P. 312 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31101-31200   *Has distinct variable group(s)

Type	Label	Description
Statement
Axiom	ax-hfvmul 31101	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 31102	Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)
Axiom	ax-hvmulass 31103	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 31104	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 31105	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 31106	Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 31122 and hvsubval 31112). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ)
20.1.5  Vector operations
Theorem	hvmulex 31107	The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
⊢ ·ℎ ∈ V
Theorem	hvaddcl 31108	Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ)
Theorem	hvmulcl 31109	Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ)
Theorem	hvmulcli 31110	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 31111	Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
⊢ −ℎ :( ℋ × ℋ)⟶ ℋ
Theorem	hvsubval 31112	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 31113	Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ)
Theorem	hvaddcli 31114	Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ
Theorem	hvcomi 31115	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 31116	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 31117	Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ
Theorem	ifhvhv0 31118	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 31119	Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)
Theorem	hvmul0 31120	Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ)
Theorem	hvmul0or 31121	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 31122	Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ)
Theorem	hvnegid 31123	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 31124	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 31125	Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    ⇒   ⊢ (0ℎ +ℎ 𝐴) = 𝐴
Theorem	hvnegidi 31126	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 31127	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 31128	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 31129	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 31130	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 31131	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 31132	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 31133	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 31134	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 31135	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 31136	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 31137	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 31138	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 31139	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 31140	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 31141	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 31142	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 31143	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 31144	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 31145	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 31146	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 31147	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 31148	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 31149	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 31150	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 31151	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 31152	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 31153	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 31154	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 31155	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 31156	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 31157	Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴))
Theorem	hvnegdii 31158	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 31159	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 31160	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 31161	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 31162	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 31163	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 31164	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 31165	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 31166	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 31167	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 31168	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 31169	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 31170	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 31171	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 31172	Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 0ℎ) = 𝐴)
Theorem	hvsubadd 31173	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 31174	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 31175	Inner product maps pairs from ℋ to ℂ. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ ·ih :( ℋ × ℋ)⟶ℂ
Theorem	hicl 31176	Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
Theorem	hicli 31177	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 31178	Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 15062 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written ⟨𝐴, 𝐵⟩, but our operation notation co 7363 allows to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4569. Physicists use ⟨𝐵 ∣ 𝐴⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 31946. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
Axiom	ax-his2 31179	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 31180	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 31946 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 31181	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 31182	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 31183	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 31184	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 31185	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 31186	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 31187	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 31188	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)))
Theorem	his2sub2 31189	Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 −ℎ 𝐶)) = ((𝐴 ·ih 𝐵) − (𝐴 ·ih 𝐶)))
Theorem	hire 31190	A necessary and sufficient condition for an inner product to be real. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) ∈ ℝ ↔ (𝐴 ·ih 𝐵) = (𝐵 ·ih 𝐴)))
Theorem	hiidrcl 31191	Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
Theorem	hi01 31192	Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0)
Theorem	hi02 31193	Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 0ℎ) = 0)
Theorem	hiidge0 31194	Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
Theorem	his6 31195	Zero inner product with self means vector is zero. Lemma 3.1(S6) of [Beran] p. 95. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → ((𝐴 ·ih 𝐴) = 0 ↔ 𝐴 = 0ℎ))
Theorem	his1i 31196	Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))
Theorem	abshicom 31197	Commuted inner products have the same absolute values. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘(𝐴 ·ih 𝐵)) = (abs‘(𝐵 ·ih 𝐴)))
Theorem	hial0 31198*	A vector whose inner product is always zero is zero. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = 0 ↔ 𝐴 = 0ℎ))
Theorem	hial02 31199*	A vector whose inner product is always zero is zero. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (∀𝑥 ∈ ℋ (𝑥 ·ih 𝐴) = 0 ↔ 𝐴 = 0ℎ))
Theorem	hisubcomi 31200	Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℋ    &   ⊢ 𝐷 ∈ ℋ    ⇒   ⊢ ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))