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Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempjopyth 29501 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐺C𝐴 ∈ ℋ) → (𝐻 ⊆ (⊥‘𝐺) → ((norm‘(((proj𝐻)‘𝐴) + ((proj𝐺)‘𝐴)))↑2) = (((norm‘((proj𝐻)‘𝐴))↑2) + ((norm‘((proj𝐺)‘𝐴))↑2))))

Theorempjnormi 29502 The norm of the projection is less than or equal to the norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       (norm‘((proj𝐻)‘𝐴)) ≤ (norm𝐴)

Theorempjpythi 29503 Pythagorean theorem for projections. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((norm𝐴)↑2) = (((norm‘((proj𝐻)‘𝐴))↑2) + ((norm‘((proj‘(⊥‘𝐻))‘𝐴))↑2))

Theorempjneli 29504 If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       𝐴𝐻 ↔ (norm‘((proj𝐻)‘𝐴)) < (norm𝐴))

Theorempjnorm 29505 The norm of the projection is less than or equal to the norm. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (norm‘((proj𝐻)‘𝐴)) ≤ (norm𝐴))

Theorempjpyth 29506 Pythagorean theorem for projectors. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((norm𝐴)↑2) = (((norm‘((proj𝐻)‘𝐴))↑2) + ((norm‘((proj‘(⊥‘𝐻))‘𝐴))↑2)))

Theorempjnel 29507 If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (¬ 𝐴𝐻 ↔ (norm‘((proj𝐻)‘𝐴)) < (norm𝐴)))

Theorempjnorm2 29508 A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch 29475 yield Theorem 26.3 of [Halmos] p. 44. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴𝐻 ↔ (norm‘((proj𝐻)‘𝐴)) = (norm𝐴)))

19.5.11  Mayet's equation E_3

Theoremmayete3i 29509 Mayet's equation E3. Part of Theorem 4.1 of [Mayet3] p. 1223. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝐹C    &   𝐺C    &   𝐴 ⊆ (⊥‘𝐶)    &   𝐴 ⊆ (⊥‘𝐹)    &   𝐶 ⊆ (⊥‘𝐹)    &   𝐴 ⊆ (⊥‘𝐵)    &   𝐶 ⊆ (⊥‘𝐷)    &   𝐹 ⊆ (⊥‘𝐺)    &   𝑋 = ((𝐴 𝐶) ∨ 𝐹)    &   𝑌 = (((𝐴 𝐵) ∩ (𝐶 𝐷)) ∩ (𝐹 𝐺))    &   𝑍 = ((𝐵 𝐷) ∨ 𝐺)       (𝑋𝑌) ⊆ 𝑍

Theoremmayetes3i 29510 Mayet's equation E^*3, derived from E3. Solution, for n = 3, to open problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. (Contributed by NM, 10-May-2009.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝐹C    &   𝐺C    &   𝑅C    &   𝐴 ⊆ (⊥‘𝐶)    &   𝐴 ⊆ (⊥‘𝐹)    &   𝐶 ⊆ (⊥‘𝐹)    &   𝐴 ⊆ (⊥‘𝐵)    &   𝐶 ⊆ (⊥‘𝐷)    &   𝐹 ⊆ (⊥‘𝐺)    &   𝑅 ⊆ (⊥‘𝑋)    &   𝑋 = ((𝐴 𝐶) ∨ 𝐹)    &   𝑌 = (((𝐴 𝐵) ∩ (𝐶 𝐷)) ∩ (𝐹 𝐺))    &   𝑍 = ((𝐵 𝐷) ∨ 𝐺)       ((𝑋 𝑅) ∩ 𝑌) ⊆ (𝑍 𝑅)

19.6  Operators on Hilbert spaces

19.6.1  Operator sum, difference, and scalar multiplication

Note on operators. Unlike some authors, we use the term "operator" to mean any function from to . This is the definition of operator in [Hughes] p. 14, the definition of operator in [AkhiezerGlazman] p. 30, and the definition of operator in [Goldberg] p. 10. For Reed and Simon, an operator is linear (definition of operator in [ReedSimon] p. 2). For Halmos, an operator is bounded and linear (definition of operator in [Halmos] p. 35). For Kalmbach and Beran, an operator is continuous and linear (definition of operator in [Kalmbach] p. 353; definition of operator in [Beran] p. 99). Note that "bounded and linear" and "continuous and linear" are equivalent by lncnbd 29819.

Definitiondf-hosum 29511* Define the sum of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
+op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))

Definitiondf-homul 29512* Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))

Definitiondf-hodif 29513* Define the difference of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) − (𝑔𝑥))))

Definitiondf-hfsum 29514* Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from to , not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
+fn = (𝑓 ∈ (ℂ ↑m ℋ), 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))

Definitiondf-hfmul 29515* Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))

Theoremhosmval 29516* Value of the sum of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))

Theoremhommval 29517* Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))

Theoremhodmval 29518* Value of the difference of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) − (𝑇𝑥))))

Theoremhfsmval 29519* Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))

Theoremhfmmval 29520* Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))

Theoremhosval 29521 Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))

Theoremhomval 29522 Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) = (𝐴 · (𝑇𝐵)))

Theoremhodval 29523 Value of the difference of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆op 𝑇)‘𝐴) = ((𝑆𝐴) − (𝑇𝐴)))

Theoremhfsval 29524 Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))

Theoremhfmval 29525 Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·fn 𝑇)‘𝐵) = (𝐴 · (𝑇𝐵)))

Theoremhoscl 29526 Closure of the sum of two Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
(((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) ∈ ℋ)

Theoremhomcl 29527 Closure of the scalar product of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) ∈ ℋ)

Theoremhodcl 29528 Closure of the difference of two Hilbert space operators. (Contributed by NM, 15-Nov-2002.) (New usage is discouraged.)
(((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆op 𝑇)‘𝐴) ∈ ℋ)

19.6.2  Zero and identity operators

Definitiondf-h0op 29529 Define the Hilbert space zero operator. See df0op2 29533 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop = (proj‘0)

Definitiondf-iop 29530 Define the Hilbert space identity operator. See dfiop2 29534 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
Iop = (proj‘ ℋ)

Theoremho0val 29531 Value of the zero Hilbert space operator (null projector). Remark in [Beran] p. 111. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ( 0hop𝐴) = 0)

Theoremho0f 29532 Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
0hop : ℋ⟶ ℋ

Theoremdf0op2 29533 Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
0hop = ( ℋ × 0)

Theoremdfiop2 29534 Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
Iop = ( I ↾ ℋ)

Theoremhoif 29535 Functionality of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
Iop : ℋ–1-1-onto→ ℋ

Theoremhoival 29536 The value of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ( Iop𝐴) = 𝐴)

Theoremhoico1 29537 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇 ∘ Iop ) = 𝑇)

Theoremhoico2 29538 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ( Iop𝑇) = 𝑇)

19.6.3  Operations on Hilbert space operators

Theoremhoaddcl 29539 The sum of Hilbert space operators is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)

Theoremhomulcl 29540 The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)

Theoremhoeq 29541* Equality of Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥) ↔ 𝑇 = 𝑈))

Theoremhoeqi 29542* Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (∀𝑥 ∈ ℋ (𝑆𝑥) = (𝑇𝑥) ↔ 𝑆 = 𝑇)

Theoremhoscli 29543 Closure of Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → ((𝑆 +op 𝑇)‘𝐴) ∈ ℋ)

Theoremhodcli 29544 Closure of Hilbert space operator difference. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → ((𝑆op 𝑇)‘𝐴) ∈ ℋ)

Theoremhocoi 29545 Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) = (𝑆‘(𝑇𝐴)))

Theoremhococli 29546 Closure of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) ∈ ℋ)

Theoremhocofi 29547 Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆𝑇): ℋ⟶ ℋ

Theoremhocofni 29548 Functionality of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆𝑇) Fn ℋ

Theoremhoaddcli 29549 Mapping of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op 𝑇): ℋ⟶ ℋ

Theoremhosubcli 29550 Mapping of difference of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆op 𝑇): ℋ⟶ ℋ

Theoremhoaddfni 29551 Functionality of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op 𝑇) Fn ℋ

Theoremhosubfni 29552 Functionality of difference of Hilbert space operators. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆op 𝑇) Fn ℋ

Theoremhoaddcomi 29553 Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op 𝑇) = (𝑇 +op 𝑆)

Theoremhosubcl 29554 Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆op 𝑇): ℋ⟶ ℋ)

Theoremhoaddcom 29555 Commutativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑇 +op 𝑆))

Theoremhodsi 29556 Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅op 𝑆) = 𝑇 ↔ (𝑆 +op 𝑇) = 𝑅)

Theoremhoaddassi 29557 Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))

Theoremhoadd12i 29558 Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇))

Theoremhoadd32i 29559 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆)

Theoremhocadddiri 29560 Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅𝑇) +op (𝑆𝑇))

Theoremhocsubdiri 29561 Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅op 𝑆) ∘ 𝑇) = ((𝑅𝑇) −op (𝑆𝑇))

Theoremho2coi 29562 Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇𝐴))))

Theoremhoaddass 29563 Associativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)))

Theoremhoadd32 29564 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆))

Theoremhoadd4 29565 Rearrangement of 4 terms in a sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 +op 𝑆) +op (𝑇 +op 𝑈)) = ((𝑅 +op 𝑇) +op (𝑆 +op 𝑈)))

Theoremhocsubdir 29566 Distributive law for Hilbert space operator difference. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅op 𝑆) ∘ 𝑇) = ((𝑅𝑇) −op (𝑆𝑇)))

Theoremhoaddid1i 29567 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (𝑇 +op 0hop ) = 𝑇

Theoremhodidi 29568 Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (𝑇op 𝑇) = 0hop

Theoremho0coi 29569 Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       ( 0hop𝑇) = 0hop

Theoremhoid1i 29570 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (𝑇 ∘ Iop ) = 𝑇

Theoremhoid1ri 29571 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       ( Iop𝑇) = 𝑇

Theoremhoaddid1 29572 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇 +op 0hop ) = 𝑇)

Theoremhodid 29573 Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇op 𝑇) = 0hop )

Theoremhon0 29574 A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)

Theoremhodseqi 29575 Subtraction and addition of equal Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op (𝑇op 𝑆)) = 𝑇

Theoremho0subi 29576 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆op 𝑇) = (𝑆 +op ( 0hopop 𝑇))

Theoremhonegsubi 29577 Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op (-1 ·op 𝑇)) = (𝑆op 𝑇)

Theoremho0sub 29578 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆op 𝑇) = (𝑆 +op ( 0hopop 𝑇)))

Theoremhosubid1 29579 The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇op 0hop ) = 𝑇)

Theoremhonegsub 29580 Relationship between Hilbert space operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 +op (-1 ·op 𝑈)) = (𝑇op 𝑈))

Theoremhomulid2 29581 An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (1 ·op 𝑇) = 𝑇)

Theoremhomco1 29582 Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇𝑈)))

Theoremhomulass 29583 Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇)))

Theoremhoadddi 29584 Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)))

Theoremhoadddir 29585 Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇) = ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)))

Theoremhomul12 29586 Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)) = (𝐵 ·op (𝐴 ·op 𝑇)))

Theoremhonegneg 29587 Double negative of a Hilbert space operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (-1 ·op (-1 ·op 𝑇)) = 𝑇)

Theoremhosubneg 29588 Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇op (-1 ·op 𝑈)) = (𝑇 +op 𝑈))

Theoremhosubdi 29589 Scalar product distributive law for operator difference. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇op 𝑈)) = ((𝐴 ·op 𝑇) −op (𝐴 ·op 𝑈)))

Theoremhonegdi 29590 Distribution of negative over addition. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇 +op 𝑈)) = ((-1 ·op 𝑇) +op (-1 ·op 𝑈)))

Theoremhonegsubdi 29591 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇op 𝑈)) = ((-1 ·op 𝑇) +op 𝑈))

Theoremhonegsubdi2 29592 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇op 𝑈)) = (𝑈op 𝑇))

Theoremhosubsub2 29593 Law for double subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆op (𝑇op 𝑈)) = (𝑆 +op (𝑈op 𝑇)))

Theoremhosub4 29594 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 +op 𝑆) −op (𝑇 +op 𝑈)) = ((𝑅op 𝑇) +op (𝑆op 𝑈)))

Theoremhosubadd4 29595 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅op 𝑆) −op (𝑇op 𝑈)) = ((𝑅 +op 𝑈) −op (𝑆 +op 𝑇)))

Theoremhoaddsubass 29596 Associative-type law for addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 +op 𝑇) −op 𝑈) = (𝑆 +op (𝑇op 𝑈)))

Theoremhoaddsub 29597 Law for operator addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 +op 𝑇) −op 𝑈) = ((𝑆op 𝑈) +op 𝑇))

Theoremhosubsub 29598 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆op (𝑇op 𝑈)) = ((𝑆op 𝑇) +op 𝑈))

Theoremhosubsub4 29599 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆op 𝑇) −op 𝑈) = (𝑆op (𝑇 +op 𝑈)))

Theoremho2times 29600 Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇))

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