P. 291 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 29001-29100 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	opsqrlem3 29001*	Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
$/\$

Theorem	opsqrlem4 29002*	Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)


Theorem	opsqrlem5 29003*	Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)


Theorem	opsqrlem6 29004*	Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)


19.6.16 Projectors as operators

Theorem	pjhmopi 29005	A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)


Theorem	pjlnopi 29006	A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)


Theorem	pjnmopi 29007	The operator norm of a projector on a nonzero closed subspace is one. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)


Theorem	pjbdlni 29008	A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)


Theorem	pjhmop 29009	A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)


Theorem	hmopidmchi 29010	An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)


Theorem	hmopidmpji 29011	An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that is a closed subspace, which is not trivial as hmopidmchi 29010 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)


Theorem	hmopidmch 29012	An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
$/\$

Theorem	hmopidmpj 29013	An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
$/\$

Theorem	pjsdii 29014	Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)


Theorem	pjddii 29015	Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.)


Theorem	pjsdi2i 29016	Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.)


Theorem	pjcoi 29017	Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.)


Theorem	pjcocli 29018	Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.)


Theorem	pjcohcli 29019	Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)


Theorem	pjadjcoi 29020	Adjoint of composition of projections. Special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.)
$/\$

Theorem	pjcofni 29021	Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)


Theorem	pjss1coi 29022	Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)


Theorem	pjss2coi 29023	Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)


Theorem	pjssmi 29024	Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)


Theorem	pjssge0i 29025	Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)


Theorem	pjdifnormi 29026	Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)


Theorem	pjnormssi 29027*	Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)


Theorem	pjorthcoi 29028	Composition of projections of orthogonal subspaces. Part (i)->(iia) of Theorem 27.4 of [Halmos] p. 45. (Contributed by NM, 6-Nov-2000.) (New usage is discouraged.)


Theorem	pjscji 29029	The projection of orthogonal subspaces is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)


Theorem	pjssumi 29030	The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)


Theorem	pjssposi 29031*	Projector ordering can be expressed by the subset relationship between their projection subspaces. (i)<->(iii) of Theorem 29.2 of [Halmos] p. 48. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)


Theorem	pjordi 29032*	The definition of projector ordering in [Halmos] p. 42 is equivalent to the definition of projector ordering in [Beran] p. 110. (We will usually express projector ordering with the even simpler equivalent ; see pjssposi 29031). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)


Theorem	pjssdif2i 29033	The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 29031). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)


Theorem	pjssdif1i 29034	A necessary and sufficient condition for the difference between two projectors to be a projector. Part 1 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 29031). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)


Theorem	pjimai 29035	The image of a projection. Lemma 5 in Daniel Lehmann, "A presentation of Quantum Logic based on an and then connective" http://www.arxiv.org/pdf/quant-ph/0701113 p. 20. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)


Theorem	pjidmcoi 29036	A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)


Theorem	pjoccoi 29037	Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)


Theorem	pjtoi 29038	Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)


Theorem	pjoci 29039	Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)


Theorem	pjidmco 29040	A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)


Theorem	dfpjop 29041	Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 28801. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
$/\$

Theorem	pjhmopidm 29042	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.)


Theorem	elpjidm 29043	A projection operator is idempotent. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)


Theorem	elpjhmop 29044	A projection operator is Hermitian. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)


Theorem	0leopj 29045	A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)


Theorem	pjadj2 29046	A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)


Theorem	pjadj3 29047	A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)


Theorem	elpjch 29048	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.)
$/\$

Theorem	elpjrn 29049*	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.)


Theorem	pjinvari 29050	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.)


Theorem	pjin1i 29051	Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)


Theorem	pjin2i 29052	Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
$/\$

Theorem	pjin3i 29053	Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
$/\$

Theorem	pjclem1 29054	Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)


Theorem	pjclem2 29055	Lemma for projection commutation theorem. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)


Theorem	pjclem3 29056	Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)


Theorem	pjclem4a 29057	Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)


Theorem	pjclem4 29058	Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)


Theorem	pjci 29059	Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)


Theorem	pjcmul1i 29060	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.)


Theorem	pjcmul2i 29061	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.)


Theorem	pjcohocli 29062	Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000.) (New usage is discouraged.)


Theorem	pjadj2coi 29063	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.)
$/\$

Theorem	pj2cocli 29064	Closure of double composition of projections. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)


Theorem	pj3lem1 29065	Lemma for projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)


Theorem	pj3si 29066	Stronger projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
$/\$

Theorem	pj3i 29067	Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
$/\$

Theorem	pj3cor1i 29068	Projection triplet corollary. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
$/\$

Theorem	pjs14i 29069	Theorem S-14 of Watanabe, p. 486. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)


19.7 States on a Hilbert lattice and Godowski's equation

19.7.1 States on a Hilbert lattice

Definition	df-st 29070*	Define the set of states on a Hilbert lattice. Definition of [Kalmbach] p. 266. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
$/\$

Definition	df-hst 29071*	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.)
$/\$ $/\$

Theorem	isst 29072*	Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
$/\$ $/\$

Theorem	ishst 29073*	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.)
$/\$ $/\$ $/\$

Theorem	sticl 29074	closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)


Theorem	stcl 29075	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.)


Theorem	hstcl 29076	Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
$/\$

Theorem	hst1a 29077	Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)


Theorem	hstel2 29078	Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	hstorth 29079	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.)
$/\$ $/\$ $/\$

Theorem	hstosum 29080	Orthogonal sum property of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	hstoc 29081	Sum of a Hilbert-space-valued state of a lattice element and its orthocomplement. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
$/\$

Theorem	hstnmoc 29082	Sum of norms of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
$/\$

Theorem	stge0 29083	The value of a state is nonnegative. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)


Theorem	stle1 29084	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.)


Theorem	hstle1 29085	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.)
$/\$

Theorem	hst1h 29086	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.)
$/\$

Theorem	hst0h 29087	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.)
$/\$

Theorem	hstpyth 29088	Pythagorean property of a Hilbert-space-valued state for orthogonal vectors and . (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	hstle 29089	Ordering property of a Hilbert-space-valued state. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	hstles 29090	Ordering property of a Hilbert-space-valued state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	hstoh 29091	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.)
$/\$ $/\$

Theorem	hst0 29092	A Hilbert-space-valued state is zero at the zero subspace. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)


Theorem	sthil 29093	The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)


Theorem	stj 29094	The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
$/\$ $/\$

Theorem	sto1i 29095	The state of a subspace plus the state of its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)


Theorem	sto2i 29096	The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)


Theorem	stge1i 29097	If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)


Theorem	stle0i 29098	If a state is less than or equal to 0, it is 0. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)


Theorem	stlei 29099	Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)


Theorem	stlesi 29100	Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

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