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
Theorem	shunssji 31401	Union is smaller than Hilbert lattice join. (Contributed by NM, 11-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵)
Theorem	shsleji 31402	Subspace sum is smaller than Hilbert lattice join. Remark in [Kalmbach] p. 65. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 +ℋ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵)
Theorem	shjcomi 31403	Commutative law for join in Sℋ. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)
Theorem	shsub1i 31404	Subspace sum is an upper bound of its arguments. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ 𝐴 ⊆ (𝐴 +ℋ 𝐵)
Theorem	shsub2i 31405	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	shub1i 31406	Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ 𝐴 ⊆ (𝐴 ∨ℋ 𝐵)
Theorem	shjcli 31407	Closure of Cℋ join. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ
Theorem	shjshcli 31408	Sℋ closure of join. (Contributed by NM, 5-May-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) ∈ Sℋ
Theorem	shlessi 31409	Subset implies subset of subspace sum. (Contributed by NM, 18-Nov-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (𝐴 +ℋ 𝐶) ⊆ (𝐵 +ℋ 𝐶))
Theorem	shlej1i 31410	Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999.) (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	shlej2i 31411	Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∨ℋ 𝐴) ⊆ (𝐶 ∨ℋ 𝐵))
Theorem	shslej 31412	Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵))
Theorem	shincl 31413	Closure of intersection of two subspaces. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∩ 𝐵) ∈ Sℋ )
Theorem	shub1 31414	Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ (𝐴 ∨ℋ 𝐵))
Theorem	shub2 31415	A subspace is a subset of its Hilbert lattice join with another. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ (𝐵 ∨ℋ 𝐴))
Theorem	shsidmi 31416	Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    ⇒   ⊢ (𝐴 +ℋ 𝐴) = 𝐴
Theorem	shslubi 31417	The least upper bound law for Hilbert subspace sum. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 +ℋ 𝐵) ⊆ 𝐶)
Theorem	shlesb1i 31418	Hilbert lattice ordering in terms of subspace sum. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 +ℋ 𝐵) = 𝐵)
Theorem	shsval2i 31419*	An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 +ℋ 𝐵) = ∩ {𝑥 ∈ Sℋ ∣ (𝐴 ∪ 𝐵) ⊆ 𝑥}
Theorem	shsval3i 31420	An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 +ℋ 𝐵) = (span‘(𝐴 ∪ 𝐵))
Theorem	shmodsi 31421	The modular law holds for subspace sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    ⇒   ⊢ (𝐴 ⊆ 𝐶 → ((𝐴 +ℋ 𝐵) ∩ 𝐶) ⊆ (𝐴 +ℋ (𝐵 ∩ 𝐶)))
Theorem	shmodi 31422	The modular law is implied by the closure of subspace sum. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    ⇒   ⊢ (((𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵) ∧ 𝐴 ⊆ 𝐶) → ((𝐴 ∨ℋ 𝐵) ∩ 𝐶) ⊆ (𝐴 ∨ℋ (𝐵 ∩ 𝐶)))
20.4.5  Projection theorem
Theorem	pjhthlem1 31423*	Lemma for pjhth 31425. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (Proof shortened by AV, 10-Jul-2022.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ (𝜑 → 𝐴 ∈ ℋ)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐻)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐻)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝐵)) ≤ (normℎ‘(𝐴 −ℎ 𝑥)))    &   ⊢ 𝑇 = (((𝐴 −ℎ 𝐵) ·ih 𝐶) / ((𝐶 ·ih 𝐶) + 1))    ⇒   ⊢ (𝜑 → ((𝐴 −ℎ 𝐵) ·ih 𝐶) = 0)
Theorem	pjhthlem2 31424*	Lemma for pjhth 31425. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ (𝜑 → 𝐴 ∈ ℋ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))
Theorem	pjhth 31425	Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely 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, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → (𝐻 +ℋ (⊥‘𝐻)) = ℋ)
Theorem	pjhtheu 31426*	Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely 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. See pjhtheu2 31448 for the uniqueness of 𝑦. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃!𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))
20.4.6  Projectors
Definition	df-pjh 31427*	Define the projection function on a Hilbert space, as a mapping from the Hilbert lattice to a function on Hilbert space. Every closed subspace is associated with a unique projection function. Remark in [Kalmbach] p. 66, adopted as a definition. (projℎ‘𝐻)‘𝐴 is the projection of vector 𝐴 onto closed subspace 𝐻. Note that the range of projℎ is the set of all projection operators, so 𝑇 ∈ ran projℎ means that 𝑇 is a projection operator. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
⊢ projℎ = (ℎ ∈ Cℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦))))
Theorem	pjhfval 31428*	The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦))))
Theorem	pjhval 31429*	Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))
Theorem	pjpreeq 31430*	Equality with a projection. This version of pjeq 31431 does not assume the Axiom of Choice via pjhth 31425. (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 31431*	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 31432	Closure of a projection in its subspace. If we consider this together with axpjpj 31452 to be axioms, the need for the ax-hcompl 31234 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 31467.) (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 31433	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 31434	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 31435	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 31436	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 31437	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 31438	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 31439	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 31440*	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 31441*	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 31442*	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 31443*	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 31444*	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 31445	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 31446	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 31447	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 31448*	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 31449	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 31450	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 31451	Decomposition of a vector into projections. This formulation of axpjpj 31452 avoids pjhth 31425. (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 31452	Decomposition of a vector into projections. See comment in axpjcl 31432. (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 31453	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 31454	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 31455	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 31456	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 31457*	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 31458*	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 31459	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 31460	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 31461	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 31462	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 31463	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 31464	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 31465	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 31466	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 31467	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 31436. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ) → 𝐴 = 𝐵)
Theorem	pjoml 31468	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 31436. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ)) → 𝐴 = 𝐵)
Theorem	pjococi 31469	Proof of orthocomplement theorem using projections. Compare ococ 31438. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (⊥‘(⊥‘𝐻)) = 𝐻
Theorem	pjoc2i 31470	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 31471	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 31472	The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)
Theorem	ch0le 31473	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 31474	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 31475	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 31476	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 31477	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 31478	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 31479	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 31480*	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 31481	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 31482	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 31483	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 31484	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 31485*	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 31486	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 31487	Join with lattice zero in Cℋ. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 0ℋ) = 𝐴
Theorem	chm1i 31488	Meet with lattice one in Cℋ. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ ℋ) = 𝐴
Theorem	chjcli 31489	Closure of Cℋ join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ
Theorem	chsleji 31490	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 31491*	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 31492	Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ
Theorem	chsscon3i 31493	Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴))
Theorem	chsscon1i 31494	Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴)
Theorem	chsscon2i 31495	Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))
Theorem	chcon2i 31496	Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴))
Theorem	chcon1i 31497	Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴)
Theorem	chcon3i 31498	Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴))
Theorem	chunssji 31499	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 31500	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)