mmtheorems102 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 10101-10200 - Page 102 of 107

Type	Label	Description
Statement
Theorem	pj2cocl 10101	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 10102	Lemma for projection triplet theorem.
Theorem	pj3s 10103	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 10104	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 10105	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 10106	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 10107	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 10108	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 10109	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 10110	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 10111	Real closure of the value of a state.
|- (S e. States -> (A e. CH -> (S` A) e. RR))
Theorem	hstclt 10112	Closure of the value of a Hilbert-space-valued state.
|- ((S e. CHStates /\ A e. CH) -> (S` A) e. H~)
Theorem	hst1t 10113	Unit value of a Hilbert-space-valued state.
|- (S e. CHStates -> (normh` (S` H~)) = 1)
Theorem	hstel2t 10114	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 10115	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 10116	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)))
Theorem	hstoct 10117	Sum of a Hilbert-space-valued state of a lattice element and its orthocomplement.
|- ((S e. CHStates /\ A e. CH) -> ((S` A) +h (S` (_|_` A))) = (S` H~))
Theorem	hstnmoct 10118	Sum of norms of a Hilbert-space-valued state.
|- ((S e. CHStates /\ A e. CH) -> (((normh` (S` A))^2) + ((normh` (S` (_|_` A)))^2)) = 1)
Theorem	stge0t 10119	The value of a state is nonnegative.
|- (S e. States -> (A e. CH -> 0 <_ (S` A)))
Theorem	stle1t 10120	The value of a state is less than or equal to one.
|- (S e. States -> (A e. CH -> (S` A) <_ 1))
Theorem	hstle1t 10121	The norm of the value of a Hilbert-space-valued state is less than or equal to one.
|- ((S e. CHStates /\ A e. CH) -> (normh` (S` A)) <_ 1)
Theorem	hst1ht 10122	The norm of a Hilbert-space-valued state equals one iff the state value equals the state value of the lattice unit.
|- ((S e. CHStates /\ A e. CH) -> ((normh` (S` A)) = 1 <-> (S` A) = (S` H~)))
Theorem	hst0ht 10123	The norm of a Hilbert-space-valued state equals zero iff the state value equals zero.
|- ((S e. CHStates /\ A e. CH) -> ((normh` (S` A)) = 0 <-> (S` A) = 0h))
Theorem	hstpytht 10124	Pythagorean property of a Hilbert-space-valued state for orthogonal vectors A and B.
|- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A (_ (_|_` B))) -> ((normh` (S` (A vH B)))^2) = (((normh` (S` A))^2) + ((normh` (S` B))^2)))
Theorem	hstlet 10125	Ordering property of a Hilbert-space-valued state.
|- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A (_ B)) -> (normh` (S` A)) <_ (normh` (S` B)))
Theorem	hstlest 10126	Ordering property of a Hilbert-space-valued state.
|- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A (_ B)) -> ((normh` (S` A)) = 1 -> (normh` (S` B)) = 1))
Theorem	hstoht 10127	A Hilbert-space-valued state orthogonal to the state of the lattice unit is zero.
|- ((S e. CHStates /\ A e. CH /\ ((S` A) .ih (S` H~)) = 0) -> (S` A) = 0h)
Theorem	hst0t 10128	A Hilbert-space-valued state is zero at the zero subspace.
|- (S e. CHStates -> (S` 0H) = 0h)
Theorem	sthil 10129	The value of a state at the full Hilbert space.
|- (S e. States -> (S` H~) = 1)
Theorem	stjt 10130	The value of a state on a join.
|- (S e. States -> (((A e. CH /\ B e. CH) /\ A (_ (_|_` B)) -> (S` (A vH B)) = ((S` A) + (S` B))))
Theorem	sto1 10131	The state of a subspace plus the state of its orthocomplement.
|- A e. CH   =>   |- (S e. States -> ((S` A) + (S` (_|_` A))) = 1)
Theorem	sto2 10132	The state of the orthocomplement.
|- A e. CH   =>   |- (S e. States -> (S` (_|_` A)) = (1 - (S` A)))
Theorem	stge1 10133	If a state is greater than or equal to 1, it is 1.
|- A e. CH   =>   |- (S e. States -> (1 <_ (S` A) <-> (S` A) = 1))
Theorem	stle0 10134	If a state is less than or equal to 0, it is 0.
|- A e. CH   =>   |- (S e. States -> ((S` A) <_ 0 <-> (S` A) = 0))
Theorem	stle 10135	Ordering law for states.
|- A e. CH   &   |- B e. CH   =>   |- (S e. States -> (A (_ B -> (S` A) <_ (S` B)))
Theorem	stles 10136	Ordering law for states.
|- A e. CH   &   |- B e. CH   =>   |- (S e. States ->