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Theorem List for Metamath Proof Explorer - 22701-22800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcvnref 22701 The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremcvntr 22702 The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremspansncv2 22703 Hilbert space has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremmdbr 22704* Binary relation expressing is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Theoremmdi 22705 Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremmdbr2 22706* Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 22704. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremmdbr3 22707* Binary relation expressing the modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)

Theoremmdbr4 22708* Binary relation expressing the modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)

Theoremdmdbr 22709* Binary relation expressing the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremdmdmd 22710 The dual modular pair property expressed in terms of the modular pair property, that hold in Hilbert lattices. Remark 29.6 of [MaedaMaeda] p. 130. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremmddmd 22711 The modular pair property expressed in terms of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremdmdi 22712 Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremdmdbr2 22713* Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 22709. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)

Theoremdmdi2 22714 Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Theoremdmdbr3 22715* Binary relation expressing the dual modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)

Theoremdmdbr4 22716* Binary relation expressing the dual modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)

Theoremdmdi4 22717 Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Theoremdmdbr5 22718* Binary relation expressing the dual modular pair property. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)

Theoremmddmd2 22719* Relationship between modular pairs and dual-modular pairs. Lemma 1.2 of [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremmdsl0 22720 A sublattice condition that transfers the modular pair property. Exercise 12 of [Kalmbach] p. 103. Also Lemma 1.5.3 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremssmd1 22721 Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremssmd2 22722 Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremssdmd1 22723 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremssdmd2 22724 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremdmdsl3 22725 Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)

Theoremmdsl3 22726 Sublattice mapping for a modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)

Theoremmdslle1i 22727 Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremmdslle2i 22728 Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremmdslj1i 22729 Join preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremmdslj2i 22730 Meet preservation of the reverse mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremmdsl1i 22731* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremmdsl2i 22732* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 28-Apr-2006.) (New usage is discouraged.)

Theoremmdsl2bi 22733* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)

Theoremcvmdi 22734 The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)

Theoremmdslmd1lem1 22735 Lemma for mdslmd1i 22739. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdslmd1lem2 22736 Lemma for mdslmd1i 22739. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdslmd1lem3 22737* Lemma for mdslmd1i 22739. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdslmd1lem4 22738* Lemma for mdslmd1i 22739. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdslmd1i 22739 Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (meet version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremmdslmd2i 22740 Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (join version). (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdsldmd1i 22741 Preservation of the dual modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdslmd3i 22742 Modular pair conditions that imply the modular pair property in a sublattice. Lemma 1.5.1 of [MaedaMaeda] p. 2. (Contributed by NM, 23-Dec-2006.) (New usage is discouraged.)

Theoremmdslmd4i 22743 Modular pair condition that implies the modular pair property in a sublattice. Lemma 1.5.2 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)

Theoremcsmdsymi 22744* Cross-symmetry implies M-symmetry. Theorem 1.9.1 of [MaedaMaeda] p. 3. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)

Theoremmdexchi 22745 An exchange lemma for modular pairs. Lemma 1.6 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremcvmd 22746 The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremcvdmd 22747 The covering property implies the dual modular pair property. Lemma 7.5.2 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

15.9.51  Atoms

Definitiondf-at 22748 Define the set of atoms in a Hilbert lattice. An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is a smallest nonzero element of the lattice. Definition of atom in [Kalmbach] p. 15. See ela 22749 and elat2 22750 for membership relations. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
HAtoms

Theoremela 22749 Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremelat2 22750* Expanded membership relation for the set of atoms, i.e. the predicate "is an atom (of the Hilbert lattice)." An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is a smallest nonzero element of the lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremelatcv0 22751 A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
HAtoms

Theorematcv0 22752 An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
HAtoms

Theorematssch 22753 Atoms are a subset of the Hilbert lattice. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
HAtoms

Theorematelch 22754 An atom is a Hilbert lattice element. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
HAtoms

Theorematne0 22755 An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
HAtoms

Theorematss 22756 A lattice element smaller than an atom is either the atom or zero. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
HAtoms

Theorematsseq 22757 Two atoms in a subset relationship are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
HAtoms HAtoms

Theorematcveq0 22758 A Hilbert lattice element covered by an atom must be the zero subspace. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremh1da 22759 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
HAtoms

Theoremspansna 22760 The span of the singleton of a vector is an atom. (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
HAtoms

Theoremsh1dle 22761 A 1-dimensional subspace is less than or equal to any subspace containing its generating vector. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Theoremch1dle 22762 A 1-dimensional subspace is less than or equal to any member of containing its generating vector. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)

Theorematom1d 22763* The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
HAtoms

15.9.52  Superposition principle

Theoremsuperpos 22764* Superposition Principle. If and are distinct atoms, there exists a third atom, distinct from and , that is the superposition of and . Definition 3.4-3(a) in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

15.9.53  Atoms, exchange and covering properties, atomicity

Theoremchcv1 22765 The Hilbert lattice has the covering property. Proposition 1(ii) of [Kalmbach] p. 140 (and its converse). (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremchcv2 22766 The Hilbert lattice has the covering property. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremchjatom 22767 The join of a closed subspace and an atom equals their subspace sum. Special case of remark in [Kalmbach] p. 65, stating that if or is finite dimensional, then this equality holds. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremshatomici 22768* The lattice of Hilbert subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
HAtoms

Theoremhatomici 22769* The Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
HAtoms

Theoremhatomic 22770* A Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. Also Definition 3.4-2 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremshatomistici 22771* The lattice of Hilbert subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
HAtoms

Theoremhatomistici 22772* is atomistic, i.e. any element is the supremum of its atoms. Remark in [Kalmbach] p. 140. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
HAtoms

Theoremchpssati 22773* Two Hilbert lattice elements in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremchrelati 22774* The Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremchrelat2i 22775* A consequence of relative atomicity. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremcvati 22776* If a Hilbert lattice element covers another, it equals the other joined with some atom. This is a consequence of the relative atomicity of Hilbert space. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
HAtoms

Theoremcvbr4i 22777* An alternate way to express the covering property. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
HAtoms

Theoremcvexchlem 22778 Lemma for cvexchi 22779. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Theoremcvexchi 22779 The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchrelat2 22780* A consequence of relative atomicity. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
HAtoms

Theoremchrelat3 22781* A consequence of relative atomicity. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
HAtoms

Theoremchrelat3i 22782* A consequence of the relative atomicity of Hilbert space: the ordering of Hilbert lattice elements is completely determined by the atoms they majorize. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremchrelat4i 22783* A consequence of relative atomicity. Extensionality principle: two lattice elements are equal iff they majorize the same atoms. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
HAtoms

Theoremcvexch 22784 The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremcvp 22785 The Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
HAtoms

Theorematnssm0 22786 The meet of a Hilbert lattice element and an incomparable atom is the zero subspace. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
HAtoms

Theorematnemeq0 22787 The meet of distinct atoms is the zero subspace. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
HAtoms HAtoms

Theorematssma 22788 The meet with an atom's superset is the atom. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
HAtoms HAtoms

Theorematcv0eq 22789 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
HAtoms HAtoms

Theorematcv1 22790 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
HAtoms HAtoms

Theorematexch 22791 The Hilbert lattice satisfies the atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem related to vector analysis was originally proved by Hermann Grassmann in 1862. Also Definition 3.4-3(b) in [MegPav2000] p. 2345 (PDF p. 8) (use atnemeq0 22787 to obtain atom inequality). (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
HAtoms HAtoms

Theorematomli 22792 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition 3.2.17 of [PtakPulmannova] p. 66. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
HAtoms HAtoms

Theorematoml2i 22793 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition P8(ii) of [BeltramettiCassinelli1] p. 400. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
HAtoms HAtoms

Theorematordi 22794 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
HAtoms

Theorematcvatlem 22795 Lemma for atcvati 22796. (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theorematcvati 22796 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (Contributed by NM, 28-Jun-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theorematcvat2i 22797 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theorematord 22798 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
HAtoms

Theorematcvat2 22799 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 29-Jun-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

15.9.54  Irreducibility

Theoremchirredlem1 22800* Lemma for chirredi 22804. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
HAtoms HAtoms

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