HomeHome Metamath Proof Explorer
Theorem List (p. 293 of 429)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27903)
  Hilbert Space Explorer  Hilbert Space Explorer
(27904-29428)
  Users' Mathboxes  Users' Mathboxes
(29429-42879)
 

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)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 42801-42879
  Copyright terms: Public domain < Previous  Next >