mmtheorems97 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 9601-9700 - Page 97 of 107

Type	Label	Description
Statement
Theorem	pjds 9601	Vector decomposition into sum of projections on orthogonal subspaces.
|- G e. CH   &   |- H e. CH   =>   |- ((A e. (G vH H) /\ G (_ (_|_` H)) -> A = (((proj` G)` A) +h ((proj` H)` A)))
Theorem	pjds3 9602	Vector decomposition into sum of projections on orthogonal subspaces.
|- F e. CH   &   |- G e. CH   &   |- H e. CH   =>   |- (((A e. ((F vH G) vH H) /\ F (_ (_|_` G)) /\ (F (_ (_|_` H) /\ G (_ (_|_` H))) -> A = ((((proj` F)` A) +h ((proj` G)` A)) +h ((proj` H)` A)))
Theorem	pj11t 9603	One-to-one correspondence of projection and subspace.
|- ((G e. CH /\ H e. CH) -> ((proj` G) = (proj` H) <-> G = H))
Theorem	pjmfn 9604	Functionality of the projection function.
|- proj Fn CH
Theorem	pjmf1 9605	The projector function maps one-to-one into the set of Hilbert space operators.
|- proj:CH-1-1->(H~ ^m H~)
Theorem	pjoi0t 9606	The inner product of projections on orthogonal subspaces vanishes.
|- (((G e. CH /\ H e. CH /\ A e. H~) /\ G (_ (_|_` H)) -> (((proj` G)` A) .ih ((proj` H)` A)) = 0)
Theorem	pjoi0 9607	The inner product of projections on orthogonal subspaces vanishes.
|- G e. CH   &   |- H e. CH   &   |- A e. H~   =>   |- (G (_ (_|_` H) -> (((proj` G)` A) .ih ((proj` H)` A)) = 0)
Theorem	pjopyth 9608	Pythagorean theorem for projections on orthogonal subspaces.
|- G e. CH   &   |- H e. CH   &   |- A e. H~   =>   |- (G (_ (_|_` H) -> ((normh` (((proj` G)` A) +h ((proj` H)` A)))^2) = (((normh` ((proj` G)` A))^2) + ((normh` ((proj` H)` A))^2)))
Theorem	pjopytht 9609	Pythagorean theorem for projections on orthogonal subspaces.
|- ((H e. CH /\ G e. CH /\ A e. H~) -> (H (_ (_|_` G) -> ((normh` (((proj` H)` A) +h ((proj` G)` A)))^2) = (((normh` ((proj` H)` A))^2) + ((normh` ((proj` G)` A))^2))))
Theorem	pjnorm 9610	The norm of the projection is less than or equal to the norm.
|- H e. CH   &   |- A e. H~   =>   |- (normh` ((proj` H)` A)) <_ (normh` A)
Theorem	pjpyth 9611	Pythagorean theorem for projections.
|- H e. CH   &   |- A e. H~   =>   |- ((normh` A)^2) = (((normh` ((proj` H)` A))^2) + ((normh` ((proj` (_|_` H))` A))^2))
Theorem	pjnel 9612	If a vector does not belong to subspace, the norm of its projection is less than its norm.
|- H e. CH   &   |- A e. H~   =>   |- (-. A e. H <-> (normh` ((proj` H)` A)) < (normh` A))
Theorem	pjnormt 9613	The norm of the projection is less than or equal to the norm.
|- ((H e. CH /\ A e. H~) -> (normh` ((proj` H)` A)) <_ (normh` A))
Theorem	pjpytht 9614	Pythagorean theorem for projectors.
|- ((H e. CH /\ A e. H~) -> ((normh` A)^2) = (((normh` ((proj` H)` A))^2) + ((normh` ((proj` (_|_` H))` A))^2)))
Theorem	pjnelt 9615	If a vector does not belong to subspace, the norm of its projection is less than its norm.
|- ((H e. CH /\ A e. H~) -> (-. A e. H <-> (normh` ((proj` H)` A)) < (normh` A)))
Theorem	pjnorm2t 9616	A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjcht 9583 yield Theorem 26.3 of [Halmos] p. 44.
|- ((H e. CH /\ A e. H~) -> (A e. H <-> (normh` ((proj` H)` A)) = (normh` A)))
Mayet's equation E_3
Theorem	mayete3 9617	Mayet's equation E3. Part of Theorem 4.1 of [Mayet3] p. 7.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- D e. CH   &   |- F e. CH   &   |- G e. CH   &   |- A (_ (_|_` B)   &   |- A (_ (_|_` C)   &   |- B (_ (_|_` C)   &   |- A (_ (_|_` D)   &   |- B (_ (_|_` F)   &   |- C (_ (_|_` G)   &   |- R = ((A vH B) vH C)   &   |- S = (((A vH D) i^i (B vH F)) i^i (C vH G))   &   |- T = ((D vH F) vH G)   =>   |- (R i^i S) (_ T
Zero and identity operators
Definition	df-h0op 9618	Define the Hilbert space zero operator. See df0op2 9622 for alternate definition.
|- 0hop = (proj` 0H)
Definition	df-iop 9619	Define the Hilbert space identity operator. See dfiop2 9623 for alternate definition.
|- Iop = (proj` H~)
Theorem	ho0valt 9620	Value of the zero Hilbert space operator (null projector). Remark in [Beran] p. 111.
|- (A e. H~ -> (0hop` A) = 0h)
Theorem	ho0f 9621	Functionality of the zero Hilbert space operator.
|- 0hop:H~-->H~
Theorem	df0op2 9622	Alternate definition of Hilbert space zero operator.
|- 0hop = (H~ X. 0H)
Theorem	dfiop2 9623	Alternate definition of Hilbert space identity operator.
|- Iop = (I |` H~)
Theorem	hoif 9624	Functionality of the Hilbert space identity operator.
|- Iop :H~-1-1-onto->H~
Theorem	hoivalt 9625	The value of the Hilbert space identity operator.
|- (A e. H~ -> ( Iop ` A) = A)
Theorem	hoico1t 9626	Composition with the Hilbert space identity operator.
|- (T:H~-->H~ -> (T o. Iop ) = T)
Theorem	hoico2t 9627	Composition with the Hilbert space identity operator.
|- (T:H~-->H~ -> ( Iop o. T) = T)
Operations on Hilbert space operators
Theorem	hoaddclt 9628	The sum of Hilbert space operators is an operator.
|- ((S:H~-->H~ /\ T:H~-->H~) -> (S +op T):H~-->H~)
Theorem	homulclt 9629	The scalar product of a Hilbert space operator is an operator.
|- ((A e. CC /\ T:H~-->H~) -> (A .op T):H~-->H~)
Theorem	hoeqt 9630	Equality of Hilbert space operators.
|- ((T:H~-->H~ /\ U:H~-->H~) -> (A.x e. H~ (T` x) = (U` x) <-> T = U))
Theorem	hoeq 9631	Equality of Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (A.x e. H~ (S` x) = (T` x) <-> S = T)
Theorem	hoscl 9632	Closure of Hilbert space operator sum.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (A e. H~ -> ((S +op T)` A) e. H~)
Theorem	hodcl 9633	Closure of Hilbert space operator difference.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (A e. H~ -> ((S -op T)` A) e. H~)
Theorem	hoco 9634	Composition of Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (A e. H~ -> ((S o. T)` A) = (S` (T` A)))
Theorem	hococl 9635	Closure of composition of Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (A e. H~ -> ((S o. T)` A) e. H~)
Theorem	hocof 9636	Mapping of composition of Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (S o. T):H~-->H~
Theorem	hocofn 9637	Functionality of composition of Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (S o. T) Fn H~
Theorem	hoaddcl 9638	Mapping of sum of Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (S +op T):H~-->H~
Theorem	hosubcl 9639	Mapping of difference of Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (S -op T):H~-->H~
Theorem	hoaddfn 9640	Functionality of sum of Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (S +op T) Fn H~
Theorem	hosubfn 9641	Functionality of difference of Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (S -op T) Fn H~
Theorem	hoaddcom 9642	Commutativity of sum of Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (S +op T) = (T +op S)
Theorem	hosubclt 9643	Mapping of difference of Hilbert space operators.
|- ((S:H~-->H~ /\ T:H~-->H~) -> (S -op T):H~-->H~)
Theorem	hoaddcomt 9644	Commutativity of sum of Hilbert space operators.
|- ((S:H~-->H~ /\ T:H~-->H~) -> (S +op T) = (T +op S))
Theorem	hods 9645	Relationship between Hilbert space operator difference and sum.
|- R:H~-->H~   &   |- S:H~-->H~   &   |- T:H~-->H~   =>   |- ((R -op S) = T <-> (S +op T) = R)
Theorem	hoaddass 9646	Associativity of sum of Hilbert space operators.
|- R:H~-->H~   &   |- S:H~-->H~   &   |- T:H~-->H~   =>   |- ((R +op S) +op T) = (R +op (S +op T))
Theorem	hoadd12 9647	Commutative/associative law for Hilbert space operator sum that swaps the first two terms.
|- R:H~-->H~   &   |- S:H~-->H~   &   |-