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Theorem List for Metamath Proof Explorer - 27601-27700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremshjcom 27601 Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))

Theoremshless 27602 Subset implies subset of subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(((𝐴S𝐵S𝐶S ) ∧ 𝐴𝐵) → (𝐴 + 𝐶) ⊆ (𝐵 + 𝐶))

Theoremshlej1 27603 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.)
(((𝐴S𝐵S𝐶S ) ∧ 𝐴𝐵) → (𝐴 𝐶) ⊆ (𝐵 𝐶))

Theoremshlej2 27604 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(((𝐴S𝐵S𝐶S ) ∧ 𝐴𝐵) → (𝐶 𝐴) ⊆ (𝐶 𝐵))

Theoremshincli 27605 Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵) ∈ S

Theoremshscomi 27606 Commutative law for subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) = (𝐵 + 𝐴)

Theoremshsvai 27607 Vector sum belongs to subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐶𝐴𝐷𝐵) → (𝐶 + 𝐷) ∈ (𝐴 + 𝐵))

Theoremshsel1i 27608 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐶𝐴𝐶 ∈ (𝐴 + 𝐵))

Theoremshsel2i 27609 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐶𝐵𝐶 ∈ (𝐴 + 𝐵))

Theoremshsvsi 27610 Vector subtraction belongs to subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐶𝐴𝐷𝐵) → (𝐶 𝐷) ∈ (𝐴 + 𝐵))

Theoremshunssi 27611 Union is smaller than subspace sum. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵) ⊆ (𝐴 + 𝐵)

Theoremshunssji 27612 Union is smaller than Hilbert lattice join. (Contributed by NM, 11-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵) ⊆ (𝐴 𝐵)

Theoremshsleji 27613 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 27614 Commutative law for join in S. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) = (𝐵 𝐴)

Theoremshsub1i 27615 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 27616 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 27617 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 27618 Closure of C join. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) ∈ C

Theoremshjshcli 27619 S closure of join. (Contributed by NM, 5-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) ∈ S

Theoremshlessi 27620 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 27621 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 27622 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 27623 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 27624 Closure of intersection of two subspaces. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴𝐵) ∈ S )

Theoremshub1 27625 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 27626 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 27627 Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
𝐴S       (𝐴 + 𝐴) = 𝐴

Theoremshslubi 27628 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 27629 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 27630* 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 27631 An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) = (span‘(𝐴𝐵))

Theoremshmodsi 27632 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 27633 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)))

20.4.5  Projection theorem

Theorempjhthlem1 27634* Lemma for pjhth 27636. (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 27635* Lemma for pjhth 27636. (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 27636 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 27637* 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 27659 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))

20.4.6  Projectors

Definitiondf-pjh 27638* 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 27639* 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 27640* 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 27641* Equality with a projection. This version of pjeq 27642 does not assume the Axiom of Choice via pjhth 27636. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ (𝐻 + (⊥‘𝐻))) → (((proj𝐻)‘𝐴) = 𝐵 ↔ (𝐵𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 + 𝑥))))

Theorempjeq 27642* 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 27643 Closure of a projection in its subspace. If we consider this together with axpjpj 27663 to be axioms, the need for the ax-hcompl 27443 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 27678.) (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) ∈ 𝐻)

Theorempjhcl 27644 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) ∈ ℋ)

20.5  Properties of Hilbert subspaces

20.5.1  Orthomodular law

Theoremomlsilem 27645 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 27646 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 27647 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 27648 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 27649 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 27650 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 27651* 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 27652* 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 27653* 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 27654* 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 27655* A subspace is the supremum of all smaller subspaces. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( ‘{𝑥C𝑥𝐴}) = 𝐴)

Theoremchsupsn 27656 Value of supremum of subset of C on a singleton. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( ‘{𝐴}) = 𝐴)

Theoremshlub 27657 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 27658 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)

20.5.2  Projectors (cont.)

Theorempjhtheu2 27659* 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 27660 Closure of a projection in its subspace. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → ((proj𝐻)‘𝐴) ∈ 𝐻)

Theorempjhcli 27661 Closure of a projection in Hilbert space. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → ((proj𝐻)‘𝐴) ∈ ℋ)

Theorempjpjpre 27662 Decomposition of a vector into projections. This formulation of axpjpj 27663 avoids pjhth 27636. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(𝜑𝐻C )    &   (𝜑𝐴 ∈ (𝐻 + (⊥‘𝐻)))       (𝜑𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴)))

Theoremaxpjpj 27663 Decomposition of a vector into projections. See comment in axpjcl 27643. (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 27664 Closure of a projection in its subspace. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) ∈ 𝐻

Theorempjhclii 27665 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) ∈ ℋ

Theorempjpj0i 27666 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 27667 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 27668* 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 27669* 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 27670 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 27671 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 27672 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 27673 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 27674 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 27675 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 27676 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 27677 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 27678 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 27647. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵S       ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) → 𝐴 = 𝐵)

Theorempjoml 27679 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 27647. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(((𝐴C𝐵S ) ∧ (𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0)) → 𝐴 = 𝐵)

Theorempjococi 27680 Proof of orthocomplement theorem using projections. Compare ococ 27649. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
𝐻C       (⊥‘(⊥‘𝐻)) = 𝐻

Theorempjoc2i 27681 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 27682 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))

20.5.3  Hilbert lattice operations

Theoremsh0le 27683 The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
(𝐴S → 0𝐴)

Theoremch0le 27684 The zero subspace is the smallest member of C. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
(𝐴C → 0𝐴)

Theoremshle0 27685 No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
(𝐴S → (𝐴 ⊆ 0𝐴 = 0))

Theoremchle0 27686 No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
(𝐴C → (𝐴 ⊆ 0𝐴 = 0))

Theoremchnlen0 27687 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 27688 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 27689 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 27690 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 27691* A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
𝐴S       (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)

Theoremshs0i 27692 Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
𝐴S       (𝐴 + 0) = 𝐴

Theoremshs00i 27693 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 27694 The closed subspace zero is the smallest member of C. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C       0𝐴

Theoremchle0i 27695 No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
𝐴C       (𝐴 ⊆ 0𝐴 = 0)

Theoremchne0i 27696* 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 27697 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 27698 Join with lattice zero in C. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 0) = 𝐴

Theoremchm1i 27699 Meet with lattice one in C. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 ∩ ℋ) = 𝐴

Theoremchjcli 27700 Closure of C join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) ∈ C

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