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Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxclo 30801 Extend class notation with set of linear Hilbert space operators.
class LinOp
 
Syntaxcbo 30802 Extend class notation with set of bounded linear operators.
class BndLinOp
 
Syntaxcuo 30803 Extend class notation with set of unitary Hilbert space operators.
class UniOp
 
Syntaxcho 30804 Extend class notation with set of Hermitian Hilbert space operators.
class HrmOp
 
Syntaxcnmf 30805 Extend class notation with the functional norm function.
class normfn
 
Syntaxcnl 30806 Extend class notation with the functional nullspace function.
class null
 
Syntaxccnfn 30807 Extend class notation with set of continuous Hilbert space functionals.
class ContFn
 
Syntaxclf 30808 Extend class notation with set of linear Hilbert space functionals.
class LinFn
 
Syntaxcado 30809 Extend class notation with Hilbert space adjoint function.
class adjβ„Ž
 
Syntaxcbr 30810 Extend class notation with the bra of a vector in Dirac bra-ket notation.
class bra
 
Syntaxck 30811 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
class ketbra
 
Syntaxcleo 30812 Extend class notation with positive operator ordering.
class ≀op
 
Syntaxcei 30813 Extend class notation with Hilbert space eigenvector function.
class eigvec
 
Syntaxcel 30814 Extend class notation with Hilbert space eigenvalue function.
class eigval
 
Syntaxcspc 30815 Extend class notation with the spectrum of an operator.
class Lambda
 
Syntaxcst 30816 Extend class notation with set of states on a Hilbert lattice.
class States
 
Syntaxchst 30817 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
class CHStates
 
Syntaxccv 30818 Extend class notation with the covers relation on a Hilbert lattice.
class β‹–β„‹
 
Syntaxcat 30819 Extend class notation with set of atoms on a Hilbert lattice.
class HAtoms
 
Syntaxcmd 30820 Extend class notation with the modular pair relation on a Hilbert lattice.
class 𝑀ℋ
 
Syntaxcdmd 30821 Extend class notation with the dual modular pair relation on a Hilbert lattice.
class 𝑀ℋ*
 
20.1.2  Preliminary ZFC lemmas
 
Definitiondf-hnorm 30822 Define the function for the norm of a vector of Hilbert space. See normval 30978 for its value and normcl 30979 for its closure. Theorems norm-i-i 30987, norm-ii-i 30991, and norm-iii-i 30993 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 30823 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 30853). 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 31021. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
β„‹ = (BaseSetβ€˜βŸ¨βŸ¨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©)
 
Definitiondf-h0v 30824 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 31022. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
0β„Ž = (0vecβ€˜βŸ¨βŸ¨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©)
 
Definitiondf-hvsub 30825* Define vector subtraction. See hvsubvali 30874 for its value and hvsubcli 30875 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 30826* Define the limit relation for Hilbert space. See hlimi 31042 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 30827* Define the set of Cauchy sequences on a Hilbert space. See hcau 31038 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 30828 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 30829 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 30830 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 30831 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 30832 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 30833 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 30834 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 30835 through axhcompl-zf 30852, 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 30822 above. See also the comment in ax-hilex 30853.

 
Theoremaxhilex-zf 30835 Derive Axiom ax-hilex 30853 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 30836 Derive Axiom ax-hfvadd 30854 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 30837 Derive Axiom ax-hvcom 30855 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 30838 Derive Axiom ax-hvass 30856 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 30839 Derive Axiom ax-hv0cl 30857 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 30840 Derive Axiom ax-hvaddid 30858 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 30841 Derive Axiom ax-hfvmul 30859 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 30842 Derive Axiom ax-hvmulid 30860 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 30843 Derive Axiom ax-hvmulass 30861 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 30844 Derive Axiom ax-hvdistr1 30862 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 30845 Derive Axiom ax-hvdistr2 30863 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 30846 Derive Axiom ax-hvmul0 30864 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 30847 Derive Axiom ax-hfi 30933 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 30848 Derive Axiom ax-his1 30936 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 30849 Derive Axiom ax-his2 30937 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 30850 Derive Axiom ax-his3 30938 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 30851 Derive Axiom ax-his4 30939 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 30852* Derive Axiom ax-hcompl 31056 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 30853, ax-hfvadd 30854, ax-hvcom 30855, ax-hvass 30856, ax-hv0cl 30857, ax-hvaddid 30858, ax-hfvmul 30859, ax-hvmulid 30860, ax-hvmulass 30861, ax-hvdistr1 30862, ax-hvdistr2 30863, ax-hvmul0 30864, ax-hfi 30933, ax-his1 30936, ax-his2 30937, ax-his3 30938, ax-his4 30939, and ax-hcompl 31056.

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 30835, axhfvadd-zf 30836, axhvcom-zf 30837, axhvass-zf 30838, axhv0cl-zf 30839, axhvaddid-zf 30840, axhfvmul-zf 30841, axhvmulid-zf 30842, axhvmulass-zf 30843, axhvdistr1-zf 30844, axhvdistr2-zf 30845, axhvmul0-zf 30846, axhfi-zf 30847, axhis1-zf 30848, axhis2-zf 30849, axhis3-zf 30850, axhis4-zf 30851, and axhcompl-zf 30852 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 30835.

 
Axiomax-hilex 30853 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 30854 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 30855 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 30856 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 30857 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
0β„Ž ∈ β„‹
 
Axiomax-hvaddid 30858 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 30859 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 30860 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 30861 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 30862 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 30863 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 30864 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 30880 and hvsubval 30870). (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 30865 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
Β·β„Ž ∈ V
 
Theoremhvaddcl 30866 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 30867 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 30868 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 30869 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 30870 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 30871 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 30872 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 30873 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 30874 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 30875 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 30876 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 30877 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 30878 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 30879 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 30880 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 30881 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 30882 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 30883 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 30884 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 30885 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 30886 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 30887 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 30888 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 30889 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 30890 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 30891 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 30892 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 30893 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 30894 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 30895 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 30896 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 30897 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 30898 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 30899 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 30900 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))
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