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Theorem List for Metamath Proof Explorer - 30901-31000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempjordi 30901* 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 𝐺 βŠ† 𝐻; see pjssposi 30900). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (βˆ€π‘₯ ∈ β„‹ 0 ≀ ((((projβ„Žβ€˜π») βˆ’op (projβ„Žβ€˜πΊ))β€˜π‘₯) Β·ih π‘₯) ↔ ((projβ„Žβ€˜πΊ) β€œ β„‹) βŠ† ((projβ„Žβ€˜π») β€œ β„‹))
 
Theorempjssdif2i 30902 The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 30900). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (𝐺 βŠ† 𝐻 ↔ ((projβ„Žβ€˜π») βˆ’op (projβ„Žβ€˜πΊ)) = (projβ„Žβ€˜(𝐻 ∩ (βŠ₯β€˜πΊ))))
 
Theorempjssdif1i 30903 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 30900). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (𝐺 βŠ† 𝐻 ↔ ((projβ„Žβ€˜π») βˆ’op (projβ„Žβ€˜πΊ)) ∈ ran projβ„Ž)
 
Theorempjimai 30904 The image of a projection. Lemma 5 in Daniel Lehmann, "A presentation of Quantum Logic based on an and then connective", https://doi.org/10.48550/arXiv.quant-ph/0701113. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
𝐴 ∈ Sβ„‹    &   π΅ ∈ Cβ„‹    β‡’   ((projβ„Žβ€˜π΅) β€œ 𝐴) = ((𝐴 +β„‹ (βŠ₯β€˜π΅)) ∩ 𝐡)
 
Theorempjidmcoi 30905 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
𝐻 ∈ Cβ„‹    β‡’   ((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜π»)) = (projβ„Žβ€˜π»)
 
Theorempjoccoi 30906 Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝐻 ∈ Cβ„‹    β‡’   ((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜(βŠ₯β€˜π»))) = 0hop
 
Theorempjtoi 30907 Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
𝐻 ∈ Cβ„‹    β‡’   ((projβ„Žβ€˜π») +op (projβ„Žβ€˜(βŠ₯β€˜π»))) = (projβ„Žβ€˜ β„‹)
 
Theorempjoci 30908 Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐻 ∈ Cβ„‹    β‡’   ((projβ„Žβ€˜ β„‹) βˆ’op (projβ„Žβ€˜π»)) = (projβ„Žβ€˜(βŠ₯β€˜π»))
 
Theorempjidmco 30909 A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻 ∈ Cβ„‹ β†’ ((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜π»)) = (projβ„Žβ€˜π»))
 
Theoremdfpjop 30910 Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 30670. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝑇 ∈ ran projβ„Ž ↔ (𝑇 ∈ HrmOp ∧ (𝑇 ∘ 𝑇) = 𝑇))
 
Theorempjhmopidm 30911 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 projβ„Ž = {𝑑 ∈ HrmOp ∣ (𝑑 ∘ 𝑑) = 𝑑}
 
Theoremelpjidm 30912 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 projβ„Ž β†’ (𝑇 ∘ 𝑇) = 𝑇)
 
Theoremelpjhmop 30913 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 projβ„Ž β†’ 𝑇 ∈ HrmOp)
 
Theorem0leopj 30914 A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)
(𝑇 ∈ ran projβ„Ž β†’ 0hop ≀op 𝑇)
 
Theorempjadj2 30915 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
(𝑇 ∈ ran projβ„Ž β†’ (adjβ„Žβ€˜π‘‡) = 𝑇)
 
Theorempjadj3 30916 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
(𝐻 ∈ Cβ„‹ β†’ (adjβ„Žβ€˜(projβ„Žβ€˜π»)) = (projβ„Žβ€˜π»))
 
Theoremelpjch 30917 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 projβ„Ž β†’ (ran 𝑇 ∈ Cβ„‹ ∧ 𝑇 = (projβ„Žβ€˜ran 𝑇)))
 
Theoremelpjrn 30918* 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 projβ„Ž β†’ ran 𝑇 = {π‘₯ ∈ β„‹ ∣ (π‘‡β€˜π‘₯) = π‘₯})
 
Theorempjinvari 30919 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: β„‹βŸΆ β„‹    &   π» ∈ Cβ„‹    &   π‘‡ = (projβ„Žβ€˜π»)    β‡’   ((𝑆 ∘ 𝑇): β„‹βŸΆπ» ↔ (𝑆 ∘ 𝑇) = (𝑇 ∘ (𝑆 ∘ 𝑇)))
 
Theorempjin1i 30920 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (projβ„Žβ€˜(𝐺 ∩ 𝐻)) = ((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜(𝐺 ∩ 𝐻)))
 
Theorempjin2i 30921 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (((projβ„Žβ€˜πΊ) = ((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) ∧ (projβ„Žβ€˜π») = ((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ))) ↔ (projβ„Žβ€˜πΊ) = (projβ„Žβ€˜π»))
 
Theorempjin3i 30922 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐹 ∈ Cβ„‹    &   πΊ ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (((projβ„Žβ€˜πΉ) = ((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∧ (projβ„Žβ€˜πΉ) = ((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜π»))) ↔ (projβ„Žβ€˜πΉ) = ((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜(𝐺 ∩ 𝐻))))
 
Theorempjclem1 30923 Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (𝐺 𝐢ℋ 𝐻 β†’ ((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) = (projβ„Žβ€˜(𝐺 ∩ 𝐻)))
 
Theorempjclem2 30924 Lemma for projection commutation theorem. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (𝐺 𝐢ℋ 𝐻 β†’ ((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) = ((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ)))
 
Theorempjclem3 30925 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) = ((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ)) β†’ ((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜(βŠ₯β€˜π»))) = ((projβ„Žβ€˜(βŠ₯β€˜π»)) ∘ (projβ„Žβ€˜πΊ)))
 
Theorempjclem4a 30926 Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (𝐴 ∈ (𝐺 ∩ 𝐻) β†’ (((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»))β€˜π΄) = 𝐴)
 
Theorempjclem4 30927 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) = ((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ)) β†’ ((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) = (projβ„Žβ€˜(𝐺 ∩ 𝐻)))
 
Theorempjci 30928 Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (𝐺 𝐢ℋ 𝐻 ↔ ((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) = ((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ)))
 
Theorempjcmul1i 30929 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.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) = ((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ)) ↔ ((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) ∈ ran projβ„Ž)
 
Theorempjcmul2i 30930 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.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) = ((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ)) ↔ ((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜π»)) = (projβ„Žβ€˜(𝐺 ∩ 𝐻)))
 
Theorempjcohocli 30931 Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000.) (New usage is discouraged.)
𝐻 ∈ Cβ„‹    &   π‘‡: β„‹βŸΆ β„‹    β‡’   (𝐴 ∈ β„‹ β†’ (((projβ„Žβ€˜π») ∘ 𝑇)β€˜π΄) ∈ 𝐻)
 
Theorempjadj2coi 30932 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.)
𝐹 ∈ Cβ„‹    &   πΊ ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (((((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»))β€˜π΄) Β·ih 𝐡) = (𝐴 Β·ih ((((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜πΉ))β€˜π΅)))
 
Theorempj2cocli 30933 Closure of double composition of projections. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹 ∈ Cβ„‹    &   πΊ ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (𝐴 ∈ β„‹ β†’ ((((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»))β€˜π΄) ∈ 𝐹)
 
Theorempj3lem1 30934 Lemma for projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹 ∈ Cβ„‹    &   πΊ ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (𝐴 ∈ ((𝐹 ∩ 𝐺) ∩ 𝐻) β†’ ((((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»))β€˜π΄) = 𝐴)
 
Theorempj3si 30935 Stronger projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹 ∈ Cβ„‹    &   πΊ ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (((((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»)) = (((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜πΉ)) ∧ ran (((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»)) βŠ† 𝐺) β†’ (((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»)) = (projβ„Žβ€˜((𝐹 ∩ 𝐺) ∩ 𝐻)))
 
Theorempj3i 30936 Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹 ∈ Cβ„‹    &   πΊ ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (((((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»)) = (((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜πΉ)) ∧ (((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»)) = (((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜πΉ)) ∘ (projβ„Žβ€˜π»))) β†’ (((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»)) = (projβ„Žβ€˜((𝐹 ∩ 𝐺) ∩ 𝐻)))
 
Theorempj3cor1i 30937 Projection triplet corollary. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹 ∈ Cβ„‹    &   πΊ ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (((((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»)) = (((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜πΉ)) ∧ (((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»)) = (((projβ„Žβ€˜πΊ) ∘ (projβ„Žβ€˜πΉ)) ∘ (projβ„Žβ€˜π»))) β†’ (((projβ„Žβ€˜πΉ) ∘ (projβ„Žβ€˜πΊ)) ∘ (projβ„Žβ€˜π»)) = (((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΉ)) ∘ (projβ„Žβ€˜πΊ)))
 
Theorempjs14i 30938 Theorem S-14 of Watanabe, p. 486. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺 ∈ Cβ„‹    &   π» ∈ Cβ„‹    β‡’   (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜(((projβ„Žβ€˜π») ∘ (projβ„Žβ€˜πΊ))β€˜π΄)) ≀ (normβ„Žβ€˜((projβ„Žβ€˜πΊ)β€˜π΄)))
 
20.7  States on a Hilbert lattice and Godowski's equation
 
20.7.1  States on a Hilbert lattice
 
Definitiondf-st 30939* 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 Cβ„‹ ) ∣ ((π‘“β€˜ β„‹) = 1 ∧ βˆ€π‘₯ ∈ Cβ„‹ βˆ€π‘¦ ∈ Cβ„‹ (π‘₯ βŠ† (βŠ₯β€˜π‘¦) β†’ (π‘“β€˜(π‘₯ βˆ¨β„‹ 𝑦)) = ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))}
 
Definitiondf-hst 30940* 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 ∈ ( β„‹ ↑m Cβ„‹ ) ∣ ((normβ„Žβ€˜(π‘“β€˜ β„‹)) = 1 ∧ βˆ€π‘₯ ∈ Cβ„‹ βˆ€π‘¦ ∈ Cβ„‹ (π‘₯ βŠ† (βŠ₯β€˜π‘¦) β†’ (((π‘“β€˜π‘₯) Β·ih (π‘“β€˜π‘¦)) = 0 ∧ (π‘“β€˜(π‘₯ βˆ¨β„‹ 𝑦)) = ((π‘“β€˜π‘₯) +β„Ž (π‘“β€˜π‘¦)))))}
 
Theoremisst 30941* Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝑆 ∈ States ↔ (𝑆: Cβ„‹ ⟢(0[,]1) ∧ (π‘†β€˜ β„‹) = 1 ∧ βˆ€π‘₯ ∈ Cβ„‹ βˆ€π‘¦ ∈ Cβ„‹ (π‘₯ βŠ† (βŠ₯β€˜π‘¦) β†’ (π‘†β€˜(π‘₯ βˆ¨β„‹ 𝑦)) = ((π‘†β€˜π‘₯) + (π‘†β€˜π‘¦)))))
 
Theoremishst 30942* 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: Cβ„‹ ⟢ β„‹ ∧ (normβ„Žβ€˜(π‘†β€˜ β„‹)) = 1 ∧ βˆ€π‘₯ ∈ Cβ„‹ βˆ€π‘¦ ∈ Cβ„‹ (π‘₯ βŠ† (βŠ₯β€˜π‘¦) β†’ (((π‘†β€˜π‘₯) Β·ih (π‘†β€˜π‘¦)) = 0 ∧ (π‘†β€˜(π‘₯ βˆ¨β„‹ 𝑦)) = ((π‘†β€˜π‘₯) +β„Ž (π‘†β€˜π‘¦))))))
 
Theoremsticl 30943 [0, 1] closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝑆 ∈ States β†’ (𝐴 ∈ Cβ„‹ β†’ (π‘†β€˜π΄) ∈ (0[,]1)))
 
Theoremstcl 30944 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 β†’ (𝐴 ∈ Cβ„‹ β†’ (π‘†β€˜π΄) ∈ ℝ))
 
Theoremhstcl 30945 Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cβ„‹ ) β†’ (π‘†β€˜π΄) ∈ β„‹)
 
Theoremhst1a 30946 Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
(𝑆 ∈ CHStates β†’ (normβ„Žβ€˜(π‘†β€˜ β„‹)) = 1)
 
Theoremhstel2 30947 Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cβ„‹ ) ∧ (𝐡 ∈ Cβ„‹ ∧ 𝐴 βŠ† (βŠ₯β€˜π΅))) β†’ (((π‘†β€˜π΄) Β·ih (π‘†β€˜π΅)) = 0 ∧ (π‘†β€˜(𝐴 βˆ¨β„‹ 𝐡)) = ((π‘†β€˜π΄) +β„Ž (π‘†β€˜π΅))))
 
Theoremhstorth 30948 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 ∧ 𝐴 ∈ Cβ„‹ ) ∧ (𝐡 ∈ Cβ„‹ ∧ 𝐴 βŠ† (βŠ₯β€˜π΅))) β†’ ((π‘†β€˜π΄) Β·ih (π‘†β€˜π΅)) = 0)
 
Theoremhstosum 30949 Orthogonal sum property of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cβ„‹ ) ∧ (𝐡 ∈ Cβ„‹ ∧ 𝐴 βŠ† (βŠ₯β€˜π΅))) β†’ (π‘†β€˜(𝐴 βˆ¨β„‹ 𝐡)) = ((π‘†β€˜π΄) +β„Ž (π‘†β€˜π΅)))
 
Theoremhstoc 30950 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 ∧ 𝐴 ∈ Cβ„‹ ) β†’ ((π‘†β€˜π΄) +β„Ž (π‘†β€˜(βŠ₯β€˜π΄))) = (π‘†β€˜ β„‹))
 
Theoremhstnmoc 30951 Sum of norms of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cβ„‹ ) β†’ (((normβ„Žβ€˜(π‘†β€˜π΄))↑2) + ((normβ„Žβ€˜(π‘†β€˜(βŠ₯β€˜π΄)))↑2)) = 1)
 
Theoremstge0 30952 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 β†’ (𝐴 ∈ Cβ„‹ β†’ 0 ≀ (π‘†β€˜π΄)))
 
Theoremstle1 30953 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 β†’ (𝐴 ∈ Cβ„‹ β†’ (π‘†β€˜π΄) ≀ 1))
 
Theoremhstle1 30954 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 ∧ 𝐴 ∈ Cβ„‹ ) β†’ (normβ„Žβ€˜(π‘†β€˜π΄)) ≀ 1)
 
Theoremhst1h 30955 The norm of a Hilbert-space-valued state equals one iff the state value equals the state value of the lattice one. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cβ„‹ ) β†’ ((normβ„Žβ€˜(π‘†β€˜π΄)) = 1 ↔ (π‘†β€˜π΄) = (π‘†β€˜ β„‹)))
 
Theoremhst0h 30956 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 ∧ 𝐴 ∈ Cβ„‹ ) β†’ ((normβ„Žβ€˜(π‘†β€˜π΄)) = 0 ↔ (π‘†β€˜π΄) = 0β„Ž))
 
Theoremhstpyth 30957 Pythagorean property of a Hilbert-space-valued state for orthogonal vectors 𝐴 and 𝐡. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cβ„‹ ) ∧ (𝐡 ∈ Cβ„‹ ∧ 𝐴 βŠ† (βŠ₯β€˜π΅))) β†’ ((normβ„Žβ€˜(π‘†β€˜(𝐴 βˆ¨β„‹ 𝐡)))↑2) = (((normβ„Žβ€˜(π‘†β€˜π΄))↑2) + ((normβ„Žβ€˜(π‘†β€˜π΅))↑2)))
 
Theoremhstle 30958 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cβ„‹ ) ∧ (𝐡 ∈ Cβ„‹ ∧ 𝐴 βŠ† 𝐡)) β†’ (normβ„Žβ€˜(π‘†β€˜π΄)) ≀ (normβ„Žβ€˜(π‘†β€˜π΅)))
 
Theoremhstles 30959 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
(((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cβ„‹ ) ∧ (𝐡 ∈ Cβ„‹ ∧ 𝐴 βŠ† 𝐡)) β†’ ((normβ„Žβ€˜(π‘†β€˜π΄)) = 1 β†’ (normβ„Žβ€˜(π‘†β€˜π΅)) = 1))
 
Theoremhstoh 30960 A Hilbert-space-valued state orthogonal to the state of the lattice one is zero. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cβ„‹ ∧ ((π‘†β€˜π΄) Β·ih (π‘†β€˜ β„‹)) = 0) β†’ (π‘†β€˜π΄) = 0β„Ž)
 
Theoremhst0 30961 A Hilbert-space-valued state is zero at the zero subspace. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
(𝑆 ∈ CHStates β†’ (π‘†β€˜0β„‹) = 0β„Ž)
 
Theoremsthil 30962 The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
(𝑆 ∈ States β†’ (π‘†β€˜ β„‹) = 1)
 
Theoremstj 30963 The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
(𝑆 ∈ States β†’ (((𝐴 ∈ Cβ„‹ ∧ 𝐡 ∈ Cβ„‹ ) ∧ 𝐴 βŠ† (βŠ₯β€˜π΅)) β†’ (π‘†β€˜(𝐴 βˆ¨β„‹ 𝐡)) = ((π‘†β€˜π΄) + (π‘†β€˜π΅))))
 
Theoremsto1i 30964 The state of a subspace plus the state of its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ ((π‘†β€˜π΄) + (π‘†β€˜(βŠ₯β€˜π΄))) = 1)
 
Theoremsto2i 30965 The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ (π‘†β€˜(βŠ₯β€˜π΄)) = (1 βˆ’ (π‘†β€˜π΄)))
 
Theoremstge1i 30966 If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ (1 ≀ (π‘†β€˜π΄) ↔ (π‘†β€˜π΄) = 1))
 
Theoremstle0i 30967 If a state is less than or equal to 0, it is 0. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ ((π‘†β€˜π΄) ≀ 0 ↔ (π‘†β€˜π΄) = 0))
 
Theoremstlei 30968 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ (𝐴 βŠ† 𝐡 β†’ (π‘†β€˜π΄) ≀ (π‘†β€˜π΅)))
 
Theoremstlesi 30969 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ (𝐴 βŠ† 𝐡 β†’ ((π‘†β€˜π΄) = 1 β†’ (π‘†β€˜π΅) = 1)))
 
Theoremstji1i 30970 Join of components of Sasaki arrow ->1. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ (π‘†β€˜((βŠ₯β€˜π΄) βˆ¨β„‹ (𝐴 ∩ 𝐡))) = ((π‘†β€˜(βŠ₯β€˜π΄)) + (π‘†β€˜(𝐴 ∩ 𝐡))))
 
Theoremstm1i 30971 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ ((π‘†β€˜(𝐴 ∩ 𝐡)) = 1 β†’ (π‘†β€˜π΄) = 1))
 
Theoremstm1ri 30972 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ ((π‘†β€˜(𝐴 ∩ 𝐡)) = 1 β†’ (π‘†β€˜π΅) = 1))
 
Theoremstm1addi 30973 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ ((π‘†β€˜(𝐴 ∩ 𝐡)) = 1 β†’ ((π‘†β€˜π΄) + (π‘†β€˜π΅)) = 2))
 
Theoremstaddi 30974 If the sum of 2 states is 2, then each state is 1. (Contributed by NM, 12-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ (((π‘†β€˜π΄) + (π‘†β€˜π΅)) = 2 β†’ (π‘†β€˜π΄) = 1))
 
Theoremstm1add3i 30975 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    &   πΆ ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ ((π‘†β€˜((𝐴 ∩ 𝐡) ∩ 𝐢)) = 1 β†’ (((π‘†β€˜π΄) + (π‘†β€˜π΅)) + (π‘†β€˜πΆ)) = 3))
 
Theoremstadd3i 30976 If the sum of 3 states is 3, then each state is 1. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    &   πΆ ∈ Cβ„‹    β‡’   (𝑆 ∈ States β†’ ((((π‘†β€˜π΄) + (π‘†β€˜π΅)) + (π‘†β€˜πΆ)) = 3 β†’ (π‘†β€˜π΄) = 1))
 
Theoremst0 30977 The state of the zero subspace. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝑆 ∈ States β†’ (π‘†β€˜0β„‹) = 0)
 
Theoremstrlem1 30978* Lemma for strong state theorem: if closed subspace 𝐴 is not contained in 𝐡, there is a unit vector 𝑒 in their difference. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (Β¬ 𝐴 βŠ† 𝐡 β†’ βˆƒπ‘’ ∈ (𝐴 βˆ– 𝐡)(normβ„Žβ€˜π‘’) = 1)
 
Theoremstrlem2 30979* Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
𝑆 = (π‘₯ ∈ Cβ„‹ ↦ ((normβ„Žβ€˜((projβ„Žβ€˜π‘₯)β€˜π‘’))↑2))    β‡’   (𝐢 ∈ Cβ„‹ β†’ (π‘†β€˜πΆ) = ((normβ„Žβ€˜((projβ„Žβ€˜πΆ)β€˜π‘’))↑2))
 
Theoremstrlem3a 30980* 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 = (π‘₯ ∈ Cβ„‹ ↦ ((normβ„Žβ€˜((projβ„Žβ€˜π‘₯)β€˜π‘’))↑2))    β‡’   ((𝑒 ∈ β„‹ ∧ (normβ„Žβ€˜π‘’) = 1) β†’ 𝑆 ∈ States)
 
Theoremstrlem3 30981* 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 = (π‘₯ ∈ Cβ„‹ ↦ ((normβ„Žβ€˜((projβ„Žβ€˜π‘₯)β€˜π‘’))↑2))    &   (πœ‘ ↔ (𝑒 ∈ (𝐴 βˆ– 𝐡) ∧ (normβ„Žβ€˜π‘’) = 1))    &   π΄ ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (πœ‘ β†’ 𝑆 ∈ States)
 
Theoremstrlem4 30982* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
𝑆 = (π‘₯ ∈ Cβ„‹ ↦ ((normβ„Žβ€˜((projβ„Žβ€˜π‘₯)β€˜π‘’))↑2))    &   (πœ‘ ↔ (𝑒 ∈ (𝐴 βˆ– 𝐡) ∧ (normβ„Žβ€˜π‘’) = 1))    &   π΄ ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (πœ‘ β†’ (π‘†β€˜π΄) = 1)
 
Theoremstrlem5 30983* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
𝑆 = (π‘₯ ∈ Cβ„‹ ↦ ((normβ„Žβ€˜((projβ„Žβ€˜π‘₯)β€˜π‘’))↑2))    &   (πœ‘ ↔ (𝑒 ∈ (𝐴 βˆ– 𝐡) ∧ (normβ„Žβ€˜π‘’) = 1))    &   π΄ ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (πœ‘ β†’ (π‘†β€˜π΅) < 1)
 
Theoremstrlem6 30984* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
𝑆 = (π‘₯ ∈ Cβ„‹ ↦ ((normβ„Žβ€˜((projβ„Žβ€˜π‘₯)β€˜π‘’))↑2))    &   (πœ‘ ↔ (𝑒 ∈ (𝐴 βˆ– 𝐡) ∧ (normβ„Žβ€˜π‘’) = 1))    &   π΄ ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (πœ‘ β†’ Β¬ ((π‘†β€˜π΄) = 1 β†’ (π‘†β€˜π΅) = 1))
 
Theoremstri 30985* 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.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (βˆ€π‘“ ∈ States ((π‘“β€˜π΄) = 1 β†’ (π‘“β€˜π΅) = 1) β†’ 𝐴 βŠ† 𝐡)
 
Theoremstrb 30986* Strong state theorem (bidirectional version). (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (βˆ€π‘“ ∈ States ((π‘“β€˜π΄) = 1 β†’ (π‘“β€˜π΅) = 1) ↔ 𝐴 βŠ† 𝐡)
 
Theoremhstrlem2 30987* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝑆 = (π‘₯ ∈ Cβ„‹ ↦ ((projβ„Žβ€˜π‘₯)β€˜π‘’))    β‡’   (𝐢 ∈ Cβ„‹ β†’ (π‘†β€˜πΆ) = ((projβ„Žβ€˜πΆ)β€˜π‘’))
 
Theoremhstrlem3a 30988* 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 = (π‘₯ ∈ Cβ„‹ ↦ ((projβ„Žβ€˜π‘₯)β€˜π‘’))    β‡’   ((𝑒 ∈ β„‹ ∧ (normβ„Žβ€˜π‘’) = 1) β†’ 𝑆 ∈ CHStates)
 
Theoremhstrlem3 30989* 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 = (π‘₯ ∈ Cβ„‹ ↦ ((projβ„Žβ€˜π‘₯)β€˜π‘’))    &   (πœ‘ ↔ (𝑒 ∈ (𝐴 βˆ– 𝐡) ∧ (normβ„Žβ€˜π‘’) = 1))    &   π΄ ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (πœ‘ β†’ 𝑆 ∈ CHStates)
 
Theoremhstrlem4 30990* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝑆 = (π‘₯ ∈ Cβ„‹ ↦ ((projβ„Žβ€˜π‘₯)β€˜π‘’))    &   (πœ‘ ↔ (𝑒 ∈ (𝐴 βˆ– 𝐡) ∧ (normβ„Žβ€˜π‘’) = 1))    &   π΄ ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (πœ‘ β†’ (normβ„Žβ€˜(π‘†β€˜π΄)) = 1)
 
Theoremhstrlem5 30991* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝑆 = (π‘₯ ∈ Cβ„‹ ↦ ((projβ„Žβ€˜π‘₯)β€˜π‘’))    &   (πœ‘ ↔ (𝑒 ∈ (𝐴 βˆ– 𝐡) ∧ (normβ„Žβ€˜π‘’) = 1))    &   π΄ ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (πœ‘ β†’ (normβ„Žβ€˜(π‘†β€˜π΅)) < 1)
 
Theoremhstrlem6 30992* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝑆 = (π‘₯ ∈ Cβ„‹ ↦ ((projβ„Žβ€˜π‘₯)β€˜π‘’))    &   (πœ‘ ↔ (𝑒 ∈ (𝐴 βˆ– 𝐡) ∧ (normβ„Žβ€˜π‘’) = 1))    &   π΄ ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (πœ‘ β†’ Β¬ ((normβ„Žβ€˜(π‘†β€˜π΄)) = 1 β†’ (normβ„Žβ€˜(π‘†β€˜π΅)) = 1))
 
Theoremhstri 30993* 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.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (βˆ€π‘“ ∈ CHStates ((normβ„Žβ€˜(π‘“β€˜π΄)) = 1 β†’ (normβ„Žβ€˜(π‘“β€˜π΅)) = 1) β†’ 𝐴 βŠ† 𝐡)
 
Theoremhstrbi 30994* Strong CH-state theorem (bidirectional version). Theorem in [Mayet3] p. 10 and its converse. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   (βˆ€π‘“ ∈ CHStates ((normβ„Žβ€˜(π‘“β€˜π΄)) = 1 β†’ (normβ„Žβ€˜(π‘“β€˜π΅)) = 1) ↔ 𝐴 βŠ† 𝐡)
 
Theoremlargei 30995* A Hilbert lattice admits a largei set of states. Remark in [Mayet] p. 370. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    β‡’   (Β¬ 𝐴 = 0β„‹ ↔ βˆƒπ‘“ ∈ States (π‘“β€˜π΄) = 1)
 
Theoremjplem1 30996 Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    β‡’   ((𝑒 ∈ β„‹ ∧ (normβ„Žβ€˜π‘’) = 1) β†’ (𝑒 ∈ 𝐴 ↔ ((normβ„Žβ€˜((projβ„Žβ€˜π΄)β€˜π‘’))↑2) = 1))
 
Theoremjplem2 30997* Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
𝑆 = (π‘₯ ∈ Cβ„‹ ↦ ((normβ„Žβ€˜((projβ„Žβ€˜π‘₯)β€˜π‘’))↑2))    &   π΄ ∈ Cβ„‹    β‡’   ((𝑒 ∈ β„‹ ∧ (normβ„Žβ€˜π‘’) = 1) β†’ (𝑒 ∈ 𝐴 ↔ (π‘†β€˜π΄) = 1))
 
Theoremjpi 30998* 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 30980 for the proof that 𝑆 is a state.) (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
𝑆 = (π‘₯ ∈ Cβ„‹ ↦ ((normβ„Žβ€˜((projβ„Žβ€˜π‘₯)β€˜π‘’))↑2))    &   π΄ ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    β‡’   ((𝑒 ∈ β„‹ ∧ (normβ„Žβ€˜π‘’) = 1) β†’ (((π‘†β€˜π΄) = 1 ∧ (π‘†β€˜π΅) = 1) ↔ (π‘†β€˜(𝐴 ∩ 𝐡)) = 1))
 
20.7.2  Godowski's equation
 
Theoremgolem1 30999 Lemma for Godowski's equation. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    &   πΆ ∈ Cβ„‹    &   πΉ = ((βŠ₯β€˜π΄) βˆ¨β„‹ (𝐴 ∩ 𝐡))    &   πΊ = ((βŠ₯β€˜π΅) βˆ¨β„‹ (𝐡 ∩ 𝐢))    &   π» = ((βŠ₯β€˜πΆ) βˆ¨β„‹ (𝐢 ∩ 𝐴))    &   π· = ((βŠ₯β€˜π΅) βˆ¨β„‹ (𝐡 ∩ 𝐴))    &   π‘… = ((βŠ₯β€˜πΆ) βˆ¨β„‹ (𝐢 ∩ 𝐡))    &   π‘† = ((βŠ₯β€˜π΄) βˆ¨β„‹ (𝐴 ∩ 𝐢))    β‡’   (𝑓 ∈ States β†’ (((π‘“β€˜πΉ) + (π‘“β€˜πΊ)) + (π‘“β€˜π»)) = (((π‘“β€˜π·) + (π‘“β€˜π‘…)) + (π‘“β€˜π‘†)))
 
Theoremgolem2 31000 Lemma for Godowski's equation. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ Cβ„‹    &   π΅ ∈ Cβ„‹    &   πΆ ∈ Cβ„‹    &   πΉ = ((βŠ₯β€˜π΄) βˆ¨β„‹ (𝐴 ∩ 𝐡))    &   πΊ = ((βŠ₯β€˜π΅) βˆ¨β„‹ (𝐡 ∩ 𝐢))    &   π» = ((βŠ₯β€˜πΆ) βˆ¨β„‹ (𝐢 ∩ 𝐴))    &   π· = ((βŠ₯β€˜π΅) βˆ¨β„‹ (𝐡 ∩ 𝐴))    &   π‘… = ((βŠ₯β€˜πΆ) βˆ¨β„‹ (𝐢 ∩ 𝐡))    &   π‘† = ((βŠ₯β€˜π΄) βˆ¨β„‹ (𝐴 ∩ 𝐢))    β‡’   (𝑓 ∈ States β†’ ((π‘“β€˜((𝐹 ∩ 𝐺) ∩ 𝐻)) = 1 β†’ (π‘“β€˜π·) = 1))
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