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Theorem List for Metamath Proof Explorer - 23601-23700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempjoccoi 23601 Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theorempjtoi 23602 Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)

Theorempjoci 23603 Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjidmco 23604 A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremdfpjop 23605 Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 23365. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theorempjhmopidm 23606 Two ways to express the set of all projection operators. (Contributed by NM, 24-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theoremelpjidm 23607 A projection operator is idempotent. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremelpjhmop 23608 A projection operator is Hermitian. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorem0leopj 23609 A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)

Theorempjadj2 23610 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjadj3 23611 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremelpjch 23612 Reconstruction of the subspace of a projection operator. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremelpjrn 23613* Reconstruction of the subspace of a projection operator. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theorempjinvari 23614 A closed subspace with projection is invariant under an operator iff . Theorem 27.1 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjin1i 23615 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)

Theorempjin2i 23616 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)

Theorempjin3i 23617 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)

Theorempjclem1 23618 Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)

Theorempjclem2 23619 Lemma for projection commutation theorem. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)

Theorempjclem3 23620 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjclem4a 23621 Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempjclem4 23622 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjci 23623 Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjcmul1i 23624 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 23625 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 23626 Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000.) (New usage is discouraged.)

Theorempjadj2coi 23627 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 23628 Closure of double composition of projections. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3lem1 23629 Lemma for projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3si 23630 Stronger projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3i 23631 Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3cor1i 23632 Projection triplet corollary. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempjs14i 23633 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 23634* 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 23635* 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 23636* Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremishst 23637* 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 23638 closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremstcl 23639 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 23640 Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhst1a 23641 Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstel2 23642 Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstorth 23643 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 23644 Orthogonal sum property of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstoc 23645 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 23646 Sum of norms of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremstge0 23647 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 23648 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 23649 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 23650 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 23651 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 23652 Pythagorean property of a Hilbert-space-valued state for orthogonal vectors and . (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)

Theoremhstle 23653 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)

Theoremhstles 23654 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstoh 23655 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 23656 A Hilbert-space-valued state is zero at the zero subspace. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremsthil 23657 The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremstj 23658 The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremsto1i 23659 The state of a subspace plus the state of its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremsto2i 23660 The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstge1i 23661 If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstle0i 23662 If a state is less than or equal to 0, it is 0. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstlei 23663 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstlesi 23664 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstji1i 23665 Join of components of Sasaki arrow ->1. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstm1i 23666 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstm1ri 23667 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstm1addi 23668 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstaddi 23669 If the sum of 2 states is 2, then each state is 1. (Contributed by NM, 12-Nov-1999.) (New usage is discouraged.)

Theoremstm1add3i 23670 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstadd3i 23671 If the sum of 3 states is 3, then each state is 1. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)

Theoremst0 23672 The state of the zero subspace. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstrlem1 23673* 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 23674* Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)

Theoremstrlem3a 23675* 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 23676* 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 23677* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremstrlem5 23678* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremstrlem6 23679* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremstri 23680* 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 23681* Strong state theorem (bidirectional version). (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

Theoremhstrlem2 23682* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem3a 23683* 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 23684* 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 23685* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem5 23686* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstrlem6 23687* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstri 23688* 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 23689* 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 23690* 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 23691 Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)

Theoremjplem2 23692* Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)

Theoremjpi 23693* 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 23675 for the proof that is a state.) (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)

18.7.2  Godowski's equation

Theoremgolem1 23694 Lemma for Godowski's equation. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)

Theoremgolem2 23695 Lemma for Godowski's equation. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)

Theoremgoeqi 23696 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 23697* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltr2i 23698* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltrlem1 23699* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstcltrlem2 23700* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

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