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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | shocel 31301* | Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Sℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))) | ||
| Theorem | ocsh 31302 | The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Sℋ ) | ||
| Theorem | shocsh 31303 | The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Sℋ ) | ||
| Theorem | ocss 31304 | An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ) | ||
| Theorem | shocss 31305 | An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ⊆ ℋ) | ||
| Theorem | occon 31306 | Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴))) | ||
| Theorem | occon2 31307 | Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵)))) | ||
| Theorem | occon2i 31308 | Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
| ⊢ 𝐴 ⊆ ℋ & ⊢ 𝐵 ⊆ ℋ ⇒ ⊢ (𝐴 ⊆ 𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵))) | ||
| Theorem | oc0 31309 | The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ (⊥‘𝐻)) | ||
| Theorem | ocorth 31310 | Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
| ⊢ (𝐻 ⊆ ℋ → ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) → (𝐴 ·ih 𝐵) = 0)) | ||
| Theorem | shocorth 31311 | Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Sℋ → ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) → (𝐴 ·ih 𝐵) = 0)) | ||
| Theorem | ococss 31312 | Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴))) | ||
| Theorem | shococss 31313 | Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Sℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴))) | ||
| Theorem | shorth 31314 | Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Sℋ → (𝐺 ⊆ (⊥‘𝐻) → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ih 𝐵) = 0))) | ||
| Theorem | ocin 31315 | Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Sℋ → (𝐴 ∩ (⊥‘𝐴)) = 0ℋ) | ||
| Theorem | occon3 31316 | Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))) | ||
| Theorem | ocnel 31317 | A nonzero vector in the complement of a subspace does not belong to the subspace. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.) |
| ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ (⊥‘𝐻) ∧ 𝐴 ≠ 0ℎ) → ¬ 𝐴 ∈ 𝐻) | ||
| Theorem | chocvali 31318* | Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of 𝐴 is the set of vectors that are orthogonal to all vectors in 𝐴. (Contributed by NM, 8-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0} | ||
| Theorem | shuni 31319 | Two subspaces with trivial intersection have a unique decomposition of the elements of the subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝐻 ∈ Sℋ ) & ⊢ (𝜑 → 𝐾 ∈ Sℋ ) & ⊢ (𝜑 → (𝐻 ∩ 𝐾) = 0ℋ) & ⊢ (𝜑 → 𝐴 ∈ 𝐻) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) & ⊢ (𝜑 → 𝐶 ∈ 𝐻) & ⊢ (𝜑 → 𝐷 ∈ 𝐾) & ⊢ (𝜑 → (𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
| Theorem | chocunii 31320 | Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part). (Contributed by NM, 23-Oct-1999.) (Proof shortened by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ (⊥‘𝐻))) → ((𝑅 = (𝐴 +ℎ 𝐵) ∧ 𝑅 = (𝐶 +ℎ 𝐷)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
| Theorem | pjhthmo 31321* | Projection Theorem, uniqueness part. Any two disjoint subspaces yield a unique decomposition of vectors into each subspace. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))) | ||
| Theorem | occllem 31322 | Lemma for occl 31323. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℋ) & ⊢ (𝜑 → 𝐹 ∈ Cauchy) & ⊢ (𝜑 → 𝐹:ℕ⟶(⊥‘𝐴)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → (( ⇝𝑣 ‘𝐹) ·ih 𝐵) = 0) | ||
| Theorem | occl 31323 | Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Cℋ ) | ||
| Theorem | shoccl 31324 | Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Cℋ ) | ||
| Theorem | choccl 31325 | Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Cℋ → (⊥‘𝐴) ∈ Cℋ ) | ||
| Theorem | choccli 31326 | Closure of Cℋ orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (⊥‘𝐴) ∈ Cℋ | ||
| Definition | df-shs 31327* | Define subspace sum in Sℋ. See shsval 31331, shsval2i 31406, and shsval3i 31407 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.) |
| ⊢ +ℋ = (𝑥 ∈ Sℋ , 𝑦 ∈ Sℋ ↦ ( +ℎ “ (𝑥 × 𝑦))) | ||
| Definition | df-span 31328* | Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 31352 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
| ⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦}) | ||
| Definition | df-chj 31329* | Define Hilbert lattice join. See chjval 31371 for its value and chjcl 31376 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to Cℋ; see sshjcl 31374. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.) |
| ⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦)))) | ||
| Definition | df-chsup 31330 | Define the supremum of a set of Hilbert lattice elements. See chsupval2 31429 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice Cℋ, to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 31358. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.) |
| ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥))) | ||
| Theorem | shsval 31331 | Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in [Kalmbach] p. 65. (Contributed by NM, 16-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = ( +ℎ “ (𝐴 × 𝐵))) | ||
| Theorem | shsss 31332 | The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ⊆ ℋ) | ||
| Theorem | shsel 31333* | Membership in the subspace sum of two Hilbert subspaces. (Contributed by NM, 14-Dec-2004.) (Revised by Mario Carneiro, 29-Jan-2014.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))) | ||
| Theorem | shsel3 31334* | Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 −ℎ 𝑦))) | ||
| Theorem | shseli 31335* | Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) | ||
| Theorem | shscli 31336 | Closure of subspace sum. (Contributed by NM, 15-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 +ℋ 𝐵) ∈ Sℋ | ||
| Theorem | shscl 31337 | Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ∈ Sℋ ) | ||
| Theorem | shscom 31338 | Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) | ||
| Theorem | shsva 31339 | Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶 +ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵))) | ||
| Theorem | shsel1 31340 | A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ 𝐴 → 𝐶 ∈ (𝐴 +ℋ 𝐵))) | ||
| Theorem | shsel2 31341 | A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 +ℋ 𝐵))) | ||
| Theorem | shsvs 31342 | Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶 −ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵))) | ||
| Theorem | shsub1 31343 | Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ (𝐴 +ℋ 𝐵)) | ||
| Theorem | shsub2 31344 | 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 | choc0 31345 | The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
| ⊢ (⊥‘0ℋ) = ℋ | ||
| Theorem | choc1 31346 | The orthocomplement of the unit subspace is the zero subspace. Does not require Axiom of Choice. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
| ⊢ (⊥‘ ℋ) = 0ℋ | ||
| Theorem | chocnul 31347 | Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.) |
| ⊢ (⊥‘∅) = ℋ | ||
| Theorem | shintcli 31348 | Closure of intersection of a nonempty subset of Sℋ. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) ⇒ ⊢ ∩ 𝐴 ∈ Sℋ | ||
| Theorem | shintcl 31349 | The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Sℋ ) | ||
| Theorem | chintcli 31350 | The intersection of a nonempty set of closed subspaces is a closed subspace. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅) ⇒ ⊢ ∩ 𝐴 ∈ Cℋ | ||
| Theorem | chintcl 31351 | The intersection (infimum) of a nonempty subset of Cℋ belongs to Cℋ. Part of Theorem 3.13 of [Beran] p. 108. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Cℋ ) | ||
| Theorem | spanval 31352* | Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in [Schechter] p. 276. (Contributed by NM, 2-Jun-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) | ||
| Theorem | hsupval 31353 | Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 31428. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) | ||
| Theorem | chsupval 31354 | The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 31429. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ Cℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) | ||
| Theorem | spancl 31355 | The span of a subset of Hilbert space is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) ∈ Sℋ ) | ||
| Theorem | elspancl 31356 | A member of a span is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ∈ (span‘𝐴)) → 𝐵 ∈ ℋ) | ||
| Theorem | shsupcl 31357 | Closure of the subspace supremum of set of subsets of Hilbert space. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ 𝒫 ℋ → (span‘∪ 𝐴) ∈ Sℋ ) | ||
| Theorem | hsupcl 31358 | Closure of supremum of set of subsets of Hilbert space. Note that the supremum belongs to Cℋ even if the subsets do not. (Contributed by NM, 10-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) ∈ Cℋ ) | ||
| Theorem | chsupcl 31359 | Closure of supremum of subset of Cℋ. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. Shows that Cℋ is a complete lattice. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 10-Nov-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ Cℋ → ( ∨ℋ ‘𝐴) ∈ Cℋ ) | ||
| Theorem | hsupss 31360 | Subset relation for supremum of Hilbert space subsets. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ 𝒫 ℋ ∧ 𝐵 ⊆ 𝒫 ℋ) → (𝐴 ⊆ 𝐵 → ( ∨ℋ ‘𝐴) ⊆ ( ∨ℋ ‘𝐵))) | ||
| Theorem | chsupss 31361 | Subset relation for supremum of subset of Cℋ. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ Cℋ ∧ 𝐵 ⊆ Cℋ ) → (𝐴 ⊆ 𝐵 → ( ∨ℋ ‘𝐴) ⊆ ( ∨ℋ ‘𝐵))) | ||
| Theorem | hsupunss 31362 | The union of a set of Hilbert space subsets is smaller than its supremum. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ 𝒫 ℋ → ∪ 𝐴 ⊆ ( ∨ℋ ‘𝐴)) | ||
| Theorem | chsupunss 31363 | The union of a set of closed subspaces is smaller than its supremum. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ Cℋ → ∪ 𝐴 ⊆ ( ∨ℋ ‘𝐴)) | ||
| Theorem | spanss2 31364 | A subset of Hilbert space is included in its span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (span‘𝐴)) | ||
| Theorem | shsupunss 31365 | The union of a set of subspaces is smaller than its supremum. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ Sℋ → ∪ 𝐴 ⊆ (span‘∪ 𝐴)) | ||
| Theorem | spanid 31366 | A subspace of Hilbert space is its own span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ Sℋ → (span‘𝐴) = 𝐴) | ||
| Theorem | spanss 31367 | Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐵 ⊆ ℋ ∧ 𝐴 ⊆ 𝐵) → (span‘𝐴) ⊆ (span‘𝐵)) | ||
| Theorem | spanssoc 31368 | The span of a subset of Hilbert space is less than or equal to its closure (double orthogonal complement). (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) ⊆ (⊥‘(⊥‘𝐴))) | ||
| Theorem | sshjval 31369 | Value of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) | ||
| Theorem | shjval 31370 | Value of join in Sℋ. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) | ||
| Theorem | chjval 31371 | Value of join in Cℋ. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) | ||
| Theorem | chjvali 31372 | Value of join in Cℋ. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))) | ||
| Theorem | sshjval3 31373 | Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice Cℋ. (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = ( ∨ℋ ‘{𝐴, 𝐵})) | ||
| Theorem | sshjcl 31374 | Closure of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.) |
| ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) ∈ Cℋ ) | ||
| Theorem | shjcl 31375 | Closure of join in Sℋ. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) ∈ Cℋ ) | ||
| Theorem | chjcl 31376 | Closure of join in Cℋ. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ 𝐵) ∈ Cℋ ) | ||
| Theorem | shjcom 31377 | Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)) | ||
| Theorem | shless 31378 | Subset implies subset of subspace sum. (Contributed 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 | shlej1 31379 | 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.) |
| ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∨ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐶)) | ||
| Theorem | shlej2 31380 | Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ∨ℋ 𝐴) ⊆ (𝐶 ∨ℋ 𝐵)) | ||
| Theorem | shincli 31381 | Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∩ 𝐵) ∈ Sℋ | ||
| Theorem | shscomi 31382 | Commutative law for subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴) | ||
| Theorem | shsvai 31383 | Vector sum belongs to subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶 +ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵)) | ||
| Theorem | shsel1i 31384 | A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ (𝐴 +ℋ 𝐵)) | ||
| Theorem | shsel2i 31385 | A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 +ℋ 𝐵)) | ||
| Theorem | shsvsi 31386 | Vector subtraction belongs to subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶 −ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵)) | ||
| Theorem | shunssi 31387 | Union is smaller than subspace sum. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 +ℋ 𝐵) | ||
| Theorem | shunssji 31388 | Union is smaller than Hilbert lattice join. (Contributed by NM, 11-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵) | ||
| Theorem | shsleji 31389 | 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.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 +ℋ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵) | ||
| Theorem | shjcomi 31390 | Commutative law for join in Sℋ. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) | ||
| Theorem | shsub1i 31391 | Subspace sum is an upper bound of its arguments. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ 𝐴 ⊆ (𝐴 +ℋ 𝐵) | ||
| Theorem | shsub2i 31392 | 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 31393 | 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 31394 | Closure of Cℋ join. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ | ||
| Theorem | shjshcli 31395 | Sℋ closure of join. (Contributed by NM, 5-May-2000.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) ∈ Sℋ | ||
| Theorem | shlessi 31396 | 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 31397 | 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 31398 | 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 31399 | 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 31400 | Closure of intersection of two subspaces. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∩ 𝐵) ∈ Sℋ ) | ||
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