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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremshsleji 27401 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.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) ⊆ (𝐴 𝐵)
 
Theoremshjcomi 27402 Commutative law for join in S. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) = (𝐵 𝐴)
 
Theoremshsub1i 27403 Subspace sum is an upper bound of its arguments. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       𝐴 ⊆ (𝐴 + 𝐵)
 
Theoremshsub2i 27404 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       𝐴 ⊆ (𝐵 + 𝐴)
 
Theoremshub1i 27405 Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       𝐴 ⊆ (𝐴 𝐵)
 
Theoremshjcli 27406 Closure of C join. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) ∈ C
 
Theoremshjshcli 27407 S closure of join. (Contributed by NM, 5-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) ∈ S
 
Theoremshlessi 27408 Subset implies subset of subspace sum. (Contributed by NM, 18-Nov-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐵 → (𝐴 + 𝐶) ⊆ (𝐵 + 𝐶))
 
Theoremshlej1i 27409 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.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐵 → (𝐴 𝐶) ⊆ (𝐵 𝐶))
 
Theoremshlej2i 27410 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐵 → (𝐶 𝐴) ⊆ (𝐶 𝐵))
 
Theoremshslej 27411 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) ⊆ (𝐴 𝐵))
 
Theoremshincl 27412 Closure of intersection of two subspaces. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴𝐵) ∈ S )
 
Theoremshub1 27413 Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐴 𝐵))
 
Theoremshub2 27414 A subspace is a subset of its Hilbert lattice join with another. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐵 𝐴))
 
Theoremshsidmi 27415 Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
𝐴S       (𝐴 + 𝐴) = 𝐴
 
Theoremshslubi 27416 The least upper bound law for Hilbert subspace sum. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       ((𝐴𝐶𝐵𝐶) ↔ (𝐴 + 𝐵) ⊆ 𝐶)
 
Theoremshlesb1i 27417 Hilbert lattice ordering in terms of subspace sum. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵 ↔ (𝐴 + 𝐵) = 𝐵)
 
Theoremshsval2i 27418* An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) = {𝑥S ∣ (𝐴𝐵) ⊆ 𝑥}
 
Theoremshsval3i 27419 An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) = (span‘(𝐴𝐵))
 
Theoremshmodsi 27420 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.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐶 → ((𝐴 + 𝐵) ∩ 𝐶) ⊆ (𝐴 + (𝐵𝐶)))
 
Theoremshmodi 27421 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.)
𝐴S    &   𝐵S    &   𝐶S       (((𝐴 + 𝐵) = (𝐴 𝐵) ∧ 𝐴𝐶) → ((𝐴 𝐵) ∩ 𝐶) ⊆ (𝐴 (𝐵𝐶)))
 
19.4.5  Projection theorem
 
Theorempjhthlem1 27422* Lemma for pjhth 27424. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐻C    &   (𝜑𝐴 ∈ ℋ)    &   (𝜑𝐵𝐻)    &   (𝜑𝐶𝐻)    &   (𝜑 → ∀𝑥𝐻 (norm‘(𝐴 𝐵)) ≤ (norm‘(𝐴 𝑥)))    &   𝑇 = (((𝐴 𝐵) ·ih 𝐶) / ((𝐶 ·ih 𝐶) + 1))       (𝜑 → ((𝐴 𝐵) ·ih 𝐶) = 0)
 
Theorempjhthlem2 27423* Lemma for pjhth 27424. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐻C    &   (𝜑𝐴 ∈ ℋ)       (𝜑 → ∃𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦))
 
Theorempjhth 27424 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.)
(𝐻C → (𝐻 + (⊥‘𝐻)) = ℋ)
 
Theorempjhtheu 27425* 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 27447 for the uniqueness of 𝑦. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ∃!𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦))
 
19.4.6  Projectors
 
Definitiondf-pjh 27426* 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)‘𝐴 is the projection of vector 𝐴 onto closed subspace 𝐻. Note that the range of proj is the set of all projection operators, so 𝑇 ∈ ran proj means that 𝑇 is a projection operator. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))
 
Theorempjhfval 27427* The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
(𝐻C → (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
 
Theorempjhval 27428* Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
 
Theorempjpreeq 27429* Equality with a projection. This version of pjeq 27430 does not assume the Axiom of Choice via pjhth 27424. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ (𝐻 + (⊥‘𝐻))) → (((proj𝐻)‘𝐴) = 𝐵 ↔ (𝐵𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 + 𝑥))))
 
Theorempjeq 27430* Equality with a projection. (Contributed by NM, 20-Jan-2007.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (((proj𝐻)‘𝐴) = 𝐵 ↔ (𝐵𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 + 𝑥))))
 
Theoremaxpjcl 27431 Closure of a projection in its subspace. If we consider this together with axpjpj 27451 to be axioms, the need for the ax-hcompl 27231 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 27466.) (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) ∈ 𝐻)
 
Theorempjhcl 27432 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) ∈ ℋ)
 
19.5  Properties of Hilbert subspaces
 
19.5.1  Orthomodular law
 
Theoremomlsilem 27433 Lemma for orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
𝐺S    &   𝐻S    &   𝐺𝐻    &   (𝐻 ∩ (⊥‘𝐺)) = 0    &   𝐴𝐻    &   𝐵𝐺    &   𝐶 ∈ (⊥‘𝐺)       (𝐴 = (𝐵 + 𝐶) → 𝐴𝐺)
 
Theoremomlsii 27434 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.)
𝐴C    &   𝐵S    &   𝐴𝐵    &   (𝐵 ∩ (⊥‘𝐴)) = 0       𝐴 = 𝐵
 
Theoremomlsi 27435 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.)
𝐴C    &   𝐵S       ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) → 𝐴 = 𝐵)
 
Theoremococi 27436 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.)
𝐴C       (⊥‘(⊥‘𝐴)) = 𝐴
 
Theoremococ 27437 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.)
(𝐴C → (⊥‘(⊥‘𝐴)) = 𝐴)
 
Theoremdfch2 27438 Alternate definition of the Hilbert lattice. (Contributed by NM, 8-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
C = {𝑥 ∈ 𝒫 ℋ ∣ (⊥‘(⊥‘𝑥)) = 𝑥}
 
Theoremococin 27439* 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)) = {𝑥C𝐴𝑥})
 
Theoremhsupval2 27440* 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 C, to allow more general theorems. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = {𝑥C 𝐴𝑥})
 
Theoremchsupval2 27441* 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.)
(𝐴C → ( 𝐴) = {𝑥C 𝐴𝑥})
 
Theoremsshjval2 27442* Value of join in the set of closed subspaces of Hilbert space C. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = {𝑥C ∣ (𝐴𝐵) ⊆ 𝑥})
 
Theoremchsupid 27443* A subspace is the supremum of all smaller subspaces. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( ‘{𝑥C𝑥𝐴}) = 𝐴)
 
Theoremchsupsn 27444 Value of supremum of subset of C on a singleton. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( ‘{𝐴}) = 𝐴)
 
Theoremshlub 27445 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.)
((𝐴S𝐵S𝐶C ) → ((𝐴𝐶𝐵𝐶) ↔ (𝐴 𝐵) ⊆ 𝐶))
 
Theoremshlubi 27446 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.)
𝐴S    &   𝐵S    &   𝐶C       ((𝐴𝐶𝐵𝐶) ↔ (𝐴 𝐵) ⊆ 𝐶)
 
19.5.2  Projectors (cont.)
 
Theorempjhtheu2 27447* Uniqueness of 𝑦 for the projection theorem. (Contributed by NM, 6-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ∃!𝑦 ∈ (⊥‘𝐻)∃𝑥𝐻 𝐴 = (𝑥 + 𝑦))
 
Theorempjcli 27448 Closure of a projection in its subspace. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → ((proj𝐻)‘𝐴) ∈ 𝐻)
 
Theorempjhcli 27449 Closure of a projection in Hilbert space. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → ((proj𝐻)‘𝐴) ∈ ℋ)
 
Theorempjpjpre 27450 Decomposition of a vector into projections. This formulation of axpjpj 27451 avoids pjhth 27424. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(𝜑𝐻C )    &   (𝜑𝐴 ∈ (𝐻 + (⊥‘𝐻)))       (𝜑𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴)))
 
Theoremaxpjpj 27451 Decomposition of a vector into projections. See comment in axpjcl 27431. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → 𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴)))
 
Theorempjclii 27452 Closure of a projection in its subspace. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) ∈ 𝐻
 
Theorempjhclii 27453 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) ∈ ℋ
 
Theorempjpj0i 27454 Decomposition of a vector into projections. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴))
 
Theorempjpji 27455 Decomposition of a vector into projections. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴))
 
Theorempjpjhth 27456* 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.)
((𝐻C𝐴 ∈ ℋ) → ∃𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦))
 
Theorempjpjhthi 27457* 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    &   𝐻C       𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)
 
Theorempjop 27458 Orthocomplement projection in terms of projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj‘(⊥‘𝐻))‘𝐴) = (𝐴 ((proj𝐻)‘𝐴)))
 
Theorempjpo 27459 Projection in terms of orthocomplement projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝐴 ((proj‘(⊥‘𝐻))‘𝐴)))
 
Theorempjopi 27460 Orthocomplement projection in terms of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj‘(⊥‘𝐻))‘𝐴) = (𝐴 ((proj𝐻)‘𝐴))
 
Theorempjpoi 27461 Projection in terms of orthocomplement projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) = (𝐴 ((proj‘(⊥‘𝐻))‘𝐴))
 
Theorempjoc1i 27462 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       (𝐴𝐻 ↔ ((proj‘(⊥‘𝐻))‘𝐴) = 0)
 
Theorempjchi 27463 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.)
𝐻C    &   𝐴 ∈ ℋ       (𝐴𝐻 ↔ ((proj𝐻)‘𝐴) = 𝐴)
 
Theorempjoccl 27464 The part of a vector that belongs to the orthocomplemented space. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴 ((proj𝐻)‘𝐴)) ∈ (⊥‘𝐻))
 
Theorempjoc1 27465 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴𝐻 ↔ ((proj‘(⊥‘𝐻))‘𝐴) = 0))
 
Theorempjomli 27466 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 27435. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵S       ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) → 𝐴 = 𝐵)
 
Theorempjoml 27467 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 27435. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(((𝐴C𝐵S ) ∧ (𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0)) → 𝐴 = 𝐵)
 
Theorempjococi 27468 Proof of orthocomplement theorem using projections. Compare ococ 27437. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
𝐻C       (⊥‘(⊥‘𝐻)) = 𝐻
 
Theorempjoc2i 27469 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.)
𝐻C    &   𝐴 ∈ ℋ       (𝐴 ∈ (⊥‘𝐻) ↔ ((proj𝐻)‘𝐴) = 0)
 
Theorempjoc2 27470 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.)
((𝐻C𝐴 ∈ ℋ) → (𝐴 ∈ (⊥‘𝐻) ↔ ((proj𝐻)‘𝐴) = 0))
 
19.5.3  Hilbert lattice operations
 
Theoremsh0le 27471 The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
(𝐴S → 0𝐴)
 
Theoremch0le 27472 The zero subspace is the smallest member of C. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
(𝐴C → 0𝐴)
 
Theoremshle0 27473 No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
(𝐴S → (𝐴 ⊆ 0𝐴 = 0))
 
Theoremchle0 27474 No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
(𝐴C → (𝐴 ⊆ 0𝐴 = 0))
 
Theoremchnlen0 27475 A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
(𝐵C → (¬ 𝐴𝐵 → ¬ 𝐴 = 0))
 
Theoremch0pss 27476 The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
(𝐴C → (0𝐴𝐴 ≠ 0))
 
Theoremorthin 27477 The intersection of orthogonal subspaces is the zero subspace. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 ⊆ (⊥‘𝐵) → (𝐴𝐵) = 0))
 
Theoremssjo 27478 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 (⊥‘𝐴)) = ℋ)
 
Theoremshne0i 27479* A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
𝐴S       (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
 
Theoremshs0i 27480 Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
𝐴S       (𝐴 + 0) = 𝐴
 
Theoremshs00i 27481 Two subspaces are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐴 = 0𝐵 = 0) ↔ (𝐴 + 𝐵) = 0)
 
Theoremch0lei 27482 The closed subspace zero is the smallest member of C. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C       0𝐴
 
Theoremchle0i 27483 No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
𝐴C       (𝐴 ⊆ 0𝐴 = 0)
 
Theoremchne0i 27484* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
𝐴C       (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
 
Theoremchocini 27485 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.)
𝐴C       (𝐴 ∩ (⊥‘𝐴)) = 0
 
Theoremchj0i 27486 Join with lattice zero in C. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 0) = 𝐴
 
Theoremchm1i 27487 Meet with lattice one in C. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 ∩ ℋ) = 𝐴
 
Theoremchjcli 27488 Closure of C join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) ∈ C
 
Theoremchsleji 27489 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 + 𝐵) ⊆ (𝐴 𝐵)
 
Theoremchseli 27490* Membership in subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
 
Theoremchincli 27491 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵) ∈ C
 
Theoremchsscon3i 27492 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴))
 
Theoremchsscon1i 27493 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴)
 
Theoremchsscon2i 27494 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))
 
Theoremchcon2i 27495 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴))
 
Theoremchcon1i 27496 Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴)
 
Theoremchcon3i 27497 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴))
 
Theoremchunssji 27498 Union is smaller than C join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵) ⊆ (𝐴 𝐵)
 
Theoremchjcomi 27499 Commutative law for join in C. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) = (𝐵 𝐴)
 
Theoremchub1i 27500 C join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 ⊆ (𝐴 𝐵)
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