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Theorem List for Metamath Proof Explorer - 29001-29100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremstrlem4 29001* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))    &   (𝜑 ↔ (𝑢 ∈ (𝐴𝐵) ∧ (norm𝑢) = 1))    &   𝐴C    &   𝐵C       (𝜑 → (𝑆𝐴) = 1)

Theoremstrlem5 29002* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))    &   (𝜑 ↔ (𝑢 ∈ (𝐴𝐵) ∧ (norm𝑢) = 1))    &   𝐴C    &   𝐵C       (𝜑 → (𝑆𝐵) < 1)

Theoremstrlem6 29003* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))    &   (𝜑 ↔ (𝑢 ∈ (𝐴𝐵) ∧ (norm𝑢) = 1))    &   𝐴C    &   𝐵C       (𝜑 → ¬ ((𝑆𝐴) = 1 → (𝑆𝐵) = 1))

Theoremstri 29004* Strong state theorem. The states on a Hilbert lattice define an ordering. Remark in [Mayet] p. 370. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (∀𝑓 ∈ States ((𝑓𝐴) = 1 → (𝑓𝐵) = 1) → 𝐴𝐵)

Theoremstrb 29005* Strong state theorem (bidirectional version). (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
𝐴C    &   𝐵C       (∀𝑓 ∈ States ((𝑓𝐴) = 1 → (𝑓𝐵) = 1) ↔ 𝐴𝐵)

Theoremhstrlem2 29006* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((proj𝑥)‘𝑢))       (𝐶C → (𝑆𝐶) = ((proj𝐶)‘𝑢))

Theoremhstrlem3a 29007* Lemma for strong set of CH states theorem: the function 𝑆, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((proj𝑥)‘𝑢))       ((𝑢 ∈ ℋ ∧ (norm𝑢) = 1) → 𝑆 ∈ CHStates)

Theoremhstrlem3 29008* Lemma for strong set of CH states theorem: the function 𝑆, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. This lemma restates the hypotheses in a more convenient form to work with. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((proj𝑥)‘𝑢))    &   (𝜑 ↔ (𝑢 ∈ (𝐴𝐵) ∧ (norm𝑢) = 1))    &   𝐴C    &   𝐵C       (𝜑𝑆 ∈ CHStates)

Theoremhstrlem4 29009* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((proj𝑥)‘𝑢))    &   (𝜑 ↔ (𝑢 ∈ (𝐴𝐵) ∧ (norm𝑢) = 1))    &   𝐴C    &   𝐵C       (𝜑 → (norm‘(𝑆𝐴)) = 1)

Theoremhstrlem5 29010* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((proj𝑥)‘𝑢))    &   (𝜑 ↔ (𝑢 ∈ (𝐴𝐵) ∧ (norm𝑢) = 1))    &   𝐴C    &   𝐵C       (𝜑 → (norm‘(𝑆𝐵)) < 1)

Theoremhstrlem6 29011* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((proj𝑥)‘𝑢))    &   (𝜑 ↔ (𝑢 ∈ (𝐴𝐵) ∧ (norm𝑢) = 1))    &   𝐴C    &   𝐵C       (𝜑 → ¬ ((norm‘(𝑆𝐴)) = 1 → (norm‘(𝑆𝐵)) = 1))

Theoremhstri 29012* Hilbert space admits a strong set of Hilbert-space-valued states (CH-states). Theorem in [Mayet3] p. 10. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (∀𝑓 ∈ CHStates ((norm‘(𝑓𝐴)) = 1 → (norm‘(𝑓𝐵)) = 1) → 𝐴𝐵)

Theoremhstrbi 29013* Strong CH-state theorem (bidirectional version). Theorem in [Mayet3] p. 10 and its converse. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (∀𝑓 ∈ CHStates ((norm‘(𝑓𝐴)) = 1 → (norm‘(𝑓𝐵)) = 1) ↔ 𝐴𝐵)

Theoremlargei 29014* A Hilbert lattice admits a largei set of states. Remark in [Mayet] p. 370. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
𝐴C       𝐴 = 0 ↔ ∃𝑓 ∈ States (𝑓𝐴) = 1)

Theoremjplem1 29015 Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
𝐴C       ((𝑢 ∈ ℋ ∧ (norm𝑢) = 1) → (𝑢𝐴 ↔ ((norm‘((proj𝐴)‘𝑢))↑2) = 1))

Theoremjplem2 29016* Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))    &   𝐴C       ((𝑢 ∈ ℋ ∧ (norm𝑢) = 1) → (𝑢𝐴 ↔ (𝑆𝐴) = 1))

Theoremjpi 29017* The function 𝑆, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a Jauch-Piron state. Remark in [Mayet] p. 370. (See strlem3a 28999 for the proof that 𝑆 is a state.) (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))    &   𝐴C    &   𝐵C       ((𝑢 ∈ ℋ ∧ (norm𝑢) = 1) → (((𝑆𝐴) = 1 ∧ (𝑆𝐵) = 1) ↔ (𝑆‘(𝐴𝐵)) = 1))

19.7.2  Godowski's equation

Theoremgolem1 29018 Lemma for Godowski's equation. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐹 = ((⊥‘𝐴) ∨ (𝐴𝐵))    &   𝐺 = ((⊥‘𝐵) ∨ (𝐵𝐶))    &   𝐻 = ((⊥‘𝐶) ∨ (𝐶𝐴))    &   𝐷 = ((⊥‘𝐵) ∨ (𝐵𝐴))    &   𝑅 = ((⊥‘𝐶) ∨ (𝐶𝐵))    &   𝑆 = ((⊥‘𝐴) ∨ (𝐴𝐶))       (𝑓 ∈ States → (((𝑓𝐹) + (𝑓𝐺)) + (𝑓𝐻)) = (((𝑓𝐷) + (𝑓𝑅)) + (𝑓𝑆)))

Theoremgolem2 29019 Lemma for Godowski's equation. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐹 = ((⊥‘𝐴) ∨ (𝐴𝐵))    &   𝐺 = ((⊥‘𝐵) ∨ (𝐵𝐶))    &   𝐻 = ((⊥‘𝐶) ∨ (𝐶𝐴))    &   𝐷 = ((⊥‘𝐵) ∨ (𝐵𝐴))    &   𝑅 = ((⊥‘𝐶) ∨ (𝐶𝐵))    &   𝑆 = ((⊥‘𝐴) ∨ (𝐴𝐶))       (𝑓 ∈ States → ((𝑓‘((𝐹𝐺) ∩ 𝐻)) = 1 → (𝑓𝐷) = 1))

Theoremgoeqi 29020 Godowski's equation, shown here as a variant equivalent to Equation SF of [Godowski] p. 730. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐹 = ((⊥‘𝐴) ∨ (𝐴𝐵))    &   𝐺 = ((⊥‘𝐵) ∨ (𝐵𝐶))    &   𝐻 = ((⊥‘𝐶) ∨ (𝐶𝐴))    &   𝐷 = ((⊥‘𝐵) ∨ (𝐵𝐴))       ((𝐹𝐺) ∩ 𝐻) ⊆ 𝐷

Theoremstcltr1i 29021* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦)))    &   𝐴C    &   𝐵C       (𝜑 → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵))

Theoremstcltr2i 29022* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦)))    &   𝐴C       (𝜑 → ((𝑆𝐴) = 1 → 𝐴 = ℋ))

Theoremstcltrlem1 29023* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦)))    &   𝐴C    &   𝐵C       (𝜑 → ((𝑆𝐵) = 1 → (𝑆‘((⊥‘𝐴) ∨ (𝐴𝐵))) = 1))

Theoremstcltrlem2 29024* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦)))    &   𝐴C    &   𝐵C       (𝜑𝐵 ⊆ ((⊥‘𝐴) ∨ (𝐴𝐵)))

Theoremstcltrthi 29025* Theorem for classically strong set of states. If there exists a "classically strong set of states" on lattice C (or actually any ortholattice, which would have an identical proof), then any two elements of the lattice commute, i.e., the lattice is distributive. (Proof due to Mladen Pavicic.) Theorem 3.3 of [MegPav2000] p. 2344. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝑠 ∈ States ∀𝑥C𝑦C (((𝑠𝑥) = 1 → (𝑠𝑦) = 1) → 𝑥𝑦)       𝐵 ⊆ ((⊥‘𝐴) ∨ (𝐴𝐵))

19.8  Cover relation, atoms, exchange axiom, and modular symmetry

19.8.1  Covers relation; modular pairs

Definitiondf-cv 29026* Define the covers relation (on the Hilbert lattice). Definition 3.2.18 of [PtakPulmannova] p. 68, whose notation we use. Ptak/Pulmannova's notation 𝐴 𝐵 is read "𝐵 covers 𝐴 " or "𝐴 is covered by 𝐵 " , and it means that 𝐵 is larger than 𝐴 and there is nothing in between. See cvbr 29029 and cvbr2 29030 for membership relations. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ (𝑥𝑦 ∧ ¬ ∃𝑧C (𝑥𝑧𝑧𝑦)))}

Definitiondf-md 29027* Define the modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M for "the ordered pair <x,y> is a modular pair." See mdbr 29041 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
𝑀 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ ∀𝑧C (𝑧𝑦 → ((𝑧 𝑥) ∩ 𝑦) = (𝑧 (𝑥𝑦))))}

Definitiondf-dmd 29028* Define the dual modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M* for "the ordered pair <x,y> is a dual modular pair." See dmdbr 29046 for membership relation. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝑀* = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ ∀𝑧C (𝑦𝑧 → ((𝑧𝑥) ∨ 𝑦) = (𝑧 ∩ (𝑥 𝑦))))}

Theoremcvbr 29029* Binary relation expressing 𝐵 covers 𝐴, which means that 𝐵 is larger than 𝐴 and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))

Theoremcvbr2 29030* Binary relation expressing 𝐵 covers 𝐴. Definition of covers in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵))))

Theoremcvcon3 29031 Contraposition law for the covers relation. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (⊥‘𝐵) ⋖ (⊥‘𝐴)))

Theoremcvpss 29032 The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))

Theoremcvnbtwn 29033 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))

Theoremcvnbtwn2 29034 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵)))

Theoremcvnbtwn3 29035 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴)))

Theoremcvnbtwn4 29036 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵))))

Theoremcvnsym 29037 The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵 → ¬ 𝐵 𝐴))

Theoremcvnref 29038 The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴C → ¬ 𝐴 𝐴)

Theoremcvntr 29039 The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → ((𝐴 𝐵𝐵 𝐶) → ¬ 𝐴 𝐶))

Theoremspansncv2 29040 Hilbert space has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ ℋ) → (¬ (span‘{𝐵}) ⊆ 𝐴𝐴 (𝐴 (span‘{𝐵}))))

Theoremmdbr 29041* Binary relation expressing 𝐴, 𝐵 is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))

Theoremmdi 29042 Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀 𝐵𝐶𝐵)) → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))

Theoremmdbr2 29043* Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 29041. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) ⊆ (𝑥 (𝐴𝐵)))))

Theoremmdbr3 29044* 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 29045* 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 29046* 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 29047 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 29048 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 29049 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 29050* Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 29046. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → (𝑥 ∩ (𝐴 𝐵)) ⊆ ((𝑥𝐴) ∨ 𝐵))))

Theoremdmdi2 29051 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 29052* 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 29053* 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 29054 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 29055* 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 29056* 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 29057 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 29058 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 29059 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 29060 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 29061 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 29062 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 29063 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 29064 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 29065 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 29066 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 29067 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 29068* 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 29069* 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 29070* 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 29071 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 29072 Lemma for mdslmd1i 29076. (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 29073 Lemma for mdslmd1i 29076. (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 29074* Lemma for mdslmd1i 29076. (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 29075* Lemma for mdslmd1i 29076. (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 29076 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 29077 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 29078 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 29079 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 29080 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 29081* 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 29082 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 29083 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 29084 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 29085 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 29086 and elat2 29087 for membership relations. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
HAtoms = {𝑥C ∣ 0 𝑥}

Theoremela 29086 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 29087* 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 29088 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 29089 An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms → 0 𝐴)

Theorematssch 29090 Atoms are a subset of the Hilbert lattice. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
HAtoms ⊆ C

Theorematelch 29091 An atom is a Hilbert lattice element. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms → 𝐴C )

Theorematne0 29092 An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴 ∈ HAtoms → 𝐴 ≠ 0)

Theorematss 29093 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 29094 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 29095 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 29096 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (⊥‘(⊥‘{𝐴})) ∈ HAtoms)

Theoremspansna 29097 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 29098 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 29099 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 29100* 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})))

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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 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