P. 311 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31001-31100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axhfvadd-zf 31001	Derive Axiom ax-hfvadd 31019 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 31002	Derive Axiom ax-hvcom 31020 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 31003	Derive Axiom ax-hvass 31021 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 31004	Derive Axiom ax-hv0cl 31022 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 31005	Derive Axiom ax-hvaddid 31023 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 31006	Derive Axiom ax-hfvmul 31024 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 31007	Derive Axiom ax-hvmulid 31025 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 31008	Derive Axiom ax-hvmulass 31026 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 31009	Derive Axiom ax-hvdistr1 31027 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 31010	Derive Axiom ax-hvdistr2 31028 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 31011	Derive Axiom ax-hvmul0 31029 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 31012	Derive Axiom ax-hfi 31098 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 31013	Derive Axiom ax-his1 31101 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 31014	Derive Axiom ax-his2 31102 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 31015	Derive Axiom ax-his3 31103 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 31016	Derive Axiom ax-his4 31104 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 31017*	Derive Axiom ax-hcompl 31221 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)
20.1.4  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 31018, ax-hfvadd 31019, ax-hvcom 31020, ax-hvass 31021, ax-hv0cl 31022, ax-hvaddid 31023, ax-hfvmul 31024, ax-hvmulid 31025, ax-hvmulass 31026, ax-hvdistr1 31027, ax-hvdistr2 31028, ax-hvmul0 31029, ax-hfi 31098, ax-his1 31101, ax-his2 31102, ax-his3 31103, ax-his4 31104, and ax-hcompl 31221. 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 31000, axhfvadd-zf 31001, axhvcom-zf 31002, axhvass-zf 31003, axhv0cl-zf 31004, axhvaddid-zf 31005, axhfvmul-zf 31006, axhvmulid-zf 31007, axhvmulass-zf 31008, axhvdistr1-zf 31009, axhvdistr2-zf 31010, axhvmul0-zf 31011, axhfi-zf 31012, axhis1-zf 31013, axhis2-zf 31014, axhis3-zf 31015, axhis4-zf 31016, and axhcompl-zf 31017 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 31000.
Axiom	ax-hilex 31018	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 31019	Vector addition is an operation on ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ +ℎ :( ℋ × ℋ)⟶ ℋ
Axiom	ax-hvcom 31020	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 31021	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 31022	The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ 0ℎ ∈ ℋ
Axiom	ax-hvaddid 31023	Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
Axiom	ax-hfvmul 31024	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 31025	Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)
Axiom	ax-hvmulass 31026	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 31027	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 31028	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 31029	Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 31045 and hvsubval 31035). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ)
20.1.5  Vector operations
Theorem	hvmulex 31030	The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
⊢ ·ℎ ∈ V
Theorem	hvaddcl 31031	Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ)
Theorem	hvmulcl 31032	Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ)
Theorem	hvmulcli 31033	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 31034	Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
⊢ −ℎ :( ℋ × ℋ)⟶ ℋ
Theorem	hvsubval 31035	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 31036	Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ)
Theorem	hvaddcli 31037	Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ
Theorem	hvcomi 31038	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 31039	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 31040	Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ
Theorem	ifhvhv0 31041	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 31042	Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)
Theorem	hvmul0 31043	Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ)
Theorem	hvmul0or 31044	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 31045	Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ)
Theorem	hvnegid 31046	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 31047	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 31048	Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    ⇒   ⊢ (0ℎ +ℎ 𝐴) = 𝐴
Theorem	hvnegidi 31049	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 31050	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 31051	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 31052	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 31053	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 31054	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 31055	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 31056	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 31057	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 31058	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 31059	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 31060	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 31061	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 31062	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 31063	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 31064	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 31065	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 31066	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 31067	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 31068	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 31069	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 31070	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 31071	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 31072	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 31073	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 31074	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 31075	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 31076	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 31077	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 31078	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 31079	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 31080	Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴))
Theorem	hvnegdii 31081	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 31082	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 31083	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 31084	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 31085	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 31086	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 31087	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 31088	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 31089	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 31090	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 31091	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))
Theorem	hvmulcan2 31092	Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0ℎ)) → ((𝐴 ·ℎ 𝐶) = (𝐵 ·ℎ 𝐶) ↔ 𝐴 = 𝐵))
Theorem	hvsubcan 31093	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	hvsubcan2 31094	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	hvsub0 31095	Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 0ℎ) = 𝐴)
Theorem	hvsubadd 31096	Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴))
Theorem	hvaddsub4 31097	Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷) ↔ (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵)))
20.1.6  Inner product postulates for a Hilbert space
Axiom	ax-hfi 31098	Inner product maps pairs from ℋ to ℂ. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ ·ih :( ℋ × ℋ)⟶ℂ
Theorem	hicl 31099	Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
Theorem	hicli 31100	Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 ·ih 𝐵) ∈ ℂ