mmtheorems98 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 9701-9800 - Page 98 of 108

Type	Label	Description
Statement
Theorem	hosubclt 9701	Mapping of difference of Hilbert space operators.
|- ((S:H~-->H~ /\ T:H~-->H~) -> (S -op T):H~-->H~)
Theorem	hoaddcomt 9702	Commutativity of sum of Hilbert space operators.
|- ((S:H~-->H~ /\ T:H~-->H~) -> (S +op T) = (T +op S))
Theorem	hods 9703	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 9704	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 9705	Commutative/associative law for Hilbert space operator sum that swaps the first two terms.
|- R:H~-->H~   &   |- S:H~-->H~   &   |- T:H~-->H~   =>   |- (R +op (S +op T)) = (S +op (R +op T))
Theorem	hoadd23 9706	Commutative/associative law for Hilbert space operator sum that swaps the second and third terms.
|- R:H~-->H~   &   |- S:H~-->H~   &   |- T:H~-->H~   =>   |- ((R +op S) +op T) = ((R +op T) +op S)
Theorem	hocadddir 9707	Distributive law for Hilbert space operator sum.
|- R:H~-->H~   &   |- S:H~-->H~   &   |- T:H~-->H~   =>   |- ((R +op S) o. T) = ((R o. T) +op (S o. T))
Theorem	hocsubdir 9708	Distributive law for Hilbert space operator difference.
|- R:H~-->H~   &   |- S:H~-->H~   &   |- T:H~-->H~   =>   |- ((R -op S) o. T) = ((R o. T) -op (S o. T))
Theorem	ho2co 9709	Double composition of Hilbert space operators.
|- R:H~-->H~   &   |- S:H~-->H~   &   |- T:H~-->H~   =>   |- (A e. H~ -> (((R o. S) o. T)` A) = (R` (S` (T` A))))
Theorem	hoaddasst 9710	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	hoadd23t 9711	Commutative/associative law for Hilbert space operator sum that swaps the second and third terms.
|- ((R:H~-->H~ /\ S:H~-->H~ /\ T:H~-->H~) -> ((R +op S) +op T) = ((R +op T) +op S))
Theorem	hoadd4t 9712	Rearrangement of 4 terms in a sum of Hilbert space operators.
|- (((R:H~-->H~ /\ S:H~-->H~) /\ (T:H~-->H~ /\ U:H~-->H~)) -> ((R +op S) +op (T +op U)) = ((R +op T) +op (S +op U)))
Theorem	hocsubdirt 9713	Distributive law for Hilbert space operator difference.
|- ((R:H~-->H~ /\ S:H~-->H~ /\ T:H~-->H~) -> ((R -op S) o. T) = ((R o. T) -op (S o. T)))
Theorem	hoaddid1 9714	Sum of a Hilbert space operator with the zero operator.
|- T:H~-->H~   =>   |- (T +op 0hop) = T
Theorem	hodid 9715	Difference of a Hilbert space operator from itself.
|- T:H~-->H~   =>   |- (T -op T) = 0hop
Theorem	ho0co 9716	Composition of the zero operator and a Hilbert space operator.
|- T:H~-->H~   =>   |- (0hop o. T) = 0hop
Theorem	hoid1 9717	Composition of Hilbert space operator with unit identity.
|- T:H~-->H~   =>   |- (T o. Iop ) = T
Theorem	hoid1r 9718	Composition of Hilbert space operator with unit identity.
|- T:H~-->H~   =>   |- ( Iop o. T) = T
Theorem	hoaddid1t 9719	Sum of a Hilbert space operator with the zero operator.
|- (T:H~-->H~ -> (T +op 0hop) = T)
Theorem	hodidt 9720	Difference of a Hilbert space operator from itself.
|- (T:H~-->H~ -> (T -op T) = 0hop)
Theorem	hon0 9721	A Hilbert space operator is not empty.
|- (T:H~-->H~ -> -. T = (/))
Theorem	hodseq 9722	Subtraction and addition of equal Hilbert space operators.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (S +op (T -op S)) = T
Theorem	ho0sub 9723	Subtraction of Hilbert space operators expressed in terms of difference from zero.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (S -op T) = (S +op (0hop -op T))
Theorem	honegsub 9724	Relationship between Hilbert operator addition and subtraction.
|- S:H~-->H~   &   |- T:H~-->H~   =>   |- (S +op (-u1 .op T)) = (S -op T)
Theorem	ho0subt 9725	Subtraction of Hilbert space operators expressed in terms of difference from zero.
|- ((S:H~-->H~ /\ T:H~-->H~) -> (S -op T) = (S +op (0hop -op T)))
Theorem	hosubid1t 9726	The zero operator subtracted from a Hilbert space operator.
|- (T:H~-->H~ -> (T -op 0hop) = T)
Theorem	honegsubt 9727	Relationship between Hilbert space operator addition and subtraction.
|- ((T:H~-->H~ /\ U:H~-->H~) -> (T +op (-u1 .op U)) = (T -op U))
Theorem	homulid2t 9728	An operator equals its scalar product with one.
|- (T:H~-->H~ -> (1 .op T) = T)
Theorem	homco1t 9729	Associative law for scalar product and composition of operators.
|- ((A e. CC /\ T:H~-->H~ /\ U:H~-->H~) -> ((A .op T) o. U) = (A .op (T o. U)))
Theorem	homulasst 9730	Scalar product associative law for Hilbert space operators.
|- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A x. B) .op T) = (A .op (B .op T)))
Theorem	hoadddit 9731	Scalar product distributive law for Hilbert space operators.
|- ((A e. CC /\ T:H~-->H~ /\ U:H~-->H~) -> (A .op (T +op U)) = ((A .op T) +op (A .op U)))
Theorem	hoadddirt 9732	Scalar product reverse distributive law for Hilbert space operators.
|- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) .op T) = ((A .op T) +op (B .op T)))
Theorem	homul12t 9733	Swap first and second factors in a nested operator scalar product.
|- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> (A .op (B .op T)) = (B .op (A .op T)))
Theorem	honegnegt 9734	Double negative of a Hilbert space operator.
|- (T:H~-->H~ -> (-u1 .op (-u1 .op T)) = T)
Theorem	hosubnegt 9735	Relationship between operator subtraction and negative.
|- ((T:H~-->H~ /\ U:H~-->H~) -> (T -op (-u1 .op U)) = (T +op U))
Theorem	hosubdit 9736	Scalar product distributive law for operator difference.
|- ((A e. CC /\ T:H~-->H~ /\ U:H~-->H~) -> (A .op (T -op U)) = ((A .op T) -op (A .op U)))
Theorem	honegdit 9737	Distribution of negative over addition.
|- ((T:H~-->H~ /\ U:H~-->H~) -> (-u1 .op (T +op U)) = ((-u1 .op T) +op (-u1 .op U)))
Theorem	honegsubdit 9738	Distribution of negative over subtraction.
|- ((T:H~-->H~ /\ U:H~-->H~) -> (-u1 .op (T -op U)) = ((-u1 .op T) +op U))
Theorem	honegsubdi2t 9739	Distribution of negative over subtraction.
|- ((T:H~-->H~ /\ U:H~-->H~) -> (-u1 .op (T -op U)) = (U -op T))
Theorem	hosubsub2t 9740	Law for double subtraction of Hilbert space operators.
|- ((S:H~-->H~ /\ T:H~-->H~ /\ U:H~-->H~) -> (S -op (T -op U)) = (S +op (U -op T)))
Theorem	hosub4t 9741	Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators.
|- (((R:H~-->H~ /\ S:H~-->H~) /\ (T:H~-->H~ /\ U:H~-->H~)) -> ((R +op S) -op (T +op U)) = ((R -op T) +op (S -op U)))
Theorem	hosubadd4t 9742	Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators.
|- (((R:H~-->H~ /\ S:H~-->H~) /\ (T:H~-->H~ /\ U:H~-->H~)) -> ((R -op S) -op (T -op U)) = ((R +op U) -op (S +op T)))
Theorem	hoaddsubasst 9743	Associative-type law for addition and subtraction of Hilbert space operators.
|- ((S:H~-->H~ /\ T:H~-->H~ /\ U:H~-->H~) -> ((S +op T) -op U) = (S +op (T -op U)))
Theorem	hoaddsubt 9744	Law for operator addition and subtraction of Hilbert space operators.
|- ((S:H~-->H~ /\ T:H~-->H~ /\ U:H~-->H~) -> ((S +op T) -op U) = ((S -op U) +op T))
Theorem	hosubsubt 9745	Law for double subtraction of Hilbert space operators.
|- ((S:H~-->H~ /\ T:H~-->H~ /\ U:H~-->H~) -> (S -op (T -op U)) = ((S -op T) +op U))
Theorem	hosubsub4t 9746	Law for double subtraction of Hilbert space operators.
|- ((S:H~-->H~ /\ T:H~-->H~ /\ U:H~-->H~) -> ((S -op T) -op U) = (S -op (T +op U)))
Theorem	ho2timest 9747	Two times a Hilbert space operator.
|-