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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Definition | df-hcau 31001* | Define the set of Cauchy sequences on a Hilbert space. See hcau 31212 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 31002 | 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 31003 | 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 31004 | 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 31005 | 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 31006 | 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 31007 | 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 31008 | 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 31009 through axhcompl-zf 31026, 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 30996 above. See also the comment in ax-hilex 31027. | ||
| Theorem | axhilex-zf 31009 | Derive Axiom ax-hilex 31027 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 31010 | Derive Axiom ax-hfvadd 31028 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 31011 | Derive Axiom ax-hvcom 31029 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 31012 | Derive Axiom ax-hvass 31030 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 31013 | Derive Axiom ax-hv0cl 31031 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 31014 | Derive Axiom ax-hvaddid 31032 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 31015 | Derive Axiom ax-hfvmul 31033 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 31016 | Derive Axiom ax-hvmulid 31034 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 31017 | Derive Axiom ax-hvmulass 31035 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 31018 | Derive Axiom ax-hvdistr1 31036 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 31019 | Derive Axiom ax-hvdistr2 31037 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 31020 | Derive Axiom ax-hvmul0 31038 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 31021 | Derive Axiom ax-hfi 31107 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 31022 | Derive Axiom ax-his1 31110 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 31023 | Derive Axiom ax-his2 31111 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 31024 | Derive Axiom ax-his3 31112 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 31025 | Derive Axiom ax-his4 31113 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 31026* | Derive Axiom ax-hcompl 31230 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 31027, ax-hfvadd 31028, ax-hvcom 31029, ax-hvass 31030, ax-hv0cl 31031, ax-hvaddid 31032, ax-hfvmul 31033, ax-hvmulid 31034, ax-hvmulass 31035, ax-hvdistr1 31036, ax-hvdistr2 31037, ax-hvmul0 31038, ax-hfi 31107, ax-his1 31110, ax-his2 31111, ax-his3 31112, ax-his4 31113, and ax-hcompl 31230. 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 31009, axhfvadd-zf 31010, axhvcom-zf 31011, axhvass-zf 31012, axhv0cl-zf 31013, axhvaddid-zf 31014, axhfvmul-zf 31015, axhvmulid-zf 31016, axhvmulass-zf 31017, axhvdistr1-zf 31018, axhvdistr2-zf 31019, axhvmul0-zf 31020, axhfi-zf 31021, axhis1-zf 31022, axhis2-zf 31023, axhis3-zf 31024, axhis4-zf 31025, and axhcompl-zf 31026 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 31009. | ||
| Axiom | ax-hilex 31027 | 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 31028 | Vector addition is an operation on ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | ||
| Axiom | ax-hvcom 31029 | 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 31030 | 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 31031 | The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ 0ℎ ∈ ℋ | ||
| Axiom | ax-hvaddid 31032 | Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | ||
| Axiom | ax-hfvmul 31033 | 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 31034 | Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | ||
| Axiom | ax-hvmulass 31035 | 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 31036 | 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 31037 | 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 31038 | Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 31054 and hvsubval 31044). (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) | ||
| Theorem | hvmulex 31039 | The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
| ⊢ ·ℎ ∈ V | ||
| Theorem | hvaddcl 31040 | Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) | ||
| Theorem | hvmulcl 31041 | Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ) | ||
| Theorem | hvmulcli 31042 | 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 31043 | Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.) |
| ⊢ −ℎ :( ℋ × ℋ)⟶ ℋ | ||
| Theorem | hvsubval 31044 | 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 31045 | Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ) | ||
| Theorem | hvaddcli 31046 | Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ | ||
| Theorem | hvcomi 31047 | 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 31048 | 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 31049 | Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ | ||
| Theorem | ifhvhv0 31050 | 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 31051 | Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴) | ||
| Theorem | hvmul0 31052 | Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ) | ||
| Theorem | hvmul0or 31053 | 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 31054 | Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ) | ||
| Theorem | hvnegid 31055 | 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 31056 | 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 31057 | Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ ⇒ ⊢ (0ℎ +ℎ 𝐴) = 𝐴 | ||
| Theorem | hvnegidi 31058 | 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 31059 | 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 31060 | 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 31061 | 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 31062 | 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 31063 | 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 31064 | 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 31065 | 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 31066 | 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 31067 | 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 31068 | 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 31069 | 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 31070 | 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 31071 | 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 31072 | 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 31073 | 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 31074 | 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 31075 | 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 31076 | 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 31077 | 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 31078 | 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 31079 | 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 31080 | 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 31081 | 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 31082 | 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 31083 | 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 31084 | 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 31085 | 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 31086 | 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)) | ||
| Theorem | hvsubsub4i 31087 | Hilbert vector space addition law. (Contributed by NM, 31-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 | hvsubsub4 31088 | 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 −ℎ 𝐵) −ℎ (𝐶 −ℎ 𝐷)) = ((𝐴 −ℎ 𝐶) −ℎ (𝐵 −ℎ 𝐷))) | ||
| Theorem | hv2times 31089 | Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴)) | ||
| Theorem | hvnegdii 31090 | Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (-1 ·ℎ (𝐴 −ℎ 𝐵)) = (𝐵 −ℎ 𝐴) | ||
| Theorem | hvsubeq0i 31091 | 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 −ℎ 𝐵) = 0ℎ ↔ 𝐴 = 𝐵) | ||
| Theorem | hvsubcan2i 31092 | Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 −ℎ 𝐵)) = (2 ·ℎ 𝐴) | ||
| Theorem | hvaddcani 31093 | Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) ↔ 𝐵 = 𝐶) | ||
| Theorem | hvsubaddi 31094 | Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) | ||
| Theorem | hvnegdi 31095 | Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ (𝐴 −ℎ 𝐵)) = (𝐵 −ℎ 𝐴)) | ||
| Theorem | hvsubeq0 31096 | 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 −ℎ 𝐵) = 0ℎ ↔ 𝐴 = 𝐵)) | ||
| Theorem | hvaddeq0 31097 | 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 +ℎ 𝐵) = 0ℎ ↔ 𝐴 = (-1 ·ℎ 𝐵))) | ||
| Theorem | hvaddcan 31098 | Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) ↔ 𝐵 = 𝐶)) | ||
| Theorem | hvaddcan2 31099 | Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐶) = (𝐵 +ℎ 𝐶) ↔ 𝐴 = 𝐵)) | ||
| Theorem | hvmulcan 31100 | Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) = (𝐴 ·ℎ 𝐶) ↔ 𝐵 = 𝐶)) | ||
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