P. 311 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31001-31100   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	ccauold 31001	Extend class notation with set of Cauchy sequences in Hilbert space.
class Cauchy
Syntax	chli 31002	Extend class notation with convergence relation in Hilbert space.
class ⇝𝑣
Syntax	csh 31003	Extend class notation with set of subspaces of a Hilbert space.
class Sℋ
Syntax	cch 31004	Extend class notation with set of closed subspaces of a Hilbert space.
class Cℋ
Syntax	cort 31005	Extend class notation with orthogonal complement in Cℋ.
class ⊥
Syntax	cph 31006	Extend class notation with subspace sum in Cℋ.
class +ℋ
Syntax	cspn 31007	Extend class notation with subspace span in Cℋ.
class span
Syntax	chj 31008	Extend class notation with join in Cℋ.
class ∨ℋ
Syntax	chsup 31009	Extend class notation with supremum of a collection in Cℋ.
class ∨ℋ
Syntax	c0h 31010	Extend class notation with zero of Cℋ.
class 0ℋ
Syntax	ccm 31011	Extend class notation with the commutes relation on a Hilbert lattice.
class 𝐶ℋ
Syntax	cpjh 31012	Extend class notation with set of projections on a Hilbert space.
class projℎ
Syntax	chos 31013	Extend class notation with sum of Hilbert space operators.
class +op
Syntax	chot 31014	Extend class notation with scalar product of a Hilbert space operator.
class ·op
Syntax	chod 31015	Extend class notation with difference of Hilbert space operators.
class −op
Syntax	chfs 31016	Extend class notation with sum of Hilbert space functionals.
class +fn
Syntax	chft 31017	Extend class notation with scalar product of Hilbert space functional.
class ·fn
Syntax	ch0o 31018	Extend class notation with the Hilbert space zero operator.
class 0hop
Syntax	chio 31019	Extend class notation with Hilbert space identity operator.
class Iop
Syntax	cnop 31020	Extend class notation with the operator norm function.
class normop
Syntax	ccop 31021	Extend class notation with set of continuous Hilbert space operators.
class ContOp
Syntax	clo 31022	Extend class notation with set of linear Hilbert space operators.
class LinOp
Syntax	cbo 31023	Extend class notation with set of bounded linear operators.
class BndLinOp
Syntax	cuo 31024	Extend class notation with set of unitary Hilbert space operators.
class UniOp
Syntax	cho 31025	Extend class notation with set of Hermitian Hilbert space operators.
class HrmOp
Syntax	cnmf 31026	Extend class notation with the functional norm function.
class normfn
Syntax	cnl 31027	Extend class notation with the functional nullspace function.
class null
Syntax	ccnfn 31028	Extend class notation with set of continuous Hilbert space functionals.
class ContFn
Syntax	clf 31029	Extend class notation with set of linear Hilbert space functionals.
class LinFn
Syntax	cado 31030	Extend class notation with Hilbert space adjoint function.
class adjℎ
Syntax	cbr 31031	Extend class notation with the bra of a vector in Dirac bra-ket notation.
class bra
Syntax	ck 31032	Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
class ketbra
Syntax	cleo 31033	Extend class notation with positive operator ordering.
class ≤op
Syntax	cei 31034	Extend class notation with Hilbert space eigenvector function.
class eigvec
Syntax	cel 31035	Extend class notation with Hilbert space eigenvalue function.
class eigval
Syntax	cspc 31036	Extend class notation with the spectrum of an operator.
class Lambda
Syntax	cst 31037	Extend class notation with set of states on a Hilbert lattice.
class States
Syntax	chst 31038	Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
class CHStates
Syntax	ccv 31039	Extend class notation with the covers relation on a Hilbert lattice.
class ⋖ℋ
Syntax	cat 31040	Extend class notation with set of atoms on a Hilbert lattice.
class HAtoms
Syntax	cmd 31041	Extend class notation with the modular pair relation on a Hilbert lattice.
class 𝑀ℋ
Syntax	cdmd 31042	Extend class notation with the dual modular pair relation on a Hilbert lattice.
class 𝑀ℋ*
20.1.2  Preliminary ZFC lemmas
Definition	df-hnorm 31043	Define the function for the norm of a vector of Hilbert space. See normval 31199 for its value and normcl 31200 for its closure. Theorems norm-i-i 31208, norm-ii-i 31212, and norm-iii-i 31214 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.)
⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
Definition	df-hba 31044	Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 31074). Note that ℋ 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 31242. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
Definition	df-h0v 31045	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 31243. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
Definition	df-hvsub 31046*	Define vector subtraction. See hvsubvali 31095 for its value and hvsubcli 31096 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦)))
Definition	df-hlim 31047*	Define the limit relation for Hilbert space. See hlimi 31263 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 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)}
Definition	df-hcau 31048*	Define the set of Cauchy sequences on a Hilbert space. See hcau 31259 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 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥}
Theorem	h2hva 31049	The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ NrmCVec    ⇒   ⊢ +ℎ = ( +𝑣 ‘𝑈)
Theorem	h2hsm 31050	The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ NrmCVec    ⇒   ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈)
Theorem	h2hnm 31051	The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ NrmCVec    ⇒   ⊢ normℎ = (normCV‘𝑈)
Theorem	h2hvs 31052	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 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ ℋ = (BaseSet‘𝑈)    ⇒   ⊢ −ℎ = ( −𝑣 ‘𝑈)
Theorem	h2hmetdval 31053	Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ ℋ = (BaseSet‘𝑈)    &   ⊢ 𝐷 = (IndMet‘𝑈)    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (normℎ‘(𝐴 −ℎ 𝐵)))
Theorem	h2hcau 31054	The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ ℋ = (BaseSet‘𝑈)    &   ⊢ 𝐷 = (IndMet‘𝑈)    ⇒   ⊢ Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ))
Theorem	h2hlm 31055	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 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ ℋ = (BaseSet‘𝑈)    &   ⊢ 𝐷 = (IndMet‘𝑈)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ⇝𝑣 = ((⇝𝑡‘𝐽) ↾ ( ℋ ↑m ℕ))
20.1.3  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 31056 through axhcompl-zf 31073, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ 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 +ℎ, ·ℎ, and ·ih before df-hnorm 31043 above. See also the comment in ax-hilex 31074.
Theorem	axhilex-zf 31056	Derive Axiom ax-hilex 31074 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ ℋ ∈ V
Theorem	axhfvadd-zf 31057	Derive Axiom ax-hfvadd 31075 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ
Theorem	axhvcom-zf 31058	Derive Axiom ax-hvcom 31076 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴))
Theorem	axhvass-zf 31059	Derive Axiom ax-hvass 31077 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)))
Theorem	axhv0cl-zf 31060	Derive Axiom ax-hv0cl 31078 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ 0ℎ ∈ ℋ
Theorem	axhvaddid-zf 31061	Derive Axiom ax-hvaddid 31079 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
Theorem	axhfvmul-zf 31062	Derive Axiom ax-hfvmul 31080 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ
Theorem	axhvmulid-zf 31063	Derive Axiom ax-hvmulid 31081 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)
Theorem	axhvmulass-zf 31064	Derive Axiom ax-hvmulass 31082 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)))
Theorem	axhvdistr1-zf 31065	Derive Axiom ax-hvdistr1 31083 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶)))
Theorem	axhvdistr2-zf 31066	Derive Axiom ax-hvdistr2 31084 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶)))
Theorem	axhvmul0-zf 31067	Derive Axiom ax-hvmul0 31085 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ)
Theorem	axhfi-zf 31068	Derive Axiom ax-hfi 31154 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    &   ⊢ ·ih = (·𝑖OLD‘𝑈)    ⇒   ⊢ ·ih :( ℋ × ℋ)⟶ℂ
Theorem	axhis1-zf 31069	Derive Axiom ax-his1 31157 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    &   ⊢ ·ih = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
Theorem	axhis2-zf 31070	Derive Axiom ax-his2 31158 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    &   ⊢ ·ih = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))
Theorem	axhis3-zf 31071	Derive Axiom ax-his3 31159 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    &   ⊢ ·ih = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))
Theorem	axhis4-zf 31072	Derive Axiom ax-his4 31160 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    &   ⊢ ·ih = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴))
Theorem	axhcompl-zf 31073*	Derive Axiom ax-hcompl 31277 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 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥)
20.1.4  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 31074, ax-hfvadd 31075, ax-hvcom 31076, ax-hvass 31077, ax-hv0cl 31078, ax-hvaddid 31079, ax-hfvmul 31080, ax-hvmulid 31081, ax-hvmulass 31082, ax-hvdistr1 31083, ax-hvdistr2 31084, ax-hvmul0 31085, ax-hfi 31154, ax-his1 31157, ax-his2 31158, ax-his3 31159, ax-his4 31160, and ax-hcompl 31277. The axioms specify the properties of 5 primitive symbols, ℋ, +ℎ, ·ℎ, 0ℎ, and ·ih. If we can prove in ZFC set theory that a class 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ is a complex Hilbert space, i.e. that 𝑈 ∈ CHilOLD, then these axioms can be proved as Theorems axhilex-zf 31056, axhfvadd-zf 31057, axhvcom-zf 31058, axhvass-zf 31059, axhv0cl-zf 31060, axhvaddid-zf 31061, axhfvmul-zf 31062, axhvmulid-zf 31063, axhvmulass-zf 31064, axhvdistr1-zf 31065, axhvdistr2-zf 31066, axhvmul0-zf 31067, axhfi-zf 31068, axhis1-zf 31069, axhis2-zf 31070, axhis3-zf 31071, axhis4-zf 31072, and axhcompl-zf 31073 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 31056.
Axiom	ax-hilex 31074	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, ℋ, which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ ℋ ∈ V
Axiom	ax-hfvadd 31075	Vector addition is an operation on ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ +ℎ :( ℋ × ℋ)⟶ ℋ
Axiom	ax-hvcom 31076	Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴))
Axiom	ax-hvass 31077	Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)))
Axiom	ax-hv0cl 31078	The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ 0ℎ ∈ ℋ
Axiom	ax-hvaddid 31079	Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
Axiom	ax-hfvmul 31080	Scalar multiplication is an operation on ℂ and ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ
Axiom	ax-hvmulid 31081	Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)
Axiom	ax-hvmulass 31082	Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)))
Axiom	ax-hvdistr1 31083	Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶)))
Axiom	ax-hvdistr2 31084	Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶)))
Axiom	ax-hvmul0 31085	Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 31101 and hvsubval 31091). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ)
20.1.5  Vector operations
Theorem	hvmulex 31086	The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
⊢ ·ℎ ∈ V
Theorem	hvaddcl 31087	Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ)
Theorem	hvmulcl 31088	Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ)
Theorem	hvmulcli 31089	Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ
Theorem	hvsubf 31090	Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
⊢ −ℎ :( ℋ × ℋ)⟶ ℋ
Theorem	hvsubval 31091	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 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)))
Theorem	hvsubcl 31092	Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ)
Theorem	hvaddcli 31093	Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ
Theorem	hvcomi 31094	Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)
Theorem	hvsubvali 31095	Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))
Theorem	hvsubcli 31096	Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ
Theorem	ifhvhv0 31097	Prove if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ. (Contributed by David A. Wheeler, 7-Dec-2018.) (New usage is discouraged.)
⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ
Theorem	hvaddlid 31098	Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)
Theorem	hvmul0 31099	Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ)
Theorem	hvmul0or 31100	If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) = 0ℎ ↔ (𝐴 = 0 ∨ 𝐵 = 0ℎ)))