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