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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Syntax | cat 30901 | Extend class notation with set of atoms on a Hilbert lattice. |
| class HAtoms | ||
| Syntax | cmd 30902 | Extend class notation with the modular pair relation on a Hilbert lattice. |
| class 𝑀ℋ | ||
| Syntax | cdmd 30903 | Extend class notation with the dual modular pair relation on a Hilbert lattice. |
| class 𝑀ℋ* | ||
| Definition | df-hnorm 30904 | Define the function for the norm of a vector of Hilbert space. See normval 31060 for its value and normcl 31061 for its closure. Theorems norm-i-i 31069, norm-ii-i 31073, and norm-iii-i 31075 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 30905 | Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 30935). 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 31103. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) | ||
| Definition | df-h0v 30906 | 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 31104. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
| ⊢ 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) | ||
| Definition | df-hvsub 30907* | Define vector subtraction. See hvsubvali 30956 for its value and hvsubcli 30957 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 30908* | Define the limit relation for Hilbert space. See hlimi 31124 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 30909* | Define the set of Cauchy sequences on a Hilbert space. See hcau 31120 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 30910 | 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 30911 | 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 30912 | 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 30913 | 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 30914 | 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 30915 | 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 30916 | 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 30917 through axhcompl-zf 30934, 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 30904 above. See also the comment in ax-hilex 30935. | ||
| Theorem | axhilex-zf 30917 | Derive Axiom ax-hilex 30935 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 30918 | Derive Axiom ax-hfvadd 30936 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 30919 | Derive Axiom ax-hvcom 30937 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 30920 | Derive Axiom ax-hvass 30938 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 30921 | Derive Axiom ax-hv0cl 30939 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 30922 | Derive Axiom ax-hvaddid 30940 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 30923 | Derive Axiom ax-hfvmul 30941 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 30924 | Derive Axiom ax-hvmulid 30942 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 30925 | Derive Axiom ax-hvmulass 30943 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 30926 | Derive Axiom ax-hvdistr1 30944 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 30927 | Derive Axiom ax-hvdistr2 30945 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 30928 | Derive Axiom ax-hvmul0 30946 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 30929 | Derive Axiom ax-hfi 31015 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 30930 | Derive Axiom ax-his1 31018 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 30931 | Derive Axiom ax-his2 31019 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 30932 | Derive Axiom ax-his3 31020 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 30933 | Derive Axiom ax-his4 31021 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 30934* | Derive Axiom ax-hcompl 31138 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 30935, ax-hfvadd 30936, ax-hvcom 30937, ax-hvass 30938, ax-hv0cl 30939, ax-hvaddid 30940, ax-hfvmul 30941, ax-hvmulid 30942, ax-hvmulass 30943, ax-hvdistr1 30944, ax-hvdistr2 30945, ax-hvmul0 30946, ax-hfi 31015, ax-his1 31018, ax-his2 31019, ax-his3 31020, ax-his4 31021, and ax-hcompl 31138. 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 30917, axhfvadd-zf 30918, axhvcom-zf 30919, axhvass-zf 30920, axhv0cl-zf 30921, axhvaddid-zf 30922, axhfvmul-zf 30923, axhvmulid-zf 30924, axhvmulass-zf 30925, axhvdistr1-zf 30926, axhvdistr2-zf 30927, axhvmul0-zf 30928, axhfi-zf 30929, axhis1-zf 30930, axhis2-zf 30931, axhis3-zf 30932, axhis4-zf 30933, and axhcompl-zf 30934 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 30917. | ||
| Axiom | ax-hilex 30935 | 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 30936 | Vector addition is an operation on ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | ||
| Axiom | ax-hvcom 30937 | 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 30938 | 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 30939 | The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ 0ℎ ∈ ℋ | ||
| Axiom | ax-hvaddid 30940 | Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | ||
| Axiom | ax-hfvmul 30941 | 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 30942 | Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | ||
| Axiom | ax-hvmulass 30943 | 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 30944 | 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 30945 | 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 30946 | Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 30962 and hvsubval 30952). (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) | ||
| Theorem | hvmulex 30947 | The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
| ⊢ ·ℎ ∈ V | ||
| Theorem | hvaddcl 30948 | Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) | ||
| Theorem | hvmulcl 30949 | Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ) | ||
| Theorem | hvmulcli 30950 | 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 30951 | Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.) |
| ⊢ −ℎ :( ℋ × ℋ)⟶ ℋ | ||
| Theorem | hvsubval 30952 | 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 30953 | Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ) | ||
| Theorem | hvaddcli 30954 | Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ | ||
| Theorem | hvcomi 30955 | 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 30956 | 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 30957 | Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ | ||
| Theorem | ifhvhv0 30958 | 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 30959 | Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴) | ||
| Theorem | hvmul0 30960 | Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ) | ||
| Theorem | hvmul0or 30961 | 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 30962 | Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ) | ||
| Theorem | hvnegid 30963 | 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 30964 | 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 30965 | Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ ⇒ ⊢ (0ℎ +ℎ 𝐴) = 𝐴 | ||
| Theorem | hvnegidi 30966 | 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 30967 | 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 30968 | 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 30969 | 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 30970 | 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 30971 | 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 30972 | 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 30973 | 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 30974 | 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 30975 | 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 30976 | 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 30977 | 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 30978 | 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 30979 | 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 30980 | 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 30981 | 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 30982 | 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 30983 | 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 30984 | 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 30985 | 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 30986 | 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 30987 | 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 30988 | 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 30989 | 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 30990 | 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 30991 | 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 30992 | 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 30993 | 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 30994 | 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 30995 | 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 30996 | 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 30997 | Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴)) | ||
| Theorem | hvnegdii 30998 | 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 30999 | 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 31000 | 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) | ||
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