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Theorem List for Metamath Proof Explorer - 30501-30600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremh2hlm 30501 The limit sequences of Hilbert space. (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.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ NrmCVec    &    β„‹ = (BaseSetβ€˜π‘ˆ)    &   π· = (IndMetβ€˜π‘ˆ)    &   π½ = (MetOpenβ€˜π·)    β‡’    ⇝𝑣 = ((β‡π‘‘β€˜π½) β†Ύ ( β„‹ ↑m β„•))
 
20.1.3  Derive the Hilbert space axioms from ZFC set theory

Before introducing the 18 axioms for Hilbert space, we first prove them as the conclusions of Theorems axhilex-zf 30502 through axhcompl-zf 30519, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ© that satisfies their hypotheses, then we can derive the Hilbert space axioms as theorems of ZFC set theory. In practice, in order to use these theorems to convert the Hilbert Space explorer to a ZFC-only subtheory, we would also have to provide definitions for the 3 (otherwise primitive) class constants +β„Ž, Β·β„Ž, and Β·ih before df-hnorm 30489 above. See also the comment in ax-hilex 30520.

 
Theoremaxhilex-zf 30502 Derive Axiom ax-hilex 30520 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’    β„‹ ∈ V
 
Theoremaxhfvadd-zf 30503 Derive Axiom ax-hfvadd 30521 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’    +β„Ž :( β„‹ Γ— β„‹)⟢ β„‹
 
Theoremaxhvcom-zf 30504 Derive Axiom ax-hvcom 30522 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž 𝐡) = (𝐡 +β„Ž 𝐴))
 
Theoremaxhvass-zf 30505 Derive Axiom ax-hvass 30523 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) +β„Ž 𝐢) = (𝐴 +β„Ž (𝐡 +β„Ž 𝐢)))
 
Theoremaxhv0cl-zf 30506 Derive Axiom ax-hv0cl 30524 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   0β„Ž ∈ β„‹
 
Theoremaxhvaddid-zf 30507 Derive Axiom ax-hvaddid 30525 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   (𝐴 ∈ β„‹ β†’ (𝐴 +β„Ž 0β„Ž) = 𝐴)
 
Theoremaxhfvmul-zf 30508 Derive Axiom ax-hfvmul 30526 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’    Β·β„Ž :(β„‚ Γ— β„‹)⟢ β„‹
 
Theoremaxhvmulid-zf 30509 Derive Axiom ax-hvmulid 30527 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   (𝐴 ∈ β„‹ β†’ (1 Β·β„Ž 𝐴) = 𝐴)
 
Theoremaxhvmulass-zf 30510 Derive Axiom ax-hvmulass 30528 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β· 𝐡) Β·β„Ž 𝐢) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)))
 
Theoremaxhvdistr1-zf 30511 Derive Axiom ax-hvdistr1 30529 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) +β„Ž (𝐴 Β·β„Ž 𝐢)))
 
Theoremaxhvdistr2-zf 30512 Derive Axiom ax-hvdistr2 30530 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 + 𝐡) Β·β„Ž 𝐢) = ((𝐴 Β·β„Ž 𝐢) +β„Ž (𝐡 Β·β„Ž 𝐢)))
 
Theoremaxhvmul0-zf 30513 Derive Axiom ax-hvmul0 30531 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   (𝐴 ∈ β„‹ β†’ (0 Β·β„Ž 𝐴) = 0β„Ž)
 
Theoremaxhfi-zf 30514 Derive Axiom ax-hfi 30600 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    &    Β·ih = (·𝑖OLDβ€˜π‘ˆ)    β‡’    Β·ih :( β„‹ Γ— β„‹)βŸΆβ„‚
 
Theoremaxhis1-zf 30515 Derive Axiom ax-his1 30603 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    &    Β·ih = (·𝑖OLDβ€˜π‘ˆ)    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 Β·ih 𝐡) = (βˆ—β€˜(𝐡 Β·ih 𝐴)))
 
Theoremaxhis2-zf 30516 Derive Axiom ax-his2 30604 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    &    Β·ih = (·𝑖OLDβ€˜π‘ˆ)    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) Β·ih 𝐢) = ((𝐴 Β·ih 𝐢) + (𝐡 Β·ih 𝐢)))
 
Theoremaxhis3-zf 30517 Derive Axiom ax-his3 30605 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    &    Β·ih = (·𝑖OLDβ€˜π‘ˆ)    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β·β„Ž 𝐡) Β·ih 𝐢) = (𝐴 Β· (𝐡 Β·ih 𝐢)))
 
Theoremaxhis4-zf 30518 Derive Axiom ax-his4 30606 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    &    Β·ih = (·𝑖OLDβ€˜π‘ˆ)    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (𝐴 Β·ih 𝐴))
 
Theoremaxhcompl-zf 30519* Derive Axiom ax-hcompl 30723 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.)
π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©    &   π‘ˆ ∈ CHilOLD    β‡’   (𝐹 ∈ Cauchy β†’ βˆƒπ‘₯ ∈ β„‹ 𝐹 ⇝𝑣 π‘₯)
 
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 30520, ax-hfvadd 30521, ax-hvcom 30522, ax-hvass 30523, ax-hv0cl 30524, ax-hvaddid 30525, ax-hfvmul 30526, ax-hvmulid 30527, ax-hvmulass 30528, ax-hvdistr1 30529, ax-hvdistr2 30530, ax-hvmul0 30531, ax-hfi 30600, ax-his1 30603, ax-his2 30604, ax-his3 30605, ax-his4 30606, and ax-hcompl 30723.

The axioms specify the properties of 5 primitive symbols, β„‹, +β„Ž, Β·β„Ž, 0β„Ž, and Β·ih.

If we can prove in ZFC set theory that a class π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ© is a complex Hilbert space, i.e. that π‘ˆ ∈ CHilOLD, then these axioms can be proved as Theorems axhilex-zf 30502, axhfvadd-zf 30503, axhvcom-zf 30504, axhvass-zf 30505, axhv0cl-zf 30506, axhvaddid-zf 30507, axhfvmul-zf 30508, axhvmulid-zf 30509, axhvmulass-zf 30510, axhvdistr1-zf 30511, axhvdistr2-zf 30512, axhvmul0-zf 30513, axhfi-zf 30514, axhis1-zf 30515, axhis2-zf 30516, axhis3-zf 30517, axhis4-zf 30518, and axhcompl-zf 30519 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 30502.

 
Axiomax-hilex 30520 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, β„‹, which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
β„‹ ∈ V
 
Axiomax-hfvadd 30521 Vector addition is an operation on β„‹. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
+β„Ž :( β„‹ Γ— β„‹)⟢ β„‹
 
Axiomax-hvcom 30522 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž 𝐡) = (𝐡 +β„Ž 𝐴))
 
Axiomax-hvass 30523 Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) +β„Ž 𝐢) = (𝐴 +β„Ž (𝐡 +β„Ž 𝐢)))
 
Axiomax-hv0cl 30524 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
0β„Ž ∈ β„‹
 
Axiomax-hvaddid 30525 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 +β„Ž 0β„Ž) = 𝐴)
 
Axiomax-hfvmul 30526 Scalar multiplication is an operation on β„‚ and β„‹. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
Β·β„Ž :(β„‚ Γ— β„‹)⟢ β„‹
 
Axiomax-hvmulid 30527 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (1 Β·β„Ž 𝐴) = 𝐴)
 
Axiomax-hvmulass 30528 Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β· 𝐡) Β·β„Ž 𝐢) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)))
 
Axiomax-hvdistr1 30529 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) +β„Ž (𝐴 Β·β„Ž 𝐢)))
 
Axiomax-hvdistr2 30530 Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 + 𝐡) Β·β„Ž 𝐢) = ((𝐴 Β·β„Ž 𝐢) +β„Ž (𝐡 Β·β„Ž 𝐢)))
 
Axiomax-hvmul0 30531 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 30547 and hvsubval 30537). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (0 Β·β„Ž 𝐴) = 0β„Ž)
 
20.1.5  Vector operations
 
Theoremhvmulex 30532 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
Β·β„Ž ∈ V
 
Theoremhvaddcl 30533 Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž 𝐡) ∈ β„‹)
 
Theoremhvmulcl 30534 Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 Β·β„Ž 𝐡) ∈ β„‹)
 
Theoremhvmulcli 30535 Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‹    β‡’   (𝐴 Β·β„Ž 𝐡) ∈ β„‹
 
Theoremhvsubf 30536 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
βˆ’β„Ž :( β„‹ Γ— β„‹)⟢ β„‹
 
Theoremhvsubval 30537 Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 βˆ’β„Ž 𝐡) = (𝐴 +β„Ž (-1 Β·β„Ž 𝐡)))
 
Theoremhvsubcl 30538 Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 βˆ’β„Ž 𝐡) ∈ β„‹)
 
Theoremhvaddcli 30539 Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 +β„Ž 𝐡) ∈ β„‹
 
Theoremhvcomi 30540 Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 +β„Ž 𝐡) = (𝐡 +β„Ž 𝐴)
 
Theoremhvsubvali 30541 Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 βˆ’β„Ž 𝐡) = (𝐴 +β„Ž (-1 Β·β„Ž 𝐡))
 
Theoremhvsubcli 30542 Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 βˆ’β„Ž 𝐡) ∈ β„‹
 
Theoremifhvhv0 30543 Prove if(𝐴 ∈ β„‹, 𝐴, 0β„Ž) ∈ β„‹. (Contributed by David A. Wheeler, 7-Dec-2018.) (New usage is discouraged.)
if(𝐴 ∈ β„‹, 𝐴, 0β„Ž) ∈ β„‹
 
Theoremhvaddlid 30544 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (0β„Ž +β„Ž 𝐴) = 𝐴)
 
Theoremhvmul0 30545 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‚ β†’ (𝐴 Β·β„Ž 0β„Ž) = 0β„Ž)
 
Theoremhvmul0or 30546 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 Β·β„Ž 𝐡) = 0β„Ž ↔ (𝐴 = 0 ∨ 𝐡 = 0β„Ž)))
 
Theoremhvsubid 30547 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 βˆ’β„Ž 𝐴) = 0β„Ž)
 
Theoremhvnegid 30548 Addition of negative of a vector to itself. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 +β„Ž (-1 Β·β„Ž 𝐴)) = 0β„Ž)
 
Theoremhv2neg 30549 Two ways to express the negative of a vector. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (0β„Ž βˆ’β„Ž 𝐴) = (-1 Β·β„Ž 𝐴))
 
Theoremhvaddlidi 30550 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   (0β„Ž +β„Ž 𝐴) = 𝐴
 
Theoremhvnegidi 30551 Addition of negative of a vector to itself. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   (𝐴 +β„Ž (-1 Β·β„Ž 𝐴)) = 0β„Ž
 
Theoremhv2negi 30552 Two ways to express the negative of a vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   (0β„Ž βˆ’β„Ž 𝐴) = (-1 Β·β„Ž 𝐴)
 
Theoremhvm1neg 30553 Convert minus one times a scalar product to the negative of the scalar. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ (-1 Β·β„Ž (𝐴 Β·β„Ž 𝐡)) = (-𝐴 Β·β„Ž 𝐡))
 
Theoremhvaddsubval 30554 Value of vector addition in terms of vector subtraction. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž 𝐡) = (𝐴 βˆ’β„Ž (-1 Β·β„Ž 𝐡)))
 
Theoremhvadd32 30555 Commutative/associative law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) +β„Ž 𝐢) = ((𝐴 +β„Ž 𝐢) +β„Ž 𝐡))
 
Theoremhvadd12 30556 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 +β„Ž (𝐡 +β„Ž 𝐢)) = (𝐡 +β„Ž (𝐴 +β„Ž 𝐢)))
 
Theoremhvadd4 30557 Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
(((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 +β„Ž 𝐡) +β„Ž (𝐢 +β„Ž 𝐷)) = ((𝐴 +β„Ž 𝐢) +β„Ž (𝐡 +β„Ž 𝐷)))
 
Theoremhvsub4 30558 Hilbert vector space addition/subtraction law. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
(((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž (𝐢 +β„Ž 𝐷)) = ((𝐴 βˆ’β„Ž 𝐢) +β„Ž (𝐡 βˆ’β„Ž 𝐷)))
 
Theoremhvaddsub12 30559 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 +β„Ž (𝐡 βˆ’β„Ž 𝐢)) = (𝐡 +β„Ž (𝐴 βˆ’β„Ž 𝐢)))
 
Theoremhvpncan 30560 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž 𝐡) = 𝐴)
 
Theoremhvpncan2 30561 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž 𝐴) = 𝐡)
 
Theoremhvaddsubass 30562 Associativity of sum and difference of Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž 𝐢) = (𝐴 +β„Ž (𝐡 βˆ’β„Ž 𝐢)))
 
Theoremhvpncan3 30563 Subtraction and addition of equal Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž (𝐡 βˆ’β„Ž 𝐴)) = 𝐡)
 
Theoremhvmulcom 30564 Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)) = (𝐡 Β·β„Ž (𝐴 Β·β„Ž 𝐢)))
 
Theoremhvsubass 30565 Hilbert vector space associative law for subtraction. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž 𝐢) = (𝐴 βˆ’β„Ž (𝐡 +β„Ž 𝐢)))
 
Theoremhvsub32 30566 Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž 𝐢) = ((𝐴 βˆ’β„Ž 𝐢) βˆ’β„Ž 𝐡))
 
Theoremhvmulassi 30567 Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‹    β‡’   ((𝐴 Β· 𝐡) Β·β„Ž 𝐢) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢))
 
Theoremhvmulcomi 30568 Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‹    β‡’   (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)) = (𝐡 Β·β„Ž (𝐴 Β·β„Ž 𝐢))
 
Theoremhvmul2negi 30569 Double negative in scalar multiplication. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‹    β‡’   (-𝐴 Β·β„Ž (-𝐡 Β·β„Ž 𝐢)) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢))
 
Theoremhvsubdistr1 30570 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 βˆ’β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) βˆ’β„Ž (𝐴 Β·β„Ž 𝐢)))
 
Theoremhvsubdistr2 30571 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’ 𝐡) Β·β„Ž 𝐢) = ((𝐴 Β·β„Ž 𝐢) βˆ’β„Ž (𝐡 Β·β„Ž 𝐢)))
 
Theoremhvdistr1i 30572 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   (𝐴 Β·β„Ž (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) +β„Ž (𝐴 Β·β„Ž 𝐢))
 
Theoremhvsubdistr1i 30573 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   (𝐴 Β·β„Ž (𝐡 βˆ’β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) βˆ’β„Ž (𝐴 Β·β„Ž 𝐢))
 
Theoremhvassi 30574 Hilbert vector space associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) +β„Ž 𝐢) = (𝐴 +β„Ž (𝐡 +β„Ž 𝐢))
 
Theoremhvadd32i 30575 Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) +β„Ž 𝐢) = ((𝐴 +β„Ž 𝐢) +β„Ž 𝐡)
 
Theoremhvsubassi 30576 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 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž 𝐢) = (𝐴 βˆ’β„Ž (𝐡 +β„Ž 𝐢))
 
Theoremhvsub32i 30577 Hilbert vector space commutative/associative law. (Contributed by NM, 7-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž 𝐢) = ((𝐴 βˆ’β„Ž 𝐢) βˆ’β„Ž 𝐡)
 
Theoremhvadd12i 30578 Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   (𝐴 +β„Ž (𝐡 +β„Ž 𝐢)) = (𝐡 +β„Ž (𝐴 +β„Ž 𝐢))
 
Theoremhvadd4i 30579 Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) +β„Ž (𝐢 +β„Ž 𝐷)) = ((𝐴 +β„Ž 𝐢) +β„Ž (𝐡 +β„Ž 𝐷))
 
Theoremhvsubsub4i 30580 Hilbert vector space addition law. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž (𝐢 βˆ’β„Ž 𝐷)) = ((𝐴 βˆ’β„Ž 𝐢) βˆ’β„Ž (𝐡 βˆ’β„Ž 𝐷))
 
Theoremhvsubsub4 30581 Hilbert vector space addition/subtraction law. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
(((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž (𝐢 βˆ’β„Ž 𝐷)) = ((𝐴 βˆ’β„Ž 𝐢) βˆ’β„Ž (𝐡 βˆ’β„Ž 𝐷)))
 
Theoremhv2times 30582 Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (2 Β·β„Ž 𝐴) = (𝐴 +β„Ž 𝐴))
 
Theoremhvnegdii 30583 Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (-1 Β·β„Ž (𝐴 βˆ’β„Ž 𝐡)) = (𝐡 βˆ’β„Ž 𝐴)
 
Theoremhvsubeq0i 30584 If the difference between two vectors is zero, they are equal. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) = 0β„Ž ↔ 𝐴 = 𝐡)
 
Theoremhvsubcan2i 30585 Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) +β„Ž (𝐴 βˆ’β„Ž 𝐡)) = (2 Β·β„Ž 𝐴)
 
Theoremhvaddcani 30586 Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) = (𝐴 +β„Ž 𝐢) ↔ 𝐡 = 𝐢)
 
Theoremhvsubaddi 30587 Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) = 𝐢 ↔ (𝐡 +β„Ž 𝐢) = 𝐴)
 
Theoremhvnegdi 30588 Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (-1 Β·β„Ž (𝐴 βˆ’β„Ž 𝐡)) = (𝐡 βˆ’β„Ž 𝐴))
 
Theoremhvsubeq0 30589 If the difference between two vectors is zero, they are equal. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) = 0β„Ž ↔ 𝐴 = 𝐡))
 
Theoremhvaddeq0 30590 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 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) = 0β„Ž ↔ 𝐴 = (-1 Β·β„Ž 𝐡)))
 
Theoremhvaddcan 30591 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) = (𝐴 +β„Ž 𝐢) ↔ 𝐡 = 𝐢))
 
Theoremhvaddcan2 30592 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐢) = (𝐡 +β„Ž 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremhvmulcan 30593 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β·β„Ž 𝐡) = (𝐴 Β·β„Ž 𝐢) ↔ 𝐡 = 𝐢))
 
Theoremhvmulcan2 30594 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ (𝐢 ∈ β„‹ ∧ 𝐢 β‰  0β„Ž)) β†’ ((𝐴 Β·β„Ž 𝐢) = (𝐡 Β·β„Ž 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremhvsubcan 30595 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) = (𝐴 βˆ’β„Ž 𝐢) ↔ 𝐡 = 𝐢))
 
Theoremhvsubcan2 30596 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐢) = (𝐡 βˆ’β„Ž 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremhvsub0 30597 Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 βˆ’β„Ž 0β„Ž) = 𝐴)
 
Theoremhvsubadd 30598 Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) = 𝐢 ↔ (𝐡 +β„Ž 𝐢) = 𝐴))
 
Theoremhvaddsub4 30599 Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 +β„Ž 𝐡) = (𝐢 +β„Ž 𝐷) ↔ (𝐴 βˆ’β„Ž 𝐢) = (𝐷 βˆ’β„Ž 𝐡)))
 
20.1.6  Inner product postulates for a Hilbert space
 
Axiomax-hfi 30600 Inner product maps pairs from β„‹ to β„‚. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
Β·ih :( β„‹ Γ— β„‹)βŸΆβ„‚
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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