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Theorem List for Metamath Proof Explorer - 29001-29100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremspansneleq 29001 Membership relation that implies equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
((𝐵 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝐴 ∈ (span‘{𝐵}) → (span‘{𝐴}) = (span‘{𝐵})))

Theoremspansnss 29002 The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵𝐴) → (span‘{𝐵}) ⊆ 𝐴)

Theoremelspansn3 29003 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵𝐴𝐶 ∈ (span‘{𝐵})) → 𝐶𝐴)

Theoremelspansn4 29004 A span membership condition implying two vectors belong to the same subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
(((𝐴S𝐵 ∈ ℋ) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ≠ 0)) → (𝐵𝐴𝐶𝐴))

Theoremelspansn5 29005 A vector belonging to both a subspace and the span of the singleton of a vector not in it must be zero. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
(𝐴S → (((𝐵 ∈ ℋ ∧ ¬ 𝐵𝐴) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶𝐴)) → 𝐶 = 0))

Theoremspansnss2 29006 The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵 ∈ ℋ) → (𝐵𝐴 ↔ (span‘{𝐵}) ⊆ 𝐴))

Theoremnormcan 29007 Cancellation-type law that "extracts" a vector 𝐴 from its inner product with a proportional vector 𝐵. (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.)
((𝐵 ∈ ℋ ∧ 𝐵 ≠ 0𝐴 ∈ (span‘{𝐵})) → (((𝐴 ·ih 𝐵) / ((norm𝐵)↑2)) · 𝐵) = 𝐴)

Theorempjspansn 29008 A projection on the span of a singleton. (The proof ws shortened by Mario Carneiro, 15-Dec-2013.) (Contributed by NM, 28-May-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0) → ((proj‘(span‘{𝐴}))‘𝐵) = (((𝐵 ·ih 𝐴) / ((norm𝐴)↑2)) · 𝐴))

Theoremspansnpji 29009 A subset of Hilbert space is orthogonal to the span of the singleton of a projection onto its orthocomplement. (Contributed by NM, 4-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ⊆ ℋ    &   𝐵 ∈ ℋ       𝐴 ⊆ (⊥‘(span‘{((proj‘(⊥‘𝐴))‘𝐵)}))

Theoremspanunsni 29010 The span of the union of a closed subspace with a singleton equals the span of its union with an orthogonal singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵 ∈ ℋ       (span‘(𝐴 ∪ {𝐵})) = (span‘(𝐴 ∪ {((proj‘(⊥‘𝐴))‘𝐵)}))

Theoremspanpr 29011 The span of a pair of vectors. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (span‘{(𝐴 + 𝐵)}) ⊆ (span‘{𝐴, 𝐵}))

Theoremh1datomi 29012 A 1-dimensional subspace is an atom. (Contributed by NM, 20-Jul-2001.) (New usage is discouraged.)
𝐴C    &   𝐵 ∈ ℋ       (𝐴 ⊆ (⊥‘(⊥‘{𝐵})) → (𝐴 = (⊥‘(⊥‘{𝐵})) ∨ 𝐴 = 0))

Theoremh1datom 29013 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
((𝐴C𝐵 ∈ ℋ) → (𝐴 ⊆ (⊥‘(⊥‘{𝐵})) → (𝐴 = (⊥‘(⊥‘{𝐵})) ∨ 𝐴 = 0)))

19.5.5  Commutes relation for Hilbert lattice elements

Definitiondf-cm 29014* Define the commutes relation (on the Hilbert lattice). Definition of commutes in [Kalmbach] p. 20, who uses the notation xCy for "x commutes with y." See cmbri 29021 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}

Theoremcmbr 29015 Binary relation expressing 𝐴 commutes with 𝐵. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))

Theorempjoml2i 29016 Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 → (𝐴 ((⊥‘𝐴) ∩ 𝐵)) = 𝐵)

Theorempjoml3i 29017 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐵𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = 𝐵)

Theorempjoml4i 29018 Variation of orthomodular law. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (𝐵 ∩ ((⊥‘𝐴) ∨ (⊥‘𝐵)))) = (𝐴 𝐵)

Theorempjoml5i 29019 The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 ((⊥‘𝐴) ∩ (𝐴 𝐵))) = (𝐴 𝐵)

Theorempjoml6i 29020* An equivalent of the orthomodular law. Theorem 29.13(e) of [MaedaMaeda] p. 132. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 → ∃𝑥C (𝐴 ⊆ (⊥‘𝑥) ∧ (𝐴 𝑥) = 𝐵))

Theoremcmbri 29021 Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))

Theoremcmcmlem 29022 Commutation is symmetric. Theorem 3.4 of [Beran] p. 45. (Contributed by NM, 3-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵𝐵 𝐶 𝐴)

Theoremcmcmi 29023 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵𝐵 𝐶 𝐴)

Theoremcmcm2i 29024 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵𝐴 𝐶 (⊥‘𝐵))

Theoremcmcm3i 29025 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵 ↔ (⊥‘𝐴) 𝐶 𝐵)

Theoremcmcm4i 29026 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵 ↔ (⊥‘𝐴) 𝐶 (⊥‘𝐵))

Theoremcmbr2i 29027 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵𝐴 = ((𝐴 𝐵) ∩ (𝐴 (⊥‘𝐵))))

Theoremcmcmii 29028 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴 𝐶 𝐵       𝐵 𝐶 𝐴

Theoremcmcm2ii 29029 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴 𝐶 𝐵       𝐴 𝐶 (⊥‘𝐵)

Theoremcmcm3ii 29030 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴 𝐶 𝐵       (⊥‘𝐴) 𝐶 𝐵

Theoremcmbr3i 29031 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵 ↔ (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = (𝐴𝐵))

Theoremcmbr4i 29032 Alternate definition for the commutes relation. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵 ↔ (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) ⊆ 𝐵)

Theoremlecmi 29033 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵𝐴 𝐶 𝐵)

Theoremlecmii 29034 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴𝐵       𝐴 𝐶 𝐵

Theoremcmj1i 29035 A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 𝐶 (𝐴 𝐵)

Theoremcmj2i 29036 A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐵 𝐶 (𝐴 𝐵)

Theoremcmm1i 29037 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 𝐶 (𝐴𝐵)

Theoremcmm2i 29038 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐵 𝐶 (𝐴𝐵)

Theoremcmbr3 29039 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐶 𝐵 ↔ (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = (𝐴𝐵)))

Theoremcm0 29040 The zero Hilbert lattice element commutes with every element. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
(𝐴C → 0 𝐶 𝐴)

Theoremcmidi 29041 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C       𝐴 𝐶 𝐴

Theorempjoml2 29042 Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C𝐴𝐵) → (𝐴 ((⊥‘𝐴) ∩ 𝐵)) = 𝐵)

Theorempjoml3 29043 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐵𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = 𝐵))

Theorempjoml5 29044 The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ((⊥‘𝐴) ∩ (𝐴 𝐵))) = (𝐴 𝐵))

Theoremcmcm 29045 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐵 𝐶 𝐴))

Theoremcmcm3 29046 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐶 𝐵 ↔ (⊥‘𝐴) 𝐶 𝐵))

Theoremcmcm2 29047 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐴 𝐶 (⊥‘𝐵)))

Theoremlecm 29048 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C𝐴𝐵) → 𝐴 𝐶 𝐵)

19.5.6  Foulis-Holland theorem

Theoremfh1 29049 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. First of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝐶 𝐵𝐴 𝐶 𝐶)) → (𝐴 ∩ (𝐵 𝐶)) = ((𝐴𝐵) ∨ (𝐴𝐶)))

Theoremfh2 29050 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝐶 𝐴𝐵 𝐶 𝐶)) → (𝐴 ∩ (𝐵 𝐶)) = ((𝐴𝐵) ∨ (𝐴𝐶)))

Theoremcm2j 29051 A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝐶 𝐵𝐴 𝐶 𝐶)) → 𝐴 𝐶 (𝐵 𝐶))

Theoremfh1i 29052 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. First of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       (𝐴 ∩ (𝐵 𝐶)) = ((𝐴𝐵) ∨ (𝐴𝐶))

Theoremfh2i 29053 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       (𝐵 ∩ (𝐴 𝐶)) = ((𝐵𝐴) ∨ (𝐵𝐶))

Theoremfh3i 29054 Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       (𝐴 (𝐵𝐶)) = ((𝐴 𝐵) ∩ (𝐴 𝐶))

Theoremfh4i 29055 Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       (𝐵 (𝐴𝐶)) = ((𝐵 𝐴) ∩ (𝐵 𝐶))

Theoremcm2ji 29056 A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 11-May-2009.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       𝐴 𝐶 (𝐵 𝐶)

Theoremcm2mi 29057 A lattice element that commutes with two others also commutes with their meet. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 11-May-2009.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       𝐴 𝐶 (𝐵𝐶)

19.5.7  Quantum Logic Explorer axioms

Theoremqlax1i 29058 One of the equations showing C is an ortholattice. (This corresponds to axiom "ax-1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C       𝐴 = (⊥‘(⊥‘𝐴))

Theoremqlax2i 29059 One of the equations showing C is an ortholattice. (This corresponds to axiom "ax-2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) = (𝐵 𝐴)

Theoremqlax3i 29060 One of the equations showing C is an ortholattice. (This corresponds to axiom "ax-3" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴 𝐵) ∨ 𝐶) = (𝐴 (𝐵 𝐶))

Theoremqlax4i 29061 One of the equations showing C is an ortholattice. (This corresponds to axiom "ax-4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (𝐵 (⊥‘𝐵))) = (𝐵 (⊥‘𝐵))

Theoremqlax5i 29062 One of the equations showing C is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (⊥‘((⊥‘𝐴) ∨ 𝐵))) = 𝐴

Theoremqlaxr1i 29063 One of the conditions showing C is an ortholattice. (This corresponds to axiom "ax-r1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴 = 𝐵       𝐵 = 𝐴

Theoremqlaxr2i 29064 One of the conditions showing C is an ortholattice. (This corresponds to axiom "ax-r2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 = 𝐵    &   𝐵 = 𝐶       𝐴 = 𝐶

Theoremqlaxr4i 29065 One of the conditions showing C is an ortholattice. (This corresponds to axiom "ax-r4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴 = 𝐵       (⊥‘𝐴) = (⊥‘𝐵)

Theoremqlaxr5i 29066 One of the conditions showing C is an ortholattice. (This corresponds to axiom "ax-r5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 = 𝐵       (𝐴 𝐶) = (𝐵 𝐶)

Theoremqlaxr3i 29067 A variation of the orthomodular law, showing C is an orthomodular lattice. (This corresponds to axiom "ax-r3" in the Quantum Logic Explorer.) (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   (𝐶 (⊥‘𝐶)) = ((⊥‘((⊥‘𝐴) ∨ (⊥‘𝐵))) ∨ (⊥‘(𝐴 𝐵)))       𝐴 = 𝐵

19.5.8  Orthogonal subspaces

Theoremchscllem1 29068* Lemma for chscl 29072. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝜑𝐴C )    &   (𝜑𝐵C )    &   (𝜑𝐵 ⊆ (⊥‘𝐴))    &   (𝜑𝐻:ℕ⟶(𝐴 + 𝐵))    &   (𝜑𝐻𝑣 𝑢)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((proj𝐴)‘(𝐻𝑛)))       (𝜑𝐹:ℕ⟶𝐴)

Theoremchscllem2 29069* Lemma for chscl 29072. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝜑𝐴C )    &   (𝜑𝐵C )    &   (𝜑𝐵 ⊆ (⊥‘𝐴))    &   (𝜑𝐻:ℕ⟶(𝐴 + 𝐵))    &   (𝜑𝐻𝑣 𝑢)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((proj𝐴)‘(𝐻𝑛)))       (𝜑𝐹 ∈ dom ⇝𝑣 )

Theoremchscllem3 29070* Lemma for chscl 29072. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝜑𝐴C )    &   (𝜑𝐵C )    &   (𝜑𝐵 ⊆ (⊥‘𝐴))    &   (𝜑𝐻:ℕ⟶(𝐴 + 𝐵))    &   (𝜑𝐻𝑣 𝑢)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((proj𝐴)‘(𝐻𝑛)))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷𝐵)    &   (𝜑 → (𝐻𝑁) = (𝐶 + 𝐷))       (𝜑𝐶 = (𝐹𝑁))

Theoremchscllem4 29071* Lemma for chscl 29072. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝜑𝐴C )    &   (𝜑𝐵C )    &   (𝜑𝐵 ⊆ (⊥‘𝐴))    &   (𝜑𝐻:ℕ⟶(𝐴 + 𝐵))    &   (𝜑𝐻𝑣 𝑢)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((proj𝐴)‘(𝐻𝑛)))    &   𝐺 = (𝑛 ∈ ℕ ↦ ((proj𝐵)‘(𝐻𝑛)))       (𝜑𝑢 ∈ (𝐴 + 𝐵))

Theoremchscl 29072 The subspace sum of two closed orthogonal spaces is closed. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝜑𝐴C )    &   (𝜑𝐵C )    &   (𝜑𝐵 ⊆ (⊥‘𝐴))       (𝜑 → (𝐴 + 𝐵) ∈ C )

Theoremosumi 29073 If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. Note that the (countable) Axiom of Choice is used for this proof via pjhth 28824, although "the hard part" of this proof, chscl 29072, requires no choice. (Contributed by NM, 28-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 ⊆ (⊥‘𝐵) → (𝐴 + 𝐵) = (𝐴 𝐵))

Theoremosumcori 29074 Corollary of osumi 29073. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) + (𝐴 ∩ (⊥‘𝐵))) = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))

Theoremosumcor2i 29075 Corollary of osumi 29073, showing it holds under the weaker hypothesis that 𝐴 and 𝐵 commute. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵 → (𝐴 + 𝐵) = (𝐴 𝐵))

Theoremosum 29076 If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)
((𝐴C𝐵C𝐴 ⊆ (⊥‘𝐵)) → (𝐴 + 𝐵) = (𝐴 𝐵))

Theoremspansnji 29077 The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Proof suggested by Eric Schechter 1-Jun-2004.) (Contributed by NM, 1-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵 ∈ ℋ       (𝐴 + (span‘{𝐵})) = (𝐴 (span‘{𝐵}))

Theoremspansnj 29078 The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ ℋ) → (𝐴 + (span‘{𝐵})) = (𝐴 (span‘{𝐵})))

Theoremspansnscl 29079 The subspace sum of a closed subspace and a one-dimensional subspace is closed. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ ℋ) → (𝐴 + (span‘{𝐵})) ∈ C )

Theoremsumspansn 29080 The sum of two vectors belong to the span of one of them iff the other vector also belongs. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 + 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴})))

Theoremspansnm0i 29081 The meet of different one-dimensional subspaces is the zero subspace. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       𝐴 ∈ (span‘{𝐵}) → ((span‘{𝐴}) ∩ (span‘{𝐵})) = 0)

Theoremnonbooli 29082 A Hilbert lattice with two or more dimensions fails the distributive law and therefore cannot be a Boolean algebra. This counterexample demonstrates a condition where ((𝐻𝐹) ∨ (𝐻𝐺)) = 0 but (𝐻 ∩ (𝐹 𝐺)) ≠ 0. The antecedent specifies that the vectors 𝐴 and 𝐵 are nonzero and non-colinear. The last three hypotheses assign one-dimensional subspaces to 𝐹, 𝐺, and 𝐻. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐹 = (span‘{𝐴})    &   𝐺 = (span‘{𝐵})    &   𝐻 = (span‘{(𝐴 + 𝐵)})       (¬ (𝐴𝐺𝐵𝐹) → (𝐻 ∩ (𝐹 𝐺)) ≠ ((𝐻𝐹) ∨ (𝐻𝐺)))

Theoremspansncvi 29083 Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶 ∈ ℋ       ((𝐴𝐵𝐵 ⊆ (𝐴 (span‘{𝐶}))) → 𝐵 = (𝐴 (span‘{𝐶})))

Theoremspansncv 29084 Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶 ∈ ℋ) → ((𝐴𝐵𝐵 ⊆ (𝐴 (span‘{𝐶}))) → 𝐵 = (𝐴 (span‘{𝐶}))))

19.5.9  Orthoarguesian laws 5OA and 3OA

Theorem5oalem1 29085 Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝑅S       ((((𝑥𝐴𝑦𝐵) ∧ 𝑣 = (𝑥 + 𝑦)) ∧ (𝑧𝐶 ∧ (𝑥 𝑧) ∈ 𝑅)) → 𝑣 ∈ (𝐵 + (𝐴 ∩ (𝐶 + 𝑅))))

Theorem5oalem2 29086 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S       ((((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝑤𝐷)) ∧ (𝑥 + 𝑦) = (𝑧 + 𝑤)) → (𝑥 𝑧) ∈ ((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)))

Theorem5oalem3 29087 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S       (((((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝑤𝐷)) ∧ (𝑓𝐹𝑔𝐺)) ∧ ((𝑥 + 𝑦) = (𝑓 + 𝑔) ∧ (𝑧 + 𝑤) = (𝑓 + 𝑔))) → (𝑥 𝑧) ∈ (((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) + ((𝐶 + 𝐹) ∩ (𝐷 + 𝐺))))

Theorem5oalem4 29088 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S       (((((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝑤𝐷)) ∧ (𝑓𝐹𝑔𝐺)) ∧ ((𝑥 + 𝑦) = (𝑓 + 𝑔) ∧ (𝑧 + 𝑤) = (𝑓 + 𝑔))) → (𝑥 𝑧) ∈ (((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) + ((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)))))

Theorem5oalem5 29089 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S    &   𝑅S    &   𝑆S       (((((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝑤𝐷)) ∧ ((𝑓𝐹𝑔𝐺) ∧ (𝑣𝑅𝑢𝑆))) ∧ (((𝑥 + 𝑦) = (𝑣 + 𝑢) ∧ (𝑧 + 𝑤) = (𝑣 + 𝑢)) ∧ (𝑓 + 𝑔) = (𝑣 + 𝑢))) → (𝑥 𝑧) ∈ ((((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)))) ∩ ((((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))) + (((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)) ∩ (((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))))))

Theorem5oalem6 29090 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S    &   𝑅S    &   𝑆S       (((((𝑥𝐴𝑦𝐵) ∧ = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝐷) ∧ = (𝑧 + 𝑤))) ∧ (((𝑓𝐹𝑔𝐺) ∧ = (𝑓 + 𝑔)) ∧ ((𝑣𝑅𝑢𝑆) ∧ = (𝑣 + 𝑢)))) → ∈ (𝐵 + (𝐴 ∩ (𝐶 + ((((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)))) ∩ ((((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))) + (((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)) ∩ (((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆))))))))))

Theorem5oalem7 29091 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) TODO: replace uses of ee4anv 2320 with 4exdistrv 1999 as in 3oalem3 29095. (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S    &   𝑅S    &   𝑆S       (((𝐴 + 𝐵) ∩ (𝐶 + 𝐷)) ∩ ((𝐹 + 𝐺) ∩ (𝑅 + 𝑆))) ⊆ (𝐵 + (𝐴 ∩ (𝐶 + ((((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)))) ∩ ((((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))) + (((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)) ∩ (((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))))))))

Theorem5oai 29092 Orthoarguesian law 5OA. This 8-variable inference is called 5OA because it can be converted to a 5-variable equation (see Quantum Logic Explorer). (Contributed by NM, 5-May-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝐹C    &   𝐺C    &   𝑅C    &   𝑆C    &   𝐴 ⊆ (⊥‘𝐵)    &   𝐶 ⊆ (⊥‘𝐷)    &   𝐹 ⊆ (⊥‘𝐺)    &   𝑅 ⊆ (⊥‘𝑆)       (((𝐴 𝐵) ∩ (𝐶 𝐷)) ∩ ((𝐹 𝐺) ∩ (𝑅 𝑆))) ⊆ (𝐵 (𝐴 ∩ (𝐶 ((((𝐴 𝐶) ∩ (𝐵 𝐷)) ∩ (((𝐴 𝑅) ∩ (𝐵 𝑆)) ∨ ((𝐶 𝑅) ∩ (𝐷 𝑆)))) ∩ ((((𝐴 𝐹) ∩ (𝐵 𝐺)) ∩ (((𝐴 𝑅) ∩ (𝐵 𝑆)) ∨ ((𝐹 𝑅) ∩ (𝐺 𝑆)))) ∨ (((𝐶 𝐹) ∩ (𝐷 𝐺)) ∩ (((𝐶 𝑅) ∩ (𝐷 𝑆)) ∨ ((𝐹 𝑅) ∩ (𝐺 𝑆)))))))))

Theorem3oalem1 29093* Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐵C    &   𝐶C    &   𝑅C    &   𝑆C       ((((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝑆) ∧ 𝑣 = (𝑧 + 𝑤))) → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)))

Theorem3oalem2 29094* Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐵C    &   𝐶C    &   𝑅C    &   𝑆C       ((((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝑆) ∧ 𝑣 = (𝑧 + 𝑤))) → 𝑣 ∈ (𝐵 + (𝑅 ∩ (𝑆 + ((𝐵 + 𝐶) ∩ (𝑅 + 𝑆))))))

Theorem3oalem3 29095 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐵C    &   𝐶C    &   𝑅C    &   𝑆C       ((𝐵 + 𝑅) ∩ (𝐶 + 𝑆)) ⊆ (𝐵 + (𝑅 ∩ (𝑆 + ((𝐵 + 𝐶) ∩ (𝑅 + 𝑆)))))

Theorem3oalem4 29096 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))       𝑅 ⊆ (⊥‘𝐵)

Theorem3oalem5 29097 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))    &   𝑆 = ((⊥‘𝐶) ∩ (𝐶 𝐴))       ((𝐵 + 𝑅) ∩ (𝐶 + 𝑆)) = ((𝐵 𝑅) ∩ (𝐶 𝑆))

Theorem3oalem6 29098 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))    &   𝑆 = ((⊥‘𝐶) ∩ (𝐶 𝐴))       (𝐵 + (𝑅 ∩ (𝑆 + ((𝐵 + 𝐶) ∩ (𝑅 + 𝑆))))) ⊆ (𝐵 (𝑅 ∩ (𝑆 ((𝐵 𝐶) ∩ (𝑅 𝑆)))))

Theorem3oai 29099 3OA (weak) orthoarguesian law. Equation IV of [GodowskiGreechie] p. 249. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))    &   𝑆 = ((⊥‘𝐶) ∩ (𝐶 𝐴))       ((𝐵 𝑅) ∩ (𝐶 𝑆)) ⊆ (𝐵 (𝑅 ∩ (𝑆 ((𝐵 𝐶) ∩ (𝑅 𝑆)))))

19.5.10  Projectors (cont.)

Theorempjorthi 29100 Projection components on orthocomplemented subspaces are orthogonal. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐻C → (((proj𝐻)‘𝐴) ·ih ((proj‘(⊥‘𝐻))‘𝐵)) = 0)

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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 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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 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