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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremissubgoilem 27501* Lemma for hhssabloilem 27502. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
((𝑥𝑌𝑦𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))       ((𝐴𝑌𝐵𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))

Theoremhhssabloilem 27502 Lemma for hhssabloi 27503. Formerly part of proof for hhssabloi 27503 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 27503 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 27504 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
(𝐻S → ( + ↾ (𝐻 × 𝐻)) ∈ AbelOp)

Theoremhhssnv 27505 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 27506 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 27507 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 27508 Lemma for hhsssh 27510. (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 27509 Lemma for hhsssh 27510. (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 27510 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 27511 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 27512 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 27513 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 27514 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

Theoremhhssph 27515 Inner product space property of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S       𝑊 ∈ CPreHilOLD

Theoremhhssims 27516 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 27517 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 27518 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 27519 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 27520 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)

Theoremhhssbn 27521 Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻C       𝑊 ∈ CBan

Theoremhhsshl 27522 Hilbert space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻C       𝑊 ∈ CHilOLD

Theoremocval 27523* 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 27524* 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 27525* 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 27526 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 27527 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 27528 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)

Theoremshocss 27529 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
(𝐴S → (⊥‘𝐴) ⊆ ℋ)

Theoremoccon 27530 Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Theoremoccon2 27531 Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵))))

Theoremoccon2i 27532 Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
𝐴 ⊆ ℋ    &   𝐵 ⊆ ℋ       (𝐴𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵)))

Theoremoc0 27533 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 27534 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 27535 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 27536 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 27537 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 27538 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 27539 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 27540 Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)))

Theoremocnel 27541 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 27542* 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 27543 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 27544 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 27545* 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 27546 Lemma for occl 27547. (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 27547 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 27548 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 27549 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 27550 Closure of C orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) ∈ C

20.4.4  Subspace sum, span, lattice join, lattice supremum

Definitiondf-shs 27551* Define subspace sum in S. See shsval 27555, shsval2i 27630, and shsval3i 27631 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
+ = (𝑥S , 𝑦S ↦ ( + “ (𝑥 × 𝑦)))

Definitiondf-span 27552* Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 27576 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
span = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦S𝑥𝑦})

Definitiondf-chj 27553* Define Hilbert lattice join. See chjval 27595 for its value and chjcl 27600 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 27598. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
= (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥𝑦))))

Definitiondf-chsup 27554 Define the supremum of a set of Hilbert lattice elements. See chsupval2 27653 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 27582. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)
= (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘ 𝑥)))

Theoremshsval 27555 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 27556 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 27557* 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 27558* 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 27559* Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))

Theoremshscli 27560 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 27561 Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) ∈ S )

Theoremshscom 27562 Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremshsva 27563 Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → ((𝐶𝐴𝐷𝐵) → (𝐶 + 𝐷) ∈ (𝐴 + 𝐵)))

Theoremshsel1 27564 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 27565 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 27566 Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → ((𝐶𝐴𝐷𝐵) → (𝐶 𝐷) ∈ (𝐴 + 𝐵)))

Theoremshsub1 27567 Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐴 + 𝐵))

Theoremshsub2 27568 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐵 + 𝐴))

Theoremchoc0 27569 The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
(⊥‘0) = ℋ

Theoremchoc1 27570 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 27571 Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
(⊥‘∅) = ℋ

Theoremshintcli 27572 Closure of intersection of a nonempty subset of S. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
(𝐴S𝐴 ≠ ∅)        𝐴S

Theoremshintcl 27573 The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐴 ≠ ∅) → 𝐴S )

Theoremchintcli 27574 The intersection of a nonempty set of closed subspaces is a closed subspace. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
(𝐴C𝐴 ≠ ∅)        𝐴C

Theoremchintcl 27575 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 27576* 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 27577 Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 27652. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))

Theoremchsupval 27578 The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 27653. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))

Theoremspancl 27579 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 27580 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 27581 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 27582 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 27583 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 27584 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 27585 Subset relation for supremum of subset of C. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 → ( 𝐴) ⊆ ( 𝐵)))

Theoremhsupunss 27586 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 27587 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 27588 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 27589 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 27590 A subspace of Hilbert space is its own span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
(𝐴S → (span‘𝐴) = 𝐴)

Theoremspanss 27591 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 27592 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 27593 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 27594 Value of join in S. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))

Theoremchjval 27595 Value of join in C. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))

Theoremchjvali 27596 Value of join in C. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵)))

Theoremsshjval3 27597 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 27598 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 27599 Closure of join in S. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 𝐵) ∈ C )

Theoremchjcl 27600 Closure of join in C. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵) ∈ C )

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