P. 313 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hhssnvt 31201	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 31202	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 31203	Lemma for hhsssh 31205. (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 31204	Lemma for hhsssh 31205. (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 31205	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 31206	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 31207	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 31208	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 31209	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 31210	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 31211	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 31212	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 31213	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 31214	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 31215	Obsolete version of cssbn 25282: 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 31216*	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 31217*	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 31218*	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 31219	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 31220	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 31221	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 31222	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 31223	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 31224	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 31225	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 31226	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 31227	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 31228	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 31229	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 31230	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 31231	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 31232	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 31233	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 31234	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 31235*	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 31236	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 31237	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 31238*	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 31239	Lemma for occl 31240. (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 31240	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 31241	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 31242	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 31243	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 31244*	Define subspace sum in Sℋ. See shsval 31248, shsval2i 31323, and shsval3i 31324 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
⊢ +ℋ = (𝑥 ∈ Sℋ , 𝑦 ∈ Sℋ ↦ ( +ℎ “ (𝑥 × 𝑦)))
Definition	df-span 31245*	Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 31269 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦})
Definition	df-chj 31246*	Define Hilbert lattice join. See chjval 31288 for its value and chjcl 31293 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 31291. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
Definition	df-chsup 31247	Define the supremum of a set of Hilbert lattice elements. See chsupval2 31346 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 31275. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)
⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
Theorem	shsval 31248	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 31249	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 31250*	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 31251*	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 31252*	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 31253	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 31254	Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ∈ Sℋ )
Theorem	shscom 31255	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 31256	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 31257	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 31258	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 31259	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 31260	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 31261	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 31262	The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ (⊥‘0ℋ) = ℋ
Theorem	choc1 31263	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 31264	Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
⊢ (⊥‘∅) = ℋ
Theorem	shintcli 31265	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 31266	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 31267	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 31268	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 31269*	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 31270	Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 31345. (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 31271	The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 31346. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ⊆ Cℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴)))
Theorem	spancl 31272	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 31273	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 31274	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 31275	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 31276	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 31277	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 31278	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 31279	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 31280	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 31281	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 31282	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 31283	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 31284	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 31285	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 31286	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 31287	Value of join in Sℋ. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
Theorem	chjval 31288	Value of join in Cℋ. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
Theorem	chjvali 31289	Value of join in Cℋ. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))
Theorem	sshjval3 31290	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 Cℋ. (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = ( ∨ℋ ‘{𝐴, 𝐵}))
Theorem	sshjcl 31291	Closure of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) ∈ Cℋ )
Theorem	shjcl 31292	Closure of join in Sℋ. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) ∈ Cℋ )
Theorem	chjcl 31293	Closure of join in Cℋ. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ 𝐵) ∈ Cℋ )
Theorem	shjcom 31294	Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴))
Theorem	shless 31295	Subset implies subset of subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +ℋ 𝐶) ⊆ (𝐵 +ℋ 𝐶))
Theorem	shlej1 31296	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.)
⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∨ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐶))
Theorem	shlej2 31297	Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ∨ℋ 𝐴) ⊆ (𝐶 ∨ℋ 𝐵))
Theorem	shincli 31298	Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 ∩ 𝐵) ∈ Sℋ
Theorem	shscomi 31299	Commutative law for subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)
Theorem	shsvai 31300	Vector sum belongs to subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶 +ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵))