P. 309 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	spansnch 30801	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 30802	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 30803	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 30804	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 30805	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 30806	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 30807*	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 30808	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 30809	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 30810	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 30811	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 30812	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 30813	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 30814	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 30815	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 30816	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 30817	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 30818	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 30819	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 30820	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 30821	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 30822	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 30823	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 30824*	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 30831 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 30825	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 30826	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 30827	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 30828	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 30829	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 30830*	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 30831	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 30832	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 30833	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 30834	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 30835	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 30836	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 30837	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 30838	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 30839	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 30840	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 30841	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 30842	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 30843	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 30844	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 30845	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 30846	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 30847	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 30848	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 30849	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 30850	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 30851	The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 𝐶ℋ 𝐴
Theorem	pjoml2 30852	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 30853	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 30854	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 30855	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 30856	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 30857	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 30858	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 30859	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 30860	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 30861	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 30862	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 30863	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 30864	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 30865	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 30866	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 30867	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 30868	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 30869	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 30870	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 30871	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 30872	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 30873	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 30874	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 30875	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 30876	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 30877	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 30878*	Lemma for chscl 30882. (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 30879*	Lemma for chscl 30882. (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 30880*	Lemma for chscl 30882. (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 30881*	Lemma for chscl 30882. (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 30882	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 30883	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 30634, although "the hard part" of this proof, chscl 30882, 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 30884	Corollary of osumi 30883. (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 30885	Corollary of osumi 30883, 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 30886	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 30887	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 30888	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 30889	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 30890	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 30891	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 30892	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 30893	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})))
Theorem	spansncv 30894	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.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ (𝐴 ∨ℋ (span‘{𝐶}))) → 𝐵 = (𝐴 ∨ℋ (span‘{𝐶}))))
20.5.9  Orthoarguesian laws 5OA and 3OA
Theorem	5oalem1 30895	Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    &   ⊢ 𝑅 ∈ Sℋ    ⇒   ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → 𝑣 ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅))))
Theorem	5oalem2 30896	Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    &   ⊢ 𝐷 ∈ Sℋ    ⇒   ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ (𝑥 +ℎ 𝑦) = (𝑧 +ℎ 𝑤)) → (𝑥 −ℎ 𝑧) ∈ ((𝐴 +ℋ 𝐶) ∩ (𝐵 +ℋ 𝐷)))
Theorem	5oalem3 30897	Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    &   ⊢ 𝐷 ∈ Sℋ    &   ⊢ 𝐹 ∈ Sℋ    &   ⊢ 𝐺 ∈ Sℋ    ⇒   ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ (𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺)) ∧ ((𝑥 +ℎ 𝑦) = (𝑓 +ℎ 𝑔) ∧ (𝑧 +ℎ 𝑤) = (𝑓 +ℎ 𝑔))) → (𝑥 −ℎ 𝑧) ∈ (((𝐴 +ℋ 𝐹) ∩ (𝐵 +ℋ 𝐺)) +ℋ ((𝐶 +ℋ 𝐹) ∩ (𝐷 +ℋ 𝐺))))
Theorem	5oalem4 30898	Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    &   ⊢ 𝐷 ∈ Sℋ    &   ⊢ 𝐹 ∈ Sℋ    &   ⊢ 𝐺 ∈ Sℋ    ⇒   ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ (𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺)) ∧ ((𝑥 +ℎ 𝑦) = (𝑓 +ℎ 𝑔) ∧ (𝑧 +ℎ 𝑤) = (𝑓 +ℎ 𝑔))) → (𝑥 −ℎ 𝑧) ∈ (((𝐴 +ℋ 𝐶) ∩ (𝐵 +ℋ 𝐷)) ∩ (((𝐴 +ℋ 𝐹) ∩ (𝐵 +ℋ 𝐺)) +ℋ ((𝐶 +ℋ 𝐹) ∩ (𝐷 +ℋ 𝐺)))))
Theorem	5oalem5 30899	Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-May-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    &   ⊢ 𝐷 ∈ Sℋ    &   ⊢ 𝐹 ∈ Sℋ    &   ⊢ 𝐺 ∈ Sℋ    &   ⊢ 𝑅 ∈ Sℋ    &   ⊢ 𝑆 ∈ Sℋ    ⇒   ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ (𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆))) ∧ (((𝑥 +ℎ 𝑦) = (𝑣 +ℎ 𝑢) ∧ (𝑧 +ℎ 𝑤) = (𝑣 +ℎ 𝑢)) ∧ (𝑓 +ℎ 𝑔) = (𝑣 +ℎ 𝑢))) → (𝑥 −ℎ 𝑧) ∈ ((((𝐴 +ℋ 𝐶) ∩ (𝐵 +ℋ 𝐷)) ∩ (((𝐴 +ℋ 𝑅) ∩ (𝐵 +ℋ 𝑆)) +ℋ ((𝐶 +ℋ 𝑅) ∩ (𝐷 +ℋ 𝑆)))) ∩ ((((𝐴 +ℋ 𝐹) ∩ (𝐵 +ℋ 𝐺)) ∩ (((𝐴 +ℋ 𝑅) ∩ (𝐵 +ℋ 𝑆)) +ℋ ((𝐹 +ℋ 𝑅) ∩ (𝐺 +ℋ 𝑆)))) +ℋ (((𝐶 +ℋ 𝐹) ∩ (𝐷 +ℋ 𝐺)) ∩ (((𝐶 +ℋ 𝑅) ∩ (𝐷 +ℋ 𝑆)) +ℋ ((𝐹 +ℋ 𝑅) ∩ (𝐺 +ℋ 𝑆)))))))
Theorem	5oalem6 30900	Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    &   ⊢ 𝐷 ∈ Sℋ    &   ⊢ 𝐹 ∈ Sℋ    &   ⊢ 𝐺 ∈ Sℋ    &   ⊢ 𝑅 ∈ Sℋ    &   ⊢ 𝑆 ∈ Sℋ    ⇒   ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) ∧ ℎ = (𝑧 +ℎ 𝑤))) ∧ (((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ ℎ = (𝑓 +ℎ 𝑔)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆) ∧ ℎ = (𝑣 +ℎ 𝑢)))) → ℎ ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ ((((𝐴 +ℋ 𝐶) ∩ (𝐵 +ℋ 𝐷)) ∩ (((𝐴 +ℋ 𝑅) ∩ (𝐵 +ℋ 𝑆)) +ℋ ((𝐶 +ℋ 𝑅) ∩ (𝐷 +ℋ 𝑆)))) ∩ ((((𝐴 +ℋ 𝐹) ∩ (𝐵 +ℋ 𝐺)) ∩ (((𝐴 +ℋ 𝑅) ∩ (𝐵 +ℋ 𝑆)) +ℋ ((𝐹 +ℋ 𝑅) ∩ (𝐺 +ℋ 𝑆)))) +ℋ (((𝐶 +ℋ 𝐹) ∩ (𝐷 +ℋ 𝐺)) ∩ (((𝐶 +ℋ 𝑅) ∩ (𝐷 +ℋ 𝑆)) +ℋ ((𝐹 +ℋ 𝑅) ∩ (𝐺 +ℋ 𝑆))))))))))