P. 317 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31601-31700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	shex 31601	The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
⊢ Sℋ ∈ V
Theorem	shssii 31602	A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ 𝐻 ⊆ ℋ
Theorem	sheli 31603	A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)
Theorem	shelii 31604	A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Sℋ    &   ⊢ 𝐴 ∈ 𝐻    ⇒   ⊢ 𝐴 ∈ ℋ
Theorem	sh0 31605	The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻)
Theorem	shaddcl 31606	Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 +ℎ 𝐵) ∈ 𝐻)
Theorem	shmulcl 31607	Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ℎ 𝐵) ∈ 𝐻)
Theorem	issh3 31608*	Subspace 𝐻 of a Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ (𝐻 ⊆ ℋ → (𝐻 ∈ Sℋ ↔ (0ℎ ∈ 𝐻 ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))))
Theorem	shsubcl 31609	Closure of vector subtraction in a subspace of a Hilbert space. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 −ℎ 𝐵) ∈ 𝐻)
20.4.2  Closed subspaces
Definition	df-ch 31610	Define the set of closed subspaces of a Hilbert space. A closed subspace is one in which the limit of every convergent sequence in the subspace belongs to the subspace. For its membership relation, see isch 31611. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 31612 and isch3 31630. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ}
Theorem	isch 31611	Closed subspace 𝐻 of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻))
Theorem	isch2 31612*	Closed subspace 𝐻 of a Hilbert space. Definition of [Beran] p. 107. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻)))
Theorem	chsh 31613	A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ )
Theorem	chsssh 31614	Closed subspaces are subspaces in a Hilbert space. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ Cℋ ⊆ Sℋ
Theorem	chex 31615	The set of closed subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
⊢ Cℋ ∈ V
Theorem	chshii 31616	A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ 𝐻 ∈ Sℋ
Theorem	ch0 31617	The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → 0ℎ ∈ 𝐻)
Theorem	chss 31618	A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ)
Theorem	chel 31619	A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)
Theorem	chssii 31620	A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ 𝐻 ⊆ ℋ
Theorem	cheli 31621	A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)
Theorem	chelii 31622	A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ 𝐻    ⇒   ⊢ 𝐴 ∈ ℋ
Theorem	chlimi 31623	The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((𝐻 ∈ Cℋ ∧ 𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻)
Theorem	hlim0 31624	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 × {0ℎ}) ⇝𝑣 0ℎ
Theorem	hlimcaui 31625	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.)
⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ Cauchy)
Theorem	hlimf 31626	Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ
Theorem	hlimuni 31627	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.)
⊢ ((𝐹 ⇝𝑣 𝐴 ∧ 𝐹 ⇝𝑣 𝐵) → 𝐴 = 𝐵)
Theorem	hlimreui 31628*	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.)
⊢ (∃𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ↔ ∃!𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥)
Theorem	hlimeui 31629*	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 𝐹 ⇝𝑣 𝑥 ↔ ∃!𝑥 𝐹 ⇝𝑣 𝑥)
Theorem	isch3 31630*	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.)
⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ∀𝑓 ∈ Cauchy (𝑓:ℕ⟶𝐻 → ∃𝑥 ∈ 𝐻 𝑓 ⇝𝑣 𝑥)))
Theorem	chcompl 31631*	Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐹 ∈ Cauchy ∧ 𝐹:ℕ⟶𝐻) → ∃𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥)
Theorem	helch 31632	The Hilbert lattice one (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 ∈ Cℋ
Theorem	ifchhv 31633	Prove if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∈ Cℋ. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.)
⊢ if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∈ Cℋ
Theorem	helsh 31634	Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
⊢ ℋ ∈ Sℋ
Theorem	shsspwh 31635	Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
⊢ Sℋ ⊆ 𝒫 ℋ
Theorem	chsspwh 31636	Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
⊢ Cℋ ⊆ 𝒫 ℋ
Theorem	hsn0elch 31637	The zero subspace belongs to the set of closed subspaces of Hilbert space. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
⊢ {0ℎ} ∈ Cℋ
Theorem	norm1 31638	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 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
Theorem	norm1exi 31639*	A normalized vector exists in a subspace iff the subspace has a nonzero vector. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Sℋ    ⇒   ⊢ (∃𝑥 ∈ 𝐻 𝑥 ≠ 0ℎ ↔ ∃𝑦 ∈ 𝐻 (normℎ‘𝑦) = 1)
Theorem	norm1hex 31640	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 ≠ 0ℎ ↔ ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
20.4.3  Orthocomplements
Definition	df-oc 31641*	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 31669 and chocvali 31688 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 31642	Define the zero for closed subspaces of Hilbert space. See h0elch 31644 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ 0ℋ = {0ℎ}
Theorem	elch0 31643	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 31644	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 31645	The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
⊢ 0ℋ ∈ Sℋ
Theorem	hhssva 31646	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 31647	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 31648	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 31649*	Lemma for hhssabloilem 31650. (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 31650	Lemma for hhssabloi 31651. Formerly part of proof for hhssabloi 31651 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 31651	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 31652	Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
⊢ (𝐻 ∈ Sℋ → ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp)
Theorem	hhssnv 31653	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 31654	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 31655	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 31656	Lemma for hhsssh 31658. (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 31657	Lemma for hhsssh 31658. (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 31658	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 31659	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 31660	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 31661	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 31662	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 31663	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 31664	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 31665	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 31666	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 31667	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 31668	Obsolete version of cssbn 25571: 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 31669*	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 31670*	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 31671*	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 31672	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 31673	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 31674	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 31675	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 31676	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 31677	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 31678	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 31679	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 31680	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 31681	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 31682	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 31683	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 31684	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 31685	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 31686	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 31687	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 31688*	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 31689	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 31690	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 31691*	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 31692	Lemma for occl 31693. (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 31693	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 31694	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 31695	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 31696	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 31697*	Define subspace sum in Sℋ. See shsval 31701, shsval2i 31776, and shsval3i 31777 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
⊢ +ℋ = (𝑥 ∈ Sℋ , 𝑦 ∈ Sℋ ↦ ( +ℎ “ (𝑥 × 𝑦)))
Definition	df-span 31698*	Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 31722 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦})
Definition	df-chj 31699*	Define Hilbert lattice join. See chjval 31741 for its value and chjcl 31746 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 31744. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
Definition	df-chsup 31700	Define the supremum of a set of Hilbert lattice elements. See chsupval2 31799 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 31728. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)
⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))