P. 323 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32201-32300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	h1de2i 32201	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) ·ℎ 𝐴) = ((𝐴 ·ih 𝐵) ·ℎ 𝐵))
Theorem	h1de2bi 32202	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 ≠ 0ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵)))
Theorem	h1de2ctlem 32203*	Lemma for h1de2ci 32204. (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 ·ℎ 𝐵))
Theorem	h1de2ci 32204*	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 ·ℎ 𝐵))
Theorem	spansni 32205	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}))
Theorem	elspansni 32206*	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 ·ℎ 𝐴))
Theorem	spansn 32207	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})))
Theorem	spansnch 32208	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}) ∈ Cℋ )
Theorem	spansnsh 32209	The span of a Hilbert space singleton is a subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (span‘{𝐴}) ∈ Sℋ )
Theorem	spansnchi 32210	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}) ∈ Cℋ
Theorem	spansnid 32211	A vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (span‘{𝐴}))
Theorem	spansnmul 32212	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 ·ℎ 𝐴) ∈ (span‘{𝐴}))
Theorem	elspansncl 32213	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)
Theorem	elspansn 32214*	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 ·ℎ 𝐴)))
Theorem	elspansn2 32215	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 ≠ 0ℎ) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵)))
Theorem	spansncol 32216	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 ·ℎ 𝐴)}) = (span‘{𝐴}))
Theorem	spansneleqi 32217	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})))
Theorem	spansneleq 32218	Membership relation that implies equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝐴 ∈ (span‘{𝐵}) → (span‘{𝐴}) = (span‘{𝐵})))
Theorem	spansnss 32219	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.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴) → (span‘{𝐵}) ⊆ 𝐴)
Theorem	elspansn3 32220	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.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ (span‘{𝐵})) → 𝐶 ∈ 𝐴)
Theorem	elspansn4 32221	A span membership condition implying two vectors belong to the same subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ≠ 0ℎ)) → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
Theorem	elspansn5 32222	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.)
⊢ (𝐴 ∈ Sℋ → (((𝐵 ∈ ℋ ∧ ¬ 𝐵 ∈ 𝐴) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ∈ 𝐴)) → 𝐶 = 0ℎ))
Theorem	spansnss2 32223	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.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ∈ 𝐴 ↔ (span‘{𝐵}) ⊆ 𝐴))
Theorem	normcan 32224	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 ≠ 0ℎ ∧ 𝐴 ∈ (span‘{𝐵})) → (((𝐴 ·ih 𝐵) / ((normℎ‘𝐵)↑2)) ·ℎ 𝐵) = 𝐴)
Theorem	pjspansn 32225	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 ≠ 0ℎ) → ((projℎ‘(span‘{𝐴}))‘𝐵) = (((𝐵 ·ih 𝐴) / ((normℎ‘𝐴)↑2)) ·ℎ 𝐴))
Theorem	spansnpji 32226	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‘{((projℎ‘(⊥‘𝐴))‘𝐵)}))
Theorem	spanunsni 32227	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.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (span‘(𝐴 ∪ {𝐵})) = (span‘(𝐴 ∪ {((projℎ‘(⊥‘𝐴))‘𝐵)}))
Theorem	spanpr 32228	The span of a pair of vectors. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (span‘{(𝐴 +ℎ 𝐵)}) ⊆ (span‘{𝐴, 𝐵}))
Theorem	h1datomi 32229	A 1-dimensional subspace is an atom. (Contributed by NM, 20-Jul-2001.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 ⊆ (⊥‘(⊥‘{𝐵})) → (𝐴 = (⊥‘(⊥‘{𝐵})) ∨ 𝐴 = 0ℋ))
Theorem	h1datom 32230	A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ⊆ (⊥‘(⊥‘{𝐵})) → (𝐴 = (⊥‘(⊥‘{𝐵})) ∨ 𝐴 = 0ℋ)))
20.5.5  Commutes relation for Hilbert lattice elements
Definition	df-cm 32231*	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 32238 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
⊢ 𝐶ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))}
Theorem	cmbr 32232	Binary relation expressing 𝐴 commutes with 𝐵. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))
Theorem	pjoml2i 32233	Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∨ℋ ((⊥‘𝐴) ∩ 𝐵)) = 𝐵)
Theorem	pjoml3i 32234	Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) = 𝐵)
Theorem	pjoml4i 32235	Variation of orthomodular law. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) = (𝐴 ∨ℋ 𝐵)
Theorem	pjoml5i 32236	The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ ((⊥‘𝐴) ∩ (𝐴 ∨ℋ 𝐵))) = (𝐴 ∨ℋ 𝐵)
Theorem	pjoml6i 32237*	An equivalent of the orthomodular law. Theorem 29.13(e) of [MaedaMaeda] p. 132. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 → ∃𝑥 ∈ Cℋ (𝐴 ⊆ (⊥‘𝑥) ∧ (𝐴 ∨ℋ 𝑥) = 𝐵))
Theorem	cmbri 32238	Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))
Theorem	cmcmlem 32239	Commutation is symmetric. Theorem 3.4 of [Beran] p. 45. (Contributed by NM, 3-Nov-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 → 𝐵 𝐶ℋ 𝐴)
Theorem	cmcmi 32240	Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐵 𝐶ℋ 𝐴)
Theorem	cmcm2i 32241	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 𝐶ℋ (⊥‘𝐵))
Theorem	cmcm3i 32242	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 ↔ (⊥‘𝐴) 𝐶ℋ 𝐵)
Theorem	cmcm4i 32243	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 ↔ (⊥‘𝐴) 𝐶ℋ (⊥‘𝐵))
Theorem	cmbr2i 32244	Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ (⊥‘𝐵))))
Theorem	cmcmii 32245	Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐴 𝐶ℋ 𝐵    ⇒   ⊢ 𝐵 𝐶ℋ 𝐴
Theorem	cmcm2ii 32246	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐴 𝐶ℋ 𝐵    ⇒   ⊢ 𝐴 𝐶ℋ (⊥‘𝐵)
Theorem	cmcm3ii 32247	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐴 𝐶ℋ 𝐵    ⇒   ⊢ (⊥‘𝐴) 𝐶ℋ 𝐵
Theorem	cmbr3i 32248	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 ↔ (𝐴 ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) = (𝐴 ∩ 𝐵))
Theorem	cmbr4i 32249	Alternate definition for the commutes relation. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 ↔ (𝐴 ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) ⊆ 𝐵)
Theorem	lecmi 32250	Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 𝐶ℋ 𝐵)
Theorem	lecmii 32251	Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 𝐶ℋ 𝐵
Theorem	cmj1i 32252	A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ 𝐴 𝐶ℋ (𝐴 ∨ℋ 𝐵)
Theorem	cmj2i 32253	A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ 𝐵 𝐶ℋ (𝐴 ∨ℋ 𝐵)
Theorem	cmm1i 32254	A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ 𝐴 𝐶ℋ (𝐴 ∩ 𝐵)
Theorem	cmm2i 32255	A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ 𝐵 𝐶ℋ (𝐴 ∩ 𝐵)
Theorem	cmbr3 32256	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ (𝐴 ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) = (𝐴 ∩ 𝐵)))
Theorem	cm0 32257	The zero Hilbert lattice element commutes with every element. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → 0ℋ 𝐶ℋ 𝐴)
Theorem	cmidi 32258	The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 𝐶ℋ 𝐴
Theorem	pjoml2 32259	Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∨ℋ ((⊥‘𝐴) ∩ 𝐵)) = 𝐵)
Theorem	pjoml3 32260	Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐵 ⊆ 𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) = 𝐵))
Theorem	pjoml5 32261	The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ ((⊥‘𝐴) ∩ (𝐴 ∨ℋ 𝐵))) = (𝐴 ∨ℋ 𝐵))
Theorem	cmcm 32262	Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐵 𝐶ℋ 𝐴))
Theorem	cmcm3 32263	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ (⊥‘𝐴) 𝐶ℋ 𝐵))
Theorem	cmcm2 32264	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 𝐶ℋ (⊥‘𝐵)))
Theorem	lecm 32265	Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵) → 𝐴 𝐶ℋ 𝐵)
20.5.6  Foulis-Holland theorem
Theorem	fh1 32266	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.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝐶ℋ 𝐵 ∧ 𝐴 𝐶ℋ 𝐶)) → (𝐴 ∩ (𝐵 ∨ℋ 𝐶)) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐶)))
Theorem	fh2 32267	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.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐵 𝐶ℋ 𝐴 ∧ 𝐵 𝐶ℋ 𝐶)) → (𝐴 ∩ (𝐵 ∨ℋ 𝐶)) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐶)))
Theorem	cm2j 32268	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.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝐶ℋ 𝐵 ∧ 𝐴 𝐶ℋ 𝐶)) → 𝐴 𝐶ℋ (𝐵 ∨ℋ 𝐶))
Theorem	fh1i 32269	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.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐴 𝐶ℋ 𝐵    &   ⊢ 𝐴 𝐶ℋ 𝐶    ⇒   ⊢ (𝐴 ∩ (𝐵 ∨ℋ 𝐶)) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐶))
Theorem	fh2i 32270	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.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐴 𝐶ℋ 𝐵    &   ⊢ 𝐴 𝐶ℋ 𝐶    ⇒   ⊢ (𝐵 ∩ (𝐴 ∨ℋ 𝐶)) = ((𝐵 ∩ 𝐴) ∨ℋ (𝐵 ∩ 𝐶))
Theorem	fh3i 32271	Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐴 𝐶ℋ 𝐵    &   ⊢ 𝐴 𝐶ℋ 𝐶    ⇒   ⊢ (𝐴 ∨ℋ (𝐵 ∩ 𝐶)) = ((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐶))
Theorem	fh4i 32272	Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐴 𝐶ℋ 𝐵    &   ⊢ 𝐴 𝐶ℋ 𝐶    ⇒   ⊢ (𝐵 ∨ℋ (𝐴 ∩ 𝐶)) = ((𝐵 ∨ℋ 𝐴) ∩ (𝐵 ∨ℋ 𝐶))
Theorem	cm2ji 32273	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.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐴 𝐶ℋ 𝐵    &   ⊢ 𝐴 𝐶ℋ 𝐶    ⇒   ⊢ 𝐴 𝐶ℋ (𝐵 ∨ℋ 𝐶)
Theorem	cm2mi 32274	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.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐴 𝐶ℋ 𝐵    &   ⊢ 𝐴 𝐶ℋ 𝐶    ⇒   ⊢ 𝐴 𝐶ℋ (𝐵 ∩ 𝐶)
20.5.7  Quantum Logic Explorer axioms
Theorem	qlax1i 32275	One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 = (⊥‘(⊥‘𝐴))
Theorem	qlax2i 32276	One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)
Theorem	qlax3i 32277	One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-3" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∨ℋ 𝐵) ∨ℋ 𝐶) = (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶))
Theorem	qlax4i 32278	One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ (𝐵 ∨ℋ (⊥‘𝐵))) = (𝐵 ∨ℋ (⊥‘𝐵))
Theorem	qlax5i 32279	One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ (⊥‘((⊥‘𝐴) ∨ℋ 𝐵))) = 𝐴
Theorem	qlaxr1i 32280	One of the conditions showing Cℋ is an ortholattice. (This corresponds to axiom "ax-r1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ 𝐵 = 𝐴
Theorem	qlaxr2i 32281	One of the conditions showing Cℋ is an ortholattice. (This corresponds to axiom "ax-r2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 = 𝐶
Theorem	qlaxr4i 32282	One of the conditions showing Cℋ is an ortholattice. (This corresponds to axiom "ax-r4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ (⊥‘𝐴) = (⊥‘𝐵)
Theorem	qlaxr5i 32283	One of the conditions showing Cℋ is an ortholattice. (This corresponds to axiom "ax-r5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∨ℋ 𝐶) = (𝐵 ∨ℋ 𝐶)
Theorem	qlaxr3i 32284	A variation of the orthomodular law, showing Cℋ 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.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ (𝐶 ∨ℋ (⊥‘𝐶)) = ((⊥‘((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) ∨ℋ (⊥‘(𝐴 ∨ℋ 𝐵)))    ⇒   ⊢ 𝐴 = 𝐵
20.5.8  Orthogonal subspaces
Theorem	chscllem1 32285*	Lemma for chscl 32289. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐵 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴))    &   ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵))    &   ⊢ (𝜑 → 𝐻 ⇝𝑣 𝑢)    &   ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛)))    ⇒   ⊢ (𝜑 → 𝐹:ℕ⟶𝐴)
Theorem	chscllem2 32286*	Lemma for chscl 32289. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐵 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴))    &   ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵))    &   ⊢ (𝜑 → 𝐻 ⇝𝑣 𝑢)    &   ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ⇝𝑣 )
Theorem	chscllem3 32287*	Lemma for chscl 32289. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐵 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴))    &   ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵))    &   ⊢ (𝜑 → 𝐻 ⇝𝑣 𝑢)    &   ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛)))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐻‘𝑁) = (𝐶 +ℎ 𝐷))    ⇒   ⊢ (𝜑 → 𝐶 = (𝐹‘𝑁))
Theorem	chscllem4 32288*	Lemma for chscl 32289. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐵 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴))    &   ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵))    &   ⊢ (𝜑 → 𝐻 ⇝𝑣 𝑢)    &   ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛)))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐵)‘(𝐻‘𝑛)))    ⇒   ⊢ (𝜑 → 𝑢 ∈ (𝐴 +ℋ 𝐵))
Theorem	chscl 32289	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.)
⊢ (𝜑 → 𝐴 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐵 ∈ Cℋ )    &   ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴))    ⇒   ⊢ (𝜑 → (𝐴 +ℋ 𝐵) ∈ Cℋ )
Theorem	osumi 32290	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 32041, although "the hard part" of this proof, chscl 32289, requires no choice. (Contributed by NM, 28-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ (⊥‘𝐵) → (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵))
Theorem	osumcori 32291	Corollary of osumi 32290. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∩ 𝐵) +ℋ (𝐴 ∩ (⊥‘𝐵))) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))
Theorem	osumcor2i 32292	Corollary of osumi 32290, showing it holds under the weaker hypothesis that 𝐴 and 𝐵 commute. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 → (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵))
Theorem	osum 32293	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.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵))
Theorem	spansnji 32294	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.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 +ℋ (span‘{𝐵})) = (𝐴 ∨ℋ (span‘{𝐵}))
Theorem	spansnj 32295	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.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℋ (span‘{𝐵})) = (𝐴 ∨ℋ (span‘{𝐵})))
Theorem	spansnscl 32296	The subspace sum of a closed subspace and a one-dimensional subspace is closed. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℋ (span‘{𝐵})) ∈ Cℋ )
Theorem	sumspansn 32297	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 +ℎ 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴})))
Theorem	spansnm0i 32298	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})) = 0ℋ)
Theorem	nonbooli 32299	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 ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ but (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ 0ℋ. 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 +ℎ 𝐵)})    ⇒   ⊢ (¬ (𝐴 ∈ 𝐺 ∨ 𝐵 ∈ 𝐹) → (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)))
Theorem	spansncvi 32300	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.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ ℋ    ⇒   ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ (𝐴 ∨ℋ (span‘{𝐶}))) → 𝐵 = (𝐴 ∨ℋ (span‘{𝐶})))