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Theorem List for Metamath Proof Explorer - 27601-27700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremspansneleq 27601 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 27602 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 27603 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 27604 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 27605 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 27606 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 27607 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 27608 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 27609 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 27610 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 27611 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 27612 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 27613 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 27614* 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 27621 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}
 
Theoremcmbr 27615 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 27616 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 27617 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐵𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = 𝐵)
 
Theorempjoml4i 27618 Variation of orthomodular law. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (𝐵 ∩ ((⊥‘𝐴) ∨ (⊥‘𝐵)))) = (𝐴 𝐵)
 
Theorempjoml5i 27619 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 27620* 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 27621 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 27622 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 27623 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 27624 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 27625 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 27626 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 27627 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 27628 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 27629 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 27630 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 27631 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 27632 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 27633 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 27634 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 27635 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 27636 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 27637 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 𝐶 (𝐴𝐵)
 
Theoremcmm2i 27638 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐵 𝐶 (𝐴𝐵)
 
Theoremcmbr3 27639 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 27640 The zero Hilbert lattice element commutes with every element. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
(𝐴C → 0 𝐶 𝐴)
 
Theoremcmidi 27641 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C       𝐴 𝐶 𝐴
 
Theorempjoml2 27642 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 27643 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐵𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = 𝐵))
 
Theorempjoml5 27644 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 27645 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 27646 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 27647 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 27648 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 27649 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 27650 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 27651 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 27652 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 27653 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 27654 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 27655 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 27656 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 27657 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 27658 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 27659 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 27660 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 27661 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 27662 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 27663 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 27664 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 27665 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 27666 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 27667 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 27668* Lemma for chscl 27672. (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 27669* Lemma for chscl 27672. (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 27670* Lemma for chscl 27672. (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 27671* Lemma for chscl 27672. (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 27672 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 27673 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 27424, although "the hard part" of this proof, chscl 27672, 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 27674 Corollary of osumi 27673. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) + (𝐴 ∩ (⊥‘𝐵))) = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))
 
Theoremosumcor2i 27675 Corollary of osumi 27673, 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 27676 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 27677 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 27678 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 27679 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 27680 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 27681 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 27682 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 27683 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 27684 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 27685 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 27686 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 27687 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 27688 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 27689 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 27690 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 27691 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (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 27692 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 27693* 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 27694* 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 27695 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 27696 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))       𝑅 ⊆ (⊥‘𝐵)
 
Theorem3oalem5 27697 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 27698 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 27699 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 27700 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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