mmtheorems100 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 9901-10000 - Page 100 of 175

Type	Label	Description
Statement
Theorem	minveclem12 9901	Lemma for minvecex 9923.
Theorem	minveclem13 9902	Lemma for minvecex 9923.
Theorem	minveclem14 9903	Lemma for minvecex 9923.
Theorem	minveclem15 9904	Lemma for minvecex 9923.
Theorem	minveclem16 9905	Lemma for minvecex 9923.
Theorem	minveclem17 9906	Lemma for minvecex 9923.
Theorem	minveclem18 9907	Lemma for minvecex 9923.
Theorem	minveclem19 9908	Lemma for minvecex 9923.
Theorem	minveclem20 9909	Lemma for minvecex 9923.
Theorem	minveclem21 9910	Lemma for minvecex 9923.
Theorem	minveclem22 9911	Lemma for minvecex 9923.
Theorem	minveclem23 9912	Lemma for minvecex 9923. Eliminate H.
Theorem	minveclem24 9913	Lemma for minvecex 9923.
Theorem	minveclem25 9914	Lemma for minvecex 9923.
Theorem	minveclem26 9915	Lemma for minvecex 9923.
Theorem	minveclem27 9916	Lemma for minvecex 9923.
Theorem	minveclem28 9917	Lemma for minvecex 9923.
Theorem	minveclem29 9918	Lemma for minvecex 9923. Sequence f is Cauchy, and since vector subspace W is complete, f therefore converges to a vector in W.
Theorem	minveclem30 9919	Lemma for minvecex 9923.
Theorem	minveclem31 9920	Lemma for minvecex 9923.
Theorem	minveclem32 9921	Lemma for minvecex 9923.
Theorem	minveclem33 9922	Lemma for minvecex 9923.
Theorem	minvecex 9923	Minimizing vector theorem (existence part). There is exactly one vector in a complete subspace W that minimizes the distance to an arbitrary vector A in a parent inner product space. Part of Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. Note that we work with the negative of supremum instead of infimum in order to use theorems we already have available.
|- R = {x | E.y e. Y x = -u(N` (AMy))}   &   |- U e. CPreHil   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- X = (BaseSet` U)   &   |- W e. (SubSp` U)   &   |- Y = (BaseSet` W)   &   |- A e. X   &   |- P = -usup(R, RR, < )   &   |- (j e. NN -> (F` j) = (N` (AM(f` j))))   &   |- D = (IndMet` W)   &   |- F e. _V   &   |- W e. CBan   =>   |- E.a e. Y (N` (AMa)) = P
Theorem	minveclem35 9924	Lemma for minveceu 9928.
Theorem	minveclem36 9925	Lemma for minveceu 9928.
Theorem	minveclem37 9926	Lemma for minveceu 9928.
Theorem	minveclem38 9927	Lemma for minveceu 9928.
Theorem	minveceu 9928	Minimizing vector theorem. There is exactly one vector in a complete subspace W that minimizes the distance to an arbitrary vector A in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. Note that we work with the negative of the supremum of negatives instead of infimum in order to use theorems we already have available.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (BaseSet` W)   &   |- R = {x | E.y e. Y x = -u(N` (AMy))}   &   |- P = -usup(R, RR, < )   &   |- U e. CPreHil   &   |- W e. ((SubSp` U) i^i CBan)   &   |- A e. X   =>   |- E!a e. Y (N` (AMa)) = P
Theorem	minveccl 9929	The minimizing vector of minveceu 9928 belongs to the subspace Y.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (BaseSet` W)   &   |- R = {x | E.y e. Y x = -u(N` (AMy))}   &   |- P = -usup(R, RR, < )   &   |- U e. CPreHil   &   |- W e. ((SubSp` U) i^i CBan)   &   |- A e. X   &   |- Q = U.{b e. Y | (N` (AMb)) = P}   =>   |- Q e. Y
Theorem	minvecdist 9930	Distance of the minimizing vector of minveceu 9928.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (BaseSet` W)   &   |- R = {x | E.y e. Y x = -u(N` (AMy))}   &   |- P = -usup(R, RR, < )   &   |- U e. CPreHil   &   |- W e. ((SubSp` U) i^i CBan)   &   |- A e. X   &   |- Q = U.{b e. Y | (N` (AMb)) = P}   =>   |- (N` (AMQ)) = P
Theorem	minvecle 9931	The minimizing vector from minveceu 9928 has the smallest distance.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (BaseSet` W)   &   |- R = {x | E.y e. Y x = -u(N` (AMy))}   &   |- P = -usup(R, RR, < )   &   |- U e. CPreHil   &   |- W e. ((SubSp` U) i^i CBan)   &   |- A e. X   &   |- Q = U.{b e. Y | (N` (AMb)) = P}   =>   |- (B e. Y -> (N` (AMQ)) <_ (N` (AMB)))
Theorem	minveclem39 9932	Lemma for minvecex2 9933.
Theorem	minvecex2 9933	Existence version of minvecle 9931.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (BaseSet` W)   &   |- U e. CPreHil   &   |- W e. ((SubSp` U) i^i CBan)   &   |- A e. X   =>   |- E.x e. Y A.y e. Y (N` (AMx)) <_ (N` (AMy))
Complex Hilbert spaces
Definition and basic properties
Syntax	chl 9934	Extend class notation with the class of all complex Hilbert spaces.
class CHil
Definition	df-hl 9935	Define the class of all complex Hilbert spaces. A Hilbert space is a Banach space which is also an inner product space.
|- CHil = (CBan i^i CPreHil)
Theorem	ishl 9936	The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- (U e. CHil <-> (U e. CBan /\ U e. CPreHil))
Theorem	hlbn 9937	Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- (U e. CHil -> U e. CBan)
Theorem	hlph 9938	Every complex Hilbert space is an inner product space (also called a pre-Hilbert space).
|- (U e. CHil -> U e. CPreHil)
Theorem	hlrel 9939	The class of all complex Hilbert spaces is a relation.
|- Rel CHil
Theorem	hlnv 9940	Every complex Hilbert space is a normed complex vector space.
|- (U e. CHil -> U e. NrmCVec)
Theorem	hlnvi 9941	Every complex Hilbert space is a normed complex vector space.
|- U e. CHil   =>   |- U e. NrmCVec
Theorem	hlvc 9942	Every complex Hilbert space is a complex vector space.
|- W = (1st` U)   =>   |- (U e. CHil -> W e. CVec)
Theorem	hlcms 9943	The induced metric on a complex Hilbert space is complete.
|- D = (IndMet` U)   =>   |- (U e. CHil -> D e. CMet)
Theorem	hlmet 9944	The induced metric on a complex Hilbert space.
|- D = (IndMet` U)   =>   |- (U e. CHil -> D e. Met)
Theorem	hlpar2 9945	The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   =>   |- ((U e. CHil /\ A e. X /\ B e. X) -> (((N` (AGB))^2) + ((N` (AMB))^2)) = (2 x. (((N` A)^2) + ((N` B)^2))))
Theorem	hlpar 9946	The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   =>   |- ((U e. CHil /\ A e. X /\ B e. X) -> (((N` (AGB))^2) + ((N` (AG(-u1SB)))^2)) = (2 x. (((N` A)^2) + ((N` B)^2))))
Standard axioms for a complex Hilbert space
Theorem	hlex 9947	The base set of a Hilbert space is a set.
|- X = (BaseSet` U)   =>   |- X e. _V
Theorem	hladdf 9948	Mapping for Hilbert space vector addition.
|- X = (BaseSet` U)   &   |- G = (+v` U)   =>   |- (U e. CHil -> G:(X X. X)-->X)
Theorem	hlcom 9949	Hilbert space vector addition is commutative.
|- X = (BaseSet` U)   &   |- G = (+v` U)   =>   |- ((U e. CHil /\ A e. X /\ B e. X) -> (AGB) = (BGA))
Theorem	hlass 9950	Hilbert space vector addition is associative.
|- X = (BaseSet` U)   &   |- G = (+v` U)   =>   |- ((U e. CHil /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = (AG(BGC)))
Theorem	hl0cl 9951	The Hilbert space zero vector.
|- X = (BaseSet` U)   &   |- Z = (0v` U)   =>   |- (U e. CHil -> Z e. X)
Theorem	hladdid 9952	Hilbert space addition with the zero vector.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- Z = (0v` U)   =>   |- ((U e. CHil /\ A e. X) -> (AGZ) = A)
Theorem	hlmulf 9953	Mapping for Hilbert space scalar multiplication.
|- X = (BaseSet` U)   &   |- S = (.s` U)   =>   |- (U e. CHil -> S:(CC X. X)-->X)
Theorem	hlmulid 9954	Hilbert space scalar multiplication by one.
|- X = (BaseSet` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ A e. X) -> (1SA) = A)
Theorem	hlmulass 9955	Hilbert space scalar multiplication associative law.
|- X = (BaseSet` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A x. B)SC) = (AS(BSC)))
Theorem	hldi 9956	Hilbert space scalar multiplication distributive law.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ (A e. CC /\ B e. X /\ C e. X)) -> (AS(BGC)) = ((ASB)G(ASC)))
Theorem	hldir 9957	Hilbert space scalar multiplication distributive law.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A + B)SC) = ((ASC)G(BSC)))
Theorem	hlmul0 9958	Hilbert space scalar multiplication by zero.
|- X = (BaseSet` U)   &   |- S = (.s` U)   &   |- Z = (0v` U)   =>   |- ((U e. CHil /\ A e. X) -> (0SA) = Z)
Theorem	hlipf 9959	Mapping for Hilbert space inner product.
|- X = (BaseSet` U)   &   |- P = (.i` U)   =>   |- (U e. CHil -> P:(X X. X)-->CC)
Theorem	hlipcj 9960	Conjugate law for Hilbert space inner product.
|- X = (BaseSet` U)   &   |- P = (.i` U)   =>   |- ((U e. CHil /\ A e. X /\ B e. X) -> (APB) = (*` (BPA)))
Theorem	hlipdir 9961	Distributive law for Hilbert space inner product.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- P = (.i` U)   =>   |- ((U e. CHil /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)PC) = ((APC) + (BPC)))
Theorem	hlipass 9962	Associative law for Hilbert space inner product.
|- X = (BaseSet` U)   &   |- S = (.s` U)   &   |- P = (.i` U)   =>   |- ((U e. CHil /\ (A e. CC /\ B e. X /\ C e. X)) -> ((ASB)PC) = (A x. (BPC)))
Theorem	hlipgt0 9963	The inner product of a Hilbert space vector by itself is positive.
|- X = (BaseSet` U)   &   |- Z = (0v` U)   &   |- P = (.i` U)   =>   |- ((U e. CHil /\ A e. X /\ A =/= Z) -> 0 < (APA))
Theorem	hlcompl 9964	Completeness of a Hilbert space.
|- X = (BaseSet` U)   &   |- D = (IndMet` U)   =>   |- ((U e. CHil /\ F e. (Cau` D)) -> E.x e. X F(~~>m` D)x)
Examples of complex Hilbert spaces
Theorem	cnhl 9965	The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- U = <.<. + , x. >., abs>.   =>   |- U e. CHil
Subspaces
Theorem	ssphl 9966	A Banach subspace of an inner product space is a Hilbert space.
|- H = (SubSp` U)   =>   |- ((U e. CPreHil /\ W e. H /\ W e. CBan) -> W e. CHil)
Hellinger-Toeplitz Theorem
Theorem	htthlem1 9967	Lemma for htthi 9979. Closure of values of an operator T on an auxiliary sequence of vectors f.
Theorem	htthlem2 9968	Lemma for htthi 9979. Elevate set variables to class variables in the self-adjoint hypothesis.
Theorem	htthlem3 9969	Lemma for htthi 9979. Construct an auxiliary sequence of functionals F` k from inner products of the given function T and auxiliary vector sequence f.
Theorem	htthlem4 9970	Lemma for htthi 9979. Value of a functional F` k.
Theorem	htthlem5 9971	Lemma for htthi 9979. Each F` k is a bounded linear functional (i.e. a bounded linear operator from the vector space to CC).
Theorem	htthlem6 9972	Lemma for htthi 9979. An upper bound of all F` k at a given vector A, when the norms of auxiliary vector sequence f are all 1 or less.
Theorem	htthlem7 9973	Lemma for htthi 9979. Convert upper bound in htthlem6 9972 to an existence condition.
Theorem	htthlem8 9974	Lemma for htthi 9979.
Theorem	htthlem9 9975	Lemma for htthi 9979.
Theorem	htthlem10 9976	Lemma for htthi 9979.
Theorem	htthlem11 9977	Lemma for htthi 9979. Use the Uniform Boundedness Theorem ubthi 9889 to show that the functional F` k is bounded.
Theorem	htthlem12 9978	Lemma for htthi 9979. Linear operator T is bounded.
Theorem	htthi 9979	Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded."
|- X = (BaseSet` U)   &   |- P = (.i` U)   &   |- L = (U LnOp U)   &   |- B = (U BLnOp U)   &   |- U e. CHil   &   |- T e. L   &   |- ((x e. X /\ y e. X) -> ((T` x)Py) = (xP(T` y)))   =>   |- T e. B
Posets and lattices
Definition and basic properties
Syntax	cps 9980	Extend class notation with the class of all posets.
class Poset
Syntax	cspw 9981	Extend class notation with the supremum of an ordered set.
class supw
Syntax	cinf 9982	Extend class notation with the infimum of an ordered set.
class infw
Syntax	cla 9983	Extend class notation with the class of all lattices.
class Lat
Definition	df-ps 9984	Define the class of all posets (partially ordered sets) with weak ordering (e.g. "less than or equal to" instead of "less than"). A poset is a relation which is transitive, reflexive, and antisymmetric.
|- Poset = {r | (Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r))}
Definition	df-spw 9985	Define suprema under weak orderings. Unlike df-sup 5664 for strong orderings, supw is evaluates to a member of the field of R iff the supremum exists. Read R supw A as the R-supremum of set A.
|- supw = {<.<.r, x>., y>. | E.z(z = {w e. U.U.r | (A.v e. x vrw /\ A.v e. U.U.r(A.u e. x urv -> wrv))} /\ y = if(z =/= (/), U.z, ~PU.U.U.r))}
Definition	df-nfw 9986	Define the class of all infima of a weak ordering relation.
|- infw = {<.<.r, x>., y>. | y = (`'r supw x)}
Definition	df-la 9987	Define the class of all lattices, which are posets in which every two elements have upper and lower bounds.
|- Lat = {r e. Poset | A.x e. dom rA.y e. dom r((r supw {x, y}) e. dom r /\ (r infw {x, y}) e. dom r)}
Theorem	isps 9988	The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation.
|- (R e. A -> (R e. Poset <-> (Rel R /\ (R o. R) C_ R /\ (R i^i `'R) = ( _I |` U.U.R))))
Theorem	psrel 9989	A poset is a relation.
|- (A e. Poset -> Rel A)
Theorem	pslem 9990	Lemma for psref 9992 and others.
Theorem	psdmrn 9991	The domain and range of a poset equal its field.
|- (R e. Poset -> (dom R = U.U.R /\ ran R = U.U.R))
Theorem	psref 9992	A poset is reflexive.
|- X = dom R   =>   |- ((R e. Poset /\ A e. X) -> ARA)
Theorem	psrn 9993	The range of a poset equals it domain.
|- X = dom R   =>   |- (R e. Poset -> X = ran R)
Theorem	psasym 9994	A poset is antisymmetric.
|- ((R e. Poset /\ ARB /\ BRA) -> A = B)
Theorem	pstr 9995	A poset is transitive.
|- ((R e. Poset /\ ARB /\ BRC) -> ARC)
Theorem	spwval2 9996	Value of supremum under a weak ordering. Read R supw A as "the R-supremum of A." U.U.R is the field of a relation R by relfld 4419. Unlike df-sup 5664 for strong orderings, the supremum exists iff R supw A belongs to the field.
|- X = U.U.R   &   |- Z = {x e. X | (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy))}   =>   |- ((R e. U /\ A e. W) -> (R supw A) = if(Z =/= (/), U.Z, ~PU.X))
Theorem	spwval3 9997	Value of a supremum.
|- X = U.U.R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. U /\ A e. W /\ E.x e. X ph) -> (R supw A) = U.{x e. X | ph})
Theorem	spwnex3 9998	When the supremum of set A doesn't exist, R supw A isn't in the the field of order relation R.
|- X = U.U.R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> -. (R supw A) e. X)
Theorem	spwmo 9999	A poset has at most one supremum.
|- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- (R e. Poset -> E*x(x e. X /\ ph))
Theorem	spweu 10000	A supremum is unique.
|- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. Poset /\ E.x e. X ph) -> E!x e. X ph)