P. 315 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31401-31500   *Has distinct variable group(s)

Type	Label	Description
Statement
20.4.3  Orthocomplements
Definition	df-oc 31401*	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 31429 and chocvali 31448 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 ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0})
Definition	df-ch0 31402	Define the zero for closed subspaces of Hilbert space. See h0elch 31404 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ 0ℋ = {0ℎ}
Theorem	elch0 31403	Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)
⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)
Theorem	h0elch 31404	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.)
⊢ 0ℋ ∈ Cℋ
Theorem	h0elsh 31405	The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
⊢ 0ℋ ∈ Sℋ
Theorem	hhssva 31406	The vector addition operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    ⇒   ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊)
Theorem	hhsssm 31407	The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    ⇒   ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) = ( ·𝑠OLD ‘𝑊)
Theorem	hhssnm 31408	The norm operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    ⇒   ⊢ (normℎ ↾ 𝐻) = (normCV‘𝑊)
Theorem	issubgoilem 31409*	Lemma for hhssabloilem 31410. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))    ⇒   ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))
Theorem	hhssabloilem 31410	Lemma for hhssabloi 31411. Formerly part of proof for hhssabloi 31411 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.)
⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ ( +ℎ ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ )
Theorem	hhssabloi 31411	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.)
⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp
Theorem	hhssablo 31412	Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
⊢ (𝐻 ∈ Sℋ → ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp)
Theorem	hhssnv 31413	Normed complex vector space property of a subspace. (Contributed by NM, 26-Mar-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ 𝑊 ∈ NrmCVec
Theorem	hhssnvt 31414	Normed complex vector space property of a subspace. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    ⇒   ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ NrmCVec)
Theorem	hhsst 31415	A member of Sℋ is a subspace. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    ⇒   ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ (SubSp‘𝑈))
Theorem	hhshsslem1 31416	Lemma for hhsssh 31418. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝑊 ∈ (SubSp‘𝑈)    &   ⊢ 𝐻 ⊆ ℋ    ⇒   ⊢ 𝐻 = (BaseSet‘𝑊)
Theorem	hhshsslem2 31417	Lemma for hhsssh 31418. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝑊 ∈ (SubSp‘𝑈)    &   ⊢ 𝐻 ⊆ ℋ    ⇒   ⊢ 𝐻 ∈ Sℋ
Theorem	hhsssh 31418	The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    ⇒   ⊢ (𝐻 ∈ Sℋ ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))
Theorem	hhsssh2 31419	The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    ⇒   ⊢ (𝐻 ∈ Sℋ ↔ (𝑊 ∈ NrmCVec ∧ 𝐻 ⊆ ℋ))
Theorem	hhssba 31420	The base set of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ 𝐻 = (BaseSet‘𝑊)
Theorem	hhssvs 31421	The vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ ( −ℎ ↾ (𝐻 × 𝐻)) = ( −𝑣 ‘𝑊)
Theorem	hhssvsf 31422	Mapping of the vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ ( −ℎ ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶𝐻
Theorem	hhssims 31423	Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝐻 ∈ Sℋ    &   ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))    ⇒   ⊢ 𝐷 = (IndMet‘𝑊)
Theorem	hhssims2 31424	Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝐷 = (IndMet‘𝑊)    &   ⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))
Theorem	hhssmet 31425	Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝐷 = (IndMet‘𝑊)    &   ⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ 𝐷 ∈ (Met‘𝐻)
Theorem	hhssmetdval 31426	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 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝐷 = (IndMet‘𝑊)    &   ⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴𝐷𝐵) = (normℎ‘(𝐴 −ℎ 𝐵)))
Theorem	hhsscms 31427	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 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝐷 = (IndMet‘𝑊)    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ 𝐷 ∈ (CMet‘𝐻)
Theorem	hhssbnOLD 31428	Obsolete version of cssbn 25417: Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ 𝑊 ∈ CBan
Theorem	ocval 31429*	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 ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0})
Theorem	ocel 31430*	Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0)))
Theorem	shocel 31431*	Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)
⊢ (𝐻 ∈ Sℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0)))
Theorem	ocsh 31432	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) ∈ Sℋ )
Theorem	shocsh 31433	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.)
⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Sℋ )
Theorem	ocss 31434	An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
Theorem	shocss 31435	An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ⊆ ℋ)
Theorem	occon 31436	Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))
Theorem	occon2 31437	Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵))))
Theorem	occon2i 31438	Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
⊢ 𝐴 ⊆ ℋ    &   ⊢ 𝐵 ⊆ ℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵)))
Theorem	oc0 31439	The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ (⊥‘𝐻))
Theorem	ocorth 31440	Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
⊢ (𝐻 ⊆ ℋ → ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) → (𝐴 ·ih 𝐵) = 0))
Theorem	shocorth 31441	Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
⊢ (𝐻 ∈ Sℋ → ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) → (𝐴 ·ih 𝐵) = 0))
Theorem	ococss 31442	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)))
Theorem	shococss 31443	Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ Sℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))
Theorem	shorth 31444	Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
⊢ (𝐻 ∈ Sℋ → (𝐺 ⊆ (⊥‘𝐻) → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ih 𝐵) = 0)))
Theorem	ocin 31445	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.)
⊢ (𝐴 ∈ Sℋ → (𝐴 ∩ (⊥‘𝐴)) = 0ℋ)
Theorem	occon3 31446	Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.)
⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)))
Theorem	ocnel 31447	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.)
⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ (⊥‘𝐻) ∧ 𝐴 ≠ 0ℎ) → ¬ 𝐴 ∈ 𝐻)
Theorem	chocvali 31448*	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.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0}
Theorem	shuni 31449	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.)
⊢ (𝜑 → 𝐻 ∈ Sℋ )    &   ⊢ (𝜑 → 𝐾 ∈ Sℋ )    &   ⊢ (𝜑 → (𝐻 ∩ 𝐾) = 0ℋ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐻)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐻)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐾)    &   ⊢ (𝜑 → (𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	chocunii 31450	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.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ (⊥‘𝐻))) → ((𝑅 = (𝐴 +ℎ 𝐵) ∧ 𝑅 = (𝐶 +ℎ 𝐷)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	pjhthmo 31451*	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.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)))
Theorem	occllem 31452	Lemma for occl 31453. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ⊆ ℋ)    &   ⊢ (𝜑 → 𝐹 ∈ Cauchy)    &   ⊢ (𝜑 → 𝐹:ℕ⟶(⊥‘𝐴))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (( ⇝𝑣 ‘𝐹) ·ih 𝐵) = 0)
Theorem	occl 31453	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) ∈ Cℋ )
Theorem	shoccl 31454	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.)
⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Cℋ )
Theorem	choccl 31455	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.)
⊢ (𝐴 ∈ Cℋ → (⊥‘𝐴) ∈ Cℋ )
Theorem	choccli 31456	Closure of Cℋ orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (⊥‘𝐴) ∈ Cℋ
20.4.4  Subspace sum, span, lattice join, lattice supremum
Definition	df-shs 31457*	Define subspace sum in Sℋ. See shsval 31461, shsval2i 31536, and shsval3i 31537 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
⊢ +ℋ = (𝑥 ∈ Sℋ , 𝑦 ∈ Sℋ ↦ ( +ℎ “ (𝑥 × 𝑦)))
Definition	df-span 31458*	Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 31482 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦})
Definition	df-chj 31459*	Define Hilbert lattice join. See chjval 31501 for its value and chjcl 31506 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 Cℋ; see sshjcl 31504. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
Definition	df-chsup 31460	Define the supremum of a set of Hilbert lattice elements. See chsupval2 31559 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice Cℋ, to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 31488. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)
⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
Theorem	shsval 31461	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.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = ( +ℎ “ (𝐴 × 𝐵)))
Theorem	shsss 31462	The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ⊆ ℋ)
Theorem	shsel 31463*	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.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)))
Theorem	shsel3 31464*	Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 −ℎ 𝑦)))
Theorem	shseli 31465*	Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))
Theorem	shscli 31466	Closure of subspace sum. (Contributed by NM, 15-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 +ℋ 𝐵) ∈ Sℋ
Theorem	shscl 31467	Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ∈ Sℋ )
Theorem	shscom 31468	Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴))
Theorem	shsva 31469	Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶 +ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵)))
Theorem	shsel1 31470	A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ 𝐴 → 𝐶 ∈ (𝐴 +ℋ 𝐵)))
Theorem	shsel2 31471	A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 +ℋ 𝐵)))
Theorem	shsvs 31472	Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶 −ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵)))
Theorem	shsub1 31473	Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ (𝐴 +ℋ 𝐵))
Theorem	shsub2 31474	Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ (𝐵 +ℋ 𝐴))
Theorem	choc0 31475	The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ (⊥‘0ℋ) = ℋ
Theorem	choc1 31476	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) = 0ℋ
Theorem	chocnul 31477	Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
⊢ (⊥‘∅) = ℋ
Theorem	shintcli 31478	Closure of intersection of a nonempty subset of Sℋ. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅)    ⇒   ⊢ ∩ 𝐴 ∈ Sℋ
Theorem	shintcl 31479	The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Sℋ )
Theorem	chintcli 31480	The intersection of a nonempty set of closed subspaces is a closed subspace. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅)    ⇒   ⊢ ∩ 𝐴 ∈ Cℋ
Theorem	chintcl 31481	The intersection (infimum) of a nonempty subset of Cℋ belongs to Cℋ. 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.)
⊢ ((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Cℋ )
Theorem	spanval 31482*	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) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥})
Theorem	hsupval 31483	Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 31558. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴)))
Theorem	chsupval 31484	The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 31559. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ⊆ Cℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴)))
Theorem	spancl 31485	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) ∈ Sℋ )
Theorem	elspancl 31486	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)
Theorem	shsupcl 31487	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) ∈ Sℋ )
Theorem	hsupcl 31488	Closure of supremum of set of subsets of Hilbert space. Note that the supremum belongs to Cℋ 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 → ( ∨ℋ ‘𝐴) ∈ Cℋ )
Theorem	chsupcl 31489	Closure of supremum of subset of Cℋ. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. Shows that Cℋ 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.)
⊢ (𝐴 ⊆ Cℋ → ( ∨ℋ ‘𝐴) ∈ Cℋ )
Theorem	hsupss 31490	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) → (𝐴 ⊆ 𝐵 → ( ∨ℋ ‘𝐴) ⊆ ( ∨ℋ ‘𝐵)))
Theorem	chsupss 31491	Subset relation for supremum of subset of Cℋ. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
⊢ ((𝐴 ⊆ Cℋ ∧ 𝐵 ⊆ Cℋ ) → (𝐴 ⊆ 𝐵 → ( ∨ℋ ‘𝐴) ⊆ ( ∨ℋ ‘𝐵)))
Theorem	hsupunss 31492	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 ⊆ ( ∨ℋ ‘𝐴))
Theorem	chsupunss 31493	The union of a set of closed subspaces is smaller than its supremum. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ⊆ Cℋ → ∪ 𝐴 ⊆ ( ∨ℋ ‘𝐴))
Theorem	spanss2 31494	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))
Theorem	shsupunss 31495	The union of a set of subspaces is smaller than its supremum. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
⊢ (𝐴 ⊆ Sℋ → ∪ 𝐴 ⊆ (span‘∪ 𝐴))
Theorem	spanid 31496	A subspace of Hilbert space is its own span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Sℋ → (span‘𝐴) = 𝐴)
Theorem	spanss 31497	Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐵 ⊆ ℋ ∧ 𝐴 ⊆ 𝐵) → (span‘𝐴) ⊆ (span‘𝐵))
Theorem	spanssoc 31498	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)))
Theorem	sshjval 31499	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 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
Theorem	shjval 31500	Value of join in Sℋ. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))