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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | shub1 31401 | 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 31402 | 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 31403 | Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ ⇒ ⊢ (𝐴 +ℋ 𝐴) = 𝐴 | ||
| Theorem | shslubi 31404 | 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 31405 | 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 31406* | 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 31407 | 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 31408 | 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 31409 | 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))) | ||
| Theorem | pjhthlem1 31410* | Lemma for pjhth 31412. (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 31411* | Lemma for pjhth 31412. (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 31412 | 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 31413* | 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 31435 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)) | ||
| Definition | df-pjh 31414* | 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 31415* | 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 31416* | 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 31417* | Equality with a projection. This version of pjeq 31418 does not assume the Axiom of Choice via pjhth 31412. (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 31418* | 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 31419 | Closure of a projection in its subspace. If we consider this together with axpjpj 31439 to be axioms, the need for the ax-hcompl 31221 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 31454.) (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 31420 | 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 | omlsilem 31421 | 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 31422 | 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 31423 | 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 31424 | 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 31425 | 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 31426 | 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 31427* | 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 31428* | 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 31429* | 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 31430* | 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 31431* | 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 31432 | 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 31433 | 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 31434 | 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 31435* | 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 31436 | 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 31437 | 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 31438 | Decomposition of a vector into projections. This formulation of axpjpj 31439 avoids pjhth 31412. (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 31439 | Decomposition of a vector into projections. See comment in axpjcl 31419. (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 31440 | 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 31441 | 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 31442 | 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 31443 | 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 31444* | 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 31445* | 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 31446 | 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 31447 | 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 31448 | 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 31449 | 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 31450 | 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 31451 | 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 31452 | 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 31453 | 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 31454 | 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 31423. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ) → 𝐴 = 𝐵) | ||
| Theorem | pjoml 31455 | 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 31423. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ)) → 𝐴 = 𝐵) | ||
| Theorem | pjococi 31456 | Proof of orthocomplement theorem using projections. Compare ococ 31425. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (⊥‘(⊥‘𝐻)) = 𝐻 | ||
| Theorem | pjoc2i 31457 | 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 31458 | 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 31459 | The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) | ||
| Theorem | ch0le 31460 | 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 31461 | 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 31462 | 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 31463 | 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 31464 | 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 31465 | 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 31466 | 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 31467* | 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 31468 | 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 31469 | 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 31470 | 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 31471 | 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 31472* | 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 31473 | 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 31474 | Join with lattice zero in Cℋ. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ 0ℋ) = 𝐴 | ||
| Theorem | chm1i 31475 | Meet with lattice one in Cℋ. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∩ ℋ) = 𝐴 | ||
| Theorem | chjcli 31476 | Closure of Cℋ join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ | ||
| Theorem | chsleji 31477 | 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 31478* | 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 31479 | Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ | ||
| Theorem | chsscon3i 31480 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴)) | ||
| Theorem | chsscon1i 31481 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴) | ||
| Theorem | chsscon2i 31482 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)) | ||
| Theorem | chcon2i 31483 | Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴)) | ||
| Theorem | chcon1i 31484 | Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴) | ||
| Theorem | chcon3i 31485 | Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴)) | ||
| Theorem | chunssji 31486 | 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 31487 | 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 31488 | 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 31489 | 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 31490 | 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 31491 | Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 31490). (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 31492 | 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 31493 | 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 31494 | 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 31495 | 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 31496 | 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 31497 | 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 31498 | 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 31499 | 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 31500 | 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)) | ||
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