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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | chsupval2 30701* | 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 30702* | 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 30703* | 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 30704 | 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 30705 | 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 30706 | 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) | ||
| Theorem | pjhtheu2 30707* | 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 30708 | 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 30709 | 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 30710 | Decomposition of a vector into projections. This formulation of axpjpj 30711 avoids pjhth 30684. (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 30711 | Decomposition of a vector into projections. See comment in axpjcl 30691. (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 30712 | 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 30713 | 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 30714 | 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 30715 | 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 30716* | 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 30717* | 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 30718 | 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 30719 | 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 30720 | 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 30721 | 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 30722 | 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 30723 | 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 30724 | 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 30725 | 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 30726 | 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 30695. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ) → 𝐴 = 𝐵) | ||
| Theorem | pjoml 30727 | 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 30695. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ)) → 𝐴 = 𝐵) | ||
| Theorem | pjococi 30728 | Proof of orthocomplement theorem using projections. Compare ococ 30697. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (⊥‘(⊥‘𝐻)) = 𝐻 | ||
| Theorem | pjoc2i 30729 | 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 30730 | 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)) | ||
| Theorem | sh0le 30731 | The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) | ||
| Theorem | ch0le 30732 | 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 30733 | 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 30734 | 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 30735 | 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 30736 | 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 30737 | 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 30738 | 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 30739* | 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 30740 | 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 30741 | 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 30742 | 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 30743 | 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 30744* | 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 30745 | 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 30746 | Join with lattice zero in Cℋ. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ 0ℋ) = 𝐴 | ||
| Theorem | chm1i 30747 | Meet with lattice one in Cℋ. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∩ ℋ) = 𝐴 | ||
| Theorem | chjcli 30748 | Closure of Cℋ join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ | ||
| Theorem | chsleji 30749 | 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 30750* | 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 30751 | Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ | ||
| Theorem | chsscon3i 30752 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴)) | ||
| Theorem | chsscon1i 30753 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴) | ||
| Theorem | chsscon2i 30754 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)) | ||
| Theorem | chcon2i 30755 | Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴)) | ||
| Theorem | chcon1i 30756 | Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴) | ||
| Theorem | chcon3i 30757 | Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴)) | ||
| Theorem | chunssji 30758 | 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 30759 | 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 30760 | 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 30761 | 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 30762 | 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 30763 | Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 30762). (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 30764 | 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 30765 | 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 30766 | 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 30767 | 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 30768 | 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 30769 | 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 30770 | 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 30771 | 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 30772 | 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 30773 | 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 30774 | 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 30775 | 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 30776 | 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 30777 | 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 30778 | 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 30779 | 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 30780 | Join with Hilbert lattice one. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ ℋ) = ℋ | ||
| Theorem | chm0i 30781 | Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∩ 0ℋ) = 0ℋ | ||
| Theorem | chm0 30782 | Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Cℋ → (𝐴 ∩ 0ℋ) = 0ℋ) | ||
| Theorem | shjshsi 30783 | 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 30784 | 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 30785* | 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 30786 | 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 30787 | 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 30788 | Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 0ℋ) = 𝐴) | ||
| Theorem | chslej 30789 | Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 +ℋ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵)) | ||
| Theorem | chincl 30790 | Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∩ 𝐵) ∈ Cℋ ) | ||
| Theorem | chsscon3 30791 | Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴))) | ||
| Theorem | chsscon1 30792 | Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴)) | ||
| Theorem | chsscon2 30793 | Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))) | ||
| Theorem | chpsscon3 30794 | Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ (⊥‘𝐴))) | ||
| Theorem | chpsscon1 30795 | Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐴) ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ 𝐴)) | ||
| Theorem | chpsscon2 30796 | Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊊ (⊥‘𝐵) ↔ 𝐵 ⊊ (⊥‘𝐴))) | ||
| Theorem | chjcom 30797 | Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)) | ||
| Theorem | chub1 30798 | Hilbert lattice join is greater than or equal to its first argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → 𝐴 ⊆ (𝐴 ∨ℋ 𝐵)) | ||
| Theorem | chub2 30799 | Hilbert lattice join is greater than or equal to its second argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → 𝐴 ⊆ (𝐵 ∨ℋ 𝐴)) | ||
| Theorem | chlub 30800 | Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∨ℋ 𝐵) ⊆ 𝐶)) | ||
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