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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdih2dimb 30701 Extend dib2dim 30700 to isomorphism H. (Contributed by NM, 22-Sep-2014.)

Theoremdih2dimbALTN 30702 Extend dia2dim 30534 to isomorphism H. (This version combines dib2dim 30700 and dih2dimb 30701 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 30703* Ordered pair member of the partial isomorphism H for atom argument not under . TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014.)

Theoremdihvalcq2 30704 Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 26-Sep-2014.)

Theoremdihopelvalcpre 30705* 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 30706* Member of value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 13-Mar-2014.)

Theoremdihlss 30707 The value of isomorphism H is a subspace. (Contributed by NM, 6-Mar-2014.)

Theoremdihss 30708 The value of isomorphism H is a set of vectors. (Contributed by NM, 14-Mar-2014.)

Theoremdihssxp 30709 An isomorphism H value is included in the vector space (expressed as ). (Contributed by NM, 26-Sep-2014.)

Theoremdihopcl 30710 Closure of an ordered pair (vector) member of a value of isomorphism H. (Contributed by NM, 26-Sep-2014.)

TheoremxihopellsmN 30711* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.)

Theoremdihopellsm 30712* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.)

Theoremdihord6apre 30713* Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord3 30714 The isomorphism H for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 6-Mar-2014.)

Theoremdihord4 30715 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.)

Theoremdihord5b 30716 Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine w/ other way to have one lhpmcvr2 (Contributed by NM, 7-Mar-2014.)

Theoremdihord6b 30717 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord6a 30718 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord5apre 30719 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord5a 30720 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord 30721 The isomorphism H is order-preserving. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdih11 30722 The isomorphism H is one-to-one. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdihf11lem 30723 Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)

Theoremdihf11 30724 The isomorphism H for a lattice is a one-to-one function. . Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdihfn 30725 Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)

Theoremdihdm 30726 Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)

Theoremdihcl 30727 Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)

Theoremdihcnvcl 30728 Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)

Theoremdihcnvid1 30729 The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)

Theoremdihcnvid2 30730 The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)

Theoremdihcnvord 30731 Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)

Theoremdihcnv11 30732 The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)

Theoremdihsslss 30733 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)

Theoremdihrnlss 30734 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)

Theoremdihrnss 30735 The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)

Theoremdihvalrel 30736 The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)

Theoremdih0 30737 The value of isomorphism H at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 9-Mar-2014.)

Theoremdih0bN 30738 A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)

Theoremdih0vbN 30739 A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)

Theoremdih0cnv 30740 The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)

Theoremdih0rn 30741 The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)

Theoremdih0sb 30742 A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)

Theoremdih1 30743 The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)

Theoremdih1rn 30744 The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)

Theoremdih1cnv 30745 The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)

TheoremdihwN 30746* Value of isomorphism H at the fiducial hyperplane . (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)

Theoremdihmeetlem1N 30747* Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem5apreN 30748* A conjunction property of isomorphism H. TODO: reduce antecedent size; general review for shorter proof. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem5aN 30749 A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem2aN 30750* Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem2N 30751* The GLB of a set of lattice elements is the same as that of the set with elements of cut down to be under . (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem3N 30752* Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem3aN 30753* Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihglblem4 30754* Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)

Theoremdihglblem5 30755* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)

Theoremdihmeetlem2N 30756 Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)

TheoremdihglbcpreN 30757* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane . (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)

TheoremdihglbcN 30758* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)

TheoremdihmeetcN 30759 Isomorphism H of a lattice meet when the meet is not under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)

TheoremdihmeetbN 30760 Isomorphism H of a lattice meet when one element is under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)

TheoremdihmeetbclemN 30761 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem3N 30762 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem4preN 30763* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem4N 30764 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem5 30765 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)

Theoremdihmeetlem6 30766 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)

Theoremdihmeetlem7N 30767 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihjatc1 30768 Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change order of here and down? (Contributed by NM, 6-Apr-2014.)

Theoremdihjatc2N 30769 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)

Theoremdihjatc3 30770 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.)

Theoremdihmeetlem8N 30771 Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change order of here and down? (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem9N 30772 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem10N 30773 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem11N 30774 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem12N 30775 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem13N 30776* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem14N 30777 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem15N 30778 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem16N 30779 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem17N 30780 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem18N 30781 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem19N 30782 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem20N 30783 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

TheoremdihmeetALTN 30784 Isomorphism H of a lattice meet. This version does not depend on the atomisticity of the constructed vector space. TODO: Delete? (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdih1dimatlem0 30785* Lemma for dih1dimat 30787. (Contributed by NM, 11-Apr-2014.)
LSAtoms                                                               Scalar

Theoremdih1dimatlem 30786* Lemma for dih1dimat 30787. (Contributed by NM, 10-Apr-2014.)
LSAtoms                                                               Scalar

Theoremdih1dimat 30787 Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
LSAtoms

Theoremdihlsprn 30788 The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)

TheoremdihlspsnssN 30789 A subspace included in a 1-dim subspace belongs to the range of isomorphism H. (Contributed by NM, 26-Apr-2014.) (New usage is discouraged.)

Theoremdihlspsnat 30790 The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)

Theoremdihatlat 30791 The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
LSAtoms

Theoremdihat 30792 There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
LSAtoms

TheoremdihpN 30793* The value of isomorphism H at the fiducial atom is determined by the vector (the zero translation ltrnid 29591 and a nonzero member of the endomorphism ring). In particular, can be replaced with the ring unit . (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)

Theoremdihlatat 30794 The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
LSAtoms

Theoremdihatexv 30795* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)

Theoremdihatexv2 30796* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)

Theoremdihglblem6 30797* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremdihglb 30798* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)

Theoremdihglb2 30799* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)

Theoremdihmeet 30800 Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)

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