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Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelspansn3 28401 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 28402 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 28403 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 28404 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 28405 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 28406 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 28407 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 28408 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 28409 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 28410 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 28411 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 28412* 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 28419 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}

Theoremcmbr 28413 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 28414 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 28415 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐵𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = 𝐵)

Theorempjoml4i 28416 Variation of orthomodular law. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (𝐵 ∩ ((⊥‘𝐴) ∨ (⊥‘𝐵)))) = (𝐴 𝐵)

Theorempjoml5i 28417 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 28418* 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 28419 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 28420 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 28421 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 28422 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 28423 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 28424 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 28425 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 28426 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 28427 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 28428 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 28429 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 28430 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 28431 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 28432 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 28433 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 28434 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 28435 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 𝐶 (𝐴𝐵)

Theoremcmm2i 28436 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐵 𝐶 (𝐴𝐵)

Theoremcmbr3 28437 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 28438 The zero Hilbert lattice element commutes with every element. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
(𝐴C → 0 𝐶 𝐴)

Theoremcmidi 28439 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C       𝐴 𝐶 𝐴

Theorempjoml2 28440 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 28441 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐵𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = 𝐵))

Theorempjoml5 28442 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 28443 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 28444 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 28445 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 28446 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 28447 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 28448 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 28449 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 28450 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 28451 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 28452 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 28453 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 28454 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 28455 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 28456 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 28457 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 28458 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 28459 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 28460 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 28461 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 28462 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 28463 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 28464 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 28465 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 28466* Lemma for chscl 28470. (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 28467* Lemma for chscl 28470. (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 28468* Lemma for chscl 28470. (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 28469* Lemma for chscl 28470. (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 28470 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 28471 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 28222, although "the hard part" of this proof, chscl 28470, 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 28472 Corollary of osumi 28471. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) + (𝐴 ∩ (⊥‘𝐵))) = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))

Theoremosumcor2i 28473 Corollary of osumi 28471, 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 28474 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 28475 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 28476 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 28477 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 28478 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 28479 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 28480 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 28481 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 28482 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 28483 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 28484 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 28485 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 28486 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 28487 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 28488 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 28489 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 28490 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 28491* 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 28492* 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 28493 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 28494 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))       𝑅 ⊆ (⊥‘𝐵)

Theorem3oalem5 28495 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 28496 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 28497 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 28498 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)

Theorempjch1 28499 Property of identity projection. Remark in [Beran] p. 111. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((proj‘ ℋ)‘𝐴) = 𝐴)

Theorempjo 28500 The orthogonal projection. Lemma 4.4(i) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj‘(⊥‘𝐻))‘𝐴) = (((proj‘ ℋ)‘𝐴) − ((proj𝐻)‘𝐴)))

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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 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