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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremchintcl 29101 The intersection (infimum) of a nonempty subset of C belongs to C. Part of Theorem 3.13 of [Beran] p. 108. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
((𝐴C𝐴 ≠ ∅) → 𝐴C )

Theoremspanval 29102* Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in [Schechter] p. 276. (Contributed by NM, 2-Jun-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (span‘𝐴) = {𝑥S𝐴𝑥})

Theoremhsupval 29103 Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 29178. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))

Theoremchsupval 29104 The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 29179. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))

Theoremspancl 29105 The span of a subset of Hilbert space is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (span‘𝐴) ∈ S )

Theoremelspancl 29106 A member of a span is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ∈ (span‘𝐴)) → 𝐵 ∈ ℋ)

Theoremshsupcl 29107 Closure of the subspace supremum of set of subsets of Hilbert space. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → (span‘ 𝐴) ∈ S )

Theoremhsupcl 29108 Closure of supremum of set of subsets of Hilbert space. Note that the supremum belongs to C even if the subsets do not. (Contributed by NM, 10-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → ( 𝐴) ∈ C )

Theoremchsupcl 29109 Closure of supremum of subset of C. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. Shows that C is a complete lattice. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 10-Nov-1999.) (New usage is discouraged.)
(𝐴C → ( 𝐴) ∈ C )

Theoremhsupss 29110 Subset relation for supremum of Hilbert space subsets. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ⊆ 𝒫 ℋ ∧ 𝐵 ⊆ 𝒫 ℋ) → (𝐴𝐵 → ( 𝐴) ⊆ ( 𝐵)))

Theoremchsupss 29111 Subset relation for supremum of subset of C. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 → ( 𝐴) ⊆ ( 𝐵)))

Theoremhsupunss 29112 The union of a set of Hilbert space subsets is smaller than its supremum. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → 𝐴 ⊆ ( 𝐴))

Theoremchsupunss 29113 The union of a set of closed subspaces is smaller than its supremum. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
(𝐴C 𝐴 ⊆ ( 𝐴))

Theoremspanss2 29114 A subset of Hilbert space is included in its span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → 𝐴 ⊆ (span‘𝐴))

Theoremshsupunss 29115 The union of a set of subspaces is smaller than its supremum. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
(𝐴S 𝐴 ⊆ (span‘ 𝐴))

Theoremspanid 29116 A subspace of Hilbert space is its own span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
(𝐴S → (span‘𝐴) = 𝐴)

Theoremspanss 29117 Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
((𝐵 ⊆ ℋ ∧ 𝐴𝐵) → (span‘𝐴) ⊆ (span‘𝐵))

Theoremspanssoc 29118 The span of a subset of Hilbert space is less than or equal to its closure (double orthogonal complement). (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (span‘𝐴) ⊆ (⊥‘(⊥‘𝐴)))

Theoremsshjval 29119 Value of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))

Theoremshjval 29120 Value of join in S. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))

Theoremchjval 29121 Value of join in C. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))

Theoremchjvali 29122 Value of join in C. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵)))

Theoremsshjval3 29123 Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice C. (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = ( ‘{𝐴, 𝐵}))

Theoremsshjcl 29124 Closure of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) ∈ C )

Theoremshjcl 29125 Closure of join in S. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 𝐵) ∈ C )

Theoremchjcl 29126 Closure of join in C. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵) ∈ C )

Theoremshjcom 29127 Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))

Theoremshless 29128 Subset implies subset of subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(((𝐴S𝐵S𝐶S ) ∧ 𝐴𝐵) → (𝐴 + 𝐶) ⊆ (𝐵 + 𝐶))

Theoremshlej1 29129 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(((𝐴S𝐵S𝐶S ) ∧ 𝐴𝐵) → (𝐴 𝐶) ⊆ (𝐵 𝐶))

Theoremshlej2 29130 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(((𝐴S𝐵S𝐶S ) ∧ 𝐴𝐵) → (𝐶 𝐴) ⊆ (𝐶 𝐵))

Theoremshincli 29131 Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵) ∈ S

Theoremshscomi 29132 Commutative law for subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) = (𝐵 + 𝐴)

Theoremshsvai 29133 Vector sum belongs to subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐶𝐴𝐷𝐵) → (𝐶 + 𝐷) ∈ (𝐴 + 𝐵))

Theoremshsel1i 29134 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐶𝐴𝐶 ∈ (𝐴 + 𝐵))

Theoremshsel2i 29135 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐶𝐵𝐶 ∈ (𝐴 + 𝐵))

Theoremshsvsi 29136 Vector subtraction belongs to subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐶𝐴𝐷𝐵) → (𝐶 𝐷) ∈ (𝐴 + 𝐵))

Theoremshunssi 29137 Union is smaller than subspace sum. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵) ⊆ (𝐴 + 𝐵)

Theoremshunssji 29138 Union is smaller than Hilbert lattice join. (Contributed by NM, 11-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵) ⊆ (𝐴 𝐵)

Theoremshsleji 29139 Subspace sum is smaller than Hilbert lattice join. Remark in [Kalmbach] p. 65. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) ⊆ (𝐴 𝐵)

Theoremshjcomi 29140 Commutative law for join in S. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) = (𝐵 𝐴)

Theoremshsub1i 29141 Subspace sum is an upper bound of its arguments. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       𝐴 ⊆ (𝐴 + 𝐵)

Theoremshsub2i 29142 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       𝐴 ⊆ (𝐵 + 𝐴)

Theoremshub1i 29143 Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       𝐴 ⊆ (𝐴 𝐵)

Theoremshjcli 29144 Closure of C join. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) ∈ C

Theoremshjshcli 29145 S closure of join. (Contributed by NM, 5-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) ∈ S

Theoremshlessi 29146 Subset implies subset of subspace sum. (Contributed by NM, 18-Nov-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐵 → (𝐴 + 𝐶) ⊆ (𝐵 + 𝐶))

Theoremshlej1i 29147 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐵 → (𝐴 𝐶) ⊆ (𝐵 𝐶))

Theoremshlej2i 29148 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐵 → (𝐶 𝐴) ⊆ (𝐶 𝐵))

Theoremshslej 29149 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) ⊆ (𝐴 𝐵))

Theoremshincl 29150 Closure of intersection of two subspaces. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴𝐵) ∈ S )

Theoremshub1 29151 Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐴 𝐵))

Theoremshub2 29152 A subspace is a subset of its Hilbert lattice join with another. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐵 𝐴))

Theoremshsidmi 29153 Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
𝐴S       (𝐴 + 𝐴) = 𝐴

Theoremshslubi 29154 The least upper bound law for Hilbert subspace sum. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       ((𝐴𝐶𝐵𝐶) ↔ (𝐴 + 𝐵) ⊆ 𝐶)

Theoremshlesb1i 29155 Hilbert lattice ordering in terms of subspace sum. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵 ↔ (𝐴 + 𝐵) = 𝐵)

Theoremshsval2i 29156* An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) = {𝑥S ∣ (𝐴𝐵) ⊆ 𝑥}

Theoremshsval3i 29157 An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) = (span‘(𝐴𝐵))

Theoremshmodsi 29158 The modular law holds for subspace sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐶 → ((𝐴 + 𝐵) ∩ 𝐶) ⊆ (𝐴 + (𝐵𝐶)))

Theoremshmodi 29159 The modular law is implied by the closure of subspace sum. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (((𝐴 + 𝐵) = (𝐴 𝐵) ∧ 𝐴𝐶) → ((𝐴 𝐵) ∩ 𝐶) ⊆ (𝐴 (𝐵𝐶)))

19.4.5  Projection theorem

Theorempjhthlem1 29160* Lemma for pjhth 29162. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (Proof shortened by AV, 10-Jul-2022.) (New usage is discouraged.)
𝐻C    &   (𝜑𝐴 ∈ ℋ)    &   (𝜑𝐵𝐻)    &   (𝜑𝐶𝐻)    &   (𝜑 → ∀𝑥𝐻 (norm‘(𝐴 𝐵)) ≤ (norm‘(𝐴 𝑥)))    &   𝑇 = (((𝐴 𝐵) ·ih 𝐶) / ((𝐶 ·ih 𝐶) + 1))       (𝜑 → ((𝐴 𝐵) ·ih 𝐶) = 0)

Theorempjhthlem2 29161* Lemma for pjhth 29162. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐻C    &   (𝜑𝐴 ∈ ℋ)       (𝜑 → ∃𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦))

Theorempjhth 29162 Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐻C → (𝐻 + (⊥‘𝐻)) = ℋ)

Theorempjhtheu 29163* Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102. See pjhtheu2 29185 for the uniqueness of 𝑦. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ∃!𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦))

19.4.6  Projectors

Definitiondf-pjh 29164* Define the projection function on a Hilbert space, as a mapping from the Hilbert lattice to a function on Hilbert space. Every closed subspace is associated with a unique projection function. Remark in [Kalmbach] p. 66, adopted as a definition. (proj𝐻)‘𝐴 is the projection of vector 𝐴 onto closed subspace 𝐻. Note that the range of proj is the set of all projection operators, so 𝑇 ∈ ran proj means that 𝑇 is a projection operator. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))

Theorempjhfval 29165* The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
(𝐻C → (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))

Theorempjhval 29166* Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))

Theorempjpreeq 29167* Equality with a projection. This version of pjeq 29168 does not assume the Axiom of Choice via pjhth 29162. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ (𝐻 + (⊥‘𝐻))) → (((proj𝐻)‘𝐴) = 𝐵 ↔ (𝐵𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 + 𝑥))))

Theorempjeq 29168* Equality with a projection. (Contributed by NM, 20-Jan-2007.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (((proj𝐻)‘𝐴) = 𝐵 ↔ (𝐵𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 + 𝑥))))

Theoremaxpjcl 29169 Closure of a projection in its subspace. If we consider this together with axpjpj 29189 to be axioms, the need for the ax-hcompl 28971 can often be avoided for the kinds of theorems we are interested in here. An interesting project is to see how far we can go by using them in place of it. In particular, we can prove the orthomodular law pjomli 29204.) (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) ∈ 𝐻)

Theorempjhcl 29170 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) ∈ ℋ)

19.5  Properties of Hilbert subspaces

19.5.1  Orthomodular law

Theoremomlsilem 29171 Lemma for orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
𝐺S    &   𝐻S    &   𝐺𝐻    &   (𝐻 ∩ (⊥‘𝐺)) = 0    &   𝐴𝐻    &   𝐵𝐺    &   𝐶 ∈ (⊥‘𝐺)       (𝐴 = (𝐵 + 𝐶) → 𝐴𝐺)

Theoremomlsii 29172 Subspace inference form of orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴C    &   𝐵S    &   𝐴𝐵    &   (𝐵 ∩ (⊥‘𝐴)) = 0       𝐴 = 𝐵

Theoremomlsi 29173 Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵S       ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) → 𝐴 = 𝐵)

Theoremococi 29174 Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
𝐴C       (⊥‘(⊥‘𝐴)) = 𝐴

Theoremococ 29175 Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
(𝐴C → (⊥‘(⊥‘𝐴)) = 𝐴)

Theoremdfch2 29176 Alternate definition of the Hilbert lattice. (Contributed by NM, 8-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
C = {𝑥 ∈ 𝒫 ℋ ∣ (⊥‘(⊥‘𝑥)) = 𝑥}

Theoremococin 29177* The double complement is the smallest closed subspace containing a subset of Hilbert space. Remark 3.12(B) of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = {𝑥C𝐴𝑥})

Theoremhsupval2 29178* Alternate definition of supremum of a subset of the Hilbert lattice. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. We actually define it on any collection of Hilbert space subsets, not just the Hilbert lattice C, to allow more general theorems. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = {𝑥C 𝐴𝑥})

Theoremchsupval2 29179* The value of the supremum of a set of closed subspaces of Hilbert space. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( 𝐴) = {𝑥C 𝐴𝑥})

Theoremsshjval2 29180* Value of join in the set of closed subspaces of Hilbert space C. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = {𝑥C ∣ (𝐴𝐵) ⊆ 𝑥})

Theoremchsupid 29181* A subspace is the supremum of all smaller subspaces. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( ‘{𝑥C𝑥𝐴}) = 𝐴)

Theoremchsupsn 29182 Value of supremum of subset of C on a singleton. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( ‘{𝐴}) = 𝐴)

Theoremshlub 29183 Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S𝐶C ) → ((𝐴𝐶𝐵𝐶) ↔ (𝐴 𝐵) ⊆ 𝐶))

Theoremshlubi 29184 Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶C       ((𝐴𝐶𝐵𝐶) ↔ (𝐴 𝐵) ⊆ 𝐶)

19.5.2  Projectors (cont.)

Theorempjhtheu2 29185* Uniqueness of 𝑦 for the projection theorem. (Contributed by NM, 6-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ∃!𝑦 ∈ (⊥‘𝐻)∃𝑥𝐻 𝐴 = (𝑥 + 𝑦))

Theorempjcli 29186 Closure of a projection in its subspace. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → ((proj𝐻)‘𝐴) ∈ 𝐻)

Theorempjhcli 29187 Closure of a projection in Hilbert space. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → ((proj𝐻)‘𝐴) ∈ ℋ)

Theorempjpjpre 29188 Decomposition of a vector into projections. This formulation of axpjpj 29189 avoids pjhth 29162. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(𝜑𝐻C )    &   (𝜑𝐴 ∈ (𝐻 + (⊥‘𝐻)))       (𝜑𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴)))

Theoremaxpjpj 29189 Decomposition of a vector into projections. See comment in axpjcl 29169. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → 𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴)))

Theorempjclii 29190 Closure of a projection in its subspace. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) ∈ 𝐻

Theorempjhclii 29191 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) ∈ ℋ

Theorempjpj0i 29192 Decomposition of a vector into projections. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴))

Theorempjpji 29193 Decomposition of a vector into projections. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴))

Theorempjpjhth 29194* Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ∃𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦))

Theorempjpjhthi 29195* Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐻C       𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)

Theorempjop 29196 Orthocomplement projection in terms of projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj‘(⊥‘𝐻))‘𝐴) = (𝐴 ((proj𝐻)‘𝐴)))

Theorempjpo 29197 Projection in terms of orthocomplement projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝐴 ((proj‘(⊥‘𝐻))‘𝐴)))

Theorempjopi 29198 Orthocomplement projection in terms of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj‘(⊥‘𝐻))‘𝐴) = (𝐴 ((proj𝐻)‘𝐴))

Theorempjpoi 29199 Projection in terms of orthocomplement projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) = (𝐴 ((proj‘(⊥‘𝐻))‘𝐴))

Theorempjoc1i 29200 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       (𝐴𝐻 ↔ ((proj‘(⊥‘𝐻))‘𝐴) = 0)

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