the vector space (expressed as ). (Contributed by NM, 26-Sep-2014.) |
Theorem | dihopcl 34892 | Closure of an ordered pair (vector) member of a value of isomorphism H. (Contributed by NM, 26-Sep-2014.) |
Theorem | xihopellsmN 34893* | Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.) |
Theorem | dihopellsm 34894* | Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) |
Theorem | dihord6apre 34895* | Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.) |
Theorem | dihord3 34896 | The isomorphism H for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 6-Mar-2014.) |
Theorem | dihord4 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.) |