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Theorem List for Metamath Proof Explorer - 31401-31500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdih1 31401 The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)

Theoremdih1rn 31402 The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)

Theoremdih1cnv 31403 The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)

TheoremdihwN 31404* Value of isomorphism H at the fiducial hyperplane . (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)

Theoremdihmeetlem1N 31405* Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem5apreN 31406* 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 31407 A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem2aN 31408* Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem2N 31409* 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 31410* Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem3aN 31411* Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihglblem4 31412* Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)

Theoremdihglblem5 31413* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)

Theoremdihmeetlem2N 31414 Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)

TheoremdihglbcpreN 31415* 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 31416* 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 31417 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 31418 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 31419 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem3N 31420 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem4preN 31421* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem4N 31422 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem5 31423 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)

Theoremdihmeetlem6 31424 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)

Theoremdihmeetlem7N 31425 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihjatc1 31426 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 31427 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)

Theoremdihjatc3 31428 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.)

Theoremdihmeetlem8N 31429 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 31430 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem10N 31431 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem11N 31432 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem12N 31433 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem13N 31434* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem14N 31435 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem15N 31436 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem16N 31437 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem17N 31438 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem18N 31439 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem19N 31440 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem20N 31441 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

TheoremdihmeetALTN 31442 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 31443* Lemma for dih1dimat 31445. (Contributed by NM, 11-Apr-2014.)
LSAtoms                                                               Scalar

Theoremdih1dimatlem 31444* Lemma for dih1dimat 31445. (Contributed by NM, 10-Apr-2014.)
LSAtoms                                                               Scalar

Theoremdih1dimat 31445 Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
LSAtoms

Theoremdihlsprn 31446 The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)

TheoremdihlspsnssN 31447 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 31448 The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)

Theoremdihatlat 31449 The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
LSAtoms

Theoremdihat 31450 There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
LSAtoms

TheoremdihpN 31451* The value of isomorphism H at the fiducial atom is determined by the vector (the zero translation ltrnid 30249 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 31452 The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
LSAtoms

Theoremdihatexv 31453* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)

Theoremdihatexv2 31454* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)

Theoremdihglblem6 31455* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremdihglb 31456* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)

Theoremdihglb2 31457* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)

Theoremdihmeet 31458 Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)

Theoremdihintcl 31459 The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)

Theoremdihmeetcl 31460 Closure of closed subspace meet for vector space. (Contributed by NM, 5-Aug-2014.)

Theoremdihmeet2 31461 Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)

Syntaxcoch 31462 Extend class notation with subspace orthocomplement for vector space.

Definitiondf-doch 31463* Define subspace orthocomplement for vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 14-Mar-2014.)

Theoremdochffval 31464* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochfval 31465* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochval 31466* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochval2 31467* Subspace orthocomplement for vector space. (Contributed by NM, 14-Apr-2014.)

Theoremdochcl 31468 Closure of subspace orthocomplement for vector space. (Contributed by NM, 9-Mar-2014.)

Theoremdochlss 31469 A subspace orthocomplement is a subspace of the vector space. (Contributed by NM, 22-Jul-2014.)

Theoremdochssv 31470 A subspace orthocomplement belongs to the vector space. (Contributed by NM, 22-Jul-2014.)

TheoremdochfN 31471 Domain and codomain of the subspace orthocomplement for the vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)

Theoremdochvalr 31472 Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)

Theoremdoch0 31473 Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)

Theoremdoch1 31474 Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)

Theoremdochoc0 31475 The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)

Theoremdochoc1 31476 The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)

Theoremdochvalr2 31477 Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)

Theoremdochvalr3 31478 Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)

Theoremdoch2val2 31479* Double orthocomplement for vector space. (Contributed by NM, 26-Jul-2014.)

Theoremdochss 31480 Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)

Theoremdochocss 31481 Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)

Theoremdochoc 31482 Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)

Theoremdochsscl 31483 If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.)

Theoremdochoccl 31484 A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.)

Theoremdochord 31485 Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)

Theoremdochord2N 31486 Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)

Theoremdochord3 31487 Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)

Theoremdoch11 31488 Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)

TheoremdochsordN 31489 Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)

Theoremdochn0nv 31490 An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.)

Theoremdihoml4c 31491 Version of dihoml4 31492 with closed subspaces. (Contributed by NM, 15-Jan-2015.)

Theoremdihoml4 31492 Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 30067 analog.) (Contributed by NM, 15-Jan-2015.)

Theoremdochspss 31493 The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)

Theoremdochocsp 31494 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)

TheoremdochspocN 31495 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)

Theoremdochocsn 31496 The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)

Theoremdochsncom 31497 Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)

Theoremdochsat 31498 The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremdochshpncl 31499 If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
LSHyp

Theoremdochlkr 31500 Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
LFnl       LSHyp       LKer

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