P. 317 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31601-31700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axhfi-zf 31601	Derive Axiom ax-hfi 31687 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 31602	Derive Axiom ax-his1 31690 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 31603	Derive Axiom ax-his2 31691 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 31604	Derive Axiom ax-his3 31692 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 31605	Derive Axiom ax-his4 31693 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 31606*	Derive Axiom ax-hcompl 31810 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 31607, ax-hfvadd 31608, ax-hvcom 31609, ax-hvass 31610, ax-hv0cl 31611, ax-hvaddid 31612, ax-hfvmul 31613, ax-hvmulid 31614, ax-hvmulass 31615, ax-hvdistr1 31616, ax-hvdistr2 31617, ax-hvmul0 31618, ax-hfi 31687, ax-his1 31690, ax-his2 31691, ax-his3 31692, ax-his4 31693, and ax-hcompl 31810. 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 31589, axhfvadd-zf 31590, axhvcom-zf 31591, axhvass-zf 31592, axhv0cl-zf 31593, axhvaddid-zf 31594, axhfvmul-zf 31595, axhvmulid-zf 31596, axhvmulass-zf 31597, axhvdistr1-zf 31598, axhvdistr2-zf 31599, axhvmul0-zf 31600, axhfi-zf 31601, axhis1-zf 31602, axhis2-zf 31603, axhis3-zf 31604, axhis4-zf 31605, and axhcompl-zf 31606 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 31589.
Axiom	ax-hilex 31607	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 31608	Vector addition is an operation on ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ +ℎ :( ℋ × ℋ)⟶ ℋ
Axiom	ax-hvcom 31609	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 31610	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 31611	The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ 0ℎ ∈ ℋ
Axiom	ax-hvaddid 31612	Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
Axiom	ax-hfvmul 31613	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 31614	Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)
Axiom	ax-hvmulass 31615	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 31616	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 31617	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 31618	Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 31634 and hvsubval 31624). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ)
20.1.5  Vector operations
Theorem	hvmulex 31619	The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
⊢ ·ℎ ∈ V
Theorem	hvaddcl 31620	Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ)
Theorem	hvmulcl 31621	Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ)
Theorem	hvmulcli 31622	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 31623	Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
⊢ −ℎ :( ℋ × ℋ)⟶ ℋ
Theorem	hvsubval 31624	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 31625	Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ)
Theorem	hvaddcli 31626	Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ
Theorem	hvcomi 31627	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 31628	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 31629	Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ
Theorem	ifhvhv0 31630	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 31631	Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)
Theorem	hvmul0 31632	Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ)
Theorem	hvmul0or 31633	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 31634	Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ)
Theorem	hvnegid 31635	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 31636	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 31637	Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    ⇒   ⊢ (0ℎ +ℎ 𝐴) = 𝐴
Theorem	hvnegidi 31638	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 31639	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 31640	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 31641	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 31642	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 31643	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 31644	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 31645	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 31646	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 31647	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 31648	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 31649	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 31650	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 31651	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 31652	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 31653	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 31654	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 31655	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 31656	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 31657	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 31658	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 31659	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 31660	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 31661	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 31662	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 31663	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 31664	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 31665	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 31666	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 31667	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 31668	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 31669	Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴))
Theorem	hvnegdii 31670	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 31671	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 31672	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 31673	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 31674	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 31675	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 31676	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 31677	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 31678	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 31679	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 31680	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 31681	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 31682	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 31683	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 31684	Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 0ℎ) = 𝐴)
Theorem	hvsubadd 31685	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 31686	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 31687	Inner product maps pairs from ℋ to ℂ. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ ·ih :( ℋ × ℋ)⟶ℂ
Theorem	hicl 31688	Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
Theorem	hicli 31689	Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 ·ih 𝐵) ∈ ℂ
Axiom	ax-his1 31690	Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 15275 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written ⟨𝐴, 𝐵⟩, but our operation notation co 7421 allows to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4835. Physicists use ⟨𝐵 ∣ 𝐴⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 32458. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
Axiom	ax-his2 31691	Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))
Axiom	ax-his3 31692	Associative law for inner product. Postulate (S3) of [Beran] p. 95. Warning: Mathematics textbooks usually use our version of the axiom. Physics textbooks, on the other hand, usually replace the left-hand side with (𝐵 ·ih (𝐴 ·ℎ 𝐶)) (e.g., Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 32458 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))
Axiom	ax-his4 31693	Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴))
20.2  Inner product and norms
20.2.1  Inner product
Theorem	his5 31694	Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = ((∗‘𝐴) · (𝐵 ·ih 𝐶)))
Theorem	his52 31695	Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih ((∗‘𝐴) ·ℎ 𝐶)) = (𝐴 · (𝐵 ·ih 𝐶)))
Theorem	his35 31696	Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·ℎ 𝐶) ·ih (𝐵 ·ℎ 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·ih 𝐷)))
Theorem	his35i 31697	Move scalar multiplication to outside of inner product. (Contributed by NM, 1-Jul-2005.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℋ    &   ⊢ 𝐷 ∈ ℋ    ⇒   ⊢ ((𝐴 ·ℎ 𝐶) ·ih (𝐵 ·ℎ 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·ih 𝐷))
Theorem	his7 31698	Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 +ℎ 𝐶)) = ((𝐴 ·ih 𝐵) + (𝐴 ·ih 𝐶)))
Theorem	hiassdi 31699	Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·ℎ 𝐵) +ℎ 𝐶) ·ih 𝐷) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷)))
Theorem	his2sub 31700	Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) − (𝐵 ·ih 𝐶)))