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Theorem List for Metamath Proof Explorer - 28001-28100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhvsubval 28001 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 𝐵) = (𝐴 + (-1 · 𝐵)))
 
Theoremhvsubcl 28002 Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) ∈ ℋ)
 
Theoremhvaddcli 28003 Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 + 𝐵) ∈ ℋ
 
Theoremhvcomi 28004 Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 + 𝐵) = (𝐵 + 𝐴)
 
Theoremhvsubvali 28005 Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 𝐵) = (𝐴 + (-1 · 𝐵))
 
Theoremhvsubcli 28006 Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 𝐵) ∈ ℋ
 
Theoremifhvhv0 28007 Prove if(𝐴 ∈ ℋ, 𝐴, 0) ∈ ℋ. (Contributed by David A. Wheeler, 7-Dec-2018.) (New usage is discouraged.)
if(𝐴 ∈ ℋ, 𝐴, 0) ∈ ℋ
 
Theoremhvaddid2 28008 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
 
Theoremhvmul0 28009 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℂ → (𝐴 · 0) = 0)
 
Theoremhvmul0or 28010 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 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0)))
 
Theoremhvsubid 28011 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 𝐴) = 0)
 
Theoremhvnegid 28012 Addition of negative of a vector to itself. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 + (-1 · 𝐴)) = 0)
 
Theoremhv2neg 28013 Two ways to express the negative of a vector. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 𝐴) = (-1 · 𝐴))
 
Theoremhvaddid2i 28014 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       (0 + 𝐴) = 𝐴
 
Theoremhvnegidi 28015 Addition of negative of a vector to itself. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       (𝐴 + (-1 · 𝐴)) = 0
 
Theoremhv2negi 28016 Two ways to express the negative of a vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       (0 𝐴) = (-1 · 𝐴)
 
Theoremhvm1neg 28017 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 · (𝐴 · 𝐵)) = (-𝐴 · 𝐵))
 
Theoremhvaddsubval 28018 Value of vector addition in terms of vector subtraction. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐴 (-1 · 𝐵)))
 
Theoremhvadd32 28019 Commutative/associative law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))
 
Theoremhvadd12 28020 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))
 
Theoremhvadd4 28021 Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)))
 
Theoremhvsub4 28022 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 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴 𝐶) + (𝐵 𝐷)))
 
Theoremhvaddsub12 28023 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 + (𝐵 𝐶)) = (𝐵 + (𝐴 𝐶)))
 
Theoremhvpncan 28024 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴)
 
Theoremhvpncan2 28025 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)
 
Theoremhvaddsubass 28026 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 + 𝐵) − 𝐶) = (𝐴 + (𝐵 𝐶)))
 
Theoremhvpncan3 28027 Subtraction and addition of equal Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + (𝐵 𝐴)) = 𝐵)
 
Theoremhvmulcom 28028 Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
 
Theoremhvsubass 28029 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 𝐵) − 𝐶) = (𝐴 (𝐵 + 𝐶)))
 
Theoremhvsub32 28030 Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 𝐵) − 𝐶) = ((𝐴 𝐶) − 𝐵))
 
Theoremhvmulassi 28031 Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℋ       ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
 
Theoremhvmulcomi 28032 Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℋ       (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))
 
Theoremhvmul2negi 28033 Double negative in scalar multiplication. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℋ       (-𝐴 · (-𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶))
 
Theoremhvsubdistr1 28034 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)))
 
Theoremhvsubdistr2 28035 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶)))
 
Theoremhvdistr1i 28036 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
 
Theoremhvsubdistr1i 28037 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       (𝐴 · (𝐵 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))
 
Theoremhvassi 28038 Hilbert vector space associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
 
Theoremhvadd32i 28039 Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)
 
Theoremhvsubassi 28040 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 𝐵) − 𝐶) = (𝐴 (𝐵 + 𝐶))
 
Theoremhvsub32i 28041 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 𝐵) − 𝐶) = ((𝐴 𝐶) − 𝐵)
 
Theoremhvadd12i 28042 Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))
 
Theoremhvadd4i 28043 Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))
 
Theoremhvsubsub4i 28044 Hilbert vector space addition law. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       ((𝐴 𝐵) − (𝐶 𝐷)) = ((𝐴 𝐶) − (𝐵 𝐷))
 
Theoremhvsubsub4 28045 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 𝐵) − (𝐶 𝐷)) = ((𝐴 𝐶) − (𝐵 𝐷)))
 
Theoremhv2times 28046 Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (2 · 𝐴) = (𝐴 + 𝐴))
 
Theoremhvnegdii 28047 Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (-1 · (𝐴 𝐵)) = (𝐵 𝐴)
 
Theoremhvsubeq0i 28048 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 𝐵) = 0𝐴 = 𝐵)
 
Theoremhvsubcan2i 28049 Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((𝐴 + 𝐵) + (𝐴 𝐵)) = (2 · 𝐴)
 
Theoremhvaddcani 28050 Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)
 
Theoremhvsubaddi 28051 Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴)
 
Theoremhvnegdi 28052 Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (-1 · (𝐴 𝐵)) = (𝐵 𝐴))
 
Theoremhvsubeq0 28053 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 𝐵) = 0𝐴 = 𝐵))
 
Theoremhvaddeq0 28054 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 + 𝐵) = 0𝐴 = (-1 · 𝐵)))
 
Theoremhvaddcan 28055 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))
 
Theoremhvaddcan2 28056 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremhvmulcan 28057 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) = (𝐴 · 𝐶) ↔ 𝐵 = 𝐶))
 
Theoremhvmulcan2 28058 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremhvsubcan 28059 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 𝐵) = (𝐴 𝐶) ↔ 𝐵 = 𝐶))
 
Theoremhvsubcan2 28060 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 𝐶) = (𝐵 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremhvsub0 28061 Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 0) = 𝐴)
 
Theoremhvsubadd 28062 Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
 
Theoremhvaddsub4 28063 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 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 𝐶) = (𝐷 𝐵)))
 
19.1.6  Inner product postulates for a Hilbert space
 
Axiomax-hfi 28064 Inner product maps pairs from to . (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
·ih :( ℋ × ℋ)⟶ℂ
 
Theoremhicl 28065 Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
 
Theoremhicli 28066 Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 ·ih 𝐵) ∈ ℂ
 
Axiomax-his1 28067 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 13886 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written 𝐴, 𝐵, but our operation notation co 6690 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4217. Physicists use 𝐵𝐴, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 28837. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
 
Axiomax-his2 28068 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 + 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))
 
Axiomax-his3 28069 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 · 𝐶)) (e.g., Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 28837 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))
 
Axiomax-his4 28070 Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (𝐴 ·ih 𝐴))
 
19.2  Inner product and norms
 
19.2.1  Inner product
 
Theoremhis5 28071 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 · 𝐶)) = ((∗‘𝐴) · (𝐵 ·ih 𝐶)))
 
Theoremhis52 28072 Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih ((∗‘𝐴) · 𝐶)) = (𝐴 · (𝐵 ·ih 𝐶)))
 
Theoremhis35 28073 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 · 𝐶) ·ih (𝐵 · 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·ih 𝐷)))
 
Theoremhis35i 28074 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 · 𝐶) ·ih (𝐵 · 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·ih 𝐷))
 
Theoremhis7 28075 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 + 𝐶)) = ((𝐴 ·ih 𝐵) + (𝐴 ·ih 𝐶)))
 
Theoremhiassdi 28076 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 · 𝐵) + 𝐶) ·ih 𝐷) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷)))
 
Theoremhis2sub 28077 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 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) − (𝐵 ·ih 𝐶)))
 
Theoremhis2sub2 28078 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 𝐶)) = ((𝐴 ·ih 𝐵) − (𝐴 ·ih 𝐶)))
 
Theoremhire 28079 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)))
 
Theoremhiidrcl 28080 Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
 
Theoremhi01 28081 Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 ·ih 𝐴) = 0)
 
Theoremhi02 28082 Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 ·ih 0) = 0)
 
Theoremhiidge0 28083 Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
 
Theoremhis6 28084 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 = 0))
 
Theoremhis1i 28085 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))
 
Theoremabshicom 28086 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)))
 
Theoremhial0 28087* 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 = 0))
 
Theoremhial02 28088* 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 = 0))
 
Theoremhisubcomi 28089 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 𝐵) ·ih (𝐶 𝐷)) = ((𝐵 𝐴) ·ih (𝐷 𝐶))
 
Theoremhi2eq 28090 Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih (𝐴 𝐵)) = (𝐵 ·ih (𝐴 𝐵)) ↔ 𝐴 = 𝐵))
 
Theoremhial2eq 28091* Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) ↔ 𝐴 = 𝐵))
 
Theoremhial2eq2 28092* Two vectors whose inner product is always equal are equal. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·ih 𝐴) = (𝑥 ·ih 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremorthcom 28093 Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 ↔ (𝐵 ·ih 𝐴) = 0))
 
Theoremnormlem0 28094 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 7-Oct-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ       ((𝐹 (𝑆 · 𝐺)) ·ih (𝐹 (𝑆 · 𝐺))) = (((𝐹 ·ih 𝐹) + (-(∗‘𝑆) · (𝐹 ·ih 𝐺))) + ((-𝑆 · (𝐺 ·ih 𝐹)) + ((𝑆 · (∗‘𝑆)) · (𝐺 ·ih 𝐺))))
 
Theoremnormlem1 28095 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 22-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝑅 ∈ ℝ    &   (abs‘𝑆) = 1       ((𝐹 ((𝑆 · 𝑅) · 𝐺)) ·ih (𝐹 ((𝑆 · 𝑅) · 𝐺))) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺))))
 
Theoremnormlem2 28096 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))       𝐵 ∈ ℝ
 
Theoremnormlem3 28097 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   𝐴 = (𝐺 ·ih 𝐺)    &   𝐶 = (𝐹 ·ih 𝐹)    &   𝑅 ∈ ℝ       (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺))))
 
Theoremnormlem4 28098 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   𝐴 = (𝐺 ·ih 𝐺)    &   𝐶 = (𝐹 ·ih 𝐹)    &   𝑅 ∈ ℝ    &   (abs‘𝑆) = 1       ((𝐹 ((𝑆 · 𝑅) · 𝐺)) ·ih (𝐹 ((𝑆 · 𝑅) · 𝐺))) = (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶)
 
Theoremnormlem5 28099 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   𝐴 = (𝐺 ·ih 𝐺)    &   𝐶 = (𝐹 ·ih 𝐹)    &   𝑅 ∈ ℝ    &   (abs‘𝑆) = 1       0 ≤ (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶)
 
Theoremnormlem6 28100 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   𝐴 = (𝐺 ·ih 𝐺)    &   𝐶 = (𝐹 ·ih 𝐹)    &   (abs‘𝑆) = 1       (abs‘𝐵) ≤ (2 · ((√‘𝐴) · (√‘𝐶)))
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