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Theorem List for Metamath Proof Explorer - 22801-22900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhstosum 22801 Orthogonal sum property of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstoc 22802 Sum of a Hilbert-space-valued state of a lattice element and its orthocomplement. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstnmoc 22803 Sum of norms of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremstge0 22804 The value of a state is nonnegative. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremstle1 22805 The value of a state is less than or equal to one. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhstle1 22806 The norm of the value of a Hilbert-space-valued state is less than or equal to one. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhst1h 22807 The norm of a Hilbert-space-valued state equals one iff the state value equals the state value of the lattice unit. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhst0h 22808 The norm of a Hilbert-space-valued state equals zero iff the state value equals zero. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstpyth 22809 Pythagorean property of a Hilbert-space-valued state for orthogonal vectors and . (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)

Theoremhstle 22810 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)

Theoremhstles 22811 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstoh 22812 A Hilbert-space-valued state orthogonal to the state of the lattice unit is zero. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhst0 22813 A Hilbert-space-valued state is zero at the zero subspace. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremsthil 22814 The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremstj 22815 The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremsto1i 22816 The state of a subspace plus the state of its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremsto2i 22817 The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstge1i 22818 If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstle0i 22819 If a state is less than or equal to 0, it is 0. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstlei 22820 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstlesi 22821 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstji1i 22822 Join of components of Sasaki arrow ->1. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstm1i 22823 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstm1ri 22824 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstm1addi 22825 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstaddi 22826 If the sum of 2 states is 2, then each state is 1. (Contributed by NM, 12-Nov-1999.) (New usage is discouraged.)

Theoremstm1add3i 22827 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstadd3i 22828 If the sum of 3 states is 3, then each state is 1. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)

Theoremst0 22829 The state of the zero subspace. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstrlem1 22830* Lemma for strong state theorem: if closed subspace is not contained in , there is a unit vector in their difference. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)

Theoremstrlem2 22831* Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)

Theoremstrlem3a 22832* Lemma for strong state theorem: the function , that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)

Theoremstrlem3 22833* Lemma for strong state theorem: the function , that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. This lemma restates the hypotheses in a more convenient form to work with. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)

Theoremstrlem4 22834* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremstrlem5 22835* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremstrlem6 22836* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremstri 22837* Strong state theorem. The states on a Hilbert lattice define an ordering. Remark in [Mayet] p. 370. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremstrb 22838* Strong state theorem (bidirectional version). (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

Theoremhstrlem2 22839* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem3a 22840* Lemma for strong set of CH states theorem: the function , that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem3 22841* Lemma for strong set of CH states theorem: the function , that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. This lemma restates the hypotheses in a more convenient form to work with. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem4 22842* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem5 22843* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem6 22844* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstri 22845* Hilbert space admits a strong set of Hilbert-space-valued states (CH-states). Theorem in [Mayet3] p. 10. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrbi 22846* Strong CH-state theorem (bidirectional version). Theorem in [Mayet3] p. 10 and its converse. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremlargei 22847* A Hilbert lattice admits a largei set of states. Remark in [Mayet] p. 370. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

Theoremjplem1 22848 Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)

Theoremjplem2 22849* Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)

Theoremjpi 22850* The function , that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a Jauch-Piron state. Remark in [Mayet] p. 370. (See strlem3a 22832 for the proof that is a state.) (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)

17.7.2  Godowski's equation

Theoremgolem1 22851 Lemma for Godowski's equation. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)

Theoremgolem2 22852 Lemma for Godowski's equation. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)

Theoremgoeqi 22853 Godowski's equation, shown here as a variant equivalent to Equation SF of [Godowski] p. 730. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)

Theoremstcltr1i 22854* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltr2i 22855* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltrlem1 22856* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltrlem2 22857* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltrthi 22858* Theorem for classically strong set of states. If there exists a "classically strong set of states" on lattice (or actually any ortholattice, which would have an identical proof), then any two elements of the lattice commute, i.e., the lattice is distributive. (Proof due to Mladen Pavicic.) Theorem 3.3 of [MegPav2000] p. 2344. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

17.8  Cover relation, atoms, exchange axiom, and modular symmetry

17.8.1  Covers relation; modular pairs

Definitiondf-cv 22859* Define the covers relation (on the Hilbert lattice). Definition 3.2.18 of [PtakPulmannova] p. 68, whose notation we use. Ptak/Pulmannova's notation is read " covers " or " is covered by " , and it means that is larger than and there is nothing in between. See cvbr 22862 and cvbr2 22863 for membership relations. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)

Definitiondf-md 22860* Define the modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M for "the ordered pair <x,y> is a modular pair." See mdbr 22874 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Definitiondf-dmd 22861* Define the dual modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M* for "the ordered pair <x,y> is a dual modular pair." See dmdbr 22879 for membership relation. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremcvbr 22862* Binary relation expressing covers , which means that is larger than and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)

Theoremcvbr2 22863* Binary relation expressing covers . Definition of covers in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremcvcon3 22864 Contraposition law for the covers relation. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremcvpss 22865 The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Theoremcvnbtwn 22866 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremcvnbtwn2 22867 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremcvnbtwn3 22868 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremcvnbtwn4 22869 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremcvnsym 22870 The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremcvnref 22871 The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremcvntr 22872 The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremspansncv2 22873 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 22874* 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 22875 Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremmdbr2 22876* Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 22874. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremmdbr3 22877* 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 22878* 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 22879* Binary relation expressing the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremdmdmd 22880 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 22881 The modular pair property expressed in terms of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremdmdi 22882 Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremdmdbr2 22883* Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 22879. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)

Theoremdmdi2 22884 Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Theoremdmdbr3 22885* 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 22886* 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 22887 Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Theoremdmdbr5 22888* Binary relation expressing the dual modular pair property. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)

Theoremmddmd2 22889* 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 22890 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 22891 Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremssmd2 22892 Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremssdmd1 22893 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremssdmd2 22894 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremdmdsl3 22895 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 22896 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 22897 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 22898 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 22899 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 22900 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.)

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