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	pjpreeq 31201*	Equality with a projection. This version of pjeq 31202 does not assume the Axiom of Choice via pjhth 31196. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ (𝐻 +ℋ (⊥‘𝐻))) → (((projℎ‘𝐻)‘𝐴) = 𝐵 ↔ (𝐵 ∈ 𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 +ℎ 𝑥))))
Theorem	pjeq 31202*	Equality with a projection. (Contributed by NM, 20-Jan-2007.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (((projℎ‘𝐻)‘𝐴) = 𝐵 ↔ (𝐵 ∈ 𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 +ℎ 𝑥))))
Theorem	axpjcl 31203	Closure of a projection in its subspace. If we consider this together with axpjpj 31223 to be axioms, the need for the ax-hcompl 31005 can often be avoided for the kinds of theorems we are interested in here. An interesting project is to see how far we can go by using them in place of it. In particular, we can prove the orthomodular law pjomli 31238.) (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) ∈ 𝐻)
Theorem	pjhcl 31204	Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) ∈ ℋ)
20.5  Properties of Hilbert subspaces
20.5.1  Orthomodular law
Theorem	omlsilem 31205	Lemma for orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
⊢ 𝐺 ∈ Sℋ    &   ⊢ 𝐻 ∈ Sℋ    &   ⊢ 𝐺 ⊆ 𝐻    &   ⊢ (𝐻 ∩ (⊥‘𝐺)) = 0ℋ    &   ⊢ 𝐴 ∈ 𝐻    &   ⊢ 𝐵 ∈ 𝐺    &   ⊢ 𝐶 ∈ (⊥‘𝐺)    ⇒   ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐴 ∈ 𝐺)
Theorem	omlsii 31206	Subspace inference form of orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ    ⇒   ⊢ 𝐴 = 𝐵
Theorem	omlsi 31207	Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ) → 𝐴 = 𝐵)
Theorem	ococi 31208	Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (⊥‘(⊥‘𝐴)) = 𝐴
Theorem	ococ 31209	Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (⊥‘(⊥‘𝐴)) = 𝐴)
Theorem	dfch2 31210	Alternate definition of the Hilbert lattice. (Contributed by NM, 8-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ Cℋ = {𝑥 ∈ 𝒫 ℋ ∣ (⊥‘(⊥‘𝑥)) = 𝑥}
Theorem	ococin 31211*	The double complement is the smallest closed subspace containing a subset of Hilbert space. Remark 3.12(B) of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥})
Theorem	hsupval2 31212*	Alternate definition of supremum of a subset of the Hilbert lattice. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. We actually define it on any collection of Hilbert space subsets, not just the Hilbert lattice Cℋ, to allow more general theorems. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = ∩ {𝑥 ∈ Cℋ ∣ ∪ 𝐴 ⊆ 𝑥})
Theorem	chsupval2 31213*	The value of the supremum of a set of closed subspaces of Hilbert space. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ⊆ Cℋ → ( ∨ℋ ‘𝐴) = ∩ {𝑥 ∈ Cℋ ∣ ∪ 𝐴 ⊆ 𝑥})
Theorem	sshjval2 31214*	Value of join in the set of closed subspaces of Hilbert space Cℋ. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = ∩ {𝑥 ∈ Cℋ ∣ (𝐴 ∪ 𝐵) ⊆ 𝑥})
Theorem	chsupid 31215*	A subspace is the supremum of all smaller subspaces. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → ( ∨ℋ ‘{𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴}) = 𝐴)
Theorem	chsupsn 31216	Value of supremum of subset of Cℋ on a singleton. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → ( ∨ℋ ‘{𝐴}) = 𝐴)
Theorem	shlub 31217	Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∨ℋ 𝐵) ⊆ 𝐶))
Theorem	shlubi 31218	Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∨ℋ 𝐵) ⊆ 𝐶)
20.5.2  Projectors (cont.)
Theorem	pjhtheu2 31219*	Uniqueness of 𝑦 for the projection theorem. (Contributed by NM, 6-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃!𝑦 ∈ (⊥‘𝐻)∃𝑥 ∈ 𝐻 𝐴 = (𝑥 +ℎ 𝑦))
Theorem	pjcli 31220	Closure of a projection in its subspace. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ℋ → ((projℎ‘𝐻)‘𝐴) ∈ 𝐻)
Theorem	pjhcli 31221	Closure of a projection in Hilbert space. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ℋ → ((projℎ‘𝐻)‘𝐴) ∈ ℋ)
Theorem	pjpjpre 31222	Decomposition of a vector into projections. This formulation of axpjpj 31223 avoids pjhth 31196. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ (𝜑 → 𝐻 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐴 ∈ (𝐻 +ℋ (⊥‘𝐻)))    ⇒   ⊢ (𝜑 → 𝐴 = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐴)))
Theorem	axpjpj 31223	Decomposition of a vector into projections. See comment in axpjcl 31203. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → 𝐴 = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐴)))
Theorem	pjclii 31224	Closure of a projection in its subspace. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ ((projℎ‘𝐻)‘𝐴) ∈ 𝐻
Theorem	pjhclii 31225	Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ ((projℎ‘𝐻)‘𝐴) ∈ ℋ
Theorem	pjpj0i 31226	Decomposition of a vector into projections. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ 𝐴 = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐴))
Theorem	pjpji 31227	Decomposition of a vector into projections. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ 𝐴 = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐴))
Theorem	pjpjhth 31228*	Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))
Theorem	pjpjhthi 31229*	Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)
Theorem	pjop 31230	Orthocomplement projection in terms of projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘(⊥‘𝐻))‘𝐴) = (𝐴 −ℎ ((projℎ‘𝐻)‘𝐴)))
Theorem	pjpo 31231	Projection in terms of orthocomplement projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) = (𝐴 −ℎ ((projℎ‘(⊥‘𝐻))‘𝐴)))
Theorem	pjopi 31232	Orthocomplement projection in terms of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ ((projℎ‘(⊥‘𝐻))‘𝐴) = (𝐴 −ℎ ((projℎ‘𝐻)‘𝐴))
Theorem	pjpoi 31233	Projection in terms of orthocomplement projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ ((projℎ‘𝐻)‘𝐴) = (𝐴 −ℎ ((projℎ‘(⊥‘𝐻))‘𝐴))
Theorem	pjoc1i 31234	Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ (𝐴 ∈ 𝐻 ↔ ((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ)
Theorem	pjchi 31235	Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ (𝐴 ∈ 𝐻 ↔ ((projℎ‘𝐻)‘𝐴) = 𝐴)
Theorem	pjoccl 31236	The part of a vector that belongs to the orthocomplemented space. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐴 −ℎ ((projℎ‘𝐻)‘𝐴)) ∈ (⊥‘𝐻))
Theorem	pjoc1 31237	Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ 𝐻 ↔ ((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ))
Theorem	pjomli 31238	Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omlsi 31207. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ) → 𝐴 = 𝐵)
Theorem	pjoml 31239	Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omlsi 31207. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ)) → 𝐴 = 𝐵)
Theorem	pjococi 31240	Proof of orthocomplement theorem using projections. Compare ococ 31209. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (⊥‘(⊥‘𝐻)) = 𝐻
Theorem	pjoc2i 31241	Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ (𝐴 ∈ (⊥‘𝐻) ↔ ((projℎ‘𝐻)‘𝐴) = 0ℎ)
Theorem	pjoc2 31242	Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (⊥‘𝐻) ↔ ((projℎ‘𝐻)‘𝐴) = 0ℎ))
20.5.3  Hilbert lattice operations
Theorem	sh0le 31243	The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)
Theorem	ch0le 31244	The zero subspace is the smallest member of Cℋ. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴)
Theorem	shle0 31245	No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
Theorem	chle0 31246	No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
Theorem	chnlen0 31247	A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
⊢ (𝐵 ∈ Cℋ → (¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 = 0ℋ))
Theorem	ch0pss 31248	The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (0ℋ ⊊ 𝐴 ↔ 𝐴 ≠ 0ℋ))
Theorem	orthin 31249	The intersection of orthogonal subspaces is the zero subspace. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ⊆ (⊥‘𝐵) → (𝐴 ∩ 𝐵) = 0ℋ))
Theorem	ssjo 31250	The lattice join of a subset with its orthocomplement is the whole space. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ (𝐴 ⊆ ℋ → (𝐴 ∨ℋ (⊥‘𝐴)) = ℋ)
Theorem	shne0i 31251*	A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    ⇒   ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)
Theorem	shs0i 31252	Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    ⇒   ⊢ (𝐴 +ℋ 0ℋ) = 𝐴
Theorem	shs00i 31253	Two subspaces are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐴 = 0ℋ ∧ 𝐵 = 0ℋ) ↔ (𝐴 +ℋ 𝐵) = 0ℋ)
Theorem	ch0lei 31254	The closed subspace zero is the smallest member of Cℋ. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 0ℋ ⊆ 𝐴
Theorem	chle0i 31255	No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)
Theorem	chne0i 31256*	A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)
Theorem	chocini 31257	Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ (⊥‘𝐴)) = 0ℋ
Theorem	chj0i 31258	Join with lattice zero in Cℋ. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 0ℋ) = 𝐴
Theorem	chm1i 31259	Meet with lattice one in Cℋ. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ ℋ) = 𝐴
Theorem	chjcli 31260	Closure of Cℋ join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ
Theorem	chsleji 31261	Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 +ℋ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵)
Theorem	chseli 31262*	Membership in subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))
Theorem	chincli 31263	Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ
Theorem	chsscon3i 31264	Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴))
Theorem	chsscon1i 31265	Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴)
Theorem	chsscon2i 31266	Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))
Theorem	chcon2i 31267	Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴))
Theorem	chcon1i 31268	Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴)
Theorem	chcon3i 31269	Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴))
Theorem	chunssji 31270	Union is smaller than Cℋ join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵)
Theorem	chjcomi 31271	Commutative law for join in Cℋ. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)
Theorem	chub1i 31272	Cℋ join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ 𝐴 ⊆ (𝐴 ∨ℋ 𝐵)
Theorem	chub2i 31273	Cℋ join is an upper bound of two elements. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ 𝐴 ⊆ (𝐵 ∨ℋ 𝐴)
Theorem	chlubi 31274	Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∨ℋ 𝐵) ⊆ 𝐶)
Theorem	chlubii 31275	Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 31274). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ∨ℋ 𝐵) ⊆ 𝐶)
Theorem	chlej1i 31276	Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∨ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐶))
Theorem	chlej2i 31277	Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∨ℋ 𝐴) ⊆ (𝐶 ∨ℋ 𝐵))
Theorem	chlej12i 31278	Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∨ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐷))
Theorem	chlejb1i 31279	Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∨ℋ 𝐵) = 𝐵)
Theorem	chdmm1i 31280	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘(𝐴 ∩ 𝐵)) = ((⊥‘𝐴) ∨ℋ (⊥‘𝐵))
Theorem	chdmm2i 31281	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘((⊥‘𝐴) ∩ 𝐵)) = (𝐴 ∨ℋ (⊥‘𝐵))
Theorem	chdmm3i 31282	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘(𝐴 ∩ (⊥‘𝐵))) = ((⊥‘𝐴) ∨ℋ 𝐵)
Theorem	chdmm4i 31283	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘((⊥‘𝐴) ∩ (⊥‘𝐵))) = (𝐴 ∨ℋ 𝐵)
Theorem	chdmj1i 31284	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘(𝐴 ∨ℋ 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵))
Theorem	chdmj2i 31285	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘((⊥‘𝐴) ∨ℋ 𝐵)) = (𝐴 ∩ (⊥‘𝐵))
Theorem	chdmj3i 31286	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘(𝐴 ∨ℋ (⊥‘𝐵))) = ((⊥‘𝐴) ∩ 𝐵)
Theorem	chdmj4i 31287	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) = (𝐴 ∩ 𝐵)
Theorem	chnlei 31288	Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∨ℋ 𝐵))
Theorem	chjassi 31289	Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∨ℋ 𝐵) ∨ℋ 𝐶) = (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶))
Theorem	chj00i 31290	Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((𝐴 = 0ℋ ∧ 𝐵 = 0ℋ) ↔ (𝐴 ∨ℋ 𝐵) = 0ℋ)
Theorem	chjoi 31291	The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ (⊥‘𝐴)) = ℋ
Theorem	chj1i 31292	Join with Hilbert lattice one. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ ℋ) = ℋ
Theorem	chm0i 31293	Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ 0ℋ) = 0ℋ
Theorem	chm0 31294	Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ∩ 0ℋ) = 0ℋ)
Theorem	shjshsi 31295	Hilbert lattice join equals the double orthocomplement of subspace sum. (Contributed by NM, 27-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 +ℋ 𝐵)))
Theorem	shjshseli 31296	A closed subspace sum equals Hilbert lattice join. Part of Lemma 31.1.5 of [MaedaMaeda] p. 136. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐴 +ℋ 𝐵) ∈ Cℋ ↔ (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵))
Theorem	chne0 31297*	A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ))
Theorem	chocin 31298	Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ∩ (⊥‘𝐴)) = 0ℋ)
Theorem	chssoc 31299	A closed subspace less than its orthocomplement is zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ (⊥‘𝐴) ↔ 𝐴 = 0ℋ))
Theorem	chj0 31300	Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 0ℋ) = 𝐴)