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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | chj0 29201 | Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 0ℋ) = 𝐴) | ||
Theorem | chslej 29202 | 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 29203 | Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∩ 𝐵) ∈ Cℋ ) | ||
Theorem | chsscon3 29204 | Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴))) | ||
Theorem | chsscon1 29205 | Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴)) | ||
Theorem | chsscon2 29206 | Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))) | ||
Theorem | chpsscon3 29207 | 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 29208 | 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 29209 | 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 29210 | 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 29211 | 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 29212 | 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 29213 | 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)) | ||
Theorem | chlej1 29214 | Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∨ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐶)) | ||
Theorem | chlej2 29215 | Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ∨ℋ 𝐴) ⊆ (𝐶 ∨ℋ 𝐵)) | ||
Theorem | chlejb1 29216 | Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∨ℋ 𝐵) = 𝐵)) | ||
Theorem | chlejb2 29217 | Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (𝐵 ∨ℋ 𝐴) = 𝐵)) | ||
Theorem | chnle 29218 | Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∨ℋ 𝐵))) | ||
Theorem | chjo 29219 | The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ (⊥‘𝐴)) = ℋ) | ||
Theorem | chabs1 29220 | Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ (𝐴 ∩ 𝐵)) = 𝐴) | ||
Theorem | chabs2 29221 | Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∩ (𝐴 ∨ℋ 𝐵)) = 𝐴) | ||
Theorem | chabs1i 29222 | Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ (𝐴 ∩ 𝐵)) = 𝐴 | ||
Theorem | chabs2i 29223 | Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∩ (𝐴 ∨ℋ 𝐵)) = 𝐴 | ||
Theorem | chjidm 29224 | Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 𝐴) = 𝐴) | ||
Theorem | chjidmi 29225 | Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐴) = 𝐴 | ||
Theorem | chj12i 29226 | A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶)) = (𝐵 ∨ℋ (𝐴 ∨ℋ 𝐶)) | ||
Theorem | chj4i 29227 | Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝐷 ∈ Cℋ ⇒ ⊢ ((𝐴 ∨ℋ 𝐵) ∨ℋ (𝐶 ∨ℋ 𝐷)) = ((𝐴 ∨ℋ 𝐶) ∨ℋ (𝐵 ∨ℋ 𝐷)) | ||
Theorem | chjjdiri 29228 | Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ ⇒ ⊢ ((𝐴 ∨ℋ 𝐵) ∨ℋ 𝐶) = ((𝐴 ∨ℋ 𝐶) ∨ℋ (𝐵 ∨ℋ 𝐶)) | ||
Theorem | chdmm1 29229 | 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 | chdmm2 29230 | 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 | chdmm3 29231 | 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 | chdmm4 29232 | 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 | chdmj1 29233 | 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 | chdmj2 29234 | 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 | chdmj3 29235 | 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 | chdmj4 29236 | 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 | chjass 29237 | 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 | chj12 29238 | A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶)) = (𝐵 ∨ℋ (𝐴 ∨ℋ 𝐶))) | ||
Theorem | chj4 29239 | Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ (𝐶 ∈ Cℋ ∧ 𝐷 ∈ Cℋ )) → ((𝐴 ∨ℋ 𝐵) ∨ℋ (𝐶 ∨ℋ 𝐷)) = ((𝐴 ∨ℋ 𝐶) ∨ℋ (𝐵 ∨ℋ 𝐷))) | ||
Theorem | ledii 29240 | An ortholattice is distributive in one ordering direction. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ ⇒ ⊢ ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∩ (𝐵 ∨ℋ 𝐶)) | ||
Theorem | lediri 29241 | An ortholattice is distributive in one ordering direction. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ ⇒ ⊢ ((𝐴 ∩ 𝐶) ∨ℋ (𝐵 ∩ 𝐶)) ⊆ ((𝐴 ∨ℋ 𝐵) ∩ 𝐶) | ||
Theorem | lejdii 29242 | An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ (𝐵 ∩ 𝐶)) ⊆ ((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐶)) | ||
Theorem | lejdiri 29243 | An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ ⇒ ⊢ ((𝐴 ∩ 𝐵) ∨ℋ 𝐶) ⊆ ((𝐴 ∨ℋ 𝐶) ∩ (𝐵 ∨ℋ 𝐶)) | ||
Theorem | ledi 29244 | An ortholattice is distributive in one ordering direction. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∩ (𝐵 ∨ℋ 𝐶))) | ||
Theorem | spansn0 29245 | The span of the singleton of the zero vector is the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.) |
⊢ (span‘{0ℎ}) = 0ℋ | ||
Theorem | span0 29246 | The span of the empty set is the zero subspace. Remark 11.6.e of [Schechter] p. 276. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
⊢ (span‘∅) = 0ℋ | ||
Theorem | elspani 29247* | Membership in the span of a subset of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ ℋ → (𝐵 ∈ (span‘𝐴) ↔ ∀𝑥 ∈ Sℋ (𝐴 ⊆ 𝑥 → 𝐵 ∈ 𝑥))) | ||
Theorem | spanuni 29248 | The span of a union is the subspace sum of spans. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ⊆ ℋ & ⊢ 𝐵 ⊆ ℋ ⇒ ⊢ (span‘(𝐴 ∪ 𝐵)) = ((span‘𝐴) +ℋ (span‘𝐵)) | ||
Theorem | spanun 29249 | The span of a union is the subspace sum of spans. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (span‘(𝐴 ∪ 𝐵)) = ((span‘𝐴) +ℋ (span‘𝐵))) | ||
Theorem | sshhococi 29250 | The join of two Hilbert space subsets (not necessarily closed subspaces) equals the join of their closures (double orthocomplements). (Contributed by NM, 1-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
⊢ 𝐴 ⊆ ℋ & ⊢ 𝐵 ⊆ ℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) = ((⊥‘(⊥‘𝐴)) ∨ℋ (⊥‘(⊥‘𝐵))) | ||
Theorem | hne0 29251 | Hilbert space has a nonzero vector iff it is not trivial. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.) |
⊢ ( ℋ ≠ 0ℋ ↔ ∃𝑥 ∈ ℋ 𝑥 ≠ 0ℎ) | ||
Theorem | chsup0 29252 | The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.) |
⊢ ( ∨ℋ ‘∅) = 0ℋ | ||
Theorem | h1deoi 29253 | Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.) |
⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0)) | ||
Theorem | h1dei 29254* | Membership in 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ ℋ ((𝐵 ·ih 𝑥) = 0 → (𝐴 ·ih 𝑥) = 0))) | ||
Theorem | h1did 29255 | A generating vector belongs to the 1-dimensional subspace it generates. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (⊥‘(⊥‘{𝐴}))) | ||
Theorem | h1dn0 29256 | A nonzero vector generates a (nonzero) 1-dimensional subspace. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (⊥‘(⊥‘{𝐴})) ≠ 0ℋ) | ||
Theorem | h1de2i 29257 | Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 17-Jul-2001.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·ih 𝐵) ·ℎ 𝐴) = ((𝐴 ·ih 𝐵) ·ℎ 𝐵)) | ||
Theorem | h1de2bi 29258 | Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐵 ≠ 0ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵))) | ||
Theorem | h1de2ctlem 29259* | Lemma for h1de2ci 29260. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵)) | ||
Theorem | h1de2ci 29260* | Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 21-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵)) | ||
Theorem | spansni 29261 | The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ ⇒ ⊢ (span‘{𝐴}) = (⊥‘(⊥‘{𝐴})) | ||
Theorem | elspansni 29262* | Membership in the span of a singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ ⇒ ⊢ (𝐵 ∈ (span‘{𝐴}) ↔ ∃𝑥 ∈ ℂ 𝐵 = (𝑥 ·ℎ 𝐴)) | ||
Theorem | spansn 29263 | The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (span‘{𝐴}) = (⊥‘(⊥‘{𝐴}))) | ||
Theorem | spansnch 29264 | The span of a Hilbert space singleton belongs to the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (span‘{𝐴}) ∈ Cℋ ) | ||
Theorem | spansnsh 29265 | The span of a Hilbert space singleton is a subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (span‘{𝐴}) ∈ Sℋ ) | ||
Theorem | spansnchi 29266 | The span of a singleton in Hilbert space is a closed subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ ⇒ ⊢ (span‘{𝐴}) ∈ Cℋ | ||
Theorem | spansnid 29267 | A vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (span‘{𝐴})) | ||
Theorem | spansnmul 29268 | A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝐵 ·ℎ 𝐴) ∈ (span‘{𝐴})) | ||
Theorem | elspansncl 29269 | A member of a span of a singleton is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ (span‘{𝐴})) → 𝐵 ∈ ℋ) | ||
Theorem | elspansn 29270* | Membership in the span of a singleton. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (𝐵 ∈ (span‘{𝐴}) ↔ ∃𝑥 ∈ ℂ 𝐵 = (𝑥 ·ℎ 𝐴))) | ||
Theorem | elspansn2 29271 | Membership in the span of a singleton. All members are collinear with the generating vector. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0ℎ) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵))) | ||
Theorem | spansncol 29272 | The singletons of collinear vectors have the same span. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (span‘{(𝐵 ·ℎ 𝐴)}) = (span‘{𝐴})) | ||
Theorem | spansneleqi 29273 | Membership relation implied by equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → ((span‘{𝐴}) = (span‘{𝐵}) → 𝐴 ∈ (span‘{𝐵}))) | ||
Theorem | spansneleq 29274 | Membership relation that implies equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝐴 ∈ (span‘{𝐵}) → (span‘{𝐴}) = (span‘{𝐵}))) | ||
Theorem | spansnss 29275 | The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴) → (span‘{𝐵}) ⊆ 𝐴) | ||
Theorem | elspansn3 29276 | A member of the span of the singleton of a vector is a member of a subspace containing the vector. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ (span‘{𝐵})) → 𝐶 ∈ 𝐴) | ||
Theorem | elspansn4 29277 | A span membership condition implying two vectors belong to the same subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ≠ 0ℎ)) → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | ||
Theorem | elspansn5 29278 | A vector belonging to both a subspace and the span of the singleton of a vector not in it must be zero. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Sℋ → (((𝐵 ∈ ℋ ∧ ¬ 𝐵 ∈ 𝐴) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ∈ 𝐴)) → 𝐶 = 0ℎ)) | ||
Theorem | spansnss2 29279 | The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ∈ 𝐴 ↔ (span‘{𝐵}) ⊆ 𝐴)) | ||
Theorem | normcan 29280 | Cancellation-type law that "extracts" a vector 𝐴 from its inner product with a proportional vector 𝐵. (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.) |
⊢ ((𝐵 ∈ ℋ ∧ 𝐵 ≠ 0ℎ ∧ 𝐴 ∈ (span‘{𝐵})) → (((𝐴 ·ih 𝐵) / ((normℎ‘𝐵)↑2)) ·ℎ 𝐵) = 𝐴) | ||
Theorem | pjspansn 29281 | A projection on the span of a singleton. (The proof ws shortened by Mario Carneiro, 15-Dec-2013.) (Contributed by NM, 28-May-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((projℎ‘(span‘{𝐴}))‘𝐵) = (((𝐵 ·ih 𝐴) / ((normℎ‘𝐴)↑2)) ·ℎ 𝐴)) | ||
Theorem | spansnpji 29282 | A subset of Hilbert space is orthogonal to the span of the singleton of a projection onto its orthocomplement. (Contributed by NM, 4-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
⊢ 𝐴 ⊆ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ 𝐴 ⊆ (⊥‘(span‘{((projℎ‘(⊥‘𝐴))‘𝐵)})) | ||
Theorem | spanunsni 29283 | The span of the union of a closed subspace with a singleton equals the span of its union with an orthogonal singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (span‘(𝐴 ∪ {𝐵})) = (span‘(𝐴 ∪ {((projℎ‘(⊥‘𝐴))‘𝐵)})) | ||
Theorem | spanpr 29284 | The span of a pair of vectors. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (span‘{(𝐴 +ℎ 𝐵)}) ⊆ (span‘{𝐴, 𝐵})) | ||
Theorem | h1datomi 29285 | A 1-dimensional subspace is an atom. (Contributed by NM, 20-Jul-2001.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ⊆ (⊥‘(⊥‘{𝐵})) → (𝐴 = (⊥‘(⊥‘{𝐵})) ∨ 𝐴 = 0ℋ)) | ||
Theorem | h1datom 29286 | A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ⊆ (⊥‘(⊥‘{𝐵})) → (𝐴 = (⊥‘(⊥‘{𝐵})) ∨ 𝐴 = 0ℋ))) | ||
Definition | df-cm 29287* | Define the commutes relation (on the Hilbert lattice). Definition of commutes in [Kalmbach] p. 20, who uses the notation xCy for "x commutes with y." See cmbri 29294 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐶ℋ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))} | ||
Theorem | cmbr 29288 | Binary relation expressing 𝐴 commutes with 𝐵. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))) | ||
Theorem | pjoml2i 29289 | Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∨ℋ ((⊥‘𝐴) ∩ 𝐵)) = 𝐵) | ||
Theorem | pjoml3i 29290 | Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) = 𝐵) | ||
Theorem | pjoml4i 29291 | Variation of orthomodular law. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) = (𝐴 ∨ℋ 𝐵) | ||
Theorem | pjoml5i 29292 | The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ ((⊥‘𝐴) ∩ (𝐴 ∨ℋ 𝐵))) = (𝐴 ∨ℋ 𝐵) | ||
Theorem | pjoml6i 29293* | An equivalent of the orthomodular law. Theorem 29.13(e) of [MaedaMaeda] p. 132. (Contributed by NM, 30-May-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ 𝐵 → ∃𝑥 ∈ Cℋ (𝐴 ⊆ (⊥‘𝑥) ∧ (𝐴 ∨ℋ 𝑥) = 𝐵)) | ||
Theorem | cmbri 29294 | Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))) | ||
Theorem | cmcmlem 29295 | Commutation is symmetric. Theorem 3.4 of [Beran] p. 45. (Contributed by NM, 3-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 𝐶ℋ 𝐵 → 𝐵 𝐶ℋ 𝐴) | ||
Theorem | cmcmi 29296 | Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐵 𝐶ℋ 𝐴) | ||
Theorem | cmcm2i 29297 | Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 𝐶ℋ (⊥‘𝐵)) | ||
Theorem | cmcm3i 29298 | Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 𝐶ℋ 𝐵 ↔ (⊥‘𝐴) 𝐶ℋ 𝐵) | ||
Theorem | cmcm4i 29299 | Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 𝐶ℋ 𝐵 ↔ (⊥‘𝐴) 𝐶ℋ (⊥‘𝐵)) | ||
Theorem | cmbr2i 29300 | Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ (⊥‘𝐵)))) |
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