P. 316 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31501-31600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elspansn5 31501	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 31502	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 31503	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 31504	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 31505	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 31506	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 31507	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 31508	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 31509	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 31510*	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 31517 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 31511	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 31512	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 31513	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 31514	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 31515	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 31516*	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 31517	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 31518	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 31519	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 31520	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 31521	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 31522	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 31523	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 31524	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 31525	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 31526	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 31527	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 31528	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 31529	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 31530	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 31531	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 31532	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 31533	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 31534	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 31535	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 31536	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 31537	The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 𝐶ℋ 𝐴
Theorem	pjoml2 31538	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 31539	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 31540	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 31541	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 31542	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 31543	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 31544	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 31545	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 31546	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 31547	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 31548	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 31549	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 31550	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 31551	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 31552	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 31553	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 31554	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 31555	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 31556	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 31557	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 31558	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 31559	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 31560	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 31561	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 31562	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 31563	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 31564*	Lemma for chscl 31568. (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 31565*	Lemma for chscl 31568. (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 31566*	Lemma for chscl 31568. (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 31567*	Lemma for chscl 31568. (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 31568	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 31569	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 31320, although "the hard part" of this proof, chscl 31568, 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 31570	Corollary of osumi 31569. (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 31571	Corollary of osumi 31569, 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 31572	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 31573	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 31574	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 31575	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 31576	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 31577	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 31578	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 31579	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 31580	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 31581	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 31582	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 31583	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 31584	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 31585	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 31586	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))))))))))
Theorem	5oalem7 31587	Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) TODO: replace uses of ee4anv 2352 with 4exdistrv 1956 as in 3oalem3 31591. (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝐶 ∈ Sℋ    &   ⊢ 𝐷 ∈ Sℋ    &   ⊢ 𝐹 ∈ Sℋ    &   ⊢ 𝐺 ∈ Sℋ    &   ⊢ 𝑅 ∈ Sℋ    &   ⊢ 𝑆 ∈ Sℋ    ⇒   ⊢ (((𝐴 +ℋ 𝐵) ∩ (𝐶 +ℋ 𝐷)) ∩ ((𝐹 +ℋ 𝐺) ∩ (𝑅 +ℋ 𝑆))) ⊆ (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)))))))))
Theorem	5oai 31588	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.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    &   ⊢ 𝐹 ∈ Cℋ    &   ⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝑅 ∈ Cℋ    &   ⊢ 𝑆 ∈ Cℋ    &   ⊢ 𝐴 ⊆ (⊥‘𝐵)    &   ⊢ 𝐶 ⊆ (⊥‘𝐷)    &   ⊢ 𝐹 ⊆ (⊥‘𝐺)    &   ⊢ 𝑅 ⊆ (⊥‘𝑆)    ⇒   ⊢ (((𝐴 ∨ℋ 𝐵) ∩ (𝐶 ∨ℋ 𝐷)) ∩ ((𝐹 ∨ℋ 𝐺) ∩ (𝑅 ∨ℋ 𝑆))) ⊆ (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)))))))))
Theorem	3oalem1 31589*	Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝑅 ∈ Cℋ    &   ⊢ 𝑆 ∈ Cℋ    ⇒   ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)))
Theorem	3oalem2 31590*	Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝑅 ∈ Cℋ    &   ⊢ 𝑆 ∈ Cℋ    ⇒   ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) → 𝑣 ∈ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))))
Theorem	3oalem3 31591	Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝑅 ∈ Cℋ    &   ⊢ 𝑆 ∈ Cℋ    ⇒   ⊢ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) ⊆ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)))))
Theorem	3oalem4 31592	Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴))    ⇒   ⊢ 𝑅 ⊆ (⊥‘𝐵)
Theorem	3oalem5 31593	Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴))    &   ⊢ 𝑆 = ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴))    ⇒   ⊢ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) = ((𝐵 ∨ℋ 𝑅) ∩ (𝐶 ∨ℋ 𝑆))
Theorem	3oalem6 31594	Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴))    &   ⊢ 𝑆 = ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴))    ⇒   ⊢ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) ⊆ (𝐵 ∨ℋ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)))))
Theorem	3oai 31595	3OA (weak) orthoarguesian law. Equation IV of [GodowskiGreechie] p. 249. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴))    &   ⊢ 𝑆 = ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴))    ⇒   ⊢ ((𝐵 ∨ℋ 𝑅) ∩ (𝐶 ∨ℋ 𝑆)) ⊆ (𝐵 ∨ℋ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)))))
20.5.10  Projectors (cont.)
Theorem	pjorthi 31596	Projection components on orthocomplemented subspaces are orthogonal. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐻 ∈ Cℋ → (((projℎ‘𝐻)‘𝐴) ·ih ((projℎ‘(⊥‘𝐻))‘𝐵)) = 0)
Theorem	pjch1 31597	Property of identity projection. Remark in [Beran] p. 111. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → ((projℎ‘ ℋ)‘𝐴) = 𝐴)
Theorem	pjo 31598	The orthogonal projection. Lemma 4.4(i) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘(⊥‘𝐻))‘𝐴) = (((projℎ‘ ℋ)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)))
Theorem	pjcompi 31599	Component of a projection. (Contributed by NM, 31-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) → ((projℎ‘𝐻)‘(𝐴 +ℎ 𝐵)) = 𝐴)
Theorem	pjidmi 31600	A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ ((projℎ‘𝐻)‘((projℎ‘𝐻)‘𝐴)) = ((projℎ‘𝐻)‘𝐴)