mmtheorems110 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 10901-11000 - Page 110 of 175

Type	Label	Description
Statement
Theorem	pjomli 10901	Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omlsi 10879.
|- A e. CH   &   |- B e. SH   =>   |- ((A C_ B /\ (B i^i (_|_` A)) = 0H) -> A = B)
Theorem	pjoml 10902	Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omlsi 10879.
|- (((A e. CH /\ B e. SH) /\ (A C_ B /\ (B i^i (_|_` A)) = 0H)) -> A = B)
Theorem	pjococi 10903	Proof of orthocomplement theorem using projections. Compare ococ 10881.
|- H e. CH   =>   |- (_|_` (_|_` H)) = H
Theorem	pjoc2i 10904	Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111.
|- H e. CH   &   |- A e. ~H   =>   |- (A e. (_|_` H) <-> ((proj` H)` A) = 0h)
Theorem	pjoc2 10905	Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111.
|- ((H e. CH /\ A e. ~H) -> (A e. (_|_` H) <-> ((proj` H)` A) = 0h))
Subspace sum, span, lattice join, lattice supremum
Definition	df-shsum 10906	Define subspace sum in SH. See shsumval 10910, shsumval2i 10993, and shsumval3i 10994 for its value.
|- +H = {<.<.x, y>., z>. | ((x e. SH /\ y e. SH) /\ z = {w e. ~H | E.v e. x E.u e. y w = (v +h u)})}
Definition	df-span 10907	Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 10932 for its value.
|- span = {<.x, y>. | (x C_ ~H /\ y = |^|{z e. SH | x C_ z})}
Definition	df-chj 10908	Define Hilbert lattice join. See chjval 10955 for its value and chjcl 10962 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to CH; see sshjcl 10960. For an alternate definition see dfchj2 10957.
|- vH = {<.<.x, y>., z>. | ((x C_ ~H /\ y C_ ~H) /\ z = (_|_` (_|_` (x u. y))))}
Definition	df-chsup 10909	Define the supremum of a set of Hilbert lattice elements. See chsupval2 10935 for its value and dfchsup2 10931 for an alternate definition. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice CH, to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 10940.
|- \/H = {<.x, y>. | (x C_ ~P~H /\ y = (_|_` (_|_` U.x)))}
Theorem	shsumval 10910	Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in [Kalmbach] p. 65.
|- ((A e. SH /\ B e. SH) -> (A +H B) = {x e. ~H | E.y e. A E.z e. B x = (y +h z)})
Theorem	shsel 10911	Membership in the subspace sum of two Hilbert subspaces.
|- ((A e. SH /\ B e. SH) -> (C e. (A +H B) <-> E.x e. A E.y e. B C = (x +h y)))
Theorem	shsel3 10912	Membership in the subspace sum of two Hilbert subspaces, using vector subtraction.
|- ((A e. SH /\ B e. SH) -> (C e. (A +H B) <-> E.x e. A E.y e. B C = (x -h y)))
Theorem	shseli 10913	Membership in subspace sum.
|- A e. SH   &   |- B e. SH   =>   |- (C e. (A +H B) <-> E.x e. A E.y e. B C = (x +h y))
Theorem	shscli 10914	Closure of subspace sum.
|- A e. SH   &   |- B e. SH   =>   |- (A +H B) e. SH
Theorem	shscl 10915	Closure of subspace sum.
|- ((A e. SH /\ B e. SH) -> (A +H B) e. SH)
Theorem	shscom 10916	Commutative law for subspace sum.
|- ((A e. SH /\ B e. SH) -> (A +H B) = (B +H A))
Theorem	shsva 10917	Vector sum belongs to subspace sum.
|- ((A e. SH /\ B e. SH) -> ((C e. A /\ D e. B) -> (C +h D) e. (A +H B)))
Theorem	shsel1 10918	A subspace sum contains a member of one of its subspaces.
|- ((A e. SH /\ B e. SH) -> (C e. A -> C e. (A +H B)))
Theorem	shsel2 10919	A subspace sum contains a member of one of its subspaces.
|- ((A e. SH /\ B e. SH) -> (C e. B -> C e. (A +H B)))
Theorem	shsvs 10920	Vector subtraction belongs to subspace sum.
|- ((A e. SH /\ B e. SH) -> ((C e. A /\ D e. B) -> (C -h D) e. (A +H B)))
Theorem	shsub1 10921	Subspace sum is an upper bound of its arguments.
|- ((A e. SH /\ B e. SH) -> A C_ (A +H B))
Theorem	shsub2 10922	Subspace sum is an upper bound of its arguments.
|- ((A e. SH /\ B e. SH) -> A C_ (B +H A))
Theorem	choc0 10923	The orthocomplement of the zero subspace is the unit subspace.
|- (_|_` 0H) = ~H
Theorem	choc1 10924	The orthocomplement of the unit subspace is the zero subspace. Does not require Axiom of Choice.
|- (_|_` ~H) = 0H
Theorem	chocnul 10925	Orthogonal complement of the empty set.
|- (_|_` (/)) = ~H
Theorem	shintcli 10926	Closure of intersection of a non-empty subset of SH.
|- (A C_ SH /\ A =/= (/))   =>   |- |^|A e. SH
Theorem	shintcl 10927	The intersection of a non-empty set of subspaces is a subspace.
|- ((A C_ SH /\ A =/= (/)) -> |^|A e. SH)
Theorem	chintcli 10928	The intersection of a non-empty set of closed subspaces is a closed subspace.
|- (A C_ CH /\ A =/= (/))   =>   |- |^|A e. CH
Theorem	chintcl 10929	The intersection (infimum) of a non-empty subset of CH belongs to CH. Part of Theorem 3.13 of [Beran] p. 108. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8).
|- ((A C_ CH /\ A =/= (/)) -> |^|A e. CH)
Theorem	ococin 10930	The double complement is the smallest closed subspace containing a subset of Hilbert space. Remark 3.12(B) of [Beran] p. 107.
|- (A C_ ~H -> (_|_` (_|_` A)) = |^|{x e. CH | A C_ x})
Theorem	dfchsup2 10931	Alternate definition of supremum of a subset of the Hilbert lattice. Definition in Proposition 1 of [Kalmbach] p. 65. We actually define it on any collection of Hilbert space subsets, not just the Hilbert lattice CH, to allow more general theorems.
|- \/H = {<.x, y>. | (x C_ ~P~H /\ y = |^|{z e. CH | U.x C_ z})}
Theorem	spanval 10932	Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in [Schechter] p. 276.
|- (A C_ ~H -> (span` A) = |^|{x e. SH | A C_ x})
Theorem	hsupval2 10933	Value of supremum of set of subsets of Hilbert space. Definition of supremum in Proposition 1 of [Kalmbach] p. 65.
|- (A C_ ~P~H -> ( \/H ` A) = |^|{x e. CH | U.A C_ x})
Theorem	hsupval 10934	Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 10933.
|- (A C_ ~P~H -> ( \/H ` A) = (_|_` (_|_` U.A)))
Theorem	chsupval2 10935	The value of the supremum of a set of closed subspaces of Hilbert space. Definition of supremum in Proposition 1 of [Kalmbach] p. 65.
|- (A C_ CH -> ( \/H ` A) = |^|{x e. CH | U.A C_ x})
Theorem	chsupval 10936	The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 10935.
|- (A C_ CH -> ( \/H ` A) = (_|_` (_|_` U.A)))
Theorem	spancl 10937	The span of a subset of Hilbert space is a subspace.
|- (A C_ ~H -> (span` A) e. SH)
Theorem	elspancl 10938	A member of a span is a vector.
|- ((A C_ ~H /\ B e. (span` A)) -> B e. ~H)
Theorem	shsupcl 10939	Closure of the subspace supremum of set of subsets of Hilbert space.
|- (A C_ ~P~H -> (span` U.A) e. SH)
Theorem	hsupcl 10940	Closure of supremum of set of subsets of Hilbert space. Note that the supremum belongs to CH even if the subsets do not.
|- (A C_ ~P~H -> ( \/H ` A) e. CH)
Theorem	chsupcl 10941	Closure of supremum of subset of CH. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. Shows that CH is a complete lattice. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8).
|- (A C_ CH -> ( \/H ` A) e. CH)
Theorem	hsupss 10942	Subset relation for supremum of Hilbert space subsets.
|- ((A C_ ~P~H /\ B C_ ~P~H) -> (A C_ B -> ( \/H ` A) C_ ( \/H ` B)))
Theorem	chsupss 10943	Subset relation for supremum of subset of CH.
|- ((A C_ CH /\ B C_ CH) -> (A C_ B -> ( \/H ` A) C_ ( \/H ` B)))
Theorem	chsupid 10944	A subspace is the supremum of all smaller subspaces.
|- (A e. CH -> ( \/H ` {x e. CH | x C_ A}) = A)
Theorem	chsupsn 10945	Value of supremum of subset of CH on a singleton.
|- (A e. CH -> ( \/H ` {A}) = A)
Theorem	hsupunss 10946	The union of a set of Hilbert space subsets is smaller than its supremum.
|- (A C_ ~P~H -> U.A C_ ( \/H ` A))
Theorem	spanss2 10947	A subset of Hilbert space is included in its span.
|- (A C_ ~H -> A C_ (span` A))
Theorem	shsupunss 10948	The union of a set of subspaces is smaller than its supremum.
|- (A C_ SH -> U.A C_ (span` U.A))
Theorem	chsupunss 10949	The union of a set of closed subspaces is smaller than its supremum.
|- (A C_ CH -> U.A C_ ( \/H ` A))
Theorem	spanid 10950	A subspace of Hilbert space is its own span.
|- (A e. SH -> (span` A) = A)
Theorem	spanss 10951	Ordering relationship for the spans of subsets of Hilbert space.
|- ((B C_ ~H /\ A C_ B) -> (span` A) C_ (span` B))
Theorem	spanssoc 10952	The span of a subset of Hilbert space is less than or equal to its closure (double orthogonal complement).
|- (A C_ ~H -> (span` A) C_ (_|_` (_|_` A)))
Theorem	sshjval 10953	Value of join for subsets of Hilbert space.
|- ((A C_ ~H /\ B C_ ~H) -> (A vH B) = (_|_` (_|_` (A u. B))))
Theorem	shjval 10954	Value of join in SH.
|- ((A e. SH /\ B e. SH) -> (A vH B) = (_|_` (_|_` (A u. B))))
Theorem	chjval 10955	Value of join in CH.
|- ((A e. CH /\ B e. CH) -> (A vH B) = (_|_` (_|_` (A u. B))))
Theorem	chjvali 10956	Value of join in CH.
|- A e. CH   &   |- B e. CH   =>   |- (A vH B) = (_|_` (_|_` (A u. B)))
Theorem	dfchj2 10957	Alternate definition of join in the set of closed subspaces of Hilbert space CH.
|- vH = {<.<.x, y>., z>. | ((x C_ ~H /\ y C_ ~H) /\ z = |^|{w e. CH | (x u. y) C_ w})}
Theorem	dfchj3 10958	Alternate definition of join in the set of closed subspaces of Hilbert space CH: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice CH.
|- vH = {<.<.x, y>., z>. | ((x C_ ~H /\ y C_ ~H) /\ z = ( \/H ` {x, y}))}
Theorem	sshjval3 10959	Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3.
|- ((A C_ ~H /\ B C_ ~H) -> (A vH B) = ( \/H ` {A, B}))
Theorem	sshjcl 10960	Closure of join for subsets of Hilbert space.
|- ((A C_ ~H /\ B C_ ~H) -> (A vH B) e. CH)
Theorem	shjcl 10961	Closure of join in SH.
|- ((A e. SH /\ B e. SH) -> (A vH B) e. CH)
Theorem	chjcl 10962	Closure of join in CH.
|- ((A e. CH /\ B e. CH) -> (A vH B) e. CH)
Theorem	shjcom 10963	Commutative law for Hilbert lattice join of subspaces.
|- ((A e. SH /\ B e. SH) -> (A vH B) = (B vH A))
Theorem	shincli 10964	Closure of intersection of two subspaces.
|- A e. SH   &   |- B e. SH   =>   |- (A i^i B) e. SH
Theorem	shscomi 10965	Commutative law for subspace sum.
|- A e. SH   &   |- B e. SH   =>   |- (A +H B) = (B +H A)
Theorem	shsvai 10966	Vector sum belongs to subspace sum.
|- A e. SH   &   |- B e. SH   =>   |- ((C e. A /\ D e. B) -> (C +h D) e. (A +H B))
Theorem	shsel1i 10967	A subspace sum contains a member of one of its subspaces.
|- A e. SH   &   |- B e. SH   =>   |- (C e. A -> C e. (A +H B))
Theorem	shsel2i 10968	A subspace sum contains a member of one of its subspaces.
|- A e. SH   &   |- B e. SH   =>   |- (C e. B -> C e. (A +H B))
Theorem	shsvsi 10969	Vector subtraction belongs to subspace sum.
|- A e. SH   &   |- B e. SH   =>   |- ((C e. A /\ D e. B) -> (C -h D) e. (A +H B))
Theorem	shunssi 10970	Union is smaller than subspace sum.
|- A e. SH   &   |- B e. SH   =>   |- (A u. B) C_ (A +H B)
Theorem	shsleji 10971	Subspace sum is smaller than Hilbert lattice join. Remark in [Kalmbach] p. 65.
|- A e. SH   &   |- B e. SH   =>   |- (A +H B) C_ (A vH B)
Theorem	shunssji 10972	Union is smaller than Hilbert lattice join.
|- A e. SH   &   |- B e. SH   =>   |- (A u. B) C_ (A vH B)
Theorem	shjcomi 10973	Commutative law for join in SH.
|- A e. SH   &   |- B e. SH   =>   |- (A vH B) = (B vH A)
Theorem	shsub1i 10974	Subspace sum is an upper bound of its arguments.
|- A e. SH   &   |- B e. SH   =>   |- A C_ (A +H B)
Theorem	shsub2i 10975	Subspace sum is an upper bound of its arguments.
|- A e. SH   &   |- B e. SH   =>   |- A C_ (B +H A)
Theorem	shub1i 10976	Hilbert lattice join is an upper bound of two subspaces.
|- A e. SH   &   |- B e. SH   =>   |- A C_ (A vH B)
Theorem	shjcli 10977	Closure of CH join.
|- A e. SH   &   |- B e. SH   =>   |- (A vH B) e. CH
Theorem	shjshcli 10978	SH closure of join.
|- A e. SH   &   |- B e. SH   =>   |- (A vH B) e. SH
Theorem	shlubi 10979	Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces.
|- A e. SH   &   |- B e. SH   &   |- C e. CH   =>   |- ((A C_ C /\ B C_ C) <-> (A vH B) C_ C)
Theorem	shlessi 10980	Subset implies subset of subspace sum.
|- A e. SH   &   |- B e. SH   &   |- C e. SH   =>   |- (A C_ B -> (A +H C) C_ (B +H C))
Theorem	shlej1i 10981	Add disjunct to both sides of Hilbert subspace ordering.
|- A e. SH   &   |- B e. SH   &   |- C e. SH   =>   |- (A C_ B -> (A vH C) C_ (B vH C))
Theorem	shlej2i 10982	Add disjunct to both sides of Hilbert subspace ordering.
|- A e. SH   &   |- B e. SH   &   |- C e. SH   =>   |- (A C_ B -> (C vH A) C_ (C vH B))
Theorem	shslej 10983	Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65.
|- ((A e. SH /\ B e. SH) -> (A +H B) C_ (A vH B))
Theorem	shincl 10984	Closure of intersection of two subspaces.
|- ((A e. SH /\ B e. SH) -> (A i^i B) e. SH)
Theorem	shub1 10985	Hilbert lattice join is an upper bound of two subspaces.
|- ((A e. SH /\ B e. SH) -> A C_ (A vH B))
Theorem	shub2 10986	A subspace is a subset of its Hilbert lattice join with another.
|- ((A e. SH /\ B e. SH) -> A C_ (B vH A))
Theorem	shlub 10987	Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces.
|- ((A e. SH /\ B e. SH /\ C e. CH) -> ((A C_ C /\ B C_ C) <-> (A vH B) C_ C))
Theorem	shlej1 10988	Add disjunct to both sides of Hilbert subspace ordering.
|- (((A e. SH /\ B e. SH /\ C e. SH) /\ A C_ B) -> (A vH C) C_ (B vH C))
Theorem	shlej2 10989	Add disjunct to both sides of Hilbert subspace ordering.
|- (((A e. SH /\ B e. SH /\ C e. SH) /\ A C_ B) -> (C vH A) C_ (C vH B))
Theorem	shsidmi 10990	Idempotent law for Hilbert subspace sum.
|- A e. SH   =>   |- (A +H A) = A
Theorem	shslubi 10991	Least upper bound law for Hilbert subspace sum.
|- A e. SH   &   |- B e. SH   &   |- C e. SH   =>   |- ((A C_ C /\ B C_ C) <-> (A +H B) C_ C)
Theorem	shlesb1i 10992	Hilbert lattice ordering in terms of subspace sum.
|- A e. SH   &   |- B e. SH   =>   |- (A C_ B <-> (A +H B) = B)
Theorem	shsumval2i 10993	An alternate way to express subspace sum.
|- A e. SH   &   |- B e. SH   =>   |- (A +H B) = |^|{x e. SH | (A u. B) C_ x}
Theorem	shsumval3i 10994	An alternate way to express subspace sum.
|- A e. SH   &   |- B e. SH   =>   |- (A +H B) = (span` (A u. B))
Theorem	shmodsi 10995	The modular law holds for subspace sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70.
|- A e. SH   &   |- B e. SH   &   |- C e. SH   =>   |- (A C_ C -> ((A +H B) i^i C) C_ (A +H (B i^i C)))
Theorem	shmodi 10996	The modular law is implied by the closure of subspace sum. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70.
|- A e. SH   &   |- B e. SH   &   |- C e. SH   =>   |- (((A +H B) = (A vH B) /\ A C_ C) -> ((A vH B) i^i C) C_ (A vH (B i^i C)))
Hilbert lattice operations
Theorem	sh0le 10997	The zero subspace is the smallest subspace.
|- (A e. SH -> 0H C_ A)
Theorem	ch0le 10998	The zero subspace is the smallest member of CH.
|- (A e. CH -> 0H C_ A)
Theorem	shle0 10999	No subspace is smaller than the zero subspace.
|- (A e. SH -> (A C_ 0H <-> A = 0H))
Theorem	chle0 11000	No Hilbert lattice element is smaller than zero.
|- (A e. CH -> (A C_ 0H <-> A = 0H))