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Theorem List for Metamath Proof Explorer - 29201-29300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempjpjhth 29201* 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.)
((𝐻C𝐴 ∈ ℋ) → ∃𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦))
 
Theorempjpjhthi 29202* 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    &   𝐻C       𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)
 
Theorempjop 29203 Orthocomplement projection in terms of projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj‘(⊥‘𝐻))‘𝐴) = (𝐴 ((proj𝐻)‘𝐴)))
 
Theorempjpo 29204 Projection in terms of orthocomplement projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝐴 ((proj‘(⊥‘𝐻))‘𝐴)))
 
Theorempjopi 29205 Orthocomplement projection in terms of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj‘(⊥‘𝐻))‘𝐴) = (𝐴 ((proj𝐻)‘𝐴))
 
Theorempjpoi 29206 Projection in terms of orthocomplement projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) = (𝐴 ((proj‘(⊥‘𝐻))‘𝐴))
 
Theorempjoc1i 29207 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       (𝐴𝐻 ↔ ((proj‘(⊥‘𝐻))‘𝐴) = 0)
 
Theorempjchi 29208 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.)
𝐻C    &   𝐴 ∈ ℋ       (𝐴𝐻 ↔ ((proj𝐻)‘𝐴) = 𝐴)
 
Theorempjoccl 29209 The part of a vector that belongs to the orthocomplemented space. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴 ((proj𝐻)‘𝐴)) ∈ (⊥‘𝐻))
 
Theorempjoc1 29210 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴𝐻 ↔ ((proj‘(⊥‘𝐻))‘𝐴) = 0))
 
Theorempjomli 29211 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 29180. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵S       ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) → 𝐴 = 𝐵)
 
Theorempjoml 29212 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 29180. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(((𝐴C𝐵S ) ∧ (𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0)) → 𝐴 = 𝐵)
 
Theorempjococi 29213 Proof of orthocomplement theorem using projections. Compare ococ 29182. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
𝐻C       (⊥‘(⊥‘𝐻)) = 𝐻
 
Theorempjoc2i 29214 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.)
𝐻C    &   𝐴 ∈ ℋ       (𝐴 ∈ (⊥‘𝐻) ↔ ((proj𝐻)‘𝐴) = 0)
 
Theorempjoc2 29215 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.)
((𝐻C𝐴 ∈ ℋ) → (𝐴 ∈ (⊥‘𝐻) ↔ ((proj𝐻)‘𝐴) = 0))
 
19.5.3  Hilbert lattice operations
 
Theoremsh0le 29216 The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
(𝐴S → 0𝐴)
 
Theoremch0le 29217 The zero subspace is the smallest member of C. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
(𝐴C → 0𝐴)
 
Theoremshle0 29218 No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
(𝐴S → (𝐴 ⊆ 0𝐴 = 0))
 
Theoremchle0 29219 No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
(𝐴C → (𝐴 ⊆ 0𝐴 = 0))
 
Theoremchnlen0 29220 A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
(𝐵C → (¬ 𝐴𝐵 → ¬ 𝐴 = 0))
 
Theoremch0pss 29221 The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
(𝐴C → (0𝐴𝐴 ≠ 0))
 
Theoremorthin 29222 The intersection of orthogonal subspaces is the zero subspace. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 ⊆ (⊥‘𝐵) → (𝐴𝐵) = 0))
 
Theoremssjo 29223 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 (⊥‘𝐴)) = ℋ)
 
Theoremshne0i 29224* A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
𝐴S       (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
 
Theoremshs0i 29225 Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
𝐴S       (𝐴 + 0) = 𝐴
 
Theoremshs00i 29226 Two subspaces are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐴 = 0𝐵 = 0) ↔ (𝐴 + 𝐵) = 0)
 
Theoremch0lei 29227 The closed subspace zero is the smallest member of C. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C       0𝐴
 
Theoremchle0i 29228 No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
𝐴C       (𝐴 ⊆ 0𝐴 = 0)
 
Theoremchne0i 29229* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
𝐴C       (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
 
Theoremchocini 29230 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.)
𝐴C       (𝐴 ∩ (⊥‘𝐴)) = 0
 
Theoremchj0i 29231 Join with lattice zero in C. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 0) = 𝐴
 
Theoremchm1i 29232 Meet with lattice one in C. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 ∩ ℋ) = 𝐴
 
Theoremchjcli 29233 Closure of C join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) ∈ C
 
Theoremchsleji 29234 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 + 𝐵) ⊆ (𝐴 𝐵)
 
Theoremchseli 29235* Membership in subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
 
Theoremchincli 29236 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵) ∈ C
 
Theoremchsscon3i 29237 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴))
 
Theoremchsscon1i 29238 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴)
 
Theoremchsscon2i 29239 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))
 
Theoremchcon2i 29240 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴))
 
Theoremchcon1i 29241 Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴)
 
Theoremchcon3i 29242 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴))
 
Theoremchunssji 29243 Union is smaller than C join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵) ⊆ (𝐴 𝐵)
 
Theoremchjcomi 29244 Commutative law for join in C. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) = (𝐵 𝐴)
 
Theoremchub1i 29245 C join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 ⊆ (𝐴 𝐵)
 
Theoremchub2i 29246 C join is an upper bound of two elements. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 ⊆ (𝐵 𝐴)
 
Theoremchlubi 29247 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐶𝐵𝐶) ↔ (𝐴 𝐵) ⊆ 𝐶)
 
Theoremchlubii 29248 Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 29247). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐶𝐵𝐶) → (𝐴 𝐵) ⊆ 𝐶)
 
Theoremchlej1i 29249 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝐴𝐵 → (𝐴 𝐶) ⊆ (𝐵 𝐶))
 
Theoremchlej2i 29250 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝐴𝐵 → (𝐶 𝐴) ⊆ (𝐶 𝐵))
 
Theoremchlej12i 29251 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐴𝐵𝐶𝐷) → (𝐴 𝐶) ⊆ (𝐵 𝐷))
 
Theoremchlejb1i 29252 Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 ↔ (𝐴 𝐵) = 𝐵)
 
Theoremchdmm1i 29253 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘(𝐴𝐵)) = ((⊥‘𝐴) ∨ (⊥‘𝐵))
 
Theoremchdmm2i 29254 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘((⊥‘𝐴) ∩ 𝐵)) = (𝐴 (⊥‘𝐵))
 
Theoremchdmm3i 29255 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘(𝐴 ∩ (⊥‘𝐵))) = ((⊥‘𝐴) ∨ 𝐵)
 
Theoremchdmm4i 29256 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘((⊥‘𝐴) ∩ (⊥‘𝐵))) = (𝐴 𝐵)
 
Theoremchdmj1i 29257 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘(𝐴 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵))
 
Theoremchdmj2i 29258 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘((⊥‘𝐴) ∨ 𝐵)) = (𝐴 ∩ (⊥‘𝐵))
 
Theoremchdmj3i 29259 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘(𝐴 (⊥‘𝐵))) = ((⊥‘𝐴) ∩ 𝐵)
 
Theoremchdmj4i 29260 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘((⊥‘𝐴) ∨ (⊥‘𝐵))) = (𝐴𝐵)
 
Theoremchnlei 29261 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐵𝐴𝐴 ⊊ (𝐴 𝐵))
 
Theoremchjassi 29262 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴 𝐵) ∨ 𝐶) = (𝐴 (𝐵 𝐶))
 
Theoremchj00i 29263 Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴 = 0𝐵 = 0) ↔ (𝐴 𝐵) = 0)
 
Theoremchjoi 29264 The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 (⊥‘𝐴)) = ℋ
 
Theoremchj1i 29265 Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C       (𝐴 ℋ) = ℋ
 
Theoremchm0i 29266 Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C       (𝐴 ∩ 0) = 0
 
Theoremchm0 29267 Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(𝐴C → (𝐴 ∩ 0) = 0)
 
Theoremshjshsi 29268 Hilbert lattice join equals the double orthocomplement of subspace sum. (Contributed by NM, 27-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) = (⊥‘(⊥‘(𝐴 + 𝐵)))
 
Theoremshjshseli 29269 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.)
𝐴S    &   𝐵S       ((𝐴 + 𝐵) ∈ C ↔ (𝐴 + 𝐵) = (𝐴 𝐵))
 
Theoremchne0 29270* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
(𝐴C → (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0))
 
Theoremchocin 29271 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.)
(𝐴C → (𝐴 ∩ (⊥‘𝐴)) = 0)
 
Theoremchssoc 29272 A closed subspace less than its orthocomplement is zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(𝐴C → (𝐴 ⊆ (⊥‘𝐴) ↔ 𝐴 = 0))
 
Theoremchj0 29273 Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(𝐴C → (𝐴 0) = 𝐴)
 
Theoremchslej 29274 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 + 𝐵) ⊆ (𝐴 𝐵))
 
Theoremchincl 29275 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵) ∈ C )
 
Theoremchsscon3 29276 Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴)))
 
Theoremchsscon1 29277 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴))
 
Theoremchsscon2 29278 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)))
 
Theoremchpsscon3 29279 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (⊥‘𝐵) ⊊ (⊥‘𝐴)))
 
Theoremchpsscon1 29280 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → ((⊥‘𝐴) ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ 𝐴))
 
Theoremchpsscon2 29281 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ⊊ (⊥‘𝐵) ↔ 𝐵 ⊊ (⊥‘𝐴)))
 
Theoremchjcom 29282 Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))
 
Theoremchub1 29283 Hilbert lattice join is greater than or equal to its first argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → 𝐴 ⊆ (𝐴 𝐵))
 
Theoremchub2 29284 Hilbert lattice join is greater than or equal to its second argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → 𝐴 ⊆ (𝐵 𝐴))
 
Theoremchlub 29285 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → ((𝐴𝐶𝐵𝐶) ↔ (𝐴 𝐵) ⊆ 𝐶))
 
Theoremchlej1 29286 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ 𝐴𝐵) → (𝐴 𝐶) ⊆ (𝐵 𝐶))
 
Theoremchlej2 29287 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ 𝐴𝐵) → (𝐶 𝐴) ⊆ (𝐶 𝐵))
 
Theoremchlejb1 29288 Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (𝐴 𝐵) = 𝐵))
 
Theoremchlejb2 29289 Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (𝐵 𝐴) = 𝐵))
 
Theoremchnle 29290 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (¬ 𝐵𝐴𝐴 ⊊ (𝐴 𝐵)))
 
Theoremchjo 29291 The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)
(𝐴C → (𝐴 (⊥‘𝐴)) = ℋ)
 
Theoremchabs1 29292 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 (𝐴𝐵)) = 𝐴)
 
Theoremchabs2 29293 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ∩ (𝐴 𝐵)) = 𝐴)
 
Theoremchabs1i 29294 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (𝐴𝐵)) = 𝐴
 
Theoremchabs2i 29295 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 ∩ (𝐴 𝐵)) = 𝐴
 
Theoremchjidm 29296 Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴C → (𝐴 𝐴) = 𝐴)
 
Theoremchjidmi 29297 Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
𝐴C       (𝐴 𝐴) = 𝐴
 
Theoremchj12i 29298 A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝐴 (𝐵 𝐶)) = (𝐵 (𝐴 𝐶))
 
Theoremchj4i 29299 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐴 𝐵) ∨ (𝐶 𝐷)) = ((𝐴 𝐶) ∨ (𝐵 𝐷))
 
Theoremchjjdiri 29300 Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴 𝐵) ∨ 𝐶) = ((𝐴 𝐶) ∨ (𝐵 𝐶))
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