![]() |
Metamath
Proof Explorer Theorem List (p. 315 of 491) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30946) |
![]() (30947-32469) |
![]() (32470-49035) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | shsub2i 31401 | 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 31402 | 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 31403 | Closure of Cℋ join. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ | ||
Theorem | shjshcli 31404 | Sℋ closure of join. (Contributed by NM, 5-May-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) ∈ Sℋ | ||
Theorem | shlessi 31405 | 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 31406 | 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 31407 | 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 31408 | 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 31409 | 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 31410 | 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 31411 | 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 31412 | Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ ⇒ ⊢ (𝐴 +ℋ 𝐴) = 𝐴 | ||
Theorem | shslubi 31413 | 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 31414 | 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 31415* | 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 31416 | 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 31417 | 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 31418 | 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 31419* | Lemma for pjhth 31421. (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 31420* | Lemma for pjhth 31421. (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 31421 | 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 31422* | 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 31444 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 31423* | 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 31424* | 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 31425* | 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 31426* | Equality with a projection. This version of pjeq 31427 does not assume the Axiom of Choice via pjhth 31421. (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 31427* | 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 31428 | Closure of a projection in its subspace. If we consider this together with axpjpj 31448 to be axioms, the need for the ax-hcompl 31230 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 31463.) (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 31429 | 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 31430 | 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 31431 | 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 31432 | 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 31433 | 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 31434 | 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 31435 | 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 31436* | 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 31437* | 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 31438* | 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 31439* | 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 31440* | 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 31441 | 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 31442 | 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 31443 | 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 31444* | 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 31445 | 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 31446 | 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 31447 | Decomposition of a vector into projections. This formulation of axpjpj 31448 avoids pjhth 31421. (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 31448 | Decomposition of a vector into projections. See comment in axpjcl 31428. (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 31449 | 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 31450 | 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 31451 | 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 31452 | 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 31453* | 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 31454* | 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 31455 | 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 31456 | 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 31457 | 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 31458 | 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 31459 | 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 31460 | 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 31461 | 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 31462 | 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 31463 | 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 31432. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ) → 𝐴 = 𝐵) | ||
Theorem | pjoml 31464 | 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 31432. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ)) → 𝐴 = 𝐵) | ||
Theorem | pjococi 31465 | Proof of orthocomplement theorem using projections. Compare ococ 31434. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (⊥‘(⊥‘𝐻)) = 𝐻 | ||
Theorem | pjoc2i 31466 | 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 31467 | 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 31468 | The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) | ||
Theorem | ch0le 31469 | 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 31470 | 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 31471 | 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 31472 | 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 31473 | 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 31474 | 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 31475 | 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 31476* | 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 31477 | 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 31478 | 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 31479 | 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 31480 | 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 31481* | 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 31482 | 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 31483 | Join with lattice zero in Cℋ. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ 0ℋ) = 𝐴 | ||
Theorem | chm1i 31484 | Meet with lattice one in Cℋ. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∩ ℋ) = 𝐴 | ||
Theorem | chjcli 31485 | Closure of Cℋ join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ | ||
Theorem | chsleji 31486 | 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 31487* | 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 31488 | Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ | ||
Theorem | chsscon3i 31489 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴)) | ||
Theorem | chsscon1i 31490 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴) | ||
Theorem | chsscon2i 31491 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)) | ||
Theorem | chcon2i 31492 | Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴)) | ||
Theorem | chcon1i 31493 | Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴) | ||
Theorem | chcon3i 31494 | Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴)) | ||
Theorem | chunssji 31495 | 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 31496 | 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 31497 | 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 31498 | 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 31499 | 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 31500 | Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 31499). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ ⇒ ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ∨ℋ 𝐵) ⊆ 𝐶) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |