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Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempjscji 28201 The projection of orthogonal subspaces is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 ⊆ (⊥‘𝐻) → (proj‘(𝐺 𝐻)) = ((proj𝐺) +op (proj𝐻)))

Theorempjssumi 28202 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 ⊆ (⊥‘𝐻) → (proj‘(𝐺 + 𝐻)) = ((proj𝐺) +op (proj𝐻)))

Theorempjssposi 28203* Projector ordering can be expressed by the subset relationship between their projection subspaces. (i)<->(iii) of Theorem 29.2 of [Halmos] p. 48. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (∀𝑥 ∈ ℋ 0 ≤ ((((proj𝐻) −op (proj𝐺))‘𝑥) ·ih 𝑥) ↔ 𝐺𝐻)

Theorempjordi 28204* The definition of projector ordering in [Halmos] p. 42 is equivalent to the definition of projector ordering in [Beran] p. 110. (We will usually express projector ordering with the even simpler equivalent 𝐺𝐻; see pjssposi 28203). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (∀𝑥 ∈ ℋ 0 ≤ ((((proj𝐻) −op (proj𝐺))‘𝑥) ·ih 𝑥) ↔ ((proj𝐺) “ ℋ) ⊆ ((proj𝐻) “ ℋ))

Theorempjssdif2i 28205 The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 28203). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐻) −op (proj𝐺)) = (proj‘(𝐻 ∩ (⊥‘𝐺))))

Theorempjssdif1i 28206 A necessary and sufficient condition for the difference between two projectors to be a projector. Part 1 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 28203). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐻) −op (proj𝐺)) ∈ ran proj)

Theorempjimai 28207 The image of a projection. Lemma 5 in Daniel Lehmann, "A presentation of Quantum Logic based on an and then connective" http://www.arxiv.org/pdf/quant-ph/0701113 p. 20. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
𝐴S    &   𝐵C       ((proj𝐵) “ 𝐴) = ((𝐴 + (⊥‘𝐵)) ∩ 𝐵)

Theorempjidmcoi 28208 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
𝐻C       ((proj𝐻) ∘ (proj𝐻)) = (proj𝐻)

Theorempjoccoi 28209 Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝐻C       ((proj𝐻) ∘ (proj‘(⊥‘𝐻))) = 0hop

Theorempjtoi 28210 Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
𝐻C       ((proj𝐻) +op (proj‘(⊥‘𝐻))) = (proj‘ ℋ)

Theorempjoci 28211 Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐻C       ((proj‘ ℋ) −op (proj𝐻)) = (proj‘(⊥‘𝐻))

Theorempjidmco 28212 A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → ((proj𝐻) ∘ (proj𝐻)) = (proj𝐻))

Theoremdfpjop 28213 Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 27973. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝑇 ∈ ran proj ↔ (𝑇 ∈ HrmOp ∧ (𝑇𝑇) = 𝑇))

Theorempjhmopidm 28214 Two ways to express the set of all projection operators. (Contributed by NM, 24-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
ran proj = {𝑡 ∈ HrmOp ∣ (𝑡𝑡) = 𝑡}

Theoremelpjidm 28215 A projection operator is idempotent. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ ran proj → (𝑇𝑇) = 𝑇)

Theoremelpjhmop 28216 A projection operator is Hermitian. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ ran proj𝑇 ∈ HrmOp)

Theorem0leopj 28217 A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)
(𝑇 ∈ ran proj → 0hopop 𝑇)

Theorempjadj2 28218 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
(𝑇 ∈ ran proj → (adj𝑇) = 𝑇)

Theorempjadj3 28219 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremelpjch 28220 Reconstruction of the subspace of a projection operator. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ ran proj → (ran 𝑇C𝑇 = (proj‘ran 𝑇)))

Theoremelpjrn 28221* Reconstruction of the subspace of a projection operator. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝑇 ∈ ran proj → ran 𝑇 = {𝑥 ∈ ℋ ∣ (𝑇𝑥) = 𝑥})

Theorempjinvari 28222 A closed subspace 𝐻 with projection 𝑇 is invariant under an operator 𝑆 iff 𝑆𝑇 = 𝑇𝑆𝑇. Theorem 27.1 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝐻C    &   𝑇 = (proj𝐻)       ((𝑆𝑇): ℋ⟶𝐻 ↔ (𝑆𝑇) = (𝑇 ∘ (𝑆𝑇)))

Theorempjin1i 28223 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (proj‘(𝐺𝐻)) = ((proj𝐺) ∘ (proj‘(𝐺𝐻)))

Theorempjin2i 28224 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (((proj𝐺) = ((proj𝐺) ∘ (proj𝐻)) ∧ (proj𝐻) = ((proj𝐻) ∘ (proj𝐺))) ↔ (proj𝐺) = (proj𝐻))

Theorempjin3i 28225 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((proj𝐹) = ((proj𝐹) ∘ (proj𝐺)) ∧ (proj𝐹) = ((proj𝐹) ∘ (proj𝐻))) ↔ (proj𝐹) = ((proj𝐹) ∘ (proj‘(𝐺𝐻))))

Theorempjclem1 28226 Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 𝐶 𝐻 → ((proj𝐺) ∘ (proj𝐻)) = (proj‘(𝐺𝐻)))

Theorempjclem2 28227 Lemma for projection commutation theorem. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 𝐶 𝐻 → ((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)))

Theorempjclem3 28228 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 28229 Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ (𝐺𝐻) → (((proj𝐺) ∘ (proj𝐻))‘𝐴) = 𝐴)

Theorempjclem4 28230 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 28231 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 28232 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 28233 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 28234 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 28235 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 28236 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 28237 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 28238 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 28239 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 28240 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 28241 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 28242* 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 28243* 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 28244* 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 28245* 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 28246 [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 28247 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 28248 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 28249 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 28250 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 28251 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 28252 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 28253 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 28254 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 28255 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 28256 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 28257 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 28258 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 28259 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 28260 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 28261 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 28262 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 28263 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 28264 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 28265 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 28266 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 28267 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 28268 The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝑆 ∈ States → (𝑆‘(⊥‘𝐴)) = (1 − (𝑆𝐴)))

Theoremstge1i 28269 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 28270 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 28271 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → (𝐴𝐵 → (𝑆𝐴) ≤ (𝑆𝐵)))

Theoremstlesi 28272 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝑆 ∈ States → (𝐴𝐵 → ((𝑆𝐴) = 1 → (𝑆𝐵) = 1)))

Theoremstji1i 28273 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 28274 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 28275 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 28276 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 28277 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 28278 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 28279 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 28280 The state of the zero subspace. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝑆 ∈ States → (𝑆‘0) = 0)

Theoremstrlem1 28281* 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 28282* 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 28283* 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 28284* 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 28285* 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 28286* 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 28287* 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 28288* 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 28289* 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 28290* 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 28291* 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 28292* 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 28293* 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 28294* 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 28295* 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 28296* 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 28297* 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 28298* 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 28299 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 28300* 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))

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