mmtheorems103 - Metamath Proof Explorer

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**Statement List for Metamath Proof Explorer -** 10201-10300 - Page 103 of 107
Type	Label	Description
Statement

Theorem	mdslj1 10201	Join preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2.
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Theorem	mdslj2 10202	Meet preservation of the reverse mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2.
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Theorem	mdsl1 10203	If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2.
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Theorem	mdsl2 10204	If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2.
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Theorem	mdsl2b 10205	If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2.
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Theorem	cvmd 10206	The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31.


Theorem	mdslmd1lem1 10207	Lemma for mdslmd1 10211.

Theorem	mdslmd1lem2 10208	Lemma for mdslmd1 10211.

Theorem	mdslmd1lem3 10209	Lemma for mdslmd1 10211.

Theorem	mdslmd1lem4 10210	Lemma for mdslmd1 10211.

Theorem	mdslmd1 10211	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).
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Theorem	mdslmd2 10212	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).
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Theorem	mdsldmd1 10213	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.
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Theorem	mdslmd3 10214	Modular pair conditions that imply the modular pair property in a sublattice. Lemma 1.5.1 of [MaedaMaeda] p. 2.
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Theorem	mdslmd4 10215	Modular pair condition that implies the modular pair property in a sublattice. Lemma 1.5.2 of [MaedaMaeda] p. 2.
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Theorem	csmdsym 10216	Cross-symmetry implies M-symmetry. Theorem 1.9.1 of [MaedaMaeda] p. 3.
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Theorem	mdexch 10217	An exchange lemma for modular pairs. Lemma 1.6 of [MaedaMaeda] p. 2.
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Theorem	cvmdt 10218	The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31.
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Theorem	cvdmdt 10219	The covering property implies the dual modular pair property. Lemma 7.5.2 of [MaedaMaeda] p. 31.
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Atoms

Definition	df-at 10220	Define the set of atoms in a Hilbert lattice. An atom is a non-zero element of a lattice such that anything less than it is zero, i.e. it is a smallest non-zero element of the lattice. Definition of atom in [Kalmbach] p. 15. See elat 10221 and elat2 10222 for membership relations.


Theorem	elat 10221	Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15.
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Theorem	elat2 10222	Expanded membership relation for the set of atoms, i.e. the predicate "is an atom (of the Hilbert lattice)." An atom is a non-zero element of a lattice such that anything less than it is zero, i.e. it is a smallest non-zero element of the lattice.
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Theorem	elatcv0 10223	A Hilbert lattice element is an atom iff it covers the zero subspace.


Theorem	atcv0 10224	An atom covers the zero subspace.


Theorem	atssch 10225	Atoms are a subset of the Hilbert lattice.


Theorem	atelch 10226	An atom is a Hilbert lattice element.


Theorem	atn0 10227	An atom is not the Hilbert lattice zero.


Theorem	atss 10228	A lattice element smaller than an atom is either the atom or zero.
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Theorem	atsseq 10229	Two atoms in a subset relationship are equal.
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Theorem	atcveq0 10230	A Hilbert lattice element covered by an atom must be the zero subspace.
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Theorem	h1dat 10231	A 1-dimensional subspace is an atom.
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Theorem	spansnat 10232	The span of the singleton of a vector is an atom.
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Theorem	sh1dle 10233	A 1-dimensional subspace is less than or equal to any subspace containing its generating vector.
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Theorem	ch1dle 10234	A 1-dimensional subspace is less than or equal to any member of containing its generating vector.
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Theorem	atom1d 10235	The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107.
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Superposition principle

Theorem	superpos 10236	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 [MegillPavicic] p. 2345 (PDF p. 8).
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Atoms, exchange and covering properties, atomicity

Theorem	chcv1t 10237	The Hilbert lattice has the covering property. Proposition 1(ii) of [Kalmbach] p. 140 (and its converse).
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Theorem	chcv2t 10238	The Hilbert lattice has the covering property.
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Theorem	chjatom 10239	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.
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Theorem	shatomic 10240	The lattice of Hilbert subspaces is atomic, i.e. any non-zero element is greater than or equal to some atom. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70.


Theorem	hatomic 10241	The Hilbert lattice is atomic, i.e. any non-zero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140.


Theorem	hatomict 10242	A Hilbert lattice is atomic, i.e. any non-zero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. Also Definition 3.4-2 in [MegillPavicic] p. 2345 (PDF p. 8).
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Theorem	shatomistic 10243	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.


Theorem	hatomistic 10244	is atomistic, i.e. any element is the supremum of its atoms. Remark in [Kalmbach] p. 140.
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Theorem	chpssat 10245	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.
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Theorem	chrelat 10246	The Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149.
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Theorem	chrelat2 10247	A consequence of relative atomicity.
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Theorem	cvat 10248	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.