mmtheorems90 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8901-9000 - Page 90 of 107

Type	Label	Description
Statement
Theorem	hvsubcan2 8901	Vector cancellation law.
|- A e. H~   &   |- B e. H~   =>   |- ((A +h B) +h (A -h B)) = (2 .h A)
Theorem	hvaddcan 8902	Cancellation law for vector addition.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   =>   |- ((A +h B) = (A +h C) <-> B = C)
Theorem	hvsubadd 8903	Relationship between vector subtraction and addition.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   =>   |- ((A -h B) = C <-> (B +h C) = A)
Theorem	hvnegdit 8904	Distribution of negative over subtraction.
|- ((A e. H~ /\ B e. H~) -> (-u1 .h (A -h B)) = (B -h A))
Theorem	hvsubeq0t 8905	If the difference between two vectors is zero, they are equal.
|- ((A e. H~ /\ B e. H~) -> ((A -h B) = 0h <-> A = B))
Theorem	hvaddeq0t 8906	If the sum of two vectors is zero, one is the negative of the other.
|- ((A e. H~ /\ B e. H~) -> ((A +h B) = 0h <-> A = (-u1 .h B)))
Theorem	hvaddcant 8907	Cancellation law for vector addition.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) = (A +h C) <-> B = C))
Theorem	hvaddcan2t 8908	Cancellation law for vector addition.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h C) = (B +h C) <-> A = B))
Theorem	hvmulcant 8909	Cancellation law for scalar multiplication.
|- (((A e. CC /\ B e. H~ /\ C e. H~) /\ A =/= 0) -> ((A .h B) = (A .h C) <-> B = C))
Theorem	hvmulcan2t 8910	Cancellation law for scalar multiplication.
|- (((A e. CC /\ B e. CC /\ C e. H~) /\ C =/= 0h) -> ((A .h C) = (B .h C) <-> A = B))
Theorem	hvsubcant 8911	Cancellation law for vector addition.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A -h B) = (A -h C) <-> B = C))
Theorem	hvsubcan2t 8912	Cancellation law for vector addition.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A -h C) = (B -h C) <-> A = B))
Theorem	hvsub0t 8913	Subtraction of a zero vector.
|- (A e. H~ -> (A -h 0h) = A)
Theorem	hvsubaddt 8914	Relationship between vector subtraction and addition.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A -h B) = C <-> (B +h C) = A))
Theorem	hvaddsub4t 8915	Hilbert vector space addition/subtraction law.
|- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> ((A +h B) = (C +h D) <-> (A -h C) = (D -h B)))
Inner product postulates for a Hilbert space
Axiom	ax-hfi 8916	Inner product maps pairs from H~ to CC.
|- .ih :(H~ X. H~)-->CC
Theorem	hiclt 8917	Closure of inner product.
|- ((A e. H~ /\ B e. H~) -> (A .ih B) e. CC)
Theorem	hicl 8918	Closure inference for inner product.
|- A e. H~   &   |- B e. H~   =>   |- (A .ih B) e. CC
Axiom	ax-his1 8919	Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that *` x is the complex conjugate cjvalt 6713 of x. In the literature, the inner product of A and B is usually written <.A, B>., but our operation notation co 3963 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 2412. Physicists use <.B | A>., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 9747.
|- ((A e. H~ /\ B e. H~) -> (A .ih B) = (*` (B .ih A)))
Axiom	ax-his2 8920	Distributive law for inner product. Postulate (S2) of [Beran] p. 95.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) .ih C) = ((A .ih C) + (B .ih C)))
Axiom	ax-his3 8921	Associative law for inner product. Postulate (S3) of [Beran] p. 95. Warning: Mathematics textbooks usually use our version of the axiom. Physics textbooks, on the other hand, usually replace the left-hand side with (B .ih (A .h C)) (e.g. Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 9747 for why the physics definition is swapped.
|- ((A e. CC /\ B e. H~ /\ C e. H~) -> ((A .h B) .ih C) = (A x. (B .ih C)))
Axiom	ax-his4 8922	Identity law for inner product. Postulate (S4) of [Beran] p. 95.
|- ((A e. H~ /\ A =/= 0h) -> 0 < (A .ih A))
Inner product
Theorem	his5t 8923	Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95.
|- ((A e. CC /\ B e. H~ /\ C e. H~) -> (B .ih (A .h C)) = ((*` A) x. (B .ih C)))
Theorem	his52t 8924	Associative law for inner product.
|- ((A e. CC /\ B e. H~ /\ C e. H~) -> (B .ih ((*` A) .h C)) = (A x. (B .ih C)))
Theorem	his35 8925	Move scalar multiplication to outside of inner product.
|- A e. CC   &   |- B e. CC   &   |- C e. H~   &   |- D e. H~   =>   |- ((A .h C) .ih (B .h D)) = ((A x. (*` B)) x. (C .ih D))
Theorem	his7t 8926	Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> (A .ih (B +h C)) = ((A .ih B) + (A .ih C)))
Theorem	hiassdit 8927	Distributive/associative law for inner product, useful for linearity proofs.
|- (((A e. CC /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> (((A .h B) +h C) .ih D) = ((A x. (B .ih D)) + (C .ih D)))
Theorem	his2subt 8928	Distributive law for inner product of vector subtraction.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A -h B) .ih C) = ((A .ih C) - (B .ih C)))
Theorem	his2sub2t 8929	Distributive law for inner product of vector subtraction.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> (A .ih (B -h C)) = ((A .ih B) - (A .ih C)))
Theorem	hiret 8930	A necessary and sufficient condition for an inner product to be real.
|- ((A e. H~ /\ B e. H~) -> ((A .ih B) e. RR <-> (A .ih B) = (B .ih A)))
Theorem	hiidrclt 8931	Real closure of inner product with self.
|- (A e. H~ -> (A .ih A) e. RR)
Theorem	hi01t 8932	Inner product with the 0 vector.
|- (A e. H~ -> (0h .ih A) = 0)
Theorem	hi02t 8933	Inner product with the 0 vector.
|- (A e. H~ -> (A .ih 0h) = 0)
Theorem	hiidge0t 8934	Inner product with self is not negative.
|- (A e. H~ -> 0 <_ (A .ih A))
Theorem	his6t 8935	Zero inner product with self means vector is zero. Lemma 3.1(S6) of [Beran] p. 95.
|- (A e. H~ -> ((A .ih A) = 0 <-> A = 0h))
Theorem	his1 8936	Conjugate law for inner product. Postulate (S1) of [Beran] p. 95.
|- A e. H~   &   |- B e. H~   =>   |- (A .ih B) = (*` (B .ih A))
Theorem	abshicomt 8937	Commuted inner products have the same absolute values.
|- ((A e. H~ /\ B e. H~) -> (abs` (A .ih B)) = (abs` (B .ih A)))
Theorem	hial0 8938	A vector whose inner product is always zero is zero.
|- (A e. H~ -> (A.x e. H~ (A .ih x) = 0 <-> A = 0h))
Theorem	hial02 8939	A vector whose inner product is always zero is zero.
|- (A e. H~ -> (A.x e. H~ (x .ih A) = 0 <-> A = 0h))
Theorem	hisubcom 8940	Two vector subtractions simultaneously commute in an inner product.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   &   |- D e. H~   =>   |- ((A -h B) .ih (C -h D)) = ((B -h A) .ih (D -h C))
Theorem	hi2eqt 8941	Lemma used to prove equality of vectors.
|- ((A e. H~ /\ B e. H~) -> ((A .ih (A -h B)) = (B .ih (A -h B)) <-> A = B))
Theorem	hial2eqt 8942	Two vectors whose inner product is always equal are equal.
|- ((A e. H~ /\ B e. H~) -> (A.x e. H~ (A .ih x) = (B .ih x) <-> A = B))
Theorem	hial2eq2t 8943	Two vectors whose inner product is always equal are equal.
|- ((A e. H~ /\ B e. H~) -> (A.x e. H~ (x .ih A) = (x .ih B) <-> A = B))
Theorem	orthcom 8944	Orthogonality commutes.
|- ((A e. H~ /\ B e. H~) -> ((A .ih B) = 0 <-> (B .ih A) = 0))
Theorem	normlem0 8945	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.
|- S e. CC   &   |- F e. H~   &   |- G e. H~   =>   |- ((F -h (S .h G)) .ih (F -h (S .h G))) = (((F .ih F) + (-u(*` S) x. (F .ih G))) + ((-uS x. (G .ih F)) + ((S x. (*` S)) x. (G .ih G))))
Theorem	normlem1 8946	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.
|- S e. CC   &   |- F e. H~   &   |- G e. H~   &   |- R e. RR   &   |- (abs` S) = 1   =>   |- ((F -h ((S x. R) .h G)) .ih (F -h ((S x. R) .h G))) = (((F .ih F) + (((*` S) x. -uR) x. (F .ih G))) + (((S x. -uR) x. (G .ih F)) + ((R^2) x.