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Theorem List for Metamath Proof Explorer - 30201-30300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcbo 30201 Extend class notation with set of bounded linear operators.
class BndLinOp
 
Syntaxcuo 30202 Extend class notation with set of unitary Hilbert space operators.
class UniOp
 
Syntaxcho 30203 Extend class notation with set of Hermitian Hilbert space operators.
class HrmOp
 
Syntaxcnmf 30204 Extend class notation with the functional norm function.
class normfn
 
Syntaxcnl 30205 Extend class notation with the functional nullspace function.
class null
 
Syntaxccnfn 30206 Extend class notation with set of continuous Hilbert space functionals.
class ContFn
 
Syntaxclf 30207 Extend class notation with set of linear Hilbert space functionals.
class LinFn
 
Syntaxcado 30208 Extend class notation with Hilbert space adjoint function.
class adjβ„Ž
 
Syntaxcbr 30209 Extend class notation with the bra of a vector in Dirac bra-ket notation.
class bra
 
Syntaxck 30210 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
class ketbra
 
Syntaxcleo 30211 Extend class notation with positive operator ordering.
class ≀op
 
Syntaxcei 30212 Extend class notation with Hilbert space eigenvector function.
class eigvec
 
Syntaxcel 30213 Extend class notation with Hilbert space eigenvalue function.
class eigval
 
Syntaxcspc 30214 Extend class notation with the spectrum of an operator.
class Lambda
 
Syntaxcst 30215 Extend class notation with set of states on a Hilbert lattice.
class States
 
Syntaxchst 30216 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
class CHStates
 
Syntaxccv 30217 Extend class notation with the covers relation on a Hilbert lattice.
class β‹–β„‹
 
Syntaxcat 30218 Extend class notation with set of atoms on a Hilbert lattice.
class HAtoms
 
Syntaxcmd 30219 Extend class notation with the modular pair relation on a Hilbert lattice.
class 𝑀ℋ
 
Syntaxcdmd 30220 Extend class notation with the dual modular pair relation on a Hilbert lattice.
class 𝑀ℋ*
 
20.1.2  Preliminary ZFC lemmas
 
Definitiondf-hnorm 30221 Define the function for the norm of a vector of Hilbert space. See normval 30377 for its value and normcl 30378 for its closure. Theorems norm-i-i 30386, norm-ii-i 30390, and norm-iii-i 30392 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.)
normβ„Ž = (π‘₯ ∈ dom dom Β·ih ↦ (βˆšβ€˜(π‘₯ Β·ih π‘₯)))
 
Definitiondf-hba 30222 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 30252). Note that β„‹ 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 30420. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
β„‹ = (BaseSetβ€˜βŸ¨βŸ¨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©)
 
Definitiondf-h0v 30223 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 30421. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
0β„Ž = (0vecβ€˜βŸ¨βŸ¨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©)
 
Definitiondf-hvsub 30224* Define vector subtraction. See hvsubvali 30273 for its value and hvsubcli 30274 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
βˆ’β„Ž = (π‘₯ ∈ β„‹, 𝑦 ∈ β„‹ ↦ (π‘₯ +β„Ž (-1 Β·β„Ž 𝑦)))
 
Definitiondf-hlim 30225* Define the limit relation for Hilbert space. See hlimi 30441 for its relational expression. Note that 𝑓:β„•βŸΆ β„‹ 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 = {βŸ¨π‘“, π‘€βŸ© ∣ ((𝑓:β„•βŸΆ β„‹ ∧ 𝑀 ∈ β„‹) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ β„• βˆ€π‘§ ∈ (β„€β‰₯β€˜π‘¦)(normβ„Žβ€˜((π‘“β€˜π‘§) βˆ’β„Ž 𝑀)) < π‘₯)}
 
Definitiondf-hcau 30226* Define the set of Cauchy sequences on a Hilbert space. See hcau 30437 for its membership relation. Note that 𝑓:β„•βŸΆ β„‹ 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 ∈ ( β„‹ ↑m β„•) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ β„• βˆ€π‘§ ∈ (β„€β‰₯β€˜π‘¦)(normβ„Žβ€˜((π‘“β€˜π‘¦) βˆ’β„Ž (π‘“β€˜π‘§))) < π‘₯}
 
Theoremh2hva 30227 The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ NrmCVec    β‡’    +β„Ž = ( +𝑣 β€˜π‘ˆ)
 
Theoremh2hsm 30228 The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ NrmCVec    β‡’    Β·β„Ž = ( ·𝑠OLD β€˜π‘ˆ)
 
Theoremh2hnm 30229 The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ NrmCVec    β‡’   normβ„Ž = (normCVβ€˜π‘ˆ)
 
Theoremh2hvs 30230 The vector subtraction operation of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ NrmCVec    &    β„‹ = (BaseSetβ€˜π‘ˆ)    β‡’    βˆ’β„Ž = ( βˆ’π‘£ β€˜π‘ˆ)
 
Theoremh2hmetdval 30231 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ NrmCVec    &    β„‹ = (BaseSetβ€˜π‘ˆ)    &   π· = (IndMetβ€˜π‘ˆ)    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴𝐷𝐡) = (normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)))
 
Theoremh2hcau 30232 The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ NrmCVec    &    β„‹ = (BaseSetβ€˜π‘ˆ)    &   π· = (IndMetβ€˜π‘ˆ)    β‡’   Cauchy = ((Cauβ€˜π·) ∩ ( β„‹ ↑m β„•))
 
Theoremh2hlm 30233 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.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ NrmCVec    &    β„‹ = (BaseSetβ€˜π‘ˆ)    &   π· = (IndMetβ€˜π‘ˆ)    &   π½ = (MetOpenβ€˜π·)    β‡’    ⇝𝑣 = ((β‡π‘‘β€˜π½) β†Ύ ( β„‹ ↑m β„•))
 
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 30234 through axhcompl-zf 30251, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ© 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 +β„Ž, Β·β„Ž, and Β·ih before df-hnorm 30221 above. See also the comment in ax-hilex 30252.

 
Theoremaxhilex-zf 30234 Derive Axiom ax-hilex 30252 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’    β„‹ ∈ V
 
Theoremaxhfvadd-zf 30235 Derive Axiom ax-hfvadd 30253 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’    +β„Ž :( β„‹ Γ— β„‹)⟢ β„‹
 
Theoremaxhvcom-zf 30236 Derive Axiom ax-hvcom 30254 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž 𝐡) = (𝐡 +β„Ž 𝐴))
 
Theoremaxhvass-zf 30237 Derive Axiom ax-hvass 30255 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) +β„Ž 𝐢) = (𝐴 +β„Ž (𝐡 +β„Ž 𝐢)))
 
Theoremaxhv0cl-zf 30238 Derive Axiom ax-hv0cl 30256 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   0β„Ž ∈ β„‹
 
Theoremaxhvaddid-zf 30239 Derive Axiom ax-hvaddid 30257 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   (𝐴 ∈ β„‹ β†’ (𝐴 +β„Ž 0β„Ž) = 𝐴)
 
Theoremaxhfvmul-zf 30240 Derive Axiom ax-hfvmul 30258 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’    Β·β„Ž :(β„‚ Γ— β„‹)⟢ β„‹
 
Theoremaxhvmulid-zf 30241 Derive Axiom ax-hvmulid 30259 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   (𝐴 ∈ β„‹ β†’ (1 Β·β„Ž 𝐴) = 𝐴)
 
Theoremaxhvmulass-zf 30242 Derive Axiom ax-hvmulass 30260 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β· 𝐡) Β·β„Ž 𝐢) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)))
 
Theoremaxhvdistr1-zf 30243 Derive Axiom ax-hvdistr1 30261 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) +β„Ž (𝐴 Β·β„Ž 𝐢)))
 
Theoremaxhvdistr2-zf 30244 Derive Axiom ax-hvdistr2 30262 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 + 𝐡) Β·β„Ž 𝐢) = ((𝐴 Β·β„Ž 𝐢) +β„Ž (𝐡 Β·β„Ž 𝐢)))
 
Theoremaxhvmul0-zf 30245 Derive Axiom ax-hvmul0 30263 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   (𝐴 ∈ β„‹ β†’ (0 Β·β„Ž 𝐴) = 0β„Ž)
 
Theoremaxhfi-zf 30246 Derive Axiom ax-hfi 30332 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    &    Β·ih = (·𝑖OLDβ€˜π‘ˆ)    β‡’    Β·ih :( β„‹ Γ— β„‹)βŸΆβ„‚
 
Theoremaxhis1-zf 30247 Derive Axiom ax-his1 30335 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    &    Β·ih = (·𝑖OLDβ€˜π‘ˆ)    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 Β·ih 𝐡) = (βˆ—β€˜(𝐡 Β·ih 𝐴)))
 
Theoremaxhis2-zf 30248 Derive Axiom ax-his2 30336 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    &    Β·ih = (·𝑖OLDβ€˜π‘ˆ)    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) Β·ih 𝐢) = ((𝐴 Β·ih 𝐢) + (𝐡 Β·ih 𝐢)))
 
Theoremaxhis3-zf 30249 Derive Axiom ax-his3 30337 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    &    Β·ih = (·𝑖OLDβ€˜π‘ˆ)    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β·β„Ž 𝐡) Β·ih 𝐢) = (𝐴 Β· (𝐡 Β·ih 𝐢)))
 
Theoremaxhis4-zf 30250 Derive Axiom ax-his4 30338 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    &    Β·ih = (·𝑖OLDβ€˜π‘ˆ)    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (𝐴 Β·ih 𝐴))
 
Theoremaxhcompl-zf 30251* Derive Axiom ax-hcompl 30455 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.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   (𝐹 ∈ Cauchy β†’ βˆƒπ‘₯ ∈ β„‹ 𝐹 ⇝𝑣 π‘₯)
 
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 30252, ax-hfvadd 30253, ax-hvcom 30254, ax-hvass 30255, ax-hv0cl 30256, ax-hvaddid 30257, ax-hfvmul 30258, ax-hvmulid 30259, ax-hvmulass 30260, ax-hvdistr1 30261, ax-hvdistr2 30262, ax-hvmul0 30263, ax-hfi 30332, ax-his1 30335, ax-his2 30336, ax-his3 30337, ax-his4 30338, and ax-hcompl 30455.

The axioms specify the properties of 5 primitive symbols, β„‹, +β„Ž, Β·β„Ž, 0β„Ž, and Β·ih.

If we can prove in ZFC set theory that a class π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ© is a complex Hilbert space, i.e. that π‘ˆ ∈ CHilOLD, then these axioms can be proved as Theorems axhilex-zf 30234, axhfvadd-zf 30235, axhvcom-zf 30236, axhvass-zf 30237, axhv0cl-zf 30238, axhvaddid-zf 30239, axhfvmul-zf 30240, axhvmulid-zf 30241, axhvmulass-zf 30242, axhvdistr1-zf 30243, axhvdistr2-zf 30244, axhvmul0-zf 30245, axhfi-zf 30246, axhis1-zf 30247, axhis2-zf 30248, axhis3-zf 30249, axhis4-zf 30250, and axhcompl-zf 30251 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 30234.

 
Axiomax-hilex 30252 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, β„‹, which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
β„‹ ∈ V
 
Axiomax-hfvadd 30253 Vector addition is an operation on β„‹. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
+β„Ž :( β„‹ Γ— β„‹)⟢ β„‹
 
Axiomax-hvcom 30254 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž 𝐡) = (𝐡 +β„Ž 𝐴))
 
Axiomax-hvass 30255 Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) +β„Ž 𝐢) = (𝐴 +β„Ž (𝐡 +β„Ž 𝐢)))
 
Axiomax-hv0cl 30256 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
0β„Ž ∈ β„‹
 
Axiomax-hvaddid 30257 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 +β„Ž 0β„Ž) = 𝐴)
 
Axiomax-hfvmul 30258 Scalar multiplication is an operation on β„‚ and β„‹. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
Β·β„Ž :(β„‚ Γ— β„‹)⟢ β„‹
 
Axiomax-hvmulid 30259 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (1 Β·β„Ž 𝐴) = 𝐴)
 
Axiomax-hvmulass 30260 Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β· 𝐡) Β·β„Ž 𝐢) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)))
 
Axiomax-hvdistr1 30261 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) +β„Ž (𝐴 Β·β„Ž 𝐢)))
 
Axiomax-hvdistr2 30262 Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 + 𝐡) Β·β„Ž 𝐢) = ((𝐴 Β·β„Ž 𝐢) +β„Ž (𝐡 Β·β„Ž 𝐢)))
 
Axiomax-hvmul0 30263 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 30279 and hvsubval 30269). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (0 Β·β„Ž 𝐴) = 0β„Ž)
 
20.1.5  Vector operations
 
Theoremhvmulex 30264 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
Β·β„Ž ∈ V
 
Theoremhvaddcl 30265 Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž 𝐡) ∈ β„‹)
 
Theoremhvmulcl 30266 Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 Β·β„Ž 𝐡) ∈ β„‹)
 
Theoremhvmulcli 30267 Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‹    β‡’   (𝐴 Β·β„Ž 𝐡) ∈ β„‹
 
Theoremhvsubf 30268 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
βˆ’β„Ž :( β„‹ Γ— β„‹)⟢ β„‹
 
Theoremhvsubval 30269 Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 βˆ’β„Ž 𝐡) = (𝐴 +β„Ž (-1 Β·β„Ž 𝐡)))
 
Theoremhvsubcl 30270 Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 βˆ’β„Ž 𝐡) ∈ β„‹)
 
Theoremhvaddcli 30271 Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 +β„Ž 𝐡) ∈ β„‹
 
Theoremhvcomi 30272 Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 +β„Ž 𝐡) = (𝐡 +β„Ž 𝐴)
 
Theoremhvsubvali 30273 Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 βˆ’β„Ž 𝐡) = (𝐴 +β„Ž (-1 Β·β„Ž 𝐡))
 
Theoremhvsubcli 30274 Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 βˆ’β„Ž 𝐡) ∈ β„‹
 
Theoremifhvhv0 30275 Prove if(𝐴 ∈ β„‹, 𝐴, 0β„Ž) ∈ β„‹. (Contributed by David A. Wheeler, 7-Dec-2018.) (New usage is discouraged.)
if(𝐴 ∈ β„‹, 𝐴, 0β„Ž) ∈ β„‹
 
Theoremhvaddlid 30276 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (0β„Ž +β„Ž 𝐴) = 𝐴)
 
Theoremhvmul0 30277 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‚ β†’ (𝐴 Β·β„Ž 0β„Ž) = 0β„Ž)
 
Theoremhvmul0or 30278 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 Β·β„Ž 𝐡) = 0β„Ž ↔ (𝐴 = 0 ∨ 𝐡 = 0β„Ž)))
 
Theoremhvsubid 30279 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 βˆ’β„Ž 𝐴) = 0β„Ž)
 
Theoremhvnegid 30280 Addition of negative of a vector to itself. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 +β„Ž (-1 Β·β„Ž 𝐴)) = 0β„Ž)
 
Theoremhv2neg 30281 Two ways to express the negative of a vector. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (0β„Ž βˆ’β„Ž 𝐴) = (-1 Β·β„Ž 𝐴))
 
Theoremhvaddlidi 30282 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   (0β„Ž +β„Ž 𝐴) = 𝐴
 
Theoremhvnegidi 30283 Addition of negative of a vector to itself. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   (𝐴 +β„Ž (-1 Β·β„Ž 𝐴)) = 0β„Ž
 
Theoremhv2negi 30284 Two ways to express the negative of a vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   (0β„Ž βˆ’β„Ž 𝐴) = (-1 Β·β„Ž 𝐴)
 
Theoremhvm1neg 30285 Convert minus one times a scalar product to the negative of the scalar. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ (-1 Β·β„Ž (𝐴 Β·β„Ž 𝐡)) = (-𝐴 Β·β„Ž 𝐡))
 
Theoremhvaddsubval 30286 Value of vector addition in terms of vector subtraction. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž 𝐡) = (𝐴 βˆ’β„Ž (-1 Β·β„Ž 𝐡)))
 
Theoremhvadd32 30287 Commutative/associative law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) +β„Ž 𝐢) = ((𝐴 +β„Ž 𝐢) +β„Ž 𝐡))
 
Theoremhvadd12 30288 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 +β„Ž (𝐡 +β„Ž 𝐢)) = (𝐡 +β„Ž (𝐴 +β„Ž 𝐢)))
 
Theoremhvadd4 30289 Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
(((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 +β„Ž 𝐡) +β„Ž (𝐢 +β„Ž 𝐷)) = ((𝐴 +β„Ž 𝐢) +β„Ž (𝐡 +β„Ž 𝐷)))
 
Theoremhvsub4 30290 Hilbert vector space addition/subtraction law. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
(((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž (𝐢 +β„Ž 𝐷)) = ((𝐴 βˆ’β„Ž 𝐢) +β„Ž (𝐡 βˆ’β„Ž 𝐷)))
 
Theoremhvaddsub12 30291 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 +β„Ž (𝐡 βˆ’β„Ž 𝐢)) = (𝐡 +β„Ž (𝐴 βˆ’β„Ž 𝐢)))
 
Theoremhvpncan 30292 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž 𝐡) = 𝐴)
 
Theoremhvpncan2 30293 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž 𝐴) = 𝐡)
 
Theoremhvaddsubass 30294 Associativity of sum and difference of Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž 𝐢) = (𝐴 +β„Ž (𝐡 βˆ’β„Ž 𝐢)))
 
Theoremhvpncan3 30295 Subtraction and addition of equal Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž (𝐡 βˆ’β„Ž 𝐴)) = 𝐡)
 
Theoremhvmulcom 30296 Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)) = (𝐡 Β·β„Ž (𝐴 Β·β„Ž 𝐢)))
 
Theoremhvsubass 30297 Hilbert vector space associative law for subtraction. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž 𝐢) = (𝐴 βˆ’β„Ž (𝐡 +β„Ž 𝐢)))
 
Theoremhvsub32 30298 Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž 𝐢) = ((𝐴 βˆ’β„Ž 𝐢) βˆ’β„Ž 𝐡))
 
Theoremhvmulassi 30299 Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‹    β‡’   ((𝐴 Β· 𝐡) Β·β„Ž 𝐢) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢))
 
Theoremhvmulcomi 30300 Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‹    β‡’   (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)) = (𝐡 Β·β„Ž (𝐴 Β·β„Ž 𝐢))
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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47852
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