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Theorem List for Metamath Proof Explorer - 28701-28800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxhis3-zf 28701 Derive axiom ax-his3 28789 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))
 
Theoremaxhis4-zf 28702 Derive axiom ax-his4 28790 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD    &    ·ih = (·𝑖OLD𝑈)       ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (𝐴 ·ih 𝐴))
 
Theoremaxhcompl-zf 28703* Derive axiom ax-hcompl 28907 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 = ⟨⟨ + , · ⟩, norm    &   𝑈 ∈ CHilOLD       (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹𝑣 𝑥)
 
19.1.4  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 28704, ax-hfvadd 28705, ax-hvcom 28706, ax-hvass 28707, ax-hv0cl 28708, ax-hvaddid 28709, ax-hfvmul 28710, ax-hvmulid 28711, ax-hvmulass 28712, ax-hvdistr1 28713, ax-hvdistr2 28714, ax-hvmul0 28715, ax-hfi 28784, ax-his1 28787, ax-his2 28788, ax-his3 28789, ax-his4 28790, and ax-hcompl 28907.

The axioms specify the properties of 5 primitive symbols, , +, ·, 0, and ·ih.

If we can prove in ZFC set theory that a class 𝑈 = ⟨⟨ + , · ⟩, norm is a complex Hilbert space, i.e. that 𝑈 ∈ CHilOLD, then these axioms can be proved as theorems axhilex-zf 28686, axhfvadd-zf 28687, axhvcom-zf 28688, axhvass-zf 28689, axhv0cl-zf 28690, axhvaddid-zf 28691, axhfvmul-zf 28692, axhvmulid-zf 28693, axhvmulass-zf 28694, axhvdistr1-zf 28695, axhvdistr2-zf 28696, axhvmul0-zf 28697, axhfi-zf 28698, axhis1-zf 28699, axhis2-zf 28700, axhis3-zf 28701, axhis4-zf 28702, and axhcompl-zf 28703 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 28686.

 
Axiomax-hilex 28704 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, , which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
ℋ ∈ V
 
Axiomax-hfvadd 28705 Vector addition is an operation on . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
+ :( ℋ × ℋ)⟶ ℋ
 
Axiomax-hvcom 28706 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Axiomax-hvass 28707 Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Axiomax-hv0cl 28708 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
0 ∈ ℋ
 
Axiomax-hvaddid 28709 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
 
Axiomax-hfvmul 28710 Scalar multiplication is an operation on and . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
· :(ℂ × ℋ)⟶ ℋ
 
Axiomax-hvmulid 28711 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (1 · 𝐴) = 𝐴)
 
Axiomax-hvmulass 28712 Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
 
Axiomax-hvdistr1 28713 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)))
 
Axiomax-hvdistr2 28714 Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
 
Axiomax-hvmul0 28715 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 28731 and hvsubval 28721). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 · 𝐴) = 0)
 
19.1.5  Vector operations
 
Theoremhvmulex 28716 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
· ∈ V
 
Theoremhvaddcl 28717 Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) ∈ ℋ)
 
Theoremhvmulcl 28718 Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 · 𝐵) ∈ ℋ)
 
Theoremhvmulcli 28719 Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℋ       (𝐴 · 𝐵) ∈ ℋ
 
Theoremhvsubf 28720 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
:( ℋ × ℋ)⟶ ℋ
 
Theoremhvsubval 28721 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 28722 Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) ∈ ℋ)
 
Theoremhvaddcli 28723 Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 + 𝐵) ∈ ℋ
 
Theoremhvcomi 28724 Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 + 𝐵) = (𝐵 + 𝐴)
 
Theoremhvsubvali 28725 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 28726 Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 𝐵) ∈ ℋ
 
Theoremifhvhv0 28727 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 28728 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
 
Theoremhvmul0 28729 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℂ → (𝐴 · 0) = 0)
 
Theoremhvmul0or 28730 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 28731 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 𝐴) = 0)
 
Theoremhvnegid 28732 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 28733 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 28734 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       (0 + 𝐴) = 𝐴
 
Theoremhvnegidi 28735 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 28736 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 28737 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 28738 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 28739 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 28740 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 28741 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 28742 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 28743 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 28744 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 28745 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 28746 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 28747 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 28748 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 28749 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 28750 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 28751 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 28752 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 28753 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 28754 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 28755 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 28756 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 28757 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 28758 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 28759 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 28760 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 28761 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 28762 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 28763 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 28764 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 28765 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 28766 Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (2 · 𝐴) = (𝐴 + 𝐴))
 
Theoremhvnegdii 28767 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 28768 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 28769 Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((𝐴 + 𝐵) + (𝐴 𝐵)) = (2 · 𝐴)
 
Theoremhvaddcani 28770 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 28771 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 28772 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 28773 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 28774 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 28775 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 28776 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 28777 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 28778 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 28779 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 28780 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 28781 Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 0) = 𝐴)
 
Theoremhvsubadd 28782 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 28783 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 28784 Inner product maps pairs from to . (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
·ih :( ℋ × ℋ)⟶ℂ
 
Theoremhicl 28785 Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
 
Theoremhicli 28786 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 28787 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 14451 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written 𝐴, 𝐵, but our operation notation co 7145 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4566. Physicists use 𝐵𝐴, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 29555. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
 
Axiomax-his2 28788 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 28789 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 29555 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 28790 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 28791 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 28792 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 28793 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 28794 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 28795 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 28796 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 28797 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 28798 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 28799 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 28800 Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
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