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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | chsscon3i 30201 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴)) | ||
Theorem | chsscon1i 30202 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴) | ||
Theorem | chsscon2i 30203 | Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)) | ||
Theorem | chcon2i 30204 | Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴)) | ||
Theorem | chcon1i 30205 | Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴) | ||
Theorem | chcon3i 30206 | Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴)) | ||
Theorem | chunssji 30207 | 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 30208 | 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 30209 | 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 30210 | 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 30211 | 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 30212 | Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 30211). (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 30213 | 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 30214 | 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 30215 | 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 30216 | 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 30217 | 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 30218 | 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 30219 | 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 30220 | 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 30221 | 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 | chdmj2i 30222 | 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 | chdmj3i 30223 | 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 | chdmj4i 30224 | 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 | chnlei 30225 | Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∨ℋ 𝐵)) | ||
Theorem | chjassi 30226 | 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 | chj00i 30227 | Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((𝐴 = 0ℋ ∧ 𝐵 = 0ℋ) ↔ (𝐴 ∨ℋ 𝐵) = 0ℋ) | ||
Theorem | chjoi 30228 | The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ (⊥‘𝐴)) = ℋ | ||
Theorem | chj1i 30229 | Join with Hilbert lattice one. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ ℋ) = ℋ | ||
Theorem | chm0i 30230 | Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∩ 0ℋ) = 0ℋ | ||
Theorem | chm0 30231 | Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Cℋ → (𝐴 ∩ 0ℋ) = 0ℋ) | ||
Theorem | shjshsi 30232 | Hilbert lattice join equals the double orthocomplement of subspace sum. (Contributed by NM, 27-Nov-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 +ℋ 𝐵))) | ||
Theorem | shjshseli 30233 | A closed subspace sum equals Hilbert lattice join. Part of Lemma 31.1.5 of [MaedaMaeda] p. 136. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ ⇒ ⊢ ((𝐴 +ℋ 𝐵) ∈ Cℋ ↔ (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵)) | ||
Theorem | chne0 30234* | 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 | chocin 30235 | Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Cℋ → (𝐴 ∩ (⊥‘𝐴)) = 0ℋ) | ||
Theorem | chssoc 30236 | A closed subspace less than its orthocomplement is zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ (⊥‘𝐴) ↔ 𝐴 = 0ℋ)) | ||
Theorem | chj0 30237 | Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 0ℋ) = 𝐴) | ||
Theorem | chslej 30238 | 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 30239 | Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∩ 𝐵) ∈ Cℋ ) | ||
Theorem | chsscon3 30240 | Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴))) | ||
Theorem | chsscon1 30241 | Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴)) | ||
Theorem | chsscon2 30242 | Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))) | ||
Theorem | chpsscon3 30243 | 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 30244 | 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 30245 | 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 30246 | 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 30247 | 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 30248 | 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 30249 | 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 30250 | 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 30251 | 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 30252 | 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 30253 | 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 30254 | 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 30255 | 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 30256 | 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 30257 | 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 30258 | 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 30259 | 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 30260 | Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 𝐴) = 𝐴) | ||
Theorem | chjidmi 30261 | Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐴) = 𝐴 | ||
Theorem | chj12i 30262 | 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 30263 | 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 30264 | 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 30265 | 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 30266 | 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 30267 | 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 30268 | 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 30269 | 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 30270 | 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 30271 | 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 30272 | 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 30273 | 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 30274 | 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 30275 | 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 30276 | 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 30277 | 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 30278 | 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 30279 | 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 30280 | 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 30281 | 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 30282 | 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 30283* | 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 30284 | 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 30285 | 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 30286 | 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 30287 | 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 30288 | The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.) |
⊢ ( ∨ℋ ‘∅) = 0ℋ | ||
Theorem | h1deoi 30289 | 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 30290* | 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 30291 | 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 30292 | 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 30293 | 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 30294 | 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 30295* | Lemma for h1de2ci 30296. (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 30296* | 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 30297 | 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 30298* | 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 30299 | 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 30300 | 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 ) |
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