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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ocin 29001 | 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 29002 | 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 29003 | 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 29004* | 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 29005 | 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 29006 | 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 29007* | 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 29008 | Lemma for occl 29009. (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 29009 | 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 29010 | 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 29011 | 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 29012 | Closure of Cℋ orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (⊥‘𝐴) ∈ Cℋ | ||
Definition | df-shs 29013* | Define subspace sum in Sℋ. See shsval 29017, shsval2i 29092, and shsval3i 29093 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.) |
⊢ +ℋ = (𝑥 ∈ Sℋ , 𝑦 ∈ Sℋ ↦ ( +ℎ “ (𝑥 × 𝑦))) | ||
Definition | df-span 29014* | Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 29038 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦}) | ||
Definition | df-chj 29015* | Define Hilbert lattice join. See chjval 29057 for its value and chjcl 29062 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 29060. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.) |
⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦)))) | ||
Definition | df-chsup 29016 | Define the supremum of a set of Hilbert lattice elements. See chsupval2 29115 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 29044. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.) |
⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥))) | ||
Theorem | shsval 29017 | 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 29018 | 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 29019* | 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 29020* | 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 29021* | 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 29022 | 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 29023 | Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ∈ Sℋ ) | ||
Theorem | shscom 29024 | 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 29025 | 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 29026 | 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 29027 | 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 29028 | 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 29029 | 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 29030 | 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 29031 | The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
⊢ (⊥‘0ℋ) = ℋ | ||
Theorem | choc1 29032 | 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 29033 | Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.) |
⊢ (⊥‘∅) = ℋ | ||
Theorem | shintcli 29034 | 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 29035 | 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 29036 | 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 29037 | 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 29038* | 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 29039 | Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 29114. (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 29040 | The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 29115. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.) |
⊢ (𝐴 ⊆ Cℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) | ||
Theorem | spancl 29041 | 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 29042 | 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 29043 | 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 29044 | 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 29045 | 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 29046 | 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 29047 | 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 29048 | 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 29049 | 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 29050 | 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 29051 | 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 29052 | 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 29053 | 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 29054 | 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 29055 | 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 29056 | 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 29057 | 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 29058 | 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 29059 | 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 29060 | 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 29061 | 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 29062 | 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 29063 | 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 29064 | 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 29065 | 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 29066 | 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 29067 | 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 29068 | 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 29069 | 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 29070 | 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 29071 | 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 29072 | 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 29073 | 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 29074 | 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 29075 | 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 29076 | 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 29077 | 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 29078 | 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 29079 | 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 29080 | Closure of Cℋ join. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ | ||
Theorem | shjshcli 29081 | Sℋ closure of join. (Contributed by NM, 5-May-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) ∈ Sℋ | ||
Theorem | shlessi 29082 | 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 29083 | 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 29084 | 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 29085 | 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 29086 | 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 29087 | 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 29088 | 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 29089 | Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ ⇒ ⊢ (𝐴 +ℋ 𝐴) = 𝐴 | ||
Theorem | shslubi 29090 | 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 29091 | 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 29092* | 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 29093 | 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 29094 | 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 29095 | 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 29096* | Lemma for pjhth 29098. (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 29097* | Lemma for pjhth 29098. (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 29098 | 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 29099* | 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 29121 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 29100* | 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)))) |
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