mmtheorems90 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8901-9000 - Page 90 of 108

Type	Label	Description
Statement
Theorem	hvnegid 8901	Addition of negative of a vector to itself.
|- A e. H~   =>   |- (A +h (-u1 .h A)) = 0h
Theorem	hv2neg 8902	Two ways to express the negative of a vector.
|- A e. H~   =>   |- (0h -h A) = (-u1 .h A)
Theorem	hvm1negt 8903	Convert minus one times a scalar product to the negative of the scalar.
|- ((A e. CC /\ B e. H~) -> (-u1 .h (A .h B)) = (-uA .h B))
Theorem	hvaddsubvalt 8904	Value of vector addition in terms of vector subtraction.
|- ((A e. H~ /\ B e. H~) -> (A +h B) = (A -h (-u1 .h B)))
Theorem	hvadd23t 8905	Commutative/associative law.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) +h C) = ((A +h C) +h B))
Theorem	hvadd12t 8906	Commutative/associative law.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> (A +h (B +h C)) = (B +h (A +h C)))
Theorem	hvadd4t 8907	Hilbert vector space addition law.
|- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> ((A +h B) +h (C +h D)) = ((A +h C) +h (B +h D)))
Theorem	hvsub4t 8908	Hilbert vector space addition/subtraction law.
|- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> ((A +h B) -h (C +h D)) = ((A -h C) +h (B -h D)))
Theorem	hvaddsub12t 8909	Commutative/associative law.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> (A +h (B -h C)) = (B +h (A -h C)))
Theorem	hvpncant 8910	Addition/subtraction cancellation law for vectors in Hilbert space.
|- ((A e. H~ /\ B e. H~) -> ((A +h B) -h B) = A)
Theorem	hvpncan2t 8911	Addition/subtraction cancellation law for vectors in Hilbert space.
|- ((A e. H~ /\ B e. H~) -> ((A +h B) -h A) = B)
Theorem	hvaddsubasst 8912	Associativity of sum and difference of Hilbert space vectors.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) -h C) = (A +h (B -h C)))
Theorem	hvpncan3t 8913	Subtraction and addition of equal Hilbert space vectors..
|- ((A e. H~ /\ B e. H~) -> (A +h (B -h A)) = B)
Theorem	hvmulcomt 8914	Scalar multiplication commutative law.
|- ((A e. CC /\ B e. CC /\ C e. H~) -> (A .h (B .h C)) = (B .h (A .h C)))
Theorem	hvmulass 8915	Scalar multiplication associative law.
|- A e. CC   &   |- B e. CC   &   |- C e. H~   =>   |- ((A x. B) .h C) = (A .h (B .h C))
Theorem	hvmulcom 8916	Scalar multiplication commutative law.
|- A e. CC   &   |- B e. CC   &   |- C e. H~   =>   |- (A .h (B .h C)) = (B .h (A .h C))
Theorem	hvmul2neg 8917	Double negative in scalar multiplication.
|- A e. CC   &   |- B e. CC   &   |- C e. H~   =>   |- (-uA .h (-uB .h C)) = (A .h (B .h C))
Theorem	hvsubdistr1t 8918	Scalar multiplication distributive law for subtraction.
|- ((A e. CC /\ B e. H~ /\ C e. H~) -> (A .h (B -h C)) = ((A .h B) -h (A .h C)))
Theorem	hvsubdistr2t 8919	Scalar multiplication distributive law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. H~) -> ((A - B) .h C) = ((A .h C) -h (B .h C)))
Theorem	hvdistr1 8920	Scalar multiplication distributive law.
|- A e. CC   &   |- B e. H~   &   |- C e. H~   =>   |- (A .h (B +h C)) = ((A .h B) +h (A .h C))
Theorem	hvsubdistr1 8921	Scalar multiplication distributive law.
|- A e. CC   &   |- B e. H~   &   |- C e. H~   =>   |- (A .h (B -h C)) = ((A .h B) -h (A .h C))
Theorem	hvass 8922	Hilbert vector space associative law.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   =>   |- ((A +h B) +h C) = (A +h (B +h C))
Theorem	hvadd23 8923	Hilbert vector space commutative/associative law.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   =>   |- ((A +h B) +h C) = ((A +h C) +h B)
Theorem	hvsubass 8924	Hilbert vector space associative law for subtraction.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   =>   |- ((A -h B) -h C) = (A -h (B +h C))
Theorem	hvsub23 8925	Hilbert vector space commutative/associative law.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   =>   |- ((A -h B) -h C) = ((A -h C) -h B)
Theorem	hvadd12 8926	Hilbert vector space commutative/associative law.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   =>   |- (A +h (B +h C)) = (B +h (A +h C))
Theorem	hvadd4 8927	Hilbert vector space addition law.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   &   |- D e. H~   =>   |- ((A +h B) +h (C +h D)) = ((A +h C) +h (B +h D))
Theorem	hvsubsub4 8928	Hilbert vector space addition law.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   &   |- D e. H~   =>   |- ((A -h B) -h (C -h D)) = ((A -h C) -h (B -h D))
Theorem	hvsubsub4t 8929	Hilbert vector space addition/subtraction law.
|- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> ((A -h B) -h (C -h D)) = ((A -h C) -h (B -h D)))
Theorem	hv2timest 8930	Two times a vector.
|- (A e. H~ -> (2 .h A) = (A +h A))
Theorem	hvnegdi 8931	Distribution of negative over subtraction.
|- A e. H~   &   |- B e. H~   =>   |- (-u1 .h (A -h B)) = (B -h A)
Theorem	hvsubeq0 8932	If the difference between two vectors is zero, they are equal.
|- A e. H~   &   |- B e. H~   =>   |- ((A -h B) = 0h <-> A = B)
Theorem	hvsubcan2 8933	Vector cancellation law.
|- A e. H~   &   |- B e. H~   =>   |- ((A +h B) +h (A -h B)) = (2 .h A)
Theorem	hvaddcan 8934	Cancellation law for vector addition.
|- A e. H~   &   |- B e. H~   &   |- C e. H~   =>   |- ((A +h B) = (A +h C) <-> B = C)
Theorem	hvsubadd 8935	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 8936	Distribution of negative over subtraction.
|- ((A e. H~ /\ B e. H~) -> (-u1 .h (A -h B)) = (B -h A))
Theorem	hvsubeq0t 8937	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 8938	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 8939	Cancellation law for vector addition.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) = (A +h C) <-> B = C))
Theorem	hvaddcan2t 8940	Cancellation law for vector addition.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h C) = (B +h C) <-> A = B))
Theorem	hvmulcant 8941	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 8942	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 8943	Cancellation law for vector addition.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A -h B) = (A -h C) <-> B = C))
Theorem	hvsubcan2t 8944	Cancellation law for vector addition.
|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A -h C) = (B -h C) <-> A = B))
Theorem	hvsub0t 8945	Subtraction of a zero vector.
|- (A e. H~ -> (A -h 0h) = A)
Theorem	hvsubaddt 8946	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 8947	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 8948	Inner product maps pairs from H~ to CC.
|- .ih :