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Theorem List for Metamath Proof Explorer - 27801-27900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcel 27801 Extend class notation with Hilbert space eigenvalue function.
class eigval

Syntaxcspc 27802 Extend class notation with the spectrum of an operator.
class Lambda

Syntaxcst 27803 Extend class notation with set of states on a Hilbert lattice.
class States

Syntaxchst 27804 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
class CHStates

Syntaxccv 27805 Extend class notation with the covers relation on a Hilbert lattice.
class

Syntaxcat 27806 Extend class notation with set of atoms on a Hilbert lattice.
class HAtoms

Syntaxcmd 27807 Extend class notation with the modular pair relation on a Hilbert lattice.
class 𝑀

Syntaxcdmd 27808 Extend class notation with the dual modular pair relation on a Hilbert lattice.
class 𝑀*

19.1.2  Preliminary ZFC lemmas

Definitiondf-hnorm 27809 Define the function for the norm of a vector of Hilbert space. See normval 27965 for its value and normcl 27966 for its closure. Theorems norm-i-i 27974, norm-ii-i 27978, and norm-iii-i 27980 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.)
norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))

Definitiondf-hba 27810 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 27840). Note that 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 28008. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)

Definitiondf-h0v 27811 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 28009. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
0 = (0vec‘⟨⟨ + , · ⟩, norm⟩)

Definitiondf-hvsub 27812* Define vector subtraction. See hvsubvali 27861 for its value and hvsubcli 27862 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
= (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 + (-1 · 𝑦)))

Definitiondf-hlim 27813* Define the limit relation for Hilbert space. See hlimi 28029 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 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥)}

Definitiondf-hcau 27814* Define the set of Cauchy sequences on a Hilbert space. See hcau 28025 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 ∈ (ℤ𝑦)(norm‘((𝑓𝑦) − (𝑓𝑧))) < 𝑥}

Theoremh2hva 27815 The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec        + = ( +𝑣𝑈)

Theoremh2hsm 27816 The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec        · = ( ·𝑠OLD𝑈)

Theoremh2hnm 27817 The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec       norm = (normCV𝑈)

Theoremh2hvs 27818 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 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec    &    ℋ = (BaseSet‘𝑈)        = ( −𝑣𝑈)

Theoremh2hmetdval 27819 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec    &    ℋ = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (norm‘(𝐴 𝐵)))

Theoremh2hcau 27820 The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec    &    ℋ = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑𝑚 ℕ))

Theoremh2hlm 27821 The limit sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec    &    ℋ = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)       𝑣 = ((⇝𝑡𝐽) ↾ ( ℋ ↑𝑚 ℕ))

19.1.3  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 27822 through axhcompl-zf 27839, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space 𝑈 = ⟨⟨ + , · ⟩, norm 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 +, ·, and ·ih before df-hnorm 27809 above. See also the comment in ax-hilex 27840.

Theoremaxhilex-zf 27822 Derive axiom ax-hilex 27840 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD        ℋ ∈ V

Theoremaxhfvadd-zf 27823 Derive axiom ax-hfvadd 27841 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD        + :( ℋ × ℋ)⟶ ℋ

Theoremaxhvcom-zf 27824 Derive axiom ax-hvcom 27842 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremaxhvass-zf 27825 Derive axiom ax-hvass 27843 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremaxhv0cl-zf 27826 Derive axiom ax-hv0cl 27844 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       0 ∈ ℋ

Theoremaxhvaddid-zf 27827 Derive axiom ax-hvaddid 27845 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)

Theoremaxhfvmul-zf 27828 Derive axiom ax-hfvmul 27846 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD        · :(ℂ × ℋ)⟶ ℋ

Theoremaxhvmulid-zf 27829 Derive axiom ax-hvmulid 27847 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       (𝐴 ∈ ℋ → (1 · 𝐴) = 𝐴)

Theoremaxhvmulass-zf 27830 Derive axiom ax-hvmulass 27848 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Theoremaxhvdistr1-zf 27831 Derive axiom ax-hvdistr1 27849 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Theoremaxhvdistr2-zf 27832 Derive axiom ax-hvdistr2 27850 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Theoremaxhvmul0-zf 27833 Derive axiom ax-hvmul0 27851 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       (𝐴 ∈ ℋ → (0 · 𝐴) = 0)

Theoremaxhfi-zf 27834 Derive axiom ax-hfi 27920 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)        ·ih :( ℋ × ℋ)⟶ℂ

Theoremaxhis1-zf 27835 Derive axiom ax-his1 27923 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))

Theoremaxhis2-zf 27836 Derive axiom ax-his2 27924 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))

Theoremaxhis3-zf 27837 Derive axiom ax-his3 27925 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))

Theoremaxhis4-zf 27838 Derive axiom ax-his4 27926 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)       ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (𝐴 ·ih 𝐴))

Theoremaxhcompl-zf 27839* Derive axiom ax-hcompl 28043 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹𝑣 𝑥)

19.1.4  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 27840, ax-hfvadd 27841, ax-hvcom 27842, ax-hvass 27843, ax-hv0cl 27844, ax-hvaddid 27845, ax-hfvmul 27846, ax-hvmulid 27847, ax-hvmulass 27848, ax-hvdistr1 27849, ax-hvdistr2 27850, ax-hvmul0 27851, ax-hfi 27920, ax-his1 27923, ax-his2 27924, ax-his3 27925, ax-his4 27926, and ax-hcompl 28043.

The axioms specify the properties of 5 primitive symbols, , +, ·, 0, and ·ih.

If we can prove in ZFC set theory that a class 𝑈 = ⟨⟨ + , · ⟩, norm is a complex Hilbert space, i.e. that 𝑈 ∈ CHilOLD, then these axioms can be proved as theorems axhilex-zf 27822, axhfvadd-zf 27823, axhvcom-zf 27824, axhvass-zf 27825, axhv0cl-zf 27826, axhvaddid-zf 27827, axhfvmul-zf 27828, axhvmulid-zf 27829, axhvmulass-zf 27830, axhvdistr1-zf 27831, axhvdistr2-zf 27832, axhvmul0-zf 27833, axhfi-zf 27834, axhis1-zf 27835, axhis2-zf 27836, axhis3-zf 27837, axhis4-zf 27838, and axhcompl-zf 27839 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 27822.

Axiomax-hilex 27840 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, , which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
ℋ ∈ V

Axiomax-hfvadd 27841 Vector addition is an operation on . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
+ :( ℋ × ℋ)⟶ ℋ

Axiomax-hvcom 27842 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Axiomax-hvass 27843 Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Axiomax-hv0cl 27844 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
0 ∈ ℋ

Axiomax-hvaddid 27845 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)

Axiomax-hfvmul 27846 Scalar multiplication is an operation on and . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
· :(ℂ × ℋ)⟶ ℋ

Axiomax-hvmulid 27847 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (1 · 𝐴) = 𝐴)

Axiomax-hvmulass 27848 Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Axiomax-hvdistr1 27849 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Axiomax-hvdistr2 27850 Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Axiomax-hvmul0 27851 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 27867 and hvsubval 27857). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 · 𝐴) = 0)

19.1.5  Vector operations

Theoremhvmulex 27852 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
· ∈ V

Theoremhvaddcl 27853 Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) ∈ ℋ)

Theoremhvmulcl 27854 Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 · 𝐵) ∈ ℋ)

Theoremhvmulcli 27855 Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℋ       (𝐴 · 𝐵) ∈ ℋ

Theoremhvsubf 27856 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
:( ℋ × ℋ)⟶ ℋ

Theoremhvsubval 27857 Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) = (𝐴 + (-1 · 𝐵)))

Theoremhvsubcl 27858 Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) ∈ ℋ)

Theoremhvaddcli 27859 Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 + 𝐵) ∈ ℋ

Theoremhvcomi 27860 Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 + 𝐵) = (𝐵 + 𝐴)

Theoremhvsubvali 27861 Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 𝐵) = (𝐴 + (-1 · 𝐵))

Theoremhvsubcli 27862 Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 𝐵) ∈ ℋ

Theoremifhvhv0 27863 Prove if(𝐴 ∈ ℋ, 𝐴, 0) ∈ ℋ (common case). (Contributed by David A. Wheeler, 7-Dec-2018.) (New usage is discouraged.)
if(𝐴 ∈ ℋ, 𝐴, 0) ∈ ℋ

Theoremhvaddid2 27864 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)

Theoremhvmul0 27865 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℂ → (𝐴 · 0) = 0)

Theoremhvmul0or 27866 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0)))

Theoremhvsubid 27867 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 𝐴) = 0)

Theoremhvnegid 27868 Addition of negative of a vector to itself. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 + (-1 · 𝐴)) = 0)

Theoremhv2neg 27869 Two ways to express the negative of a vector. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 𝐴) = (-1 · 𝐴))

Theoremhvaddid2i 27870 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       (0 + 𝐴) = 𝐴

Theoremhvnegidi 27871 Addition of negative of a vector to itself. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       (𝐴 + (-1 · 𝐴)) = 0

Theoremhv2negi 27872 Two ways to express the negative of a vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       (0 𝐴) = (-1 · 𝐴)

Theoremhvm1neg 27873 Convert minus one times a scalar product to the negative of the scalar. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 · (𝐴 · 𝐵)) = (-𝐴 · 𝐵))

Theoremhvaddsubval 27874 Value of vector addition in terms of vector subtraction. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐴 (-1 · 𝐵)))

Theoremhvadd32 27875 Commutative/associative law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))

Theoremhvadd12 27876 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))

Theoremhvadd4 27877 Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)))

Theoremhvsub4 27878 Hilbert vector space addition/subtraction law. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴 𝐶) + (𝐵 𝐷)))

Theoremhvaddsub12 27879 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 + (𝐵 𝐶)) = (𝐵 + (𝐴 𝐶)))

Theoremhvpncan 27880 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴)

Theoremhvpncan2 27881 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)

Theoremhvaddsubass 27882 Associativity of sum and difference of Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 𝐶)))

Theoremhvpncan3 27883 Subtraction and addition of equal Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + (𝐵 𝐴)) = 𝐵)

Theoremhvmulcom 27884 Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))

Theoremhvsubass 27885 Hilbert vector space associative law for subtraction. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 𝐵) − 𝐶) = (𝐴 (𝐵 + 𝐶)))

Theoremhvsub32 27886 Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 𝐵) − 𝐶) = ((𝐴 𝐶) − 𝐵))

Theoremhvmulassi 27887 Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℋ       ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))

Theoremhvmulcomi 27888 Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℋ       (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))

Theoremhvmul2negi 27889 Double negative in scalar multiplication. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℋ       (-𝐴 · (-𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶))

Theoremhvsubdistr1 27890 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)))

Theoremhvsubdistr2 27891 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶)))

Theoremhvdistr1i 27892 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))

Theoremhvsubdistr1i 27893 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       (𝐴 · (𝐵 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))

Theoremhvassi 27894 Hilbert vector space associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))

Theoremhvadd32i 27895 Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)

Theoremhvsubassi 27896 Hilbert vector space associative law for subtraction. (Contributed by NM, 7-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((𝐴 𝐵) − 𝐶) = (𝐴 (𝐵 + 𝐶))

Theoremhvsub32i 27897 Hilbert vector space commutative/associative law. (Contributed by NM, 7-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((𝐴 𝐵) − 𝐶) = ((𝐴 𝐶) − 𝐵)

Theoremhvadd12i 27898 Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))

Theoremhvadd4i 27899 Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))

Theoremhvsubsub4i 27900 Hilbert vector space addition law. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       ((𝐴 𝐵) − (𝐶 𝐷)) = ((𝐴 𝐶) − (𝐵 𝐷))

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