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Theorem List for Metamath Proof Explorer - 28501-28600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremchnle 28501 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (¬ 𝐵𝐴𝐴 ⊊ (𝐴 𝐵)))

Theoremchjo 28502 The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)
(𝐴C → (𝐴 (⊥‘𝐴)) = ℋ)

Theoremchabs1 28503 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 (𝐴𝐵)) = 𝐴)

Theoremchabs2 28504 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ∩ (𝐴 𝐵)) = 𝐴)

Theoremchabs1i 28505 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (𝐴𝐵)) = 𝐴

Theoremchabs2i 28506 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 ∩ (𝐴 𝐵)) = 𝐴

Theoremchjidm 28507 Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴C → (𝐴 𝐴) = 𝐴)

Theoremchjidmi 28508 Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
𝐴C       (𝐴 𝐴) = 𝐴

Theoremchj12i 28509 A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝐴 (𝐵 𝐶)) = (𝐵 (𝐴 𝐶))

Theoremchj4i 28510 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐴 𝐵) ∨ (𝐶 𝐷)) = ((𝐴 𝐶) ∨ (𝐵 𝐷))

Theoremchjjdiri 28511 Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴 𝐵) ∨ 𝐶) = ((𝐴 𝐶) ∨ (𝐵 𝐶))

Theoremchdmm1 28512 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘(𝐴𝐵)) = ((⊥‘𝐴) ∨ (⊥‘𝐵)))

Theoremchdmm2 28513 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘((⊥‘𝐴) ∩ 𝐵)) = (𝐴 (⊥‘𝐵)))

Theoremchdmm3 28514 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘(𝐴 ∩ (⊥‘𝐵))) = ((⊥‘𝐴) ∨ 𝐵))

Theoremchdmm4 28515 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘((⊥‘𝐴) ∩ (⊥‘𝐵))) = (𝐴 𝐵))

Theoremchdmj1 28516 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘(𝐴 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵)))

Theoremchdmj2 28517 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘((⊥‘𝐴) ∨ 𝐵)) = (𝐴 ∩ (⊥‘𝐵)))

Theoremchdmj3 28518 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘(𝐴 (⊥‘𝐵))) = ((⊥‘𝐴) ∩ 𝐵))

Theoremchdmj4 28519 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘((⊥‘𝐴) ∨ (⊥‘𝐵))) = (𝐴𝐵))

Theoremchjass 28520 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → ((𝐴 𝐵) ∨ 𝐶) = (𝐴 (𝐵 𝐶)))

Theoremchj12 28521 A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 (𝐵 𝐶)) = (𝐵 (𝐴 𝐶)))

Theoremchj4 28522 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
(((𝐴C𝐵C ) ∧ (𝐶C𝐷C )) → ((𝐴 𝐵) ∨ (𝐶 𝐷)) = ((𝐴 𝐶) ∨ (𝐵 𝐷)))

Theoremledii 28523 An ortholattice is distributive in one ordering direction. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐵) ∨ (𝐴𝐶)) ⊆ (𝐴 ∩ (𝐵 𝐶))

Theoremlediri 28524 An ortholattice is distributive in one ordering direction. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐶) ∨ (𝐵𝐶)) ⊆ ((𝐴 𝐵) ∩ 𝐶)

Theoremlejdii 28525 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝐴 (𝐵𝐶)) ⊆ ((𝐴 𝐵) ∩ (𝐴 𝐶))

Theoremlejdiri 28526 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐵) ∨ 𝐶) ⊆ ((𝐴 𝐶) ∩ (𝐵 𝐶))

Theoremledi 28527 An ortholattice is distributive in one ordering direction. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → ((𝐴𝐵) ∨ (𝐴𝐶)) ⊆ (𝐴 ∩ (𝐵 𝐶)))

19.5.4  Span (cont.) and one-dimensional subspaces

Theoremspansn0 28528 The span of the singleton of the zero vector is the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
(span‘{0}) = 0

Theoremspan0 28529 The span of the empty set is the zero subspace. Remark 11.6.e of [Schechter] p. 276. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
(span‘∅) = 0

Theoremelspani 28530* Membership in the span of a subset of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
𝐵 ∈ V       (𝐴 ⊆ ℋ → (𝐵 ∈ (span‘𝐴) ↔ ∀𝑥S (𝐴𝑥𝐵𝑥)))

Theoremspanuni 28531 The span of a union is the subspace sum of spans. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
𝐴 ⊆ ℋ    &   𝐵 ⊆ ℋ       (span‘(𝐴𝐵)) = ((span‘𝐴) + (span‘𝐵))

Theoremspanun 28532 The span of a union is the subspace sum of spans. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (span‘(𝐴𝐵)) = ((span‘𝐴) + (span‘𝐵)))

Theoremsshhococi 28533 The join of two Hilbert space subsets (not necessarily closed subspaces) equals the join of their closures (double orthocomplements). (Contributed by NM, 1-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ⊆ ℋ    &   𝐵 ⊆ ℋ       (𝐴 𝐵) = ((⊥‘(⊥‘𝐴)) ∨ (⊥‘(⊥‘𝐵)))

Theoremhne0 28534 Hilbert space has a nonzero vector iff it is not trivial. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
( ℋ ≠ 0 ↔ ∃𝑥 ∈ ℋ 𝑥 ≠ 0)

Theoremchsup0 28535 The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
( ‘∅) = 0

Theoremh1deoi 28536 Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.)
𝐵 ∈ ℋ       (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0))

Theoremh1dei 28537* Membership in 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐵 ∈ ℋ       (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ ℋ ((𝐵 ·ih 𝑥) = 0 → (𝐴 ·ih 𝑥) = 0)))

Theoremh1did 28538 A generating vector belongs to the 1-dimensional subspace it generates. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
(𝐴 ∈ ℋ → 𝐴 ∈ (⊥‘(⊥‘{𝐴})))

Theoremh1dn0 28539 A nonzero vector generates a (nonzero) 1-dimensional subspace. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (⊥‘(⊥‘{𝐴})) ≠ 0)

Theoremh1de2i 28540 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 17-Jul-2001.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·ih 𝐵) · 𝐴) = ((𝐴 ·ih 𝐵) · 𝐵))

Theoremh1de2bi 28541 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐵 ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) · 𝐵)))

Theoremh1de2ctlem 28542* Lemma for h1de2ci 28543. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∃𝑥 ∈ ℂ 𝐴 = (𝑥 · 𝐵))

Theoremh1de2ci 28543* Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 21-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐵 ∈ ℋ       (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∃𝑥 ∈ ℂ 𝐴 = (𝑥 · 𝐵))

Theoremspansni 28544 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
𝐴 ∈ ℋ       (span‘{𝐴}) = (⊥‘(⊥‘{𝐴}))

Theoremelspansni 28545* Membership in the span of a singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
𝐴 ∈ ℋ       (𝐵 ∈ (span‘{𝐴}) ↔ ∃𝑥 ∈ ℂ 𝐵 = (𝑥 · 𝐴))

Theoremspansn 28546 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (span‘{𝐴}) = (⊥‘(⊥‘{𝐴})))

Theoremspansnch 28547 The span of a Hilbert space singleton belongs to the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (span‘{𝐴}) ∈ C )

Theoremspansnsh 28548 The span of a Hilbert space singleton is a subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (span‘{𝐴}) ∈ S )

Theoremspansnchi 28549 The span of a singleton in Hilbert space is a closed subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
𝐴 ∈ ℋ       (span‘{𝐴}) ∈ C

Theoremspansnid 28550 A vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → 𝐴 ∈ (span‘{𝐴}))

Theoremspansnmul 28551 A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝐵 · 𝐴) ∈ (span‘{𝐴}))

Theoremelspansncl 28552 A member of a span of a singleton is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ (span‘{𝐴})) → 𝐵 ∈ ℋ)

Theoremelspansn 28553* Membership in the span of a singleton. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐵 ∈ (span‘{𝐴}) ↔ ∃𝑥 ∈ ℂ 𝐵 = (𝑥 · 𝐴)))

Theoremelspansn2 28554 Membership in the span of a singleton. All members are collinear with the generating vector. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) · 𝐵)))

Theoremspansncol 28555 The singletons of collinear vectors have the same span. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (span‘{(𝐵 · 𝐴)}) = (span‘{𝐴}))

Theoremspansneleqi 28556 Membership relation implied by equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((span‘{𝐴}) = (span‘{𝐵}) → 𝐴 ∈ (span‘{𝐵})))

Theoremspansneleq 28557 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 28558 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 28559 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 28560 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 28561 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 28562 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 28563 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 28564 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 28565 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 28566 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 28567 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 28568 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 28569 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 28570* 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 28577 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}

Theoremcmbr 28571 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 28572 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 28573 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐵𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = 𝐵)

Theorempjoml4i 28574 Variation of orthomodular law. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (𝐵 ∩ ((⊥‘𝐴) ∨ (⊥‘𝐵)))) = (𝐴 𝐵)

Theorempjoml5i 28575 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 28576* 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 28577 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 28578 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 28579 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 28580 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 28581 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 28582 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 28583 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 28584 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 28585 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 28586 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 28587 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 28588 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 28589 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 28590 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 28591 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 28592 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 28593 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 𝐶 (𝐴𝐵)

Theoremcmm2i 28594 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐵 𝐶 (𝐴𝐵)

Theoremcmbr3 28595 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 28596 The zero Hilbert lattice element commutes with every element. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
(𝐴C → 0 𝐶 𝐴)

Theoremcmidi 28597 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C       𝐴 𝐶 𝐴

Theorempjoml2 28598 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 28599 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐵𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = 𝐵))

Theorempjoml5 28600 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))

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