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Theorem List for Metamath Proof Explorer - 29001-29100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremchss 29001 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
(𝐻C𝐻 ⊆ ℋ)

Theoremchel 29002 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐻C𝐴𝐻) → 𝐴 ∈ ℋ)

Theoremchssii 29003 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
𝐻C       𝐻 ⊆ ℋ

Theoremcheli 29004 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
𝐻C       (𝐴𝐻𝐴 ∈ ℋ)

Theoremchelii 29005 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴𝐻       𝐴 ∈ ℋ

Theoremchlimi 29006 The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ V       ((𝐻C𝐹:ℕ⟶𝐻𝐹𝑣 𝐴) → 𝐴𝐻)

Theoremhlim0 29007 The zero sequence in Hilbert space converges to the zero vector. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(ℕ × {0}) ⇝𝑣 0

Theoremhlimcaui 29008 If a sequence in Hilbert space subset converges to a limit, it is a Cauchy sequence. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐹𝑣 𝐴𝐹 ∈ Cauchy)

Theoremhlimf 29009 Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
𝑣 :dom ⇝𝑣 ⟶ ℋ

Theoremhlimuni 29010 A Hilbert space sequence converges to at most one limit. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 2-May-2015.) (New usage is discouraged.)
((𝐹𝑣 𝐴𝐹𝑣 𝐵) → 𝐴 = 𝐵)

Theoremhlimreui 29011* The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(∃𝑥𝐻 𝐹𝑣 𝑥 ↔ ∃!𝑥𝐻 𝐹𝑣 𝑥)

Theoremhlimeui 29012* The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(∃𝑥 𝐹𝑣 𝑥 ↔ ∃!𝑥 𝐹𝑣 𝑥)

Theoremisch3 29013* A Hilbert subspace is closed iff it is complete. A complete subspace is one in which every Cauchy sequence of vectors in the subspace converges to a member of the subspace (Definition of complete subspace in [Beran] p. 96). Remark 3.12 of [Beran] p. 107. (Contributed by NM, 24-Dec-2001.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐻C ↔ (𝐻S ∧ ∀𝑓 ∈ Cauchy (𝑓:ℕ⟶𝐻 → ∃𝑥𝐻 𝑓𝑣 𝑥)))

Theoremchcompl 29014* Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐹 ∈ Cauchy ∧ 𝐹:ℕ⟶𝐻) → ∃𝑥𝐻 𝐹𝑣 𝑥)

Theoremhelch 29015 The unit Hilbert lattice element (which is all of Hilbert space) belongs to the Hilbert lattice. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 6-Sep-1999.) (New usage is discouraged.)
ℋ ∈ C

Theoremifchhv 29016 Prove if(𝐴C , 𝐴, ℋ) ∈ C. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.)
if(𝐴C , 𝐴, ℋ) ∈ C

Theoremhelsh 29017 Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
ℋ ∈ S

Theoremshsspwh 29018 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
S ⊆ 𝒫 ℋ

Theoremchsspwh 29019 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
C ⊆ 𝒫 ℋ

Theoremhsn0elch 29020 The zero subspace belongs to the set of closed subspaces of Hilbert space. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
{0} ∈ C

Theoremnorm1 29021 From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)

Theoremnorm1exi 29022* A normalized vector exists in a subspace iff the subspace has a nonzero vector. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
𝐻S       (∃𝑥𝐻 𝑥 ≠ 0 ↔ ∃𝑦𝐻 (norm𝑦) = 1)

Theoremnorm1hex 29023 A normalized vector can exist only iff the Hilbert space has a nonzero vector. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
(∃𝑥 ∈ ℋ 𝑥 ≠ 0 ↔ ∃𝑦 ∈ ℋ (norm𝑦) = 1)

19.4.3  Orthocomplements

Definitiondf-oc 29024* Define orthogonal complement of a subset (usually a subspace) of Hilbert space. The orthogonal complement is the set of all vectors orthogonal to all vectors in the subset. See ocval 29052 and chocvali 29071 for its value. Textbooks usually denote this unary operation with the symbol as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation) rather than introducing a new syntactic form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})

Definitiondf-ch0 29025 Define the zero for closed subspaces of Hilbert space. See h0elch 29027 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
0 = {0}

Theoremelch0 29026 Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)
(𝐴 ∈ 0𝐴 = 0)

Theoremh0elch 29027 The zero subspace is a closed subspace. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
0C

Theoremh0elsh 29028 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
0S

Theoremhhssva 29029 The vector addition operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       ( + ↾ (𝐻 × 𝐻)) = ( +𝑣𝑊)

Theoremhhsssm 29030 The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       ( · ↾ (ℂ × 𝐻)) = ( ·𝑠OLD𝑊)

Theoremhhssnm 29031 The norm operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       (norm𝐻) = (normCV𝑊)

Theoremissubgoilem 29032* Lemma for hhssabloilem 29033. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
((𝑥𝑌𝑦𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))       ((𝐴𝑌𝐵𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))

Theoremhhssabloilem 29033 Lemma for hhssabloi 29034. Formerly part of proof for hhssabloi 29034 which was based on the deprecated definition "SubGrpOp" for subgroups. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 27-Aug-2021.) (New usage is discouraged.)
𝐻S       ( + ∈ GrpOp ∧ ( + ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( + ↾ (𝐻 × 𝐻)) ⊆ + )

Theoremhhssabloi 29034 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Proof shortened by AV, 27-Aug-2021.) (New usage is discouraged.)
𝐻S       ( + ↾ (𝐻 × 𝐻)) ∈ AbelOp

Theoremhhssablo 29035 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
(𝐻S → ( + ↾ (𝐻 × 𝐻)) ∈ AbelOp)

Theoremhhssnv 29036 Normed complex vector space property of a subspace. (Contributed by NM, 26-Mar-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S       𝑊 ∈ NrmCVec

Theoremhhssnvt 29037 Normed complex vector space property of a subspace. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       (𝐻S𝑊 ∈ NrmCVec)

Theoremhhsst 29038 A member of S is a subspace. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       (𝐻S𝑊 ∈ (SubSp‘𝑈))

Theoremhhshsslem1 29039 Lemma for hhsssh 29041. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝑊 ∈ (SubSp‘𝑈)    &   𝐻 ⊆ ℋ       𝐻 = (BaseSet‘𝑊)

Theoremhhshsslem2 29040 Lemma for hhsssh 29041. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝑊 ∈ (SubSp‘𝑈)    &   𝐻 ⊆ ℋ       𝐻S

Theoremhhsssh 29041 The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       (𝐻S ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))

Theoremhhsssh2 29042 The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       (𝐻S ↔ (𝑊 ∈ NrmCVec ∧ 𝐻 ⊆ ℋ))

Theoremhhssba 29043 The base set of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S       𝐻 = (BaseSet‘𝑊)

Theoremhhssvs 29044 The vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S       ( − ↾ (𝐻 × 𝐻)) = ( −𝑣𝑊)

Theoremhhssvsf 29045 Mapping of the vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S       ( − ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶𝐻

Theoremhhssims 29046 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S    &   𝐷 = ((norm ∘ − ) ↾ (𝐻 × 𝐻))       𝐷 = (IndMet‘𝑊)

Theoremhhssims2 29047 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐷 = (IndMet‘𝑊)    &   𝐻S       𝐷 = ((norm ∘ − ) ↾ (𝐻 × 𝐻))

Theoremhhssmet 29048 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐷 = (IndMet‘𝑊)    &   𝐻S       𝐷 ∈ (Met‘𝐻)

Theoremhhssmetdval 29049 Value of the distance function of the metric space of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐷 = (IndMet‘𝑊)    &   𝐻S       ((𝐴𝐻𝐵𝐻) → (𝐴𝐷𝐵) = (norm‘(𝐴 𝐵)))

Theoremhhsscms 29050 The induced metric of a closed subspace is complete. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐷 = (IndMet‘𝑊)    &   𝐻C       𝐷 ∈ (CMet‘𝐻)

TheoremhhssbnOLD 29051 Obsolete version of cssbn 23968: Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻C       𝑊 ∈ CBan

Theoremocval 29052* Value of orthogonal complement of a subset of Hilbert space. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})

Theoremocel 29053* Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
(𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))

Theoremshocel 29054* Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)
(𝐻S → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))

Theoremocsh 29055 The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ S )

Theoremshocsh 29056 The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
(𝐴S → (⊥‘𝐴) ∈ S )

Theoremocss 29057 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)

Theoremshocss 29058 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
(𝐴S → (⊥‘𝐴) ⊆ ℋ)

Theoremoccon 29059 Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Theoremoccon2 29060 Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵))))

Theoremoccon2i 29061 Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
𝐴 ⊆ ℋ    &   𝐵 ⊆ ℋ       (𝐴𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵)))

Theoremoc0 29062 The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
(𝐻S → 0 ∈ (⊥‘𝐻))

Theoremocorth 29063 Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
(𝐻 ⊆ ℋ → ((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) → (𝐴 ·ih 𝐵) = 0))

Theoremshocorth 29064 Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
(𝐻S → ((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) → (𝐴 ·ih 𝐵) = 0))

Theoremococss 29065 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))

Theoremshococss 29066 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
(𝐴S𝐴 ⊆ (⊥‘(⊥‘𝐴)))

Theoremshorth 29067 Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
(𝐻S → (𝐺 ⊆ (⊥‘𝐻) → ((𝐴𝐺𝐵𝐻) → (𝐴 ·ih 𝐵) = 0)))

Theoremocin 29068 Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
(𝐴S → (𝐴 ∩ (⊥‘𝐴)) = 0)

Theoremoccon3 29069 Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)))

Theoremocnel 29070 A nonzero vector in the complement of a subspace does not belong to the subspace. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
((𝐻S𝐴 ∈ (⊥‘𝐻) ∧ 𝐴 ≠ 0) → ¬ 𝐴𝐻)

Theoremchocvali 29071* Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of 𝐴 is the set of vectors that are orthogonal to all vectors in 𝐴. (Contributed by NM, 8-Aug-2004.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0}

Theoremshuni 29072 Two subspaces with trivial intersection have a unique decomposition of the elements of the subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(𝜑𝐻S )    &   (𝜑𝐾S )    &   (𝜑 → (𝐻𝐾) = 0)    &   (𝜑𝐴𝐻)    &   (𝜑𝐵𝐾)    &   (𝜑𝐶𝐻)    &   (𝜑𝐷𝐾)    &   (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremchocunii 29073 Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part). (Contributed by NM, 23-Oct-1999.) (Proof shortened by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐻C       (((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) ∧ (𝐶𝐻𝐷 ∈ (⊥‘𝐻))) → ((𝑅 = (𝐴 + 𝐵) ∧ 𝑅 = (𝐶 + 𝐷)) → (𝐴 = 𝐶𝐵 = 𝐷)))

Theorempjhthmo 29074* Projection Theorem, uniqueness part. Any two disjoint subspaces yield a unique decomposition of vectors into each subspace. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)))

Theoremoccllem 29075 Lemma for occl 29076. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝜑𝐴 ⊆ ℋ)    &   (𝜑𝐹 ∈ Cauchy)    &   (𝜑𝐹:ℕ⟶(⊥‘𝐴))    &   (𝜑𝐵𝐴)       (𝜑 → (( ⇝𝑣𝐹) ·ih 𝐵) = 0)

Theoremoccl 29076 Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ C )

Theoremshoccl 29077 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
(𝐴S → (⊥‘𝐴) ∈ C )

Theoremchoccl 29078 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
(𝐴C → (⊥‘𝐴) ∈ C )

Theoremchoccli 29079 Closure of C orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) ∈ C

19.4.4  Subspace sum, span, lattice join, lattice supremum

Definitiondf-shs 29080* Define subspace sum in S. See shsval 29084, shsval2i 29159, and shsval3i 29160 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
+ = (𝑥S , 𝑦S ↦ ( + “ (𝑥 × 𝑦)))

Definitiondf-span 29081* Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 29105 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
span = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦S𝑥𝑦})

Definitiondf-chj 29082* Define Hilbert lattice join. See chjval 29124 for its value and chjcl 29129 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to C; see sshjcl 29127. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
= (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥𝑦))))

Definitiondf-chsup 29083 Define the supremum of a set of Hilbert lattice elements. See chsupval2 29182 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice C, to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 29111. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)
= (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘ 𝑥)))

Theoremshsval 29084 Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in [Kalmbach] p. 65. (Contributed by NM, 16-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) = ( + “ (𝐴 × 𝐵)))

Theoremshsss 29085 The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) ⊆ ℋ)

Theoremshsel 29086* Membership in the subspace sum of two Hilbert subspaces. (Contributed by NM, 14-Dec-2004.) (Revised by Mario Carneiro, 29-Jan-2014.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦)))

Theoremshsel3 29087* Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 𝑦)))

Theoremshseli 29088* Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))

Theoremshscli 29089 Closure of subspace sum. (Contributed by NM, 15-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) ∈ S

Theoremshscl 29090 Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) ∈ S )

Theoremshscom 29091 Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremshsva 29092 Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → ((𝐶𝐴𝐷𝐵) → (𝐶 + 𝐷) ∈ (𝐴 + 𝐵)))

Theoremshsel1 29093 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐶𝐴𝐶 ∈ (𝐴 + 𝐵)))

Theoremshsel2 29094 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐶𝐵𝐶 ∈ (𝐴 + 𝐵)))

Theoremshsvs 29095 Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → ((𝐶𝐴𝐷𝐵) → (𝐶 𝐷) ∈ (𝐴 + 𝐵)))

Theoremshsub1 29096 Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐴 + 𝐵))

Theoremshsub2 29097 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐵 + 𝐴))

Theoremchoc0 29098 The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
(⊥‘0) = ℋ

Theoremchoc1 29099 The orthocomplement of the unit subspace is the zero subspace. Does not require Axiom of Choice. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(⊥‘ ℋ) = 0

Theoremchocnul 29100 Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
(⊥‘∅) = ℋ

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