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
Theorem	hlcompl 31001	Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
⊢ 𝐷 = (IndMet‘𝑈)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡‘𝐽))
19.7.3  Examples of complex Hilbert spaces
Theorem	cnchl 31002	The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩    ⇒   ⊢ 𝑈 ∈ CHilOLD
19.7.4  Hellinger-Toeplitz Theorem
Theorem	htthlem 31003*	Lemma for htth 31004. The collection 𝐾, which consists of functions 𝐹(𝑧)(𝑤) = ⟨𝑤 ∣ 𝑇(𝑧)⟩ = ⟨𝑇(𝑤) ∣ 𝑧⟩ for each 𝑧 in the unit ball, is a collection of bounded linear functions by ipblnfi 30941, so by the Uniform Boundedness theorem ubth 30959, there is a uniform bound 𝑦 on ∥ 𝐹(𝑥) ∥ for all 𝑥 in the unit ball. Then ∣ 𝑇(𝑥) ∣ ↑2 = ⟨𝑇(𝑥) ∣ 𝑇(𝑥)⟩ = 𝐹(𝑥)( 𝑇(𝑥)) ≤ 𝑦 ∣ 𝑇(𝑥) ∣, so ∣ 𝑇(𝑥) ∣ ≤ 𝑦 and 𝑇 is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑈)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑈 ∈ CHilOLD    &   ⊢ 𝑊 = ⟨⟨ + , · ⟩, abs⟩    &   ⊢ (𝜑 → 𝑇 ∈ 𝐿)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦))    &   ⊢ 𝐹 = (𝑧 ∈ 𝑋 ↦ (𝑤 ∈ 𝑋 ↦ (𝑤𝑃(𝑇‘𝑧))))    &   ⊢ 𝐾 = (𝐹 “ {𝑧 ∈ 𝑋 ∣ (𝑁‘𝑧) ≤ 1})    ⇒   ⊢ (𝜑 → 𝑇 ∈ 𝐵)
Theorem	htth 31004*	Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑈)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵)
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED) This part contains the definitions and theorems used by the Hilbert Space Explorer (HSE), mmhil.html. Because it axiomatizes a single complex Hilbert space whose existence is assumed, its usefulness is limited. For example, it cannot work with real or quaternion Hilbert spaces and it cannot study relationships between two Hilbert spaces. More information can be found on the Hilbert Space Explorer page. Future development should instead work with general Hilbert spaces as defined by df-hil 21694; note that df-hil 21694 uses extensible structures. The intent is for this deprecated section to be deleted once all its theorems have been translated into extensible structure versions (or are not useful). Many of the theorems in this section have already been translated to extensible structure versions, but there is still a lot in this section that might be useful for future reference. It is much easier to translate these by hand from this section than to start from scratch from textbook proofs, since the HSE omits no details.
20.1  Axiomatization of complex pre-Hilbert spaces
20.1.1  Basic Hilbert space definitions
Syntax	chba 31005	Extend class notation with Hilbert vector space.
class ℋ
Syntax	cva 31006	Extend class notation with vector addition in Hilbert space. In the literature, the subscript "h" is omitted, but we need it to avoid ambiguity with complex number addition + caddc 11032.
class +ℎ
Syntax	csm 31007	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 ·ℎ
Syntax	csp 31008	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 to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 4575.
class ·ih
Syntax	cno 31009	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 normℎ
Syntax	c0v 31010	Extend class notation with zero vector in Hilbert space.
class 0ℎ
Syntax	cmv 31011	Extend class notation with vector subtraction in Hilbert space.
class −ℎ
Syntax	ccauold 31012	Extend class notation with set of Cauchy sequences in Hilbert space.
class Cauchy
Syntax	chli 31013	Extend class notation with convergence relation in Hilbert space.
class ⇝𝑣
Syntax	csh 31014	Extend class notation with set of subspaces of a Hilbert space.
class Sℋ
Syntax	cch 31015	Extend class notation with set of closed subspaces of a Hilbert space.
class Cℋ
Syntax	cort 31016	Extend class notation with orthogonal complement in Cℋ.
class ⊥
Syntax	cph 31017	Extend class notation with subspace sum in Cℋ.
class +ℋ
Syntax	cspn 31018	Extend class notation with subspace span in Cℋ.
class span
Syntax	chj 31019	Extend class notation with join in Cℋ.
class ∨ℋ
Syntax	chsup 31020	Extend class notation with supremum of a collection in Cℋ.
class ∨ℋ
Syntax	c0h 31021	Extend class notation with zero of Cℋ.
class 0ℋ
Syntax	ccm 31022	Extend class notation with the commutes relation on a Hilbert lattice.
class 𝐶ℋ
Syntax	cpjh 31023	Extend class notation with set of projections on a Hilbert space.
class projℎ
Syntax	chos 31024	Extend class notation with sum of Hilbert space operators.
class +op
Syntax	chot 31025	Extend class notation with scalar product of a Hilbert space operator.
class ·op
Syntax	chod 31026	Extend class notation with difference of Hilbert space operators.
class −op
Syntax	chfs 31027	Extend class notation with sum of Hilbert space functionals.
class +fn
Syntax	chft 31028	Extend class notation with scalar product of Hilbert space functional.
class ·fn
Syntax	ch0o 31029	Extend class notation with the Hilbert space zero operator.
class 0hop
Syntax	chio 31030	Extend class notation with Hilbert space identity operator.
class Iop
Syntax	cnop 31031	Extend class notation with the operator norm function.
class normop
Syntax	ccop 31032	Extend class notation with set of continuous Hilbert space operators.
class ContOp
Syntax	clo 31033	Extend class notation with set of linear Hilbert space operators.
class LinOp
Syntax	cbo 31034	Extend class notation with set of bounded linear operators.
class BndLinOp
Syntax	cuo 31035	Extend class notation with set of unitary Hilbert space operators.
class UniOp
Syntax	cho 31036	Extend class notation with set of Hermitian Hilbert space operators.
class HrmOp
Syntax	cnmf 31037	Extend class notation with the functional norm function.
class normfn
Syntax	cnl 31038	Extend class notation with the functional nullspace function.
class null
Syntax	ccnfn 31039	Extend class notation with set of continuous Hilbert space functionals.
class ContFn
Syntax	clf 31040	Extend class notation with set of linear Hilbert space functionals.
class LinFn
Syntax	cado 31041	Extend class notation with Hilbert space adjoint function.
class adjℎ
Syntax	cbr 31042	Extend class notation with the bra of a vector in Dirac bra-ket notation.
class bra
Syntax	ck 31043	Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
class ketbra
Syntax	cleo 31044	Extend class notation with positive operator ordering.
class ≤op
Syntax	cei 31045	Extend class notation with Hilbert space eigenvector function.
class eigvec
Syntax	cel 31046	Extend class notation with Hilbert space eigenvalue function.
class eigval
Syntax	cspc 31047	Extend class notation with the spectrum of an operator.
class Lambda
Syntax	cst 31048	Extend class notation with set of states on a Hilbert lattice.
class States
Syntax	chst 31049	Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
class CHStates
Syntax	ccv 31050	Extend class notation with the covers relation on a Hilbert lattice.
class ⋖ℋ
Syntax	cat 31051	Extend class notation with set of atoms on a Hilbert lattice.
class HAtoms
Syntax	cmd 31052	Extend class notation with the modular pair relation on a Hilbert lattice.
class 𝑀ℋ
Syntax	cdmd 31053	Extend class notation with the dual modular pair relation on a Hilbert lattice.
class 𝑀ℋ*
20.1.2  Preliminary ZFC lemmas
Definition	df-hnorm 31054	Define the function for the norm of a vector of Hilbert space. See normval 31210 for its value and normcl 31211 for its closure. Theorems norm-i-i 31219, norm-ii-i 31223, and norm-iii-i 31225 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 31055	Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 31085). 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 31253. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
Definition	df-h0v 31056	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 31254. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
Definition	df-hvsub 31057*	Define vector subtraction. See hvsubvali 31106 for its value and hvsubcli 31107 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 31058*	Define the limit relation for Hilbert space. See hlimi 31274 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 31059*	Define the set of Cauchy sequences on a Hilbert space. See hcau 31270 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 31060	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 31061	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 31062	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 31063	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 31064	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 31065	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 31066	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 31067 through axhcompl-zf 31084, 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 31054 above. See also the comment in ax-hilex 31085.
Theorem	axhilex-zf 31067	Derive Axiom ax-hilex 31085 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 31068	Derive Axiom ax-hfvadd 31086 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 31069	Derive Axiom ax-hvcom 31087 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 31070	Derive Axiom ax-hvass 31088 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 31071	Derive Axiom ax-hv0cl 31089 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 31072	Derive Axiom ax-hvaddid 31090 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 31073	Derive Axiom ax-hfvmul 31091 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 31074	Derive Axiom ax-hvmulid 31092 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 31075	Derive Axiom ax-hvmulass 31093 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 31076	Derive Axiom ax-hvdistr1 31094 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 31077	Derive Axiom ax-hvdistr2 31095 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 31078	Derive Axiom ax-hvmul0 31096 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 31079	Derive Axiom ax-hfi 31165 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 31080	Derive Axiom ax-his1 31168 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 31081	Derive Axiom ax-his2 31169 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 31082	Derive Axiom ax-his3 31170 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 31083	Derive Axiom ax-his4 31171 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 31084*	Derive Axiom ax-hcompl 31288 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 31085, ax-hfvadd 31086, ax-hvcom 31087, ax-hvass 31088, ax-hv0cl 31089, ax-hvaddid 31090, ax-hfvmul 31091, ax-hvmulid 31092, ax-hvmulass 31093, ax-hvdistr1 31094, ax-hvdistr2 31095, ax-hvmul0 31096, ax-hfi 31165, ax-his1 31168, ax-his2 31169, ax-his3 31170, ax-his4 31171, and ax-hcompl 31288. 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 31067, axhfvadd-zf 31068, axhvcom-zf 31069, axhvass-zf 31070, axhv0cl-zf 31071, axhvaddid-zf 31072, axhfvmul-zf 31073, axhvmulid-zf 31074, axhvmulass-zf 31075, axhvdistr1-zf 31076, axhvdistr2-zf 31077, axhvmul0-zf 31078, axhfi-zf 31079, axhis1-zf 31080, axhis2-zf 31081, axhis3-zf 31082, axhis4-zf 31083, and axhcompl-zf 31084 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 31067.
Axiom	ax-hilex 31085	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 31086	Vector addition is an operation on ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ +ℎ :( ℋ × ℋ)⟶ ℋ
Axiom	ax-hvcom 31087	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 31088	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 31089	The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ 0ℎ ∈ ℋ
Axiom	ax-hvaddid 31090	Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
Axiom	ax-hfvmul 31091	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 31092	Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)
Axiom	ax-hvmulass 31093	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 31094	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 31095	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 31096	Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 31112 and hvsubval 31102). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ)
20.1.5  Vector operations
Theorem	hvmulex 31097	The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
⊢ ·ℎ ∈ V
Theorem	hvaddcl 31098	Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ)
Theorem	hvmulcl 31099	Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ)
Theorem	hvmulcli 31100	Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ