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Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-hlim 27001* Define the limit relation for Hilbert space. See hlimi 27217 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 27002* Define the set of Cauchy sequences on a Hilbert space. See hcau 27213 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 27003 The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec        + = ( +𝑣𝑈)
 
Theoremh2hsm 27004 The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec        · = ( ·𝑠OLD𝑈)
 
Theoremh2hnm 27005 The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ NrmCVec       norm = (normCV𝑈)
 
Theoremh2hvs 27006 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 27007 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 27008 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 27009 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 27010 through axhcompl-zf 27027, 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 26997 above. See also the comment in ax-hilex 27028.

 
Theoremaxhilex-zf 27010 Derive axiom ax-hilex 27028 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 27011 Derive axiom ax-hfvadd 27029 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 27012 Derive axiom ax-hvcom 27030 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 27013 Derive axiom ax-hvass 27031 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 27014 Derive axiom ax-hv0cl 27032 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 27015 Derive axiom ax-hvaddid 27033 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 27016 Derive axiom ax-hfvmul 27034 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 27017 Derive axiom ax-hvmulid 27035 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 27018 Derive axiom ax-hvmulass 27036 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 27019 Derive axiom ax-hvdistr1 27037 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 27020 Derive axiom ax-hvdistr2 27038 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 27021 Derive axiom ax-hvmul0 27039 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 27022 Derive axiom ax-hfi 27108 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 27023 Derive axiom ax-his1 27111 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 27024 Derive axiom ax-his2 27112 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 27025 Derive axiom ax-his3 27113 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 27026 Derive axiom ax-his4 27114 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 27027* Derive axiom ax-hcompl 27231 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 27028, ax-hfvadd 27029, ax-hvcom 27030, ax-hvass 27031, ax-hv0cl 27032, ax-hvaddid 27033, ax-hfvmul 27034, ax-hvmulid 27035, ax-hvmulass 27036, ax-hvdistr1 27037, ax-hvdistr2 27038, ax-hvmul0 27039, ax-hfi 27108, ax-his1 27111, ax-his2 27112, ax-his3 27113, ax-his4 27114, and ax-hcompl 27231.

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 27010, axhfvadd-zf 27011, axhvcom-zf 27012, axhvass-zf 27013, axhv0cl-zf 27014, axhvaddid-zf 27015, axhfvmul-zf 27016, axhvmulid-zf 27017, axhvmulass-zf 27018, axhvdistr1-zf 27019, axhvdistr2-zf 27020, axhvmul0-zf 27021, axhfi-zf 27022, axhis1-zf 27023, axhis2-zf 27024, axhis3-zf 27025, axhis4-zf 27026, and axhcompl-zf 27027 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 27010.

 
Axiomax-hilex 27028 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 27029 Vector addition is an operation on . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
+ :( ℋ × ℋ)⟶ ℋ
 
Axiomax-hvcom 27030 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Axiomax-hvass 27031 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 27032 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
0 ∈ ℋ
 
Axiomax-hvaddid 27033 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
 
Axiomax-hfvmul 27034 Scalar multiplication is an operation on and . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
· :(ℂ × ℋ)⟶ ℋ
 
Axiomax-hvmulid 27035 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (1 · 𝐴) = 𝐴)
 
Axiomax-hvmulass 27036 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 27037 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 27038 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 27039 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 27055 and hvsubval 27045). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 · 𝐴) = 0)
 
19.1.5  Vector operations
 
Theoremhvmulex 27040 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
· ∈ V
 
Theoremhvaddcl 27041 Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) ∈ ℋ)
 
Theoremhvmulcl 27042 Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 · 𝐵) ∈ ℋ)
 
Theoremhvmulcli 27043 Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℋ       (𝐴 · 𝐵) ∈ ℋ
 
Theoremhvsubf 27044 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
:( ℋ × ℋ)⟶ ℋ
 
Theoremhvsubval 27045 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 27046 Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) ∈ ℋ)
 
Theoremhvaddcli 27047 Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 + 𝐵) ∈ ℋ
 
Theoremhvcomi 27048 Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 + 𝐵) = (𝐵 + 𝐴)
 
Theoremhvsubvali 27049 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 27050 Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 𝐵) ∈ ℋ
 
Theoremifhvhv0 27051 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 27052 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
 
Theoremhvmul0 27053 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℂ → (𝐴 · 0) = 0)
 
Theoremhvmul0or 27054 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 27055 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 𝐴) = 0)
 
Theoremhvnegid 27056 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 27057 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 27058 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       (0 + 𝐴) = 𝐴
 
Theoremhvnegidi 27059 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 27060 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 27061 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 27062 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 27063 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 27064 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 27065 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 27066 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 27067 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 27068 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 27069 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 27070 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 27071 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 27072 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 27073 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 27074 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 27075 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 27076 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 27077 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 27078 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 27079 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 27080 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 27081 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 27082 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 27083 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 27084 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 27085 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 27086 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 27087 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 27088 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))
 
Theoremhvsubsub4 27089 Hilbert vector space addition/subtraction law. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 𝐵) − (𝐶 𝐷)) = ((𝐴 𝐶) − (𝐵 𝐷)))
 
Theoremhv2times 27090 Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (2 · 𝐴) = (𝐴 + 𝐴))
 
Theoremhvnegdii 27091 Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (-1 · (𝐴 𝐵)) = (𝐵 𝐴)
 
Theoremhvsubeq0i 27092 If the difference between two vectors is zero, they are equal. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((𝐴 𝐵) = 0𝐴 = 𝐵)
 
Theoremhvsubcan2i 27093 Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((𝐴 + 𝐵) + (𝐴 𝐵)) = (2 · 𝐴)
 
Theoremhvaddcani 27094 Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)
 
Theoremhvsubaddi 27095 Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴)
 
Theoremhvnegdi 27096 Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (-1 · (𝐴 𝐵)) = (𝐵 𝐴))
 
Theoremhvsubeq0 27097 If the difference between two vectors is zero, they are equal. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 𝐵) = 0𝐴 = 𝐵))
 
Theoremhvaddeq0 27098 If the sum of two vectors is zero, one is the negative of the other. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 + 𝐵) = 0𝐴 = (-1 · 𝐵)))
 
Theoremhvaddcan 27099 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))
 
Theoremhvaddcan2 27100 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
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