mmtheorems119 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 11801-11900 - Page 119 of 175

Type	Label	Description
Statement
Theorem	hstpyth 11801	Pythagorean property of a Hilbert-space-valued state for orthogonal vectors A and B.
|- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A C_ (_|_` B))) -> ((normh` (S` (A vH B)))^2) = (((normh` (S` A))^2) + ((normh` (S` B))^2)))
Theorem	hstle 11802	Ordering property of a Hilbert-space-valued state.
|- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A C_ B)) -> (normh` (S` A)) <_ (normh` (S` B)))
Theorem	hstles 11803	Ordering property of a Hilbert-space-valued state.
|- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A C_ B)) -> ((normh` (S` A)) = 1 -> (normh` (S` B)) = 1))
Theorem	hstoh 11804	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	hst0 11805	A Hilbert-space-valued state is zero at the zero subspace.
|- (S e. CHStates -> (S` 0H) = 0h)
Theorem	sthil 11806	The value of a state at the full Hilbert space.
|- (S e. States -> (S` ~H) = 1)
Theorem	stj 11807	The value of a state on a join.
|- (S e. States -> (((A e. CH /\ B e. CH) /\ A C_ (_|_` B)) -> (S` (A vH B)) = ((S` A) + (S` B))))
Theorem	sto1i 11808	The state of a subspace plus the state of its orthocomplement.
|- A e. CH   =>   |- (S e. States -> ((S` A) + (S` (_|_` A))) = 1)
Theorem	sto2i 11809	The state of the orthocomplement.
|- A e. CH   =>   |- (S e. States -> (S` (_|_` A)) = (1 - (S` A)))
Theorem	stge1i 11810	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	stle0i 11811	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	stlei 11812	Ordering law for states.
|- A e. CH   &   |- B e. CH   =>   |- (S e. States -> (A C_ B -> (S` A) <_ (S` B)))
Theorem	stlesi 11813	Ordering law for states.
|- A e. CH   &   |- B e. CH   =>   |- (S e. States -> (A C_ B -> ((S` A) = 1 -> (S` B) = 1)))
Theorem	stji1i 11814	Join of components of Sasaki arrow ->1.
|- A e. CH   &   |- B e. CH   =>   |- (S e. States -> (S` ((_|_` A) vH (A i^i B))) = ((S` (_|_` A)) + (S` (A i^i B))))
Theorem	stm1i 11815	State of component of unit meet.
|- A e. CH   &   |- B e. CH   =>   |- (S e. States -> ((S` (A i^i B)) = 1 -> (S` A) = 1))
Theorem	stm1ri 11816	State of component of unit meet.
|- A e. CH   &   |- B e. CH   =>   |- (S e. States -> ((S` (A i^i B)) = 1 -> (S` B) = 1))
Theorem	stm1addi 11817	Sum of states whose meet is 1.
|- A e. CH   &   |- B e. CH   =>   |- (S e. States -> ((S` (A i^i B)) = 1 -> ((S` A) + (S` B)) = 2))
Theorem	staddi 11818	If the sum of 2 states is 2, then each state is 1.
|- A e. CH   &   |- B e. CH   =>   |- (S e. States -> (((S` A) + (S` B)) = 2 -> (S` A) = 1))
Theorem	stm1add3i 11819	Sum of states whose meet is 1.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   =>   |- (S e. States -> ((S` ((A i^i B) i^i C)) = 1 -> (((S` A) + (S` B)) + (S` C)) = 3))
Theorem	stadd3i 11820	If the sum of 3 states is 3, then each state is 1.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   =>   |- (S e. States -> ((((S` A) + (S` B)) + (S` C)) = 3 -> (S` A) = 1))
Theorem	st0 11821	The state of the zero subspace.
|- (S e. States -> (S` 0H) = 0)
Theorem	strlem1 11822	Lemma for strong state theorem: if closed subspace A is not contained in B, there is a unit vector u in their difference.
Theorem	strlem2 11823	Lemma for strong state theorem.
Theorem	strlem3a 11824	Lemma for strong state theorem: the function S, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state.
Theorem	strlem3 11825	Lemma for strong state theorem: the function S, 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.
Theorem	strlem4 11826	Lemma for strong state theorem.
Theorem	strlem5 11827	Lemma for strong state theorem.
Theorem	strlem6 11828	Lemma for strong state theorem.
Theorem	stri 11829	Strong state theorem. The states on a Hilbert lattice define an ordering. Remark in [Mayet] p. 370.
|- A e. CH   &   |- B e. CH   =>   |- (A.f e. States ((f` A) = 1 -> (f` B) = 1) -> A C_ B)
Theorem	strb 11830	Strong state theorem (bidirectional version).
|- A e. CH   &   |- B e. CH   =>   |- (A.f e. States ((f` A) = 1 -> (f` B) = 1) <-> A C_ B)
Theorem	hstrlem2 11831	Lemma for strong set of CH states theorem.
Theorem	hstrlem3a 11832	Lemma for strong set of CH states theorem: the function S, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state.
Theorem	hstrlem3 11833	Lemma for strong set of CH states theorem: the function S, 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.
Theorem	hstrlem4 11834	Lemma for strong set of CH states theorem.
Theorem	hstrlem5 11835	Lemma for strong set of CH states theorem.
Theorem	hstrlem6 11836	Lemma for strong set of CH states theorem.
Theorem	hstri 11837	Hilbert space admits a strong set of Hilbert-space-valued states (CH-states). Theorem in [Mayet3] p. 10.
|- A e. CH   &   |- B e. CH   =>   |- (A.f e. CHStates ((normh` (f` A)) = 1 -> (normh` (f` B)) = 1) -> A C_ B)
Theorem	hstrbi 11838	Strong CH-state theorem (bidirectional version). Theorem in [Mayet3] p. 10 and its converse.
|- A e. CH   &   |- B e. CH   =>   |- (A.f e. CHStates ((normh` (f` A)) = 1 -> (normh` (f` B)) = 1) <-> A C_ B)
Theorem	largei 11839	A Hilbert lattice admits a largei set of states. Remark in [Mayet] p. 370.
|- A e. CH   =>   |- (-. A = 0H <-> E.f e. States (f` A) = 1)
Theorem	jplem1 11840	Lemma for Jauch-Piron theorem.
Theorem	jplem2 11841	Lemma for Jauch-Piron theorem.
Theorem	jpi 11842	The function S, 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 11824 for the proof that S is a state.)
|- S = {<.x, y>. | (x e. CH /\ y = ((normh` ((proj` x)` u))^2))}   &   |- A e. CH   &   |- B e. CH   =>   |- ((u e. ~H /\ (normh` u) = 1) -> (((S` A) = 1 /\ (S` B) = 1) <-> (S` (A i^i B)) = 1))
Godowski's equation
Theorem	golem1 11843	Lemma for Godowski's equation.
Theorem	golem2 11844	Lemma for Godowski's equation.
Theorem	goeqi 11845	Godowski's equation, shown here as a variant equivalent to Equation SF of [Godowski] p. 730.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- F = ((_|_` A) vH (A i^i B))   &   |- G = ((_|_` B) vH (B i^i C))   &   |- H = ((_|_` C) vH (C i^i A))   &   |- D = ((_|_` B) vH (B i^i A))   =>   |- ((F i^i G) i^i H) C_ D
Theorem	stcltr1i 11846	Property of a strong classical state.
|- (ph <-> (S e. States /\ A.x e. CH A.y e. CH (((S` x) = 1 -> (S` y) = 1) -> x C_ y)))   &   |- A e. CH   &   |- B e. CH   =>   |- (ph -> (((S` A) = 1 -> (S` B) = 1) -> A C_ B))
Theorem	stcltr2i 11847	Property of a strong classical state.
|- (ph <-> (S e. States /\ A.x e. CH A.y e. CH (((S` x) = 1 -> (S` y) = 1) -> x C_ y)))   &   |- A e. CH   =>   |- (ph -> ((S` A) = 1 -> A = ~H))
Theorem	stcltrlem1 11848	Lemma for strong classical state theorem.
Theorem	stcltrlem2 11849	Lemma for strong classical state theorem.
Theorem	stcltrthi 11850	Theorem for classically strong set of states. If there exists a "classically strong set of states" on lattice CH (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.
|- A e. CH   &   |- B e. CH   &   |- E.s e. States A.x e. CH A.y e. CH (((s` x) = 1 -> (s` y) = 1) -> x C_ y)   =>   |- B C_ ((_|_` A) vH (A i^i B))
Covers relation; modular pairs
Definition	df-cv 11851	Define the covers relation (on the Hilbert lattice). Definition 3.2.18 of [PtakPulmannova] p. 68, whose notation we use. Ptak/Pulmannova's notation A <o B is read "B covers A " or "A is covered by B " , and it means that B is larger than A and there is nothing in between. See cvbr 11854 and cvbr2 11855 for membership relations.
|- <o = {<.x, y>. | ((x e. CH /\ y e. CH) /\ (x C. y /\ -. E.z e. CH (x C. z /\ z C. y)))}
Definition	df-md 11852	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 11866 for membership relation.
|- MH = {<.x, y>. | ((x e. CH /\ y e. CH) /\ A.z e. CH (z C_ y -> ((z vH x) i^i y) = (z vH (x i^i y))))}
Definition	df-dmd 11853	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 11871 for membership relation.
|- MH* = {<.x, y>. | ((x e. CH /\ y e. CH) /\ A.z e. CH (y C_ z -> ((z i^i x) vH y) = (z i^i (x vH y))))}
Theorem	cvbr 11854	Binary relation expressing B covers A, which means that B is larger than A and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68.
|- ((A e. CH /\ B e. CH) -> (A <o B <-> (A C. B /\ -. E.x e. CH (A C. x /\ x C. B))))
Theorem	cvbr2 11855	Binary relation expressing B covers A. Definition of covers in [Kalmbach] p. 15.
|- ((A e. CH /\ B e. CH) -> (A <o B <-> (A C. B /\ A.x e. CH ((A C. x /\ x C_ B) -> x = B))))
Theorem	cvcon3 11856	Contraposition law for the covers relation.
|- ((A e. CH /\ B e. CH) -> (A <o B <-> (_|_` B) <o (_|_` A)))
Theorem	cvpss 11857	The covers relation implies proper subset.
|- ((A e. CH /\ B e. CH) -> (A <o B -> A C. B))
Theorem	cvnbtwn 11858	The covers relation implies no in-betweenness.
|- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> -. (A C. C /\ C C. B)))
Theorem	cvnbtwn2 11859	The covers relation implies no in-betweenness.
|- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> ((A C. C /\ C C_ B) -> C = B)))
Theorem	cvnbtwn3 11860	The covers relation implies no in-betweenness.
|- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> ((A C_ C /\ C C. B) -> C = A)))
Theorem	cvnbtwn4 11861	The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31.
|- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> ((A C_ C /\ C C_ B) -> (C = A \/ C = B))))
Theorem	cvnsym 11862	The covers relation is not symmetric.
|- ((A e. CH /\ B e. CH) -> (A <o B -> -. B <o A))
Theorem	cvnref 11863	The covers relation is not reflexive.
|- (A e. CH -> -. A <o A)
Theorem	cvntr 11864	The covers relation is not transitive.
|- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A <o B /\ B <o C) -> -. A <o C))
Theorem	spansncv2 11865	Hilbert space has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153.
|- ((A e. CH /\ B e. ~H) -> (-. (span` {B}) C_ A -> A <o (A vH (span` {B}))))
Theorem	mdbr 11866	Binary relation expressing <.A, B>. is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1.
|- ((A e. CH /\ B e. CH) -> (A MH B <-> A.x e. CH (x C_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))
Theorem	mdi 11867	Consequence of the modular pair property.
|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A MH B /\ C C_ B)) -> ((C vH A) i^i B) = (C vH (A i^i B)))
Theorem	mdbr2 11868	Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 11866.
|- ((A e. CH /\ B e. CH) -> (A MH B <-> A.x e. CH (x C_ B -> ((x vH A) i^i B) C_ (x vH (A i^i B)))))
Theorem	mdbr3 11869	Binary relation expressing the modular pair property. This version quantifies an equality instead of an inference.
|- ((A e. CH /\ B e. CH) -> (A MH B <-> A.x e. CH (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B))))
Theorem	mdbr4 11870	Binary relation expressing the modular pair property. This version quantifies an ordering instead of an inference.
|- ((A e. CH /\ B e. CH) -> (A MH B <-> A.x e. CH (((x i^i B) vH A) i^i B) C_ ((x i^i B) vH (A i^i B))))
Theorem	dmdbr 11871	Binary relation expressing the dual modular pair property.
|- ((A e. CH /\ B e. CH) -> (A MH* B <-> A.x e. CH (B C_ x -> ((x i^i A) vH B) = (x i^i (A vH B)))))
Theorem	dmdmd 11872	The dual modular pair property expressed in terms of the modular pair property, that hold in Hilbert lattices. Remark 29.6 of [MaedaMaeda] p. 130.
|- ((A e. CH /\ B e. CH) -> (A MH* B <-> (_|_` A) MH (_|_` B)))
Theorem	mddmd 11873	The modular pair property expressed in terms of the dual modular pair property.
|- ((A e. CH /\ B e. CH) -> (A MH B <-> (_|_` A) MH* (_|_` B)))
Theorem	dmdi 11874	Consequence of the dual modular pair property.
|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A MH* B /\ B C_ C)) -> ((C i^i A) vH B) = (C i^i (A vH B)))
Theorem	dmdbr2 11875	Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 11871.
|- ((A e. CH /\ B e. CH) -> (A MH* B <-> A.x e. CH (B C_ x -> (x i^i (A vH B)) C_ ((x i^i A) vH B))))
Theorem	dmdi2 11876	Consequence of the dual modular pair property.
|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A MH* B /\ B C_ C)) -> (C i^i (A vH B)) C_ ((C i^i A) vH B))
Theorem	dmdbr3 11877	Binary relation expressing the dual modular pair property. This version quantifies an equality instead of an inference.
|- ((A e. CH /\ B e. CH) -> (A MH* B <-> A.x e. CH (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B))))
Theorem	dmdbr4 11878	Binary relation expressing the dual modular pair property. This version quantifies an ordering instead of an inference.
|- ((A e. CH /\ B e. CH) -> (A MH* B <-> A.x e. CH ((x vH B) i^i (A vH B)) C_ (((x vH B) i^i A) vH B)))
Theorem	dmdi4 11879	Consequence of the dual modular pair property.
|- ((A e. CH /\ B e. CH /\ C e. CH) -> (A MH* B -> ((C vH B) i^i (A vH B)) C_ (((C vH B) i^i A) vH B)))
Theorem	dmdbr5 11880	Binary relation expressing the dual modular pair property.
|- ((A e. CH /\ B e. CH) -> (A MH* B <-> A.x e. CH (x C_ (A vH B) -> x C_ (((x vH B) i^i A) vH B))))
Theorem	mddmd2 11881	Relationship between modular pairs and dual-modular pairs. Lemma 1.2 of [MaedaMaeda] p. 1.
|- (A e. CH -> (A.x e. CH A MH x <-> A.x e. CH A MH* x))
Theorem	mdsl0 11882	A sublattice condition that transfers the modular pair property. Exercise 12 of [Kalmbach] p. 103. Also Lemma 1.5.3 of [MaedaMaeda] p. 2.
|- (((A e. CH /\ B e. CH) /\ (C e. CH /\ D e. CH)) -> ((((C C_ A /\ D C_ B) /\ (A i^i B) = 0H) /\ A MH B) -> C MH D))
Theorem	ssmd1 11883	Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1.
|- ((A e. CH /\ B e. CH /\ A C_ B) -> A MH B)
Theorem	ssmd2 11884	Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1.
|- ((A e. CH /\ B e. CH /\ A C_ B) -> B MH A)
Theorem	ssdmd1 11885	Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1.
|- ((A e. CH /\ B e. CH /\ A C_ B) -> A MH* B)
Theorem	ssdmd2 11886	Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1.
|- ((A e. CH /\ B e. CH /\ A C_ B) -> (_|_` B) MH (_|_` A))
Theorem	dmdsl3 11887	Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2.
|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B MH* A /\ A C_ C /\ C C_ (A vH B))) -> ((C i^i B) vH A) = C)
Theorem	mdsl3 11888	Sublattice mapping for a modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2.
|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A MH B /\ (A i^i B) C_ C /\ C C_ B)) -> ((C vH A) i^i B) = C)
Theorem	mdslle1i 11889	Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- D e. CH   =>   |- ((B MH* A /\ A C_ (C i^i D) /\ (C vH D) C_ (A vH B)) -> (C C_ D <-> (C i^i B) C_ (D i^i B)))
Theorem	mdslle2i 11890	Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- D e. CH   =>   |- ((A MH B /\ (A i^i B) C_ (C i^i D) /\ (C vH D) C_ B) -> (C C_ D <-> (C vH A) C_ (D vH A)))
Theorem	mdslj1i 11891	Join preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- D e. CH   =>   |- (((A MH B /\ B MH* A) /\ (A C_ (C i^i D) /\ (C vH D) C_ (A vH B))) -> ((C vH D) i^i B) = ((C i^i B) vH (D i^i B)))
Theorem	mdslj2i 11892	Meet preservation of the reverse mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- D e. CH   =>   |- (((A MH B /\ B MH* A) /\ ((A i^i B) C_ (C i^i D) /\ (C vH D) C_ B)) -> ((C i^i D) vH A) = ((C vH A) i^i (D vH A)))
Theorem	mdsl1i 11893	If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2.
|- A e. CH   &   |- B e. CH   =>   |- (A.x e. CH (((A i^i B) C_ x /\ x C_ (A vH B)) -> (x C_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))) <-> A MH B)
Theorem	mdsl2i 11894	If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2.
|- A e. CH   &   |- B e. CH   =>   |- (A MH B <-> A.x e. CH (((A i^i B) C_ x /\ x C_ B) -> ((x vH A) i^i B) C_ (x vH (A i^i B))))
Theorem	mdsl2bi 11895	If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2.
|- A e. CH   &   |- B e. CH   =>   |- (A MH B <-> A.x e. CH (((A i^i B) C_ x /\ x C_ B) -> ((x vH A) i^i B) = (x vH (A i^i B))))
Theorem	cvmdi 11896	The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31.
|- A e. CH   &   |- B e. CH   =>   |- ((A i^i B) <o B -> A MH B)
Theorem	mdslmd1lem1 11897	Lemma for mdslmd1i 11901.
Theorem	mdslmd1lem2 11898	Lemma for mdslmd1i 11901.
Theorem	mdslmd1lem3 11899	Lemma for mdslmd1i 11901.
Theorem	mdslmd1lem4 11900	Lemma for mdslmd1i 11901.