P. 316 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31501-31600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hlvc 31501	Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑊 = (1st ‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝑊 ∈ CVecOLD)
Theorem	hlcmet 31502	The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐷 = (IndMet‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))
Theorem	hlmet 31503	The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐷 = (IndMet‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (Met‘𝑋))
Theorem	hlpar2 31504	The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))
Theorem	hlpar 31505	The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺( -1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))
19.7.2  Standard axioms for a complex Hilbert space
Theorem	hlex 31506	The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    ⇒   ⊢ 𝑋 ∈ V
Theorem	hladdf 31507	Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝐺:(𝑋 × 𝑋)⟶𝑋)
Theorem	hlcom 31508	Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
Theorem	hlass 31509	Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))
Theorem	hl0cl 31510	The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝑍 ∈ 𝑋)
Theorem	hladdid 31511	Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)
Theorem	hlmulf 31512	Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝑆:(ℂ × 𝑋)⟶𝑋)
Theorem	hlmulid 31513	Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
Theorem	hlmulass 31514	Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))
Theorem	hldi 31515	Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
Theorem	hldir 31516	Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
Theorem	hlmul0 31517	Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍)
Theorem	hlipf 31518	Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝑃:(𝑋 × 𝑋)⟶ℂ)
Theorem	hlipcj 31519	Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (∗‘(𝐵𝑃𝐴)))
Theorem	hlipdir 31520	Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))
Theorem	hlipass 31521	Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))
Theorem	hlipgt0 31522	The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 0 < (𝐴𝑃𝐴))
Theorem	hlcompl 31523	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 31524	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 31525*	Lemma for htth 31526. 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 31463, so by the Uniform Boundedness theorem ubth 31481, 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 31526*	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 22016; note that df-hil 22016 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 31527	Extend class notation with Hilbert vector space.
class ℋ
Syntax	cva 31528	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 11207.
class +ℎ
Syntax	csm 31529	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 31530	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 4835.
class ·ih
Syntax	cno 31531	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 31532	Extend class notation with zero vector in Hilbert space.
class 0ℎ
Syntax	cmv 31533	Extend class notation with vector subtraction in Hilbert space.
class −ℎ
Syntax	ccauold 31534	Extend class notation with set of Cauchy sequences in Hilbert space.
class Cauchy
Syntax	chli 31535	Extend class notation with convergence relation in Hilbert space.
class ⇝𝑣
Syntax	csh 31536	Extend class notation with set of subspaces of a Hilbert space.
class Sℋ
Syntax	cch 31537	Extend class notation with set of closed subspaces of a Hilbert space.
class Cℋ
Syntax	cort 31538	Extend class notation with orthogonal complement in Cℋ.
class ⊥
Syntax	cph 31539	Extend class notation with subspace sum in Cℋ.
class +ℋ
Syntax	cspn 31540	Extend class notation with subspace span in Cℋ.
class span
Syntax	chj 31541	Extend class notation with join in Cℋ.
class ∨ℋ
Syntax	chsup 31542	Extend class notation with supremum of a collection in Cℋ.
class ∨ℋ
Syntax	c0h 31543	Extend class notation with zero of Cℋ.
class 0ℋ
Syntax	ccm 31544	Extend class notation with the commutes relation on a Hilbert lattice.
class 𝐶ℋ
Syntax	cpjh 31545	Extend class notation with set of projections on a Hilbert space.
class projℎ
Syntax	chos 31546	Extend class notation with sum of Hilbert space operators.
class +op
Syntax	chot 31547	Extend class notation with scalar product of a Hilbert space operator.
class ·op
Syntax	chod 31548	Extend class notation with difference of Hilbert space operators.
class −op
Syntax	chfs 31549	Extend class notation with sum of Hilbert space functionals.
class +fn
Syntax	chft 31550	Extend class notation with scalar product of Hilbert space functional.
class ·fn
Syntax	ch0o 31551	Extend class notation with the Hilbert space zero operator.
class 0hop
Syntax	chio 31552	Extend class notation with Hilbert space identity operator.
class Iop
Syntax	cnop 31553	Extend class notation with the operator norm function.
class normop
Syntax	ccop 31554	Extend class notation with set of continuous Hilbert space operators.
class ContOp
Syntax	clo 31555	Extend class notation with set of linear Hilbert space operators.
class LinOp
Syntax	cbo 31556	Extend class notation with set of bounded linear operators.
class BndLinOp
Syntax	cuo 31557	Extend class notation with set of unitary Hilbert space operators.
class UniOp
Syntax	cho 31558	Extend class notation with set of Hermitian Hilbert space operators.
class HrmOp
Syntax	cnmf 31559	Extend class notation with the functional norm function.
class normfn
Syntax	cnl 31560	Extend class notation with the functional nullspace function.
class null
Syntax	ccnfn 31561	Extend class notation with set of continuous Hilbert space functionals.
class ContFn
Syntax	clf 31562	Extend class notation with set of linear Hilbert space functionals.
class LinFn
Syntax	cado 31563	Extend class notation with Hilbert space adjoint function.
class adjℎ
Syntax	cbr 31564	Extend class notation with the bra of a vector in Dirac bra-ket notation.
class bra
Syntax	ck 31565	Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
class ketbra
Syntax	cleo 31566	Extend class notation with positive operator ordering.
class ≤op
Syntax	cei 31567	Extend class notation with Hilbert space eigenvector function.
class eigvec
Syntax	cel 31568	Extend class notation with Hilbert space eigenvalue function.
class eigval
Syntax	cspc 31569	Extend class notation with the spectrum of an operator.
class Lambda
Syntax	cst 31570	Extend class notation with set of states on a Hilbert lattice.
class States
Syntax	chst 31571	Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
class CHStates
Syntax	ccv 31572	Extend class notation with the covers relation on a Hilbert lattice.
class ⋖ℋ
Syntax	cat 31573	Extend class notation with set of atoms on a Hilbert lattice.
class HAtoms
Syntax	cmd 31574	Extend class notation with the modular pair relation on a Hilbert lattice.
class 𝑀ℋ
Syntax	cdmd 31575	Extend class notation with the dual modular pair relation on a Hilbert lattice.
class 𝑀ℋ*
20.1.2  Preliminary ZFC lemmas
Definition	df-hnorm 31576	Define the function for the norm of a vector of Hilbert space. See normval 31732 for its value and normcl 31733 for its closure. Theorems norm-i-i 31741, norm-ii-i 31745, and norm-iii-i 31747 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 31577	Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 31607). 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 31775. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
Definition	df-h0v 31578	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 31776. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
⊢ 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
Definition	df-hvsub 31579*	Define vector subtraction. See hvsubvali 31628 for its value and hvsubcli 31629 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 31580*	Define the limit relation for Hilbert space. See hlimi 31796 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 31581*	Define the set of Cauchy sequences on a Hilbert space. See hcau 31792 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 31582	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 31583	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 31584	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 31585	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 31586	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 31587	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 31588	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 31589 through axhcompl-zf 31606, 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 31576 above. See also the comment in ax-hilex 31607.
Theorem	axhilex-zf 31589	Derive Axiom ax-hilex 31607 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 31590	Derive Axiom ax-hfvadd 31608 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 31591	Derive Axiom ax-hvcom 31609 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 31592	Derive Axiom ax-hvass 31610 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 31593	Derive Axiom ax-hv0cl 31611 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 31594	Derive Axiom ax-hvaddid 31612 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 31595	Derive Axiom ax-hfvmul 31613 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 31596	Derive Axiom ax-hvmulid 31614 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 31597	Derive Axiom ax-hvmulass 31615 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 31598	Derive Axiom ax-hvdistr1 31616 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 31599	Derive Axiom ax-hvdistr2 31617 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 31600	Derive Axiom ax-hvmul0 31618 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)