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Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempjclem4a 23701 Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempjclem4 23702 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjci 23703 Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjcmul1i 23704 A necessary and sufficient condition for the product of two projectors to be a projector is that the projectors commute. Part 1 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjcmul2i 23705 The projection subspace of the difference between two projectors. Part 2 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjcohocli 23706 Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000.) (New usage is discouraged.)

Theorempjadj2coi 23707 Adjoint of double composition of projections. Generalization of special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)

Theorempj2cocli 23708 Closure of double composition of projections. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3lem1 23709 Lemma for projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3si 23710 Stronger projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3i 23711 Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3cor1i 23712 Projection triplet corollary. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempjs14i 23713 Theorem S-14 of Watanabe, p. 486. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

18.7  States on a Hilbert lattice and Godowski's equation

18.7.1  States on a Hilbert lattice

Definitiondf-st 23714* Define the set of states on a Hilbert lattice. Definition of [Kalmbach] p. 266. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Definitiondf-hst 23715* Define the set of complex Hilbert-space-valued states on a Hilbert lattice. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremisst 23716* Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremishst 23717* Property of a complex Hilbert-space-valued state. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremsticl 23718 closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremstcl 23719 Real closure of the value of a state. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhstcl 23720 Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhst1a 23721 Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstel2 23722 Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstorth 23723 Orthogonality property of a Hilbert-space-valued state. This is a key feature distinguishing it from a real-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstosum 23724 Orthogonal sum property of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstoc 23725 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 23726 Sum of norms of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremstge0 23727 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 23728 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 23729 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 23730 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 23731 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 23732 Pythagorean property of a Hilbert-space-valued state for orthogonal vectors and . (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)

Theoremhstle 23733 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)

Theoremhstles 23734 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstoh 23735 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 23736 A Hilbert-space-valued state is zero at the zero subspace. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremsthil 23737 The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremstj 23738 The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremsto1i 23739 The state of a subspace plus the state of its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremsto2i 23740 The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstge1i 23741 If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstle0i 23742 If a state is less than or equal to 0, it is 0. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstlei 23743 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstlesi 23744 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstji1i 23745 Join of components of Sasaki arrow ->1. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstm1i 23746 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstm1ri 23747 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstm1addi 23748 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstaddi 23749 If the sum of 2 states is 2, then each state is 1. (Contributed by NM, 12-Nov-1999.) (New usage is discouraged.)

Theoremstm1add3i 23750 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstadd3i 23751 If the sum of 3 states is 3, then each state is 1. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)

Theoremst0 23752 The state of the zero subspace. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstrlem1 23753* 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 23754* Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)

Theoremstrlem3a 23755* 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 23756* 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 23757* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremstrlem5 23758* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremstrlem6 23759* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremstri 23760* 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 23761* Strong state theorem (bidirectional version). (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

Theoremhstrlem2 23762* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem3a 23763* 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 23764* 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 23765* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem5 23766* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem6 23767* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstri 23768* 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 23769* 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 23770* 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 23771 Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)

Theoremjplem2 23772* Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)

Theoremjpi 23773* 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 23755 for the proof that is a state.) (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)

18.7.2  Godowski's equation

Theoremgolem1 23774 Lemma for Godowski's equation. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)

Theoremgolem2 23775 Lemma for Godowski's equation. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)

Theoremgoeqi 23776 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 23777* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltr2i 23778* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltrlem1 23779* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltrlem2 23780* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltrthi 23781* 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.)

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

18.8.1  Covers relation; modular pairs

Definitiondf-cv 23782* 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 23785 and cvbr2 23786 for membership relations. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)

Definitiondf-md 23783* 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 23797 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Definitiondf-dmd 23784* 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 23802 for membership relation. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremcvbr 23785* 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 23786* Binary relation expressing covers . Definition of covers in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremcvcon3 23787 Contraposition law for the covers relation. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremcvpss 23788 The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Theoremcvnbtwn 23789 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremcvnbtwn2 23790 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremcvnbtwn3 23791 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremcvnbtwn4 23792 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 23793 The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremcvnref 23794 The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremcvntr 23795 The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremspansncv2 23796 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 23797* 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 23798 Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremmdbr2 23799* Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 23797. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremmdbr3 23800* 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.)

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