P. 376 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37501-37600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hlatexch3N 37501	Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑅))
Theorem	hlatexch4 37502	Exchange 2 atoms. (Contributed by NM, 13-May-2013.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑆))
Theorem	ps-2b 37503	Variation of projective geometry axiom ps-2 37499. (Contributed by NM, 3-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ≠ 0 )
Theorem	3atlem1 37504	Lemma for 3at 37511. (Contributed by NM, 22-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3atlem2 37505	Lemma for 3at 37511. (Contributed by NM, 22-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3atlem3 37506	Lemma for 3at 37511. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑈 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3atlem4 37507	Lemma for 3at 37511. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑅))
Theorem	3atlem5 37508	Lemma for 3at 37511. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3atlem6 37509	Lemma for 3at 37511. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄 ∧ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3atlem7 37510	Lemma for 3at 37511. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3at 37511	Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 37498 for lines and 4at 37634 for volumes. I could not find this proof in the literature on projective geometry (where it is either given as an axiom or stated as an unproved fact), but it is similar to Theorem 15 of Veblen, "The Foundations of Geometry" (1911), p. 18, which uses different axioms. This proof was written before I became aware of Veblen's, and it is possible that a shorter proof could be obtained by using Veblen's proof for hints. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄)) → (((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈)))
20.24.12  Projective geometries based on Hilbert lattices
Syntax	clln 37512	Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.
class LLines
Syntax	clpl 37513	Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.
class LPlanes
Syntax	clvol 37514	Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.
class LVols
Syntax	clines 37515	Extend class notation with set of all projective lines for a Hilbert lattice.
class Lines
Syntax	cpointsN 37516	Extend class notation with set of all projective points.
class Points
Syntax	cpsubsp 37517	Extend class notation with set of all projective subspaces.
class PSubSp
Syntax	cpmap 37518	Extend class notation with projective map.
class pmap
Definition	df-llines 37519*	Define the set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice 𝑘, in other words all elements of height 2 (or lattice dimension 2 or projective dimension 1). (Contributed by NM, 16-Jun-2012.)
⊢ LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
Definition	df-lplanes 37520*	Define the set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice 𝑘, in other words all elements of height 3 (or lattice dimension 3 or projective dimension 2). (Contributed by NM, 16-Jun-2012.)
⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
Definition	df-lvols 37521*	Define the set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice 𝑘, in other words all elements of height 4 (or lattice dimension 4 or projective dimension 3). (Contributed by NM, 1-Jul-2012.)
⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LPlanes‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
Definition	df-lines 37522*	Define set of all projective lines for a Hilbert lattice (actually in any set at all, for simplicity). The join of two distinct atoms equals a line. Definition of lines in item 1 of [Holland95] p. 222. (Contributed by NM, 19-Sep-2011.)
⊢ Lines = (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})})
Definition	df-pointsN 37523*	Define set of all projective points in a Hilbert lattice (actually in any set at all, for simplicity). A projective point is the singleton of a lattice atom. Definition 15.1 of [MaedaMaeda] p. 61. Note that item 1 in [Holland95] p. 222 defines a point as the atom itself, but this leads to a complicated subspace ordering that may be either membership or inclusion based on its arguments. (Contributed by NM, 2-Oct-2011.)
⊢ Points = (𝑘 ∈ V ↦ {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}})
Definition	df-psubsp 37524*	Define set of all projective subspaces. Based on definition of subspace in [Holland95] p. 212. (Contributed by NM, 2-Oct-2011.)
⊢ PSubSp = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠))})
Definition	df-pmap 37525*	Define projective map for 𝑘 at 𝑎. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)
⊢ pmap = (𝑘 ∈ V ↦ (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎}))
Theorem	llnset 37526*	The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})
Theorem	islln 37527*	The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)))
Theorem	islln4 37528*	The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))
Theorem	llni 37529	Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝑋 ∈ 𝑁)
Theorem	llnbase 37530	A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵)
Theorem	islln3 37531*	The predicate "is a lattice line". (Contributed by NM, 17-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞))))
Theorem	islln2 37532*	The predicate "is a lattice line". (Contributed by NM, 23-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞)))))
Theorem	llni2 37533	The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∨ 𝑄) ∈ 𝑁)
Theorem	llnnleat 37534	An atom cannot majorize a lattice line. (Contributed by NM, 8-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑋 ≤ 𝑃)
Theorem	llnneat 37535	A lattice line is not an atom. (Contributed by NM, 19-Jun-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → ¬ 𝑋 ∈ 𝐴)
Theorem	2atneat 37536	The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ¬ (𝑃 ∨ 𝑄) ∈ 𝐴)
Theorem	llnn0 37537	A lattice line is nonzero. (Contributed by NM, 15-Jul-2012.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋 ≠ 0 )
Theorem	islln2a 37538	The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ∨ 𝑄) ∈ 𝑁 ↔ 𝑃 ≠ 𝑄))
Theorem	llnle 37539*	Any element greater than 0 and not an atom majorizes a lattice line. (Contributed by NM, 28-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑋 ≠ 0 ∧ ¬ 𝑋 ∈ 𝐴)) → ∃𝑦 ∈ 𝑁 𝑦 ≤ 𝑋)
Theorem	atcvrlln2 37540	An atom under a line is covered by it. (Contributed by NM, 2-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁) ∧ 𝑃 ≤ 𝑋) → 𝑃𝐶𝑋)
Theorem	atcvrlln 37541	An element covering an atom is a lattice line and vice-versa. (Contributed by NM, 18-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 ∈ 𝐴 ↔ 𝑌 ∈ 𝑁))
Theorem	llnexatN 37542*	Given an atom on a line, there is another atom whose join equals the line. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → ∃𝑞 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ 𝑋 = (𝑃 ∨ 𝑞)))
Theorem	llncmp 37543	If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
Theorem	llnnlt 37544	Two lattice lines cannot satisfy the less than relation. (Contributed by NM, 26-Jun-2012.)
⊢ < = (lt‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 < 𝑌)
Theorem	2llnmat 37545	Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑋 ≠ 𝑌 ∧ (𝑋 ∧ 𝑌) ≠ 0 )) → (𝑋 ∧ 𝑌) ∈ 𝐴)
Theorem	2at0mat0 37546	Special case of 2atmat0 37547 where one atom could be zero. (Contributed by NM, 30-May-2013.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ (𝑆 ∈ 𝐴 ∨ 𝑆 = 0 ) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → (((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ 𝐴 ∨ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) = 0 ))
Theorem	2atmat0 37547	The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → (((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ 𝐴 ∨ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) = 0 ))
Theorem	2atm 37548	An atom majorized by two different atom joins (which could be atoms or lines) is equal to their intersection. (Contributed by NM, 30-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑇 = ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)))
Theorem	ps-2c 37549	Variation of projective geometry axiom ps-2 37499. (Contributed by NM, 3-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ ((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑃 ∨ 𝑅) ≠ (𝑆 ∨ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ∈ 𝐴)
Theorem	lplnset 37550*	The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐴 → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
Theorem	islpln 37551*	The predicate "is a lattice plane". (Contributed by NM, 16-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋)))
Theorem	islpln4 37552*	The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑃 ↔ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋))
Theorem	lplni 37553	Condition implying a lattice plane. (Contributed by NM, 20-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝑃)
Theorem	islpln3 37554*	The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑃 ↔ ∃𝑦 ∈ 𝑁 ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝))))
Theorem	lplnbase 37555	A lattice plane is a lattice element. (Contributed by NM, 17-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝐵)
Theorem	islpln5 37556*	The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 24-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑃 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝 ∨ 𝑞) ∧ 𝑋 = ((𝑝 ∨ 𝑞) ∨ 𝑟))))
Theorem	islpln2 37557*	The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 25-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝 ∨ 𝑞) ∧ 𝑋 = ((𝑝 ∨ 𝑞) ∨ 𝑟)))))
Theorem	lplni2 37558	The join of 3 different atoms is a lattice plane. (Contributed by NM, 4-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))) → ((𝑄 ∨ 𝑅) ∨ 𝑆) ∈ 𝑃)
Theorem	lvolex3N 37559*	There is an atom outside of a lattice plane i.e. a 3-dimensional lattice volume exists. (Contributed by NM, 28-Jul-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋)
Theorem	llnmlplnN 37560	The intersection of a line with a plane not containing it is an atom. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ (¬ 𝑋 ≤ 𝑌 ∧ (𝑋 ∧ 𝑌) ≠ 0 )) → (𝑋 ∧ 𝑌) ∈ 𝐴)
Theorem	lplnle 37561*	Any element greater than 0 and not an atom and not a lattice line majorizes a lattice plane. (Contributed by NM, 28-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑋 ≠ 0 ∧ ¬ 𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑁)) → ∃𝑦 ∈ 𝑃 𝑦 ≤ 𝑋)
Theorem	lplnnle2at 37562	A lattice line (or atom) cannot majorize a lattice plane. (Contributed by NM, 8-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))
Theorem	lplnnleat 37563	A lattice plane cannot majorize an atom. (Contributed by NM, 14-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑋 ≤ 𝑄)
Theorem	lplnnlelln 37564	A lattice plane is not less than or equal to a lattice line. (Contributed by NM, 14-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 ≤ 𝑌)
Theorem	2atnelpln 37565	The join of two atoms is not a lattice plane. (Contributed by NM, 16-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ¬ (𝑄 ∨ 𝑅) ∈ 𝑃)
Theorem	lplnneat 37566	No lattice plane is an atom. (Contributed by NM, 15-Jul-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ¬ 𝑋 ∈ 𝐴)
Theorem	lplnnelln 37567	No lattice plane is a lattice line. (Contributed by NM, 19-Jun-2012.)
⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ¬ 𝑋 ∈ 𝑁)
Theorem	lplnn0N 37568	A lattice plane is nonzero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → 𝑋 ≠ 0 )
Theorem	islpln2a 37569	The predicate "is a lattice plane" for join of atoms. (Contributed by NM, 16-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (((𝑄 ∨ 𝑅) ∨ 𝑆) ∈ 𝑃 ↔ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))))
Theorem	islpln2ah 37570	The predicate "is a lattice plane" for join of atoms. Version of islpln2a 37569 expressed with an abbreviation hypothesis. (Contributed by NM, 30-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑌 ∈ 𝑃 ↔ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))))
Theorem	lplnriaN 37571	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → ¬ 𝑄 ≤ (𝑅 ∨ 𝑆))
Theorem	lplnribN 37572	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → ¬ 𝑅 ≤ (𝑄 ∨ 𝑆))
Theorem	lplnric 37573	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))
Theorem	lplnri1 37574	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → 𝑄 ≠ 𝑅)
Theorem	lplnri2N 37575	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → 𝑄 ≠ 𝑆)
Theorem	lplnri3N 37576	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → 𝑅 ≠ 𝑆)
Theorem	lplnllnneN 37577	Two lattice lines defined by atoms defining a lattice plane are not equal. (Contributed by NM, 9-Oct-2012.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → (𝑄 ∨ 𝑆) ≠ (𝑅 ∨ 𝑆))
Theorem	llncvrlpln2 37578	A lattice line under a lattice plane is covered by it. (Contributed by NM, 24-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → 𝑋𝐶𝑌)
Theorem	llncvrlpln 37579	An element covering a lattice line is a lattice plane and vice-versa. (Contributed by NM, 26-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 ∈ 𝑁 ↔ 𝑌 ∈ 𝑃))
Theorem	2lplnmN 37580	If the join of two lattice planes covers one of them, their meet is a lattice line. (Contributed by NM, 30-Jun-2012.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝑁)
Theorem	2llnmj 37581	The meet of two lattice lines is an atom iff their join is a lattice plane. (Contributed by NM, 27-Jun-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → ((𝑋 ∧ 𝑌) ∈ 𝐴 ↔ (𝑋 ∨ 𝑌) ∈ 𝑃))
Theorem	2atmat 37582	The meet of two intersecting lines (expressed as joins of atoms) is an atom. (Contributed by NM, 21-Nov-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑆 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ 𝐴)
Theorem	lplncmp 37583	If two lattice planes are comparable, they are equal. (Contributed by NM, 24-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
Theorem	lplnexatN 37584*	Given a lattice line on a lattice plane, there is an atom whose join with the line equals the plane. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑌 ∧ 𝑋 = (𝑌 ∨ 𝑞)))
Theorem	lplnexllnN 37585*	Given an atom on a lattice plane, there is a lattice line whose join with the atom equals the plane. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
Theorem	lplnnlt 37586	Two lattice planes cannot satisfy the less than relation. (Contributed by NM, 7-Jul-2012.)
⊢ < = (lt‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ¬ 𝑋 < 𝑌)
Theorem	2llnjaN 37587	The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 37588 in terms of atoms). (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝑃) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ≠ 𝑅) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ ((𝑄 ∨ 𝑅) ≤ 𝑊 ∧ (𝑆 ∨ 𝑇) ≤ 𝑊 ∧ (𝑄 ∨ 𝑅) ≠ (𝑆 ∨ 𝑇))) → ((𝑄 ∨ 𝑅) ∨ (𝑆 ∨ 𝑇)) = 𝑊)
Theorem	2llnjN 37588	The join of two different lattice lines in a lattice plane equals the plane. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∨ 𝑌) = 𝑊)
Theorem	2llnm2N 37589	The meet of two different lattice lines in a lattice plane is an atom. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝐴)
Theorem	2llnm3N 37590	Two lattice lines in a lattice plane always meet. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≠ 0 )
Theorem	2llnm4 37591	Two lattice lines that majorize the same atom always meet. (Contributed by NM, 20-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ≠ 0 )
Theorem	2llnmeqat 37592	An atom equals the intersection of two majorizing lines. (Contributed by NM, 3-Apr-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝑃 = (𝑋 ∧ 𝑌))
Theorem	lvolset 37593*	The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐴 → 𝑉 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥})
Theorem	islvol 37594*	The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋)))
Theorem	islvol4 37595*	The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑉 ↔ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋))
Theorem	lvoli 37596	Condition implying a 3-dim lattice volume. (Contributed by NM, 1-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝑉)
Theorem	islvol3 37597*	The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑉 ↔ ∃𝑦 ∈ 𝑃 ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝))))
Theorem	lvoli3 37598	Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) ∈ 𝑉)
Theorem	lvolbase 37599	A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐵)
Theorem	islvol5 37600*	The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑉 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ((𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑝 ∨ 𝑞) ∨ 𝑟)) ∧ 𝑋 = (((𝑝 ∨ 𝑞) ∨ 𝑟) ∨ 𝑠))))