HomeTheorem List (p. 312 of 504)
GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 312 of 504) | < Previous Next > |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | Metamath Proof Explorer
(1-31060) |
Hilbert Space Explorer
(31061-32583) |
Users' Mathboxes
(32584-50374) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Syntax | cei 31101 | Extend class notation with Hilbert space eigenvector function. |
| class eigvec | ||
| Syntax | cel 31102 | Extend class notation with Hilbert space eigenvalue function. |
| class eigval | ||
| Syntax | cspc 31103 | Extend class notation with the spectrum of an operator. |
| class Lambda | ||
| Syntax | cst 31104 | Extend class notation with set of states on a Hilbert lattice. |
| class States | ||
| Syntax | chst 31105 | Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice. |
| class CHStates | ||
| Syntax | ccv 31106 | Extend class notation with the covers relation on a Hilbert lattice. |
| class ⋖ℋ | ||
| Syntax | cat 31107 | Extend class notation with set of atoms on a Hilbert lattice. |
| class HAtoms | ||
| Syntax | cmd 31108 | Extend class notation with the modular pair relation on a Hilbert lattice. |
| class 𝑀ℋ | ||
| Syntax | cdmd 31109 | Extend class notation with the dual modular pair relation on a Hilbert lattice. |
| class 𝑀ℋ* | ||
| Definition | df-hnorm 31110 | Define the function for the norm of a vector of Hilbert space. See normval 31266 for its value and normcl 31267 for its closure. Theorems norm-i-i 31275, norm-ii-i 31279, and norm-iii-i 31281 show it has the expected properties of a norm. In the literature, the norm of 𝐴 is usually written "|| 𝐴 ||", but we use function notation to take advantage of our existing theorems about functions. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
| ⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥))) | ||
| Definition | df-hba 31111 | Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 31141). Note that ℋ is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. This definition can be proved independently from those axioms as Theorem hhba 31309. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) | ||
| Definition | df-h0v 31112 | 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 31310. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) | ||
| Definition | df-hvsub 31113* | Define vector subtraction. See hvsubvali 31162 for its value and hvsubcli 31163 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
| ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦))) | ||
| Definition | df-hlim 31114* | Define the limit relation for Hilbert space. See hlimi 31330 for its relational expression. Note that 𝑓:ℕ⟶ ℋ is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of converge in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
| ⊢ ⇝𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)} | ||
| Definition | df-hcau 31115* | Define the set of Cauchy sequences on a Hilbert space. See hcau 31326 for its membership relation. Note that 𝑓:ℕ⟶ ℋ is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of Cauchy sequence in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
| ⊢ Cauchy = {𝑓 ∈ ( ℋ ↑m ℕ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥} | ||
| Theorem | h2hva 31116 | The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ +ℎ = ( +𝑣 ‘𝑈) | ||
| Theorem | h2hsm 31117 | The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) | ||
| Theorem | h2hnm 31118 | The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ normℎ = (normCV‘𝑈) | ||
| Theorem | h2hvs 31119 | The vector subtraction operation of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ NrmCVec & ⊢ ℋ = (BaseSet‘𝑈) ⇒ ⊢ −ℎ = ( −𝑣 ‘𝑈) | ||
| Theorem | h2hmetdval 31120 | Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ NrmCVec & ⊢ ℋ = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (normℎ‘(𝐴 −ℎ 𝐵))) | ||
| Theorem | h2hcau 31121 | The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ NrmCVec & ⊢ ℋ = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) | ||
| Theorem | h2hlm 31122 | The limit sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ NrmCVec & ⊢ ℋ = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ⇝𝑣 = ((⇝𝑡‘𝐽) ↾ ( ℋ ↑m ℕ)) | ||
Before introducing the 18 axioms for Hilbert space, we first prove them as the conclusions of Theorems axhilex-zf 31123 through axhcompl-zf 31140, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ that satisfies their hypotheses, then we can derive the Hilbert space axioms as theorems of ZFC set theory. In practice, in order to use these theorems to convert the Hilbert Space explorer to a ZFC-only subtheory, we would also have to provide definitions for the 3 (otherwise primitive) class constants +ℎ, ·ℎ, and ·ih before df-hnorm 31110 above. See also the comment in ax-hilex 31141. | ||
| Theorem | axhilex-zf 31123 | Derive Axiom ax-hilex 31141 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ℋ ∈ V | ||
| Theorem | axhfvadd-zf 31124 | Derive Axiom ax-hfvadd 31142 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | ||
| Theorem | axhvcom-zf 31125 | Derive Axiom ax-hvcom 31143 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)) | ||
| Theorem | axhvass-zf 31126 | Derive Axiom ax-hvass 31144 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))) | ||
| Theorem | axhv0cl-zf 31127 | Derive Axiom ax-hv0cl 31145 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ 0ℎ ∈ ℋ | ||
| Theorem | axhvaddid-zf 31128 | Derive Axiom ax-hvaddid 31146 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | ||
| Theorem | axhfvmul-zf 31129 | Derive Axiom ax-hfvmul 31147 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | ||
| Theorem | axhvmulid-zf 31130 | Derive Axiom ax-hvmulid 31148 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | ||
| Theorem | axhvmulass-zf 31131 | Derive Axiom ax-hvmulass 31149 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶))) | ||
| Theorem | axhvdistr1-zf 31132 | Derive Axiom ax-hvdistr1 31150 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶))) | ||
| Theorem | axhvdistr2-zf 31133 | Derive Axiom ax-hvdistr2 31151 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶))) | ||
| Theorem | axhvmul0-zf 31134 | Derive Axiom ax-hvmul0 31152 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) | ||
| Theorem | axhfi-zf 31135 | Derive Axiom ax-hfi 31221 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD & ⊢ ·ih = (·𝑖OLD‘𝑈) ⇒ ⊢ ·ih :( ℋ × ℋ)⟶ℂ | ||
| Theorem | axhis1-zf 31136 | Derive Axiom ax-his1 31224 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD & ⊢ ·ih = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))) | ||
| Theorem | axhis2-zf 31137 | Derive Axiom ax-his2 31225 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD & ⊢ ·ih = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶))) | ||
| Theorem | axhis3-zf 31138 | Derive Axiom ax-his3 31226 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD & ⊢ ·ih = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶))) | ||
| Theorem | axhis4-zf 31139 | Derive Axiom ax-his4 31227 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD & ⊢ ·ih = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴)) | ||
| Theorem | axhcompl-zf 31140* | Derive Axiom ax-hcompl 31344 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.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
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 31141, ax-hfvadd 31142, ax-hvcom 31143, ax-hvass 31144, ax-hv0cl 31145, ax-hvaddid 31146, ax-hfvmul 31147, ax-hvmulid 31148, ax-hvmulass 31149, ax-hvdistr1 31150, ax-hvdistr2 31151, ax-hvmul0 31152, ax-hfi 31221, ax-his1 31224, ax-his2 31225, ax-his3 31226, ax-his4 31227, and ax-hcompl 31344. The axioms specify the properties of 5 primitive symbols, ℋ, +ℎ, ·ℎ, 0ℎ, and ·ih. If we can prove in ZFC set theory that a class 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ is a complex Hilbert space, i.e. that 𝑈 ∈ CHilOLD, then these axioms can be proved as Theorems axhilex-zf 31123, axhfvadd-zf 31124, axhvcom-zf 31125, axhvass-zf 31126, axhv0cl-zf 31127, axhvaddid-zf 31128, axhfvmul-zf 31129, axhvmulid-zf 31130, axhvmulass-zf 31131, axhvdistr1-zf 31132, axhvdistr2-zf 31133, axhvmul0-zf 31134, axhfi-zf 31135, axhis1-zf 31136, axhis2-zf 31137, axhis3-zf 31138, axhis4-zf 31139, and axhcompl-zf 31140 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 31123. | ||
| Axiom | ax-hilex 31141 | 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, ℋ, which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ ℋ ∈ V | ||
| Axiom | ax-hfvadd 31142 | Vector addition is an operation on ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | ||
| Axiom | ax-hvcom 31143 | Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)) | ||
| Axiom | ax-hvass 31144 | Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))) | ||
| Axiom | ax-hv0cl 31145 | The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ 0ℎ ∈ ℋ | ||
| Axiom | ax-hvaddid 31146 | Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | ||
| Axiom | ax-hfvmul 31147 | Scalar multiplication is an operation on ℂ and ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | ||
| Axiom | ax-hvmulid 31148 | Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | ||
| Axiom | ax-hvmulass 31149 | Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶))) | ||
| Axiom | ax-hvdistr1 31150 | Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶))) | ||
| Axiom | ax-hvdistr2 31151 | Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶))) | ||
| Axiom | ax-hvmul0 31152 | Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 31168 and hvsubval 31158). (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) | ||
| Theorem | hvmulex 31153 | The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
| ⊢ ·ℎ ∈ V | ||
| Theorem | hvaddcl 31154 | Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) | ||
| Theorem | hvmulcl 31155 | Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ) | ||
| Theorem | hvmulcli 31156 | Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ | ||
| Theorem | hvsubf 31157 | Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.) |
| ⊢ −ℎ :( ℋ × ℋ)⟶ ℋ | ||
| Theorem | hvsubval 31158 | 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 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) | ||
| Theorem | hvsubcl 31159 | Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ) | ||
| Theorem | hvaddcli 31160 | Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ | ||
| Theorem | hvcomi 31161 | Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) | ||
| Theorem | hvsubvali 31162 | Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) | ||
| Theorem | hvsubcli 31163 | Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ | ||
| Theorem | ifhvhv0 31164 | Prove if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ. (Contributed by David A. Wheeler, 7-Dec-2018.) (New usage is discouraged.) |
| ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ | ||
| Theorem | hvaddlid 31165 | Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴) | ||
| Theorem | hvmul0 31166 | Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ) | ||
| Theorem | hvmul0or 31167 | 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 ·ℎ 𝐵) = 0ℎ ↔ (𝐴 = 0 ∨ 𝐵 = 0ℎ))) | ||
| Theorem | hvsubid 31168 | Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ) | ||
| Theorem | hvnegid 31169 | Addition of negative of a vector to itself. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ (-1 ·ℎ 𝐴)) = 0ℎ) | ||
| Theorem | hv2neg 31170 | Two ways to express the negative of a vector. (Contributed by NM, 23-May-2005.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (0ℎ −ℎ 𝐴) = (-1 ·ℎ 𝐴)) | ||
| Theorem | hvaddlidi 31171 | Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ ⇒ ⊢ (0ℎ +ℎ 𝐴) = 𝐴 | ||
| Theorem | hvnegidi 31172 | Addition of negative of a vector to itself. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ ⇒ ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐴)) = 0ℎ | ||
| Theorem | hv2negi 31173 | Two ways to express the negative of a vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ ⇒ ⊢ (0ℎ −ℎ 𝐴) = (-1 ·ℎ 𝐴) | ||
| Theorem | hvm1neg 31174 | 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 ·ℎ (𝐴 ·ℎ 𝐵)) = (-𝐴 ·ℎ 𝐵)) | ||
| Theorem | hvaddsubval 31175 | Value of vector addition in terms of vector subtraction. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐴 −ℎ (-1 ·ℎ 𝐵))) | ||
| Theorem | hvadd32 31176 | Commutative/associative law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵)) | ||
| Theorem | hvadd12 31177 | Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))) | ||
| Theorem | hvadd4 31178 | Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷)) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷))) | ||
| Theorem | hvsub4 31179 | 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 +ℎ 𝐵) −ℎ (𝐶 +ℎ 𝐷)) = ((𝐴 −ℎ 𝐶) +ℎ (𝐵 −ℎ 𝐷))) | ||
| Theorem | hvaddsub12 31180 | Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 −ℎ 𝐶)) = (𝐵 +ℎ (𝐴 −ℎ 𝐶))) | ||
| Theorem | hvpncan 31181 | Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) −ℎ 𝐵) = 𝐴) | ||
| Theorem | hvpncan2 31182 | Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) −ℎ 𝐴) = 𝐵) | ||
| Theorem | hvaddsubass 31183 | 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 +ℎ 𝐵) −ℎ 𝐶) = (𝐴 +ℎ (𝐵 −ℎ 𝐶))) | ||
| Theorem | hvpncan3 31184 | Subtraction and addition of equal Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ (𝐵 −ℎ 𝐴)) = 𝐵) | ||
| Theorem | hvmulcom 31185 | Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))) | ||
| Theorem | hvsubass 31186 | 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 −ℎ 𝐵) −ℎ 𝐶) = (𝐴 −ℎ (𝐵 +ℎ 𝐶))) | ||
| Theorem | hvsub32 31187 | Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) −ℎ 𝐶) = ((𝐴 −ℎ 𝐶) −ℎ 𝐵)) | ||
| Theorem | hvmulassi 31188 | Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) | ||
| Theorem | hvmulcomi 31189 | Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)) | ||
| Theorem | hvmul2negi 31190 | Double negative in scalar multiplication. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ (-𝐴 ·ℎ (-𝐵 ·ℎ 𝐶)) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) | ||
| Theorem | hvsubdistr1 31191 | Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 −ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) −ℎ (𝐴 ·ℎ 𝐶))) | ||
| Theorem | hvsubdistr2 31192 | Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 − 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) −ℎ (𝐵 ·ℎ 𝐶))) | ||
| Theorem | hvdistr1i 31193 | Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶)) | ||
| Theorem | hvsubdistr1i 31194 | Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ (𝐴 ·ℎ (𝐵 −ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) −ℎ (𝐴 ·ℎ 𝐶)) | ||
| Theorem | hvassi 31195 | Hilbert vector space associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)) | ||
| Theorem | hvadd32i 31196 | Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵) | ||
| Theorem | hvsubassi 31197 | 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 −ℎ 𝐵) −ℎ 𝐶) = (𝐴 −ℎ (𝐵 +ℎ 𝐶)) | ||
| Theorem | hvsub32i 31198 | 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 −ℎ 𝐵) −ℎ 𝐶) = ((𝐴 −ℎ 𝐶) −ℎ 𝐵) | ||
| Theorem | hvadd12i 31199 | Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) | ||
| Theorem | hvadd4i 31200 | Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ & ⊢ 𝐷 ∈ ℋ ⇒ ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷)) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷)) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |