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Theorem List for Metamath Proof Explorer - 28701-28800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcsm 28701 Extend class notation with scalar multiplication in Hilbert space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.
class ·

Syntaxcsp 28702 Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of 𝐴 and 𝐵 is usually written 𝐴, 𝐵 but our operation notation allows us to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 4577.
class ·ih

Syntaxcno 28703 Extend class notation with the norm function in Hilbert space. In the literature, the norm of 𝐴 is usually written "|| 𝐴 ||", but we use function notation to take advantage of our existing theorems about functions.
class norm

Syntaxc0v 28704 Extend class notation with zero vector in Hilbert space.
class 0

Syntaxcmv 28705 Extend class notation with vector subtraction in Hilbert space.
class

Syntaxccauold 28706 Extend class notation with set of Cauchy sequences in Hilbert space.
class Cauchy

Syntaxchli 28707 Extend class notation with convergence relation in Hilbert space.
class 𝑣

Syntaxcsh 28708 Extend class notation with set of subspaces of a Hilbert space.
class S

Syntaxcch 28709 Extend class notation with set of closed subspaces of a Hilbert space.
class C

Syntaxcort 28710 Extend class notation with orthogonal complement in C.
class

Syntaxcph 28711 Extend class notation with subspace sum in C.
class +

Syntaxcspn 28712 Extend class notation with subspace span in C.
class span

Syntaxchj 28713 Extend class notation with join in C.
class

Syntaxchsup 28714 Extend class notation with supremum of a collection in C.
class

Syntaxc0h 28715 Extend class notation with zero of C.
class 0

Syntaxccm 28716 Extend class notation with the commutes relation on a Hilbert lattice.
class 𝐶

Syntaxcpjh 28717 Extend class notation with set of projections on a Hilbert space.
class proj

Syntaxchos 28718 Extend class notation with sum of Hilbert space operators.
class +op

Syntaxchot 28719 Extend class notation with scalar product of a Hilbert space operator.
class ·op

Syntaxchod 28720 Extend class notation with difference of Hilbert space operators.
class op

Syntaxchfs 28721 Extend class notation with sum of Hilbert space functionals.
class +fn

Syntaxchft 28722 Extend class notation with scalar product of Hilbert space functional.
class ·fn

Syntaxch0o 28723 Extend class notation with the Hilbert space zero operator.
class 0hop

Syntaxchio 28724 Extend class notation with Hilbert space identity operator.
class Iop

Syntaxcnop 28725 Extend class notation with the operator norm function.
class normop

Syntaxccop 28726 Extend class notation with set of continuous Hilbert space operators.
class ContOp

Syntaxclo 28727 Extend class notation with set of linear Hilbert space operators.
class LinOp

Syntaxcbo 28728 Extend class notation with set of bounded linear operators.
class BndLinOp

Syntaxcuo 28729 Extend class notation with set of unitary Hilbert space operators.
class UniOp

Syntaxcho 28730 Extend class notation with set of Hermitian Hilbert space operators.
class HrmOp

Syntaxcnmf 28731 Extend class notation with the functional norm function.
class normfn

Syntaxcnl 28732 Extend class notation with the functional nullspace function.
class null

Syntaxccnfn 28733 Extend class notation with set of continuous Hilbert space functionals.
class ContFn

Syntaxclf 28734 Extend class notation with set of linear Hilbert space functionals.
class LinFn

Syntaxcbr 28736 Extend class notation with the bra of a vector in Dirac bra-ket notation.
class bra

Syntaxck 28737 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
class ketbra

Syntaxcleo 28738 Extend class notation with positive operator ordering.
class op

Syntaxcei 28739 Extend class notation with Hilbert space eigenvector function.
class eigvec

Syntaxcel 28740 Extend class notation with Hilbert space eigenvalue function.
class eigval

Syntaxcspc 28741 Extend class notation with the spectrum of an operator.
class Lambda

Syntaxcst 28742 Extend class notation with set of states on a Hilbert lattice.
class States

Syntaxchst 28743 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
class CHStates

Syntaxccv 28744 Extend class notation with the covers relation on a Hilbert lattice.
class

Syntaxcat 28745 Extend class notation with set of atoms on a Hilbert lattice.
class HAtoms

Syntaxcmd 28746 Extend class notation with the modular pair relation on a Hilbert lattice.
class 𝑀

Syntaxcdmd 28747 Extend class notation with the dual modular pair relation on a Hilbert lattice.
class 𝑀*

19.1.2  Preliminary ZFC lemmas

Definitiondf-hnorm 28748 Define the function for the norm of a vector of Hilbert space. See normval 28904 for its value and normcl 28905 for its closure. Theorems norm-i-i 28913, norm-ii-i 28917, and norm-iii-i 28919 show it has the expected properties of a norm. In the literature, the norm of 𝐴 is usually written "|| 𝐴 ||", but we use function notation to take advantage of our existing theorems about functions. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))

Definitiondf-hba 28749 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 28779). Note that is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. This definition can be proved independently from those axioms as theorem hhba 28947. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)

Definitiondf-h0v 28750 Define the zero vector of Hilbert space. Note that 0vec is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. It is proved from the axioms as theorem hh0v 28948. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
0 = (0vec‘⟨⟨ + , · ⟩, norm⟩)

Definitiondf-hvsub 28751* Define vector subtraction. See hvsubvali 28800 for its value and hvsubcli 28801 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
= (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 + (-1 · 𝑦)))

Definitiondf-hlim 28752* Define the limit relation for Hilbert space. See hlimi 28968 for its relational expression. Note that 𝑓:ℕ⟶ ℋ is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of converge in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥)}

Definitiondf-hcau 28753* Define the set of Cauchy sequences on a Hilbert space. See hcau 28964 for its membership relation. Note that 𝑓:ℕ⟶ ℋ is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of Cauchy sequence in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
Cauchy = {𝑓 ∈ ( ℋ ↑m ℕ) ∣ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑦) − (𝑓𝑧))) < 𝑥}

Theoremh2hva 28754 The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec        + = ( +𝑣𝑈)

Theoremh2hsm 28755 The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec        · = ( ·𝑠OLD𝑈)

Theoremh2hnm 28756 The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec       norm = (normCV𝑈)

Theoremh2hvs 28757 The vector subtraction operation of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec    &    ℋ = (BaseSet‘𝑈)        = ( −𝑣𝑈)

Theoremh2hmetdval 28758 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec    &    ℋ = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (norm‘(𝐴 𝐵)))

Theoremh2hcau 28759 The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec    &    ℋ = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ))

Theoremh2hlm 28760 The limit sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec    &    ℋ = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)       𝑣 = ((⇝𝑡𝐽) ↾ ( ℋ ↑m ℕ))

19.1.3  Derive the Hilbert space axioms from ZFC set theory

Before introducing the 18 axioms for Hilbert space, we first prove them as the conclusions of theorems axhilex-zf 28761 through axhcompl-zf 28778, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space 𝑈 = ⟨⟨ + , · ⟩, norm that satisfies their hypotheses, then we can derive the Hilbert space axioms as theorems of ZFC set theory. In practice, in order to use these theorems to convert the Hilbert Space explorer to a ZFC-only subtheory, we would also have to provide definitions for the 3 (otherwise primitive) class constants +, ·, and ·ih before df-hnorm 28748 above. See also the comment in ax-hilex 28779.

Theoremaxhilex-zf 28761 Derive axiom ax-hilex 28779 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD        ℋ ∈ V

Theoremaxhfvadd-zf 28762 Derive axiom ax-hfvadd 28780 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD        + :( ℋ × ℋ)⟶ ℋ

Theoremaxhvcom-zf 28763 Derive axiom ax-hvcom 28781 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremaxhvass-zf 28764 Derive axiom ax-hvass 28782 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremaxhv0cl-zf 28765 Derive axiom ax-hv0cl 28783 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       0 ∈ ℋ

Theoremaxhvaddid-zf 28766 Derive axiom ax-hvaddid 28784 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)

Theoremaxhfvmul-zf 28767 Derive axiom ax-hfvmul 28785 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD        · :(ℂ × ℋ)⟶ ℋ

Theoremaxhvmulid-zf 28768 Derive axiom ax-hvmulid 28786 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       (𝐴 ∈ ℋ → (1 · 𝐴) = 𝐴)

Theoremaxhvmulass-zf 28769 Derive axiom ax-hvmulass 28787 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Theoremaxhvdistr1-zf 28770 Derive axiom ax-hvdistr1 28788 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Theoremaxhvdistr2-zf 28771 Derive axiom ax-hvdistr2 28789 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Theoremaxhvmul0-zf 28772 Derive axiom ax-hvmul0 28790 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       (𝐴 ∈ ℋ → (0 · 𝐴) = 0)

Theoremaxhfi-zf 28773 Derive axiom ax-hfi 28859 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)        ·ih :( ℋ × ℋ)⟶ℂ

Theoremaxhis1-zf 28774 Derive axiom ax-his1 28862 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))

Theoremaxhis2-zf 28775 Derive axiom ax-his2 28863 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))

Theoremaxhis3-zf 28776 Derive axiom ax-his3 28864 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))

Theoremaxhis4-zf 28777 Derive axiom ax-his4 28865 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)       ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (𝐴 ·ih 𝐴))

Theoremaxhcompl-zf 28778* Derive axiom ax-hcompl 28982 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹𝑣 𝑥)

19.1.4  Introduce the vector space axioms for a Hilbert space

Here we introduce the axioms a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. The 18 axioms for a complex Hilbert space consist of ax-hilex 28779, ax-hfvadd 28780, ax-hvcom 28781, ax-hvass 28782, ax-hv0cl 28783, ax-hvaddid 28784, ax-hfvmul 28785, ax-hvmulid 28786, ax-hvmulass 28787, ax-hvdistr1 28788, ax-hvdistr2 28789, ax-hvmul0 28790, ax-hfi 28859, ax-his1 28862, ax-his2 28863, ax-his3 28864, ax-his4 28865, and ax-hcompl 28982.

The axioms specify the properties of 5 primitive symbols, , +, ·, 0, and ·ih.

If we can prove in ZFC set theory that a class 𝑈 = ⟨⟨ + , · ⟩, norm is a complex Hilbert space, i.e. that 𝑈 ∈ CHilOLD, then these axioms can be proved as theorems axhilex-zf 28761, axhfvadd-zf 28762, axhvcom-zf 28763, axhvass-zf 28764, axhv0cl-zf 28765, axhvaddid-zf 28766, axhfvmul-zf 28767, axhvmulid-zf 28768, axhvmulass-zf 28769, axhvdistr1-zf 28770, axhvdistr2-zf 28771, axhvmul0-zf 28772, axhfi-zf 28773, axhis1-zf 28774, axhis2-zf 28775, axhis3-zf 28776, axhis4-zf 28777, and axhcompl-zf 28778 respectively. In that case, the theorems of the Hilbert Space Explorer will become theorems of ZFC set theory. See also the comments in axhilex-zf 28761.

Axiomax-hilex 28779 This is our first axiom for a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. We assume that there exists a primitive class, , which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
ℋ ∈ V

Axiomax-hfvadd 28780 Vector addition is an operation on . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
+ :( ℋ × ℋ)⟶ ℋ

Axiomax-hvcom 28781 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Axiomax-hvass 28782 Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Axiomax-hv0cl 28783 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
0 ∈ ℋ

Axiomax-hvaddid 28784 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)

Axiomax-hfvmul 28785 Scalar multiplication is an operation on and . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
· :(ℂ × ℋ)⟶ ℋ

Axiomax-hvmulid 28786 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (1 · 𝐴) = 𝐴)

Axiomax-hvmulass 28787 Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Axiomax-hvdistr1 28788 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Axiomax-hvdistr2 28789 Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Axiomax-hvmul0 28790 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 28806 and hvsubval 28796). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 · 𝐴) = 0)

19.1.5  Vector operations

Theoremhvmulex 28791 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
· ∈ V

Theoremhvaddcl 28792 Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) ∈ ℋ)

Theoremhvmulcl 28793 Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 · 𝐵) ∈ ℋ)

Theoremhvmulcli 28794 Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℋ       (𝐴 · 𝐵) ∈ ℋ

Theoremhvsubf 28795 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
:( ℋ × ℋ)⟶ ℋ

Theoremhvsubval 28796 Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) = (𝐴 + (-1 · 𝐵)))

Theoremhvsubcl 28797 Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) ∈ ℋ)

Theoremhvaddcli 28798 Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 + 𝐵) ∈ ℋ

Theoremhvcomi 28799 Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 + 𝐵) = (𝐵 + 𝐴)

Theoremhvsubvali 28800 Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 𝐵) = (𝐴 + (-1 · 𝐵))

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