mmtheorems102 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 10101-10200 - Page 102 of 108

Type	Label	Description
Statement
Theorem	pjocco 10101	Composition of projections of a subspace and its orthocomplement.
|- H e. CH   =>   |- ((proj` H) o. (proj` (_|_` H))) = 0hop
Theorem	pjtot 10102	Subspace sum of projection and projection of orthocomplement.
|- H e. CH   =>   |- ((proj` H) +op (proj` (_|_` H))) = (proj` H~)
Theorem	pjoc 10103	Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45.
|- H e. CH   =>   |- ((proj` H~) -op (proj` H)) = (proj` (_|_` H))
Theorem	pjidmcot 10104	A projection operator is idempotent. Property (ii) of [Beran] p. 109.
|- (H e. CH -> ((proj` H) o. (proj` H)) = (proj` H))
Theorem	pjhmopidm 10105	Two ways to express the set of all projection operators.
|- ran proj = (HrmOp i^i {t | (t o. t) = t})
Theorem	dfpjopt 10106	Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplint 9861.
|- (T e. ran proj <-> (T e. HrmOp /\ (T o. T) = T))
Theorem	elpjidmt 10107	A projection operator is idempotent. Part of Theorem 26.1 of [Halmos] p. 43.
|- (T e. ran proj -> (T o. T) = T)
Theorem	elpjhmopt 10108	A projection operator is Hermitian. Part of Theorem 26.1 of [Halmos] p. 43.
|- (T e. ran proj -> T e. HrmOp)
Theorem	pjadj2t 10109	A projector is self-adjoint. Property (i) of [Beran] p. 109.
|- (T e. ran proj -> (adjh` T) = T)
Theorem	pjadj3t 10110	A projector is self-adjoint. Property (i) of [Beran] p. 109.
|- (H e. CH -> (adjh` (proj` H)) = (proj` H))
Theorem	elpjcht 10111	Reconstruction of the subspace of a projection operator. Part of Theorem 26.2 of [Halmos] p. 44.
|- (T e. ran proj -> ({x e. H~ | (T` x) = x} e. CH /\ T = (proj` {x e. H~ | (T` x) = x})))
Theorem	elpjrnt 10112	Reconstruction of the subspace of a projection operator.
|- (T e. ran proj -> ran T = {x e. H~ | (T` x) = x})
Theorem	elpjrncht 10113	Reconstruction of the subspace of a projection operator.
|- (T e. ran proj -> (ran T e. CH /\ T = (proj` ran T)))
Theorem	pjinvar 10114	A closed subspace H with projection T is invariant under an operator S iff ST = TST. Theorem 27.1 of [Halmos] p. 45.
|- S:H~-->H~   &   |- H e. CH   &   |- T = (proj` H)   =>   |- ((S o. T):H~-->H <-> (S o. T) = (T o. (S o. T)))
Theorem	pjin1 10115	Lemma for Theorem 1.22 of Mittelstaedt, p. 20.
Theorem	pjin2 10116	Lemma for Theorem 1.22 of Mittelstaedt, p. 20.
Theorem	pjin3 10117	Lemma for Theorem 1.22 of Mittelstaedt, p. 20.
Theorem	pjclem1 10118	Lemma for projection commutation theorem.
Theorem	pjclem2 10119	Lemma for projection commutation theorem.
Theorem	pjclem3 10120	Lemma for projection commutation theorem.
Theorem	pjclem4a 10121	Lemma for projection commutation theorem.
Theorem	pjclem4 10122	Lemma for projection commutation theorem.
Theorem	pjc 10123	Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67.
|- G e. CH   &   |- H e. CH   =>   |- (G C_H H <-> ((proj` G) o. (proj` H)) = ((proj` H) o. (proj` G)))
Theorem	pjcmmul1 10124	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.
|- G e. CH   &   |- H e. CH   =>   |- (((proj` G) o. (proj` H)) = ((proj` H) o. (proj` G)) <-> ((proj` G) o. (proj` H)) e. ran proj)
Theorem	pjcmmul2 10125	The projection subspace of the difference between two projectors. Part 2 of Theorem 1 of [AkhiezerGlazman] p. 65.
|- G e. CH   &   |- H e. CH   =>   |- (((proj` G) o. (proj` H)) = ((proj` H) o. (proj` G)) <-> ((proj` G) o. (proj` H)) = (proj` (G i^i H)))
Theorem	pjcohocl 10126	Closure of composition of projection and Hilbert space operator.
|- H e. CH   &   |- T:H~-->H~   =>   |- (A e. H~ -> (((proj` H) o. T)` A) e. H)
Theorem	pjadj2co 10127	Adjoint of double composition of projections. Generalization of special case of Theorem 3.11(viii) of [Beran] p. 106.
|- F e. CH   &   |- G e. CH   &   |- H e. CH   =>   |- ((A e. H~ /\ B e. H~) -> (((((proj` F) o. (proj` G)) o. (proj` H))` A) .ih B) = (A .ih ((((proj` H) o. (proj` G)) o. (proj` F))` B)))
Theorem	pj2cocl 10128	Closure of double composition of projections.
|- F e. CH   &   |- G e. CH   &   |- H e. CH   =>   |- (A e. H~ -> ((((proj` F) o. (proj` G)) o. (proj` H))` A) e. F)
Theorem	pj3lem1 10129	Lemma for projection triplet theorem.
Theorem	pj3s 10130	Stronger projection triplet theorem.
|- F e. CH   &   |- G e. CH   &   |- H e. CH   =>   |- (((((proj` F) o. (proj` G)) o. (proj` H)) = (((proj` H) o. (proj` G)) o. (proj` F)) /\ ran (((proj` F) o. (proj` G)) o. (proj` H)) (_ G) -> (((proj` F) o. (proj` G)) o. (proj` H)) = (proj` ((F i^i G) i^i H)))
Theorem	pj3 10131	Projection triplet theorem.
|- F e. CH   &   |- G e. CH   &   |- H e. CH   =>   |- (((((proj` F) o. (proj` G)) o. (proj` H)) = (((proj` H) o. (proj` G)) o. (proj` F)) /\ (((proj` F) o. (proj` G)) o. (proj` H)) = (((proj` G) o. (proj` F)) o. (proj` H))) -> (((proj` F) o. (proj` G)) o. (proj` H)) = (proj` ((F i^i G) i^i H)))
Theorem	pj3cor1 10132	Projection triplet corollary.
|- F e. CH   &   |- G e. CH   &   |- H e. CH   =>   |- (((((proj` F) o. (proj` G)) o. (proj` H)) = (((proj` H) o. (proj` G)) o. (proj` F)) /\ (((proj` F) o. (proj` G)) o. (proj` H)) = (((proj` G) o. (proj` F)) o. (proj` H))) -> (((proj` F) o. (proj` G)) o. (proj` H)) = (((proj` H) o. (proj` F)) o. (proj` G)))
Theorem	pjs14 10133	Theorem S-14 of Watanabe, p. 486.
|- G e. CH   &   |- H e. CH   =>   |- (A e. H~ -> (normh` (((proj` H) o. (proj` G))` A)) <_ (normh` ((proj` G)` A)))
States on a Hilbert lattice
Definition	df-st 10134	Define the set of states on a Hilbert lattice. Definition of [Kalmbach] p. 266.
|- States = {f | ((f:CH-->RR /\ A.x e. CH (0 <_ (f` x) /\ (f` x) <_ 1)) /\ ((f` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (f` (x vH y)) = ((f` x) + (f` y)))))}
Definition	df-hst 10135	Define the set of complex Hilbert-space-valued states on a Hilbert lattice. Definition of CH-states in [Mayet3] p. 9.
|- CHStates = {f | (f:CH-->H~ /\ (normh` (f` H~)) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (((f` x) .ih (f` y)) = 0 /\ (f` (x vH y)) = ((f` x) +h (f` y)))))}
Theorem	stelt 10136	Property of a state.
|- (S e. States <-> ((S:CH-->RR /\ A.x e. CH (0 <_ (S` x) /\ (S` x) <_ 1)) /\ ((S` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y))))))
Theorem	hstelt 10137	Property of a complex Hilbert-space-valued state. Definition of CH-states in [Mayet3] p. 9.
|- (S e. CHStates <-> (S:CH-->H~ /\ (normh` (S` H~)) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y))))))
Theorem	stclt 10138	Real closure of the value of a state.
|- (S e. States -> (A e. CH -> (S` A) e. RR))
Theorem	hstclt 10139	Closure of the value of a Hilbert-space-valued state.
|- ((S e. CHStates /\ A e. CH) -> (S` A) e. H~)
Theorem	hst1t 10140	Unit value of a Hilbert-space-valued state.
|- (S e. CHStates -> (normh` (S` H~)) = 1)
Theorem	hstel2t 10141	Properties of a Hilbert-space-valued state.
|- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A (_ (_|_` B))) -> (((S` A) .ih (S` B)) = 0 /\ (S` (A vH B)) = ((S` A) +h (S` B))))
Theorem	hstortht 10142	Orthogonality property of a Hilbert-space-valued state. This is a key feature distinguishing it from a real-valued state.
|- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A (_ (_|_` B))) -> ((S` A) .ih (S` B)) = 0)
Theorem	hstosumt 10143	Orthogonal sum property of a Hilbert-space-valued state.
|- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A (_ (_|_` B))) -> (S` (A vH B)) = ((S` A) +h (S` B)))