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Theorem List for Metamath Proof Explorer - 29201-29300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremishst 29201* Property of a complex Hilbert-space-valued state. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
(𝑆 ∈ CHStates ↔ (𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))

Theoremsticl 29202 [0, 1] closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝑆 ∈ States → (𝐴C → (𝑆𝐴) ∈ (0[,]1)))

Theoremstcl 29203 Real closure of the value of a state. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝑆 ∈ States → (𝐴C → (𝑆𝐴) ∈ ℝ))

Theoremhstcl 29204 Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴C ) → (𝑆𝐴) ∈ ℋ)

Theoremhst1a 29205 Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
(𝑆 ∈ CHStates → (norm‘(𝑆‘ ℋ)) = 1)

Theoremhstel2 29206 Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))

Theoremhstorth 29207 Orthogonality property of a Hilbert-space-valued state. This is a key feature distinguishing it from a real-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → ((𝑆𝐴) ·ih (𝑆𝐵)) = 0)

Theoremhstosum 29208 Orthogonal sum property of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))

Theoremhstoc 29209 Sum of a Hilbert-space-valued state of a lattice element and its orthocomplement. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴C ) → ((𝑆𝐴) + (𝑆‘(⊥‘𝐴))) = (𝑆‘ ℋ))

Theoremhstnmoc 29210 Sum of norms of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴C ) → (((norm‘(𝑆𝐴))↑2) + ((norm‘(𝑆‘(⊥‘𝐴)))↑2)) = 1)

Theoremstge0 29211 The value of a state is nonnegative. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝑆 ∈ States → (𝐴C → 0 ≤ (𝑆𝐴)))

Theoremstle1 29212 The value of a state is less than or equal to one. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝑆 ∈ States → (𝐴C → (𝑆𝐴) ≤ 1))

Theoremhstle1 29213 The norm of the value of a Hilbert-space-valued state is less than or equal to one. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴C ) → (norm‘(𝑆𝐴)) ≤ 1)

Theoremhst1h 29214 The norm of a Hilbert-space-valued state equals one iff the state value equals the state value of the lattice unit. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴C ) → ((norm‘(𝑆𝐴)) = 1 ↔ (𝑆𝐴) = (𝑆‘ ℋ)))

Theoremhst0h 29215 The norm of a Hilbert-space-valued state equals zero iff the state value equals zero. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴C ) → ((norm‘(𝑆𝐴)) = 0 ↔ (𝑆𝐴) = 0))

Theoremhstpyth 29216 Pythagorean property of a Hilbert-space-valued state for orthogonal vectors 𝐴 and 𝐵. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → ((norm‘(𝑆‘(𝐴 𝐵)))↑2) = (((norm‘(𝑆𝐴))↑2) + ((norm‘(𝑆𝐵))↑2)))

Theoremhstle 29217 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴𝐵)) → (norm‘(𝑆𝐴)) ≤ (norm‘(𝑆𝐵)))

Theoremhstles 29218 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴𝐵)) → ((norm‘(𝑆𝐴)) = 1 → (norm‘(𝑆𝐵)) = 1))

Theoremhstoh 29219 A Hilbert-space-valued state orthogonal to the state of the lattice unit is zero. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴C ∧ ((𝑆𝐴) ·ih (𝑆‘ ℋ)) = 0) → (𝑆𝐴) = 0)

Theoremhst0 29220 A Hilbert-space-valued state is zero at the zero subspace. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
(𝑆 ∈ CHStates → (𝑆‘0) = 0)

Theoremsthil 29221 The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
(𝑆 ∈ States → (𝑆‘ ℋ) = 1)

Theoremstj 29222 The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
(𝑆 ∈ States → (((𝐴C𝐵C ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))

Theoremsto1i 29223 The state of a subspace plus the state of its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝑆 ∈ States → ((𝑆𝐴) + (𝑆‘(⊥‘𝐴))) = 1)

Theoremsto2i 29224 The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝑆 ∈ States → (𝑆‘(⊥‘𝐴)) = (1 − (𝑆𝐴)))

Theoremstge1i 29225 If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴C       (𝑆 ∈ States → (1 ≤ (𝑆𝐴) ↔ (𝑆𝐴) = 1))

Theoremstle0i 29226 If a state is less than or equal to 0, it is 0. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴C       (𝑆 ∈ States → ((𝑆𝐴) ≤ 0 ↔ (𝑆𝐴) = 0))

Theoremstlei 29227 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → (𝐴𝐵 → (𝑆𝐴) ≤ (𝑆𝐵)))

Theoremstlesi 29228 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → (𝐴𝐵 → ((𝑆𝐴) = 1 → (𝑆𝐵) = 1)))

Theoremstji1i 29229 Join of components of Sasaki arrow ->1. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → (𝑆‘((⊥‘𝐴) ∨ (𝐴𝐵))) = ((𝑆‘(⊥‘𝐴)) + (𝑆‘(𝐴𝐵))))

Theoremstm1i 29230 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → ((𝑆‘(𝐴𝐵)) = 1 → (𝑆𝐴) = 1))

Theoremstm1ri 29231 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → ((𝑆‘(𝐴𝐵)) = 1 → (𝑆𝐵) = 1))

Theoremstm1addi 29232 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → ((𝑆‘(𝐴𝐵)) = 1 → ((𝑆𝐴) + (𝑆𝐵)) = 2))

Theoremstaddi 29233 If the sum of 2 states is 2, then each state is 1. (Contributed by NM, 12-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → (((𝑆𝐴) + (𝑆𝐵)) = 2 → (𝑆𝐴) = 1))

Theoremstm1add3i 29234 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝑆 ∈ States → ((𝑆‘((𝐴𝐵) ∩ 𝐶)) = 1 → (((𝑆𝐴) + (𝑆𝐵)) + (𝑆𝐶)) = 3))

Theoremstadd3i 29235 If the sum of 3 states is 3, then each state is 1. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝑆 ∈ States → ((((𝑆𝐴) + (𝑆𝐵)) + (𝑆𝐶)) = 3 → (𝑆𝐴) = 1))

Theoremst0 29236 The state of the zero subspace. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝑆 ∈ States → (𝑆‘0) = 0)

Theoremstrlem1 29237* Lemma for strong state theorem: if closed subspace 𝐴 is not contained in 𝐵, there is a unit vector 𝑢 in their difference. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴𝐵 → ∃𝑢 ∈ (𝐴𝐵)(norm𝑢) = 1)

Theoremstrlem2 29238* Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))       (𝐶C → (𝑆𝐶) = ((norm‘((proj𝐶)‘𝑢))↑2))

Theoremstrlem3a 29239* Lemma for strong state 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, 28-Oct-1999.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))       ((𝑢 ∈ ℋ ∧ (norm𝑢) = 1) → 𝑆 ∈ States)

Theoremstrlem3 29240* Lemma for strong state 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, 28-Oct-1999.) (New usage is discouraged.)
𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))    &   (𝜑 ↔ (𝑢 ∈ (𝐴𝐵) ∧ (norm𝑢) = 1))    &   𝐴C    &   𝐵C       (𝜑𝑆 ∈ States)

Theoremstrlem4 29241* 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 29242* 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 29243* 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 29244* 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 29245* 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 29246* 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 29247* 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 29248* 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 29249* 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 29250* 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 29251* 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 29252* 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 29253* 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 29254* 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 29255 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 29256* 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 29257* 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 29239 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 29258 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 29259 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 29260 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 29261* 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 29262* 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 29263* 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 29264* 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 29265* 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 29266* 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 29269 and cvbr2 29270 for membership relations. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ (𝑥𝑦 ∧ ¬ ∃𝑧C (𝑥𝑧𝑧𝑦)))}

Definitiondf-md 29267* 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 29281 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 29268* 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 29286 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 29269* 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 29270* 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 29271 Contraposition law for the covers relation. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (⊥‘𝐵) ⋖ (⊥‘𝐴)))

Theoremcvpss 29272 The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))

Theoremcvnbtwn 29273 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))

Theoremcvnbtwn2 29274 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵)))

Theoremcvnbtwn3 29275 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴)))

Theoremcvnbtwn4 29276 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 29277 The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵 → ¬ 𝐵 𝐴))

Theoremcvnref 29278 The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴C → ¬ 𝐴 𝐴)

Theoremcvntr 29279 The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → ((𝐴 𝐵𝐵 𝐶) → ¬ 𝐴 𝐶))

Theoremspansncv2 29280 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 29281* 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 29282 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 29283* Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 29281. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) ⊆ (𝑥 (𝐴𝐵)))))

Theoremmdbr3 29284* 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 29285* 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 29286* 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 29287 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 29288 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 29289 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 29290* Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 29286. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → (𝑥 ∩ (𝐴 𝐵)) ⊆ ((𝑥𝐴) ∨ 𝐵))))

Theoremdmdi2 29291 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 29292* 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 29293* 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 29294 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 29295* 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 29296* 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 29297 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 29298 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 29299 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 29300 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)

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