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Theorem List for Metamath Proof Explorer - 29901-30000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempjclem3 29901 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)) → ((proj𝐺) ∘ (proj‘(⊥‘𝐻))) = ((proj‘(⊥‘𝐻)) ∘ (proj𝐺)))
 
Theorempjclem4a 29902 Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ (𝐺𝐻) → (((proj𝐺) ∘ (proj𝐻))‘𝐴) = 𝐴)
 
Theorempjclem4 29903 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)) → ((proj𝐺) ∘ (proj𝐻)) = (proj‘(𝐺𝐻)))
 
Theorempjci 29904 Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 𝐶 𝐻 ↔ ((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)))
 
Theorempjcmul1i 29905 A necessary and sufficient condition for the product of two projectors to be a projector is that the projectors commute. Part 1 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)) ↔ ((proj𝐺) ∘ (proj𝐻)) ∈ ran proj)
 
Theorempjcmul2i 29906 The projection subspace of the difference between two projectors. Part 2 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)) ↔ ((proj𝐺) ∘ (proj𝐻)) = (proj‘(𝐺𝐻)))
 
Theorempjcohocli 29907 Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000.) (New usage is discouraged.)
𝐻C    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → (((proj𝐻) ∘ 𝑇)‘𝐴) ∈ 𝐻)
 
Theorempjadj2coi 29908 Adjoint of double composition of projections. Generalization of special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻))‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((((proj𝐻) ∘ (proj𝐺)) ∘ (proj𝐹))‘𝐵)))
 
Theorempj2cocli 29909 Closure of double composition of projections. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (𝐴 ∈ ℋ → ((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻))‘𝐴) ∈ 𝐹)
 
Theorempj3lem1 29910 Lemma for projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (𝐴 ∈ ((𝐹𝐺) ∩ 𝐻) → ((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻))‘𝐴) = 𝐴)
 
Theorempj3si 29911 Stronger projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐻) ∘ (proj𝐺)) ∘ (proj𝐹)) ∧ ran (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) ⊆ 𝐺) → (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (proj‘((𝐹𝐺) ∩ 𝐻)))
 
Theorempj3i 29912 Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐻) ∘ (proj𝐺)) ∘ (proj𝐹)) ∧ (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐺) ∘ (proj𝐹)) ∘ (proj𝐻))) → (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (proj‘((𝐹𝐺) ∩ 𝐻)))
 
Theorempj3cor1i 29913 Projection triplet corollary. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐻) ∘ (proj𝐺)) ∘ (proj𝐹)) ∧ (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐺) ∘ (proj𝐹)) ∘ (proj𝐻))) → (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐻) ∘ (proj𝐹)) ∘ (proj𝐺)))
 
Theorempjs14i 29914 Theorem S-14 of Watanabe, p. 486. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (norm‘(((proj𝐻) ∘ (proj𝐺))‘𝐴)) ≤ (norm‘((proj𝐺)‘𝐴)))
 
19.7  States on a Hilbert lattice and Godowski's equation
 
19.7.1  States on a Hilbert lattice
 
Definitiondf-st 29915* Define the set of states on a Hilbert lattice. Definition of [Kalmbach] p. 266. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
States = {𝑓 ∈ ((0[,]1) ↑m C ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))))}
 
Definitiondf-hst 29916* Define the set of complex Hilbert-space-valued states on a Hilbert lattice. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
CHStates = {𝑓 ∈ ( ℋ ↑m C ) ∣ ((norm‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))))}
 
Theoremisst 29917* Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝑆 ∈ States ↔ (𝑆: C ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
 
Theoremishst 29918* 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 29919 [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 29920 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 29921 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 29922 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 29923 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 29924 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 29925 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 29926 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 29927 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 29928 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 29929 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 29930 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 29931 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 29932 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 29933 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 29934 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 29935 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 29936 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 29937 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 29938 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 29939 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 29940 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 29941 The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝑆 ∈ States → (𝑆‘(⊥‘𝐴)) = (1 − (𝑆𝐴)))
 
Theoremstge1i 29942 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 29943 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 29944 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → (𝐴𝐵 → (𝑆𝐴) ≤ (𝑆𝐵)))
 
Theoremstlesi 29945 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → (𝐴𝐵 → ((𝑆𝐴) = 1 → (𝑆𝐵) = 1)))
 
Theoremstji1i 29946 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 29947 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 29948 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 29949 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 29950 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 29951 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 29952 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 29953 The state of the zero subspace. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝑆 ∈ States → (𝑆‘0) = 0)
 
Theoremstrlem1 29954* 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 29955* 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 29956* 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 29957* 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 29958* 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 29959* 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 29960* 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 29961* 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 29962* 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 29963* 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 29964* 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 29965* 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 29966* 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 29967* 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 29968* 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 29969* 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 29970* 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 29971* 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 29972 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 29973* 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 29974* 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 29956 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 29975 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 29976 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 29977 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 29978* 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 29979* 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 29980* 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 29981* 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 29982* 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 29983* 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 29986 and cvbr2 29987 for membership relations. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ (𝑥𝑦 ∧ ¬ ∃𝑧C (𝑥𝑧𝑧𝑦)))}
 
Definitiondf-md 29984* 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 29998 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 29985* 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 30003 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 29986* 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 29987* 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 29988 Contraposition law for the covers relation. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (⊥‘𝐵) ⋖ (⊥‘𝐴)))
 
Theoremcvpss 29989 The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
 
Theoremcvnbtwn 29990 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
 
Theoremcvnbtwn2 29991 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵)))
 
Theoremcvnbtwn3 29992 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴)))
 
Theoremcvnbtwn4 29993 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 29994 The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵 → ¬ 𝐵 𝐴))
 
Theoremcvnref 29995 The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴C → ¬ 𝐴 𝐴)
 
Theoremcvntr 29996 The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → ((𝐴 𝐵𝐵 𝐶) → ¬ 𝐴 𝐶))
 
Theoremspansncv2 29997 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 29998* 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 29999 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 30000* Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 29998. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) ⊆ (𝑥 (𝐴𝐵)))))
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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 44801-44834
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