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Theorem List for Metamath Proof Explorer - 30001-30100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmdbr2 30001* Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 29999. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) ⊆ (𝑥 (𝐴𝐵)))))
 
Theoremmdbr3 30002* Binary relation expressing the modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
 
Theoremmdbr4 30003* Binary relation expressing the modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) ⊆ ((𝑥𝐵) ∨ (𝐴𝐵))))
 
Theoremdmdbr 30004* Binary relation expressing the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
 
Theoremdmdmd 30005 The dual modular pair property expressed in terms of the modular pair property, that hold in Hilbert lattices. Remark 29.6 of [MaedaMaeda] p. 130. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ (⊥‘𝐴) 𝑀 (⊥‘𝐵)))
 
Theoremmddmd 30006 The modular pair property expressed in terms of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ (⊥‘𝐴) 𝑀* (⊥‘𝐵)))
 
Theoremdmdi 30007 Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))
 
Theoremdmdbr2 30008* Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 30004. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → (𝑥 ∩ (𝐴 𝐵)) ⊆ ((𝑥𝐴) ∨ 𝐵))))
 
Theoremdmdi2 30009 Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → (𝐶 ∩ (𝐴 𝐵)) ⊆ ((𝐶𝐴) ∨ 𝐵))
 
Theoremdmdbr3 30010* Binary relation expressing the dual modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (((𝑥 𝐵) ∩ 𝐴) ∨ 𝐵) = ((𝑥 𝐵) ∩ (𝐴 𝐵))))
 
Theoremdmdbr4 30011* Binary relation expressing the dual modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C ((𝑥 𝐵) ∩ (𝐴 𝐵)) ⊆ (((𝑥 𝐵) ∩ 𝐴) ∨ 𝐵)))
 
Theoremdmdi4 30012 Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝑀* 𝐵 → ((𝐶 𝐵) ∩ (𝐴 𝐵)) ⊆ (((𝐶 𝐵) ∩ 𝐴) ∨ 𝐵)))
 
Theoremdmdbr5 30013* Binary relation expressing the dual modular pair property. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝑥 ⊆ (𝐴 𝐵) → 𝑥 ⊆ (((𝑥 𝐵) ∩ 𝐴) ∨ 𝐵))))
 
Theoremmddmd2 30014* Relationship between modular pairs and dual-modular pairs. Lemma 1.2 of [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
(𝐴C → (∀𝑥C 𝐴 𝑀 𝑥 ↔ ∀𝑥C 𝐴 𝑀* 𝑥))
 
Theoremmdsl0 30015 A sublattice condition that transfers the modular pair property. Exercise 12 of [Kalmbach] p. 103. Also Lemma 1.5.3 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(((𝐴C𝐵C ) ∧ (𝐶C𝐷C )) → ((((𝐶𝐴𝐷𝐵) ∧ (𝐴𝐵) = 0) ∧ 𝐴 𝑀 𝐵) → 𝐶 𝑀 𝐷))
 
Theoremssmd1 30016 Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐴𝐵) → 𝐴 𝑀 𝐵)
 
Theoremssmd2 30017 Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐴𝐵) → 𝐵 𝑀 𝐴)
 
Theoremssdmd1 30018 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐴𝐵) → 𝐴 𝑀* 𝐵)
 
Theoremssdmd2 30019 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐴𝐵) → (⊥‘𝐵) 𝑀 (⊥‘𝐴))
 
Theoremdmdsl3 30020 Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = 𝐶)
 
Theoremmdsl3 30021 Sublattice mapping for a modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀 𝐵 ∧ (𝐴𝐵) ⊆ 𝐶𝐶𝐵)) → ((𝐶 𝐴) ∩ 𝐵) = 𝐶)
 
Theoremmdslle1i 30022 Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐵 𝑀* 𝐴𝐴 ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ (𝐴 𝐵)) → (𝐶𝐷 ↔ (𝐶𝐵) ⊆ (𝐷𝐵)))
 
Theoremmdslle2i 30023 Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐴 𝑀 𝐵 ∧ (𝐴𝐵) ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ 𝐵) → (𝐶𝐷 ↔ (𝐶 𝐴) ⊆ (𝐷 𝐴)))
 
Theoremmdslj1i 30024 Join preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ (𝐴 ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ (𝐴 𝐵))) → ((𝐶 𝐷) ∩ 𝐵) = ((𝐶𝐵) ∨ (𝐷𝐵)))
 
Theoremmdslj2i 30025 Meet preservation of the reverse mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐵) ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ 𝐵)) → ((𝐶𝐷) ∨ 𝐴) = ((𝐶 𝐴) ∩ (𝐷 𝐴)))
 
Theoremmdsl1i 30026* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (∀𝑥C (((𝐴𝐵) ⊆ 𝑥𝑥 ⊆ (𝐴 𝐵)) → (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))) ↔ 𝐴 𝑀 𝐵)
 
Theoremmdsl2i 30027* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 28-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀 𝐵 ↔ ∀𝑥C (((𝐴𝐵) ⊆ 𝑥𝑥𝐵) → ((𝑥 𝐴) ∩ 𝐵) ⊆ (𝑥 (𝐴𝐵))))
 
Theoremmdsl2bi 30028* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀 𝐵 ↔ ∀𝑥C (((𝐴𝐵) ⊆ 𝑥𝑥𝐵) → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))
 
Theoremcvmdi 30029 The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) ⋖ 𝐵𝐴 𝑀 𝐵)
 
Theoremmdslmd1lem1 30030 Lemma for mdslmd1i 30034. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝑅C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐶𝐴𝐷) ∧ (𝐶 ⊆ (𝐴 𝐵) ∧ 𝐷 ⊆ (𝐴 𝐵)))) → (((𝑅 𝐴) ⊆ 𝐷 → (((𝑅 𝐴) ∨ 𝐶) ∩ 𝐷) ⊆ ((𝑅 𝐴) ∨ (𝐶𝐷))) → ((((𝐶𝐵) ∩ (𝐷𝐵)) ⊆ 𝑅𝑅 ⊆ (𝐷𝐵)) → ((𝑅 (𝐶𝐵)) ∩ (𝐷𝐵)) ⊆ (𝑅 ((𝐶𝐵) ∩ (𝐷𝐵))))))
 
Theoremmdslmd1lem2 30031 Lemma for mdslmd1i 30034. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝑅C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐶𝐴𝐷) ∧ (𝐶 ⊆ (𝐴 𝐵) ∧ 𝐷 ⊆ (𝐴 𝐵)))) → (((𝑅𝐵) ⊆ (𝐷𝐵) → (((𝑅𝐵) ∨ (𝐶𝐵)) ∩ (𝐷𝐵)) ⊆ ((𝑅𝐵) ∨ ((𝐶𝐵) ∩ (𝐷𝐵)))) → (((𝐶𝐷) ⊆ 𝑅𝑅𝐷) → ((𝑅 𝐶) ∩ 𝐷) ⊆ (𝑅 (𝐶𝐷)))))
 
Theoremmdslmd1lem3 30032* Lemma for mdslmd1i 30034. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝑥C ∧ ((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐶𝐴𝐷) ∧ (𝐶 ⊆ (𝐴 𝐵) ∧ 𝐷 ⊆ (𝐴 𝐵))))) → (((𝑥 𝐴) ⊆ 𝐷 → (((𝑥 𝐴) ∨ 𝐶) ∩ 𝐷) ⊆ ((𝑥 𝐴) ∨ (𝐶𝐷))) → ((((𝐶𝐵) ∩ (𝐷𝐵)) ⊆ 𝑥𝑥 ⊆ (𝐷𝐵)) → ((𝑥 (𝐶𝐵)) ∩ (𝐷𝐵)) ⊆ (𝑥 ((𝐶𝐵) ∩ (𝐷𝐵))))))
 
Theoremmdslmd1lem4 30033* Lemma for mdslmd1i 30034. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝑥C ∧ ((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐶𝐴𝐷) ∧ (𝐶 ⊆ (𝐴 𝐵) ∧ 𝐷 ⊆ (𝐴 𝐵))))) → (((𝑥𝐵) ⊆ (𝐷𝐵) → (((𝑥𝐵) ∨ (𝐶𝐵)) ∩ (𝐷𝐵)) ⊆ ((𝑥𝐵) ∨ ((𝐶𝐵) ∩ (𝐷𝐵)))) → (((𝐶𝐷) ⊆ 𝑥𝑥𝐷) → ((𝑥 𝐶) ∩ 𝐷) ⊆ (𝑥 (𝐶𝐷)))))
 
Theoremmdslmd1i 30034 Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (meet version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ (𝐴 ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ (𝐴 𝐵))) → (𝐶 𝑀 𝐷 ↔ (𝐶𝐵) 𝑀 (𝐷𝐵)))
 
Theoremmdslmd2i 30035 Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (join version). (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐵) ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ 𝐵)) → (𝐶 𝑀 𝐷 ↔ (𝐶 𝐴) 𝑀 (𝐷 𝐴)))
 
Theoremmdsldmd1i 30036 Preservation of the dual modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ (𝐴 ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ (𝐴 𝐵))) → (𝐶 𝑀* 𝐷 ↔ (𝐶𝐵) 𝑀* (𝐷𝐵)))
 
Theoremmdslmd3i 30037 Modular pair conditions that imply the modular pair property in a sublattice. Lemma 1.5.1 of [MaedaMaeda] p. 2. (Contributed by NM, 23-Dec-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵 ∧ (𝐴𝐵) 𝑀 𝐶) ∧ ((𝐴𝐶) ⊆ 𝐷𝐷𝐴)) → 𝐷 𝑀 (𝐵𝐶))
 
Theoremmdslmd4i 30038 Modular pair condition that implies the modular pair property in a sublattice. Lemma 1.5.2 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐴 𝑀 𝐵 ∧ ((𝐴𝐵) ⊆ 𝐶𝐶𝐴) ∧ ((𝐴𝐵) ⊆ 𝐷𝐷𝐵)) → 𝐶 𝑀 𝐷)
 
Theoremcsmdsymi 30039* Cross-symmetry implies M-symmetry. Theorem 1.9.1 of [MaedaMaeda] p. 3. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((∀𝑐C (𝑐 𝑀 𝐵𝐵 𝑀* 𝑐) ∧ 𝐴 𝑀 𝐵) → 𝐵 𝑀 𝐴)
 
Theoremmdexchi 30040 An exchange lemma for modular pairs. Lemma 1.6 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴 𝑀 𝐵𝐶 𝑀 (𝐴 𝐵) ∧ (𝐶 ∩ (𝐴 𝐵)) ⊆ 𝐴) → ((𝐶 𝐴) 𝑀 𝐵 ∧ ((𝐶 𝐴) ∩ 𝐵) = (𝐴𝐵)))
 
Theoremcvmd 30041 The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ∧ (𝐴𝐵) ⋖ 𝐵) → 𝐴 𝑀 𝐵)
 
Theoremcvdmd 30042 The covering property implies the dual modular pair property. Lemma 7.5.2 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐵 (𝐴 𝐵)) → 𝐴 𝑀* 𝐵)
 
19.8.2  Atoms
 
Definitiondf-at 30043 Define the set of atoms in a Hilbert lattice. An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. Definition of atom in [Kalmbach] p. 15. See ela 30044 and elat2 30045 for membership relations. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
HAtoms = {𝑥C ∣ 0 𝑥}
 
Theoremela 30044 Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
 
Theoremelat2 30045* Expanded membership relation for the set of atoms, i.e. the predicate "is an atom (of the Hilbert lattice)." An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms ↔ (𝐴C ∧ (𝐴 ≠ 0 ∧ ∀𝑥C (𝑥𝐴 → (𝑥 = 𝐴𝑥 = 0)))))
 
Theoremelatcv0 30046 A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴C → (𝐴 ∈ HAtoms ↔ 0 𝐴))
 
Theorematcv0 30047 An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms → 0 𝐴)
 
Theorematssch 30048 Atoms are a subset of the Hilbert lattice. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
HAtoms ⊆ C
 
Theorematelch 30049 An atom is a Hilbert lattice element. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms → 𝐴C )
 
Theorematne0 30050 An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴 ∈ HAtoms → 𝐴 ≠ 0)
 
Theorematss 30051 A lattice element smaller than an atom is either the atom or zero. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (𝐴𝐵 → (𝐴 = 𝐵𝐴 = 0)))
 
Theorematsseq 30052 Two atoms in a subset relationship are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (𝐴𝐵𝐴 = 𝐵))
 
Theorematcveq0 30053 A Hilbert lattice element covered by an atom must be the zero subspace. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (𝐴 𝐵𝐴 = 0))
 
Theoremh1da 30054 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (⊥‘(⊥‘{𝐴})) ∈ HAtoms)
 
Theoremspansna 30055 The span of the singleton of a vector is an atom. (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (span‘{𝐴}) ∈ HAtoms)
 
Theoremsh1dle 30056 A 1-dimensional subspace is less than or equal to any subspace containing its generating vector. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
((𝐴S𝐵𝐴) → (⊥‘(⊥‘{𝐵})) ⊆ 𝐴)
 
Theoremch1dle 30057 A 1-dimensional subspace is less than or equal to any member of C containing its generating vector. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)
((𝐴C𝐵𝐴) → (⊥‘(⊥‘{𝐵})) ⊆ 𝐴)
 
Theorematom1d 30058* The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms ↔ ∃𝑥 ∈ ℋ (𝑥 ≠ 0𝐴 = (span‘{𝑥})))
 
19.8.3  Superposition principle
 
Theoremsuperpos 30059* Superposition Principle. If 𝐴 and 𝐵 are distinct atoms, there exists a third atom, distinct from 𝐴 and 𝐵, that is the superposition of 𝐴 and 𝐵. Definition 3.4-3(a) in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ∧ 𝐴𝐵) → ∃𝑥 ∈ HAtoms (𝑥𝐴𝑥𝐵𝑥 ⊆ (𝐴 𝐵)))
 
19.8.4  Atoms, exchange and covering properties, atomicity
 
Theoremchcv1 30060 The Hilbert lattice has the covering property. Proposition 1(ii) of [Kalmbach] p. 140 (and its converse). (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (¬ 𝐵𝐴𝐴 (𝐴 𝐵)))
 
Theoremchcv2 30061 The Hilbert lattice has the covering property. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (𝐴 ⊊ (𝐴 𝐵) ↔ 𝐴 (𝐴 𝐵)))
 
Theoremchjatom 30062 The join of a closed subspace and an atom equals their subspace sum. Special case of remark in [Kalmbach] p. 65, stating that if 𝐴 or 𝐵 is finite-dimensional, then this equality holds. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (𝐴 + 𝐵) = (𝐴 𝐵))
 
Theoremshatomici 30063* The lattice of Hilbert subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
𝐴S       (𝐴 ≠ 0 → ∃𝑥 ∈ HAtoms 𝑥𝐴)
 
Theoremhatomici 30064* The Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
𝐴C       (𝐴 ≠ 0 → ∃𝑥 ∈ HAtoms 𝑥𝐴)
 
Theoremhatomic 30065* A Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. Also Definition 3.4-2 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐴 ≠ 0) → ∃𝑥 ∈ HAtoms 𝑥𝐴)
 
Theoremshatomistici 30066* The lattice of Hilbert subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
𝐴S       𝐴 = (span‘ {𝑥 ∈ HAtoms ∣ 𝑥𝐴})
 
Theoremhatomistici 30067* C is atomistic, i.e. any element is the supremum of its atoms. Remark in [Kalmbach] p. 140. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
𝐴C       𝐴 = ( ‘{𝑥 ∈ HAtoms ∣ 𝑥𝐴})
 
Theoremchpssati 30068* Two Hilbert lattice elements in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 → ∃𝑥 ∈ HAtoms (𝑥𝐵 ∧ ¬ 𝑥𝐴))
 
Theoremchrelati 30069* The Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 → ∃𝑥 ∈ HAtoms (𝐴 ⊊ (𝐴 𝑥) ∧ (𝐴 𝑥) ⊆ 𝐵))
 
Theoremchrelat2i 30070* A consequence of relative atomicity. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥𝐴 ∧ ¬ 𝑥𝐵))
 
Theoremcvati 30071* If a Hilbert lattice element covers another, it equals the other joined with some atom. This is a consequence of the relative atomicity of Hilbert space. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵 → ∃𝑥 ∈ HAtoms (𝐴 𝑥) = 𝐵)
 
Theoremcvbr4i 30072* An alternate way to express the covering property. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ∃𝑥 ∈ HAtoms (𝐴 𝑥) = 𝐵))
 
Theoremcvexchlem 30073 Lemma for cvexchi 30074. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) ⋖ 𝐵𝐴 (𝐴 𝐵))
 
Theoremcvexchi 30074 The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) ⋖ 𝐵𝐴 (𝐴 𝐵))
 
Theoremchrelat2 30075* A consequence of relative atomicity. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (¬ 𝐴𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
 
Theoremchrelat3 30076* A consequence of relative atomicity. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥𝐴𝑥𝐵)))
 
Theoremchrelat3i 30077* A consequence of the relative atomicity of Hilbert space: the ordering of Hilbert lattice elements is completely determined by the atoms they majorize. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥𝐴𝑥𝐵))
 
Theoremchrelat4i 30078* A consequence of relative atomicity. Extensionality principle: two lattice elements are equal iff they majorize the same atoms. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 = 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥𝐴𝑥𝐵))
 
Theoremcvexch 30079 The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → ((𝐴𝐵) ⋖ 𝐵𝐴 (𝐴 𝐵)))
 
Theoremcvp 30080 The Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → ((𝐴𝐵) = 0𝐴 (𝐴 𝐵)))
 
Theorematnssm0 30081 The meet of a Hilbert lattice element and an incomparable atom is the zero subspace. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (¬ 𝐵𝐴 ↔ (𝐴𝐵) = 0))
 
Theorematnemeq0 30082 The meet of distinct atoms is the zero subspace. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (𝐴𝐵 ↔ (𝐴𝐵) = 0))
 
Theorematssma 30083 The meet with an atom's superset is the atom. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵C ) → (𝐴𝐵 ↔ (𝐴𝐵) ∈ HAtoms))
 
Theorematcv0eq 30084 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (0 (𝐴 𝐵) ↔ 𝐴 = 𝐵))
 
Theorematcv1 30085 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(((𝐴C𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐴 (𝐵 𝐶)) → (𝐴 = 0𝐵 = 𝐶))
 
Theorematexch 30086 The Hilbert lattice satisfies the atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem related to vector analysis was originally proved by Hermann Grassmann in 1862. Also Definition 3.4-3(b) in [MegPav2000] p. 2345 (PDF p. 8) (use atnemeq0 30082 to obtain atom inequality). (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐵 ⊆ (𝐴 𝐶) ∧ (𝐴𝐵) = 0) → 𝐶 ⊆ (𝐴 𝐵)))
 
Theorematomli 30087 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition 3.2.17 of [PtakPulmannova] p. 66. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C       (𝐵 ∈ HAtoms → ((𝐴 𝐵) ∩ (⊥‘𝐴)) ∈ (HAtoms ∪ {0}))
 
Theorematoml2i 30088 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition P8(ii) of [BeltramettiCassinelli1] p. 400. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
𝐴C       ((𝐵 ∈ HAtoms ∧ ¬ 𝐵𝐴) → ((𝐴 𝐵) ∩ (⊥‘𝐴)) ∈ HAtoms)
 
Theorematordi 30089 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
𝐴C       ((𝐵 ∈ HAtoms ∧ 𝐴 𝐶 𝐵) → (𝐵𝐴𝐵 ⊆ (⊥‘𝐴)))
 
Theorematcvatlem 30090 Lemma for atcvati 30091. (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
𝐴C       (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (𝐴 ≠ 0𝐴 ⊊ (𝐵 𝐶))) → (¬ 𝐵𝐴𝐴 ∈ HAtoms))
 
Theorematcvati 30091 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (Contributed by NM, 28-Jun-2004.) (New usage is discouraged.)
𝐴C       ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0𝐴 ⊊ (𝐵 𝐶)) → 𝐴 ∈ HAtoms))
 
Theorematcvat2i 30092 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
𝐴C       ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((¬ 𝐵 = 𝐶𝐴 (𝐵 𝐶)) → 𝐴 ∈ HAtoms))
 
Theorematord 30093 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms ∧ 𝐴 𝐶 𝐵) → (𝐵𝐴𝐵 ⊆ (⊥‘𝐴)))
 
Theorematcvat2 30094 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 29-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((¬ 𝐵 = 𝐶𝐴 (𝐵 𝐶)) → 𝐴 ∈ HAtoms))
 
19.8.5  Irreducibility
 
Theoremchirredlem1 30095* Lemma for chirredi 30099. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
𝐴C       (((𝑝 ∈ HAtoms ∧ (𝑞C𝑞 ⊆ (⊥‘𝐴))) ∧ ((𝑟 ∈ HAtoms ∧ 𝑟𝐴) ∧ 𝑟 ⊆ (𝑝 𝑞))) → (𝑝 ∩ (⊥‘𝑟)) = 0)
 
Theoremchirredlem2 30096* Lemma for chirredi 30099. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C       ((((𝑝 ∈ HAtoms ∧ 𝑝𝐴) ∧ (𝑞C𝑞 ⊆ (⊥‘𝐴))) ∧ ((𝑟 ∈ HAtoms ∧ 𝑟𝐴) ∧ 𝑟 ⊆ (𝑝 𝑞))) → ((⊥‘𝑟) ∩ (𝑝 𝑞)) = 𝑞)
 
Theoremchirredlem3 30097* Lemma for chirredi 30099. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C    &   (𝑥C𝐴 𝐶 𝑥)       ((((𝑝 ∈ HAtoms ∧ 𝑝𝐴) ∧ (𝑞 ∈ HAtoms ∧ 𝑞 ⊆ (⊥‘𝐴))) ∧ (𝑟 ∈ HAtoms ∧ 𝑟 ⊆ (𝑝 𝑞))) → (𝑟𝐴𝑟 = 𝑝))
 
Theoremchirredlem4 30098* Lemma for chirredi 30099. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C    &   (𝑥C𝐴 𝐶 𝑥)       ((((𝑝 ∈ HAtoms ∧ 𝑝𝐴) ∧ (𝑞 ∈ HAtoms ∧ 𝑞 ⊆ (⊥‘𝐴))) ∧ (𝑟 ∈ HAtoms ∧ 𝑟 ⊆ (𝑝 𝑞))) → (𝑟 = 𝑝𝑟 = 𝑞))
 
Theoremchirredi 30099* The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C    &   (𝑥C𝐴 𝐶 𝑥)       (𝐴 = 0𝐴 = ℋ)
 
Theoremchirred 30100* The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
((𝐴C ∧ ∀𝑥C 𝐴 𝐶 𝑥) → (𝐴 = 0𝐴 = ℋ))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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