P. 396 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39501-39600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	atcvrlln2 39501	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 39502	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 39503*	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 39504	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 39505	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 39506	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 39507	Special case of 2atmat0 39508 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 39508	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 39509	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 39510	Variation of projective geometry axiom ps-2 39460. (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 39511*	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 39512*	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 39513*	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 39514	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 39515*	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 39516	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 39517*	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 39518*	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 39519	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 39520*	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 39521	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 39522*	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 39523	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 39524	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 39525	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 39526	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 39527	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 39528	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 39529	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 39530	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 39531	The predicate "is a lattice plane" for join of atoms. Version of islpln2a 39530 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 39532	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 39533	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 39534	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 39535	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 39536	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 39537	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 39538	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 39539	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 39540	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 39541	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 39542	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 39543	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 39544	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 39545*	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 39546*	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 39547	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 39548	The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 39549 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 39549	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 39550	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 39551	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 39552	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 39553	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 39554*	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 39555*	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 39556*	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 39557	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 39558*	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 39559	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 39560	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 39561*	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))))
Theorem	islvol2 39562*	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 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ((𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑝 ∨ 𝑞) ∨ 𝑟)) ∧ 𝑋 = (((𝑝 ∨ 𝑞) ∨ 𝑟) ∨ 𝑠)))))
Theorem	lvoli2 39563	The join of 4 different atoms is a lattice volume. (Contributed by NM, 8-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (((𝑃 ∨ 𝑄) ∨ 𝑅) ∨ 𝑆) ∈ 𝑉)
Theorem	lvolnle3at 39564	A lattice plane (or lattice line or atom) cannot majorize a lattice volume. (Contributed by NM, 8-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
Theorem	lvolnleat 39565	An atom cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑋 ≤ 𝑃)
Theorem	lvolnlelln 39566	A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 ≤ 𝑌)
Theorem	lvolnlelpln 39567	A lattice plane cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) → ¬ 𝑋 ≤ 𝑌)
Theorem	3atnelvolN 39568	The join of 3 atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑉)
Theorem	2atnelvolN 39569	The join of two atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ (𝑃 ∨ 𝑄) ∈ 𝑉)
Theorem	lvolneatN 39570	No lattice volume is an atom. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 ∈ 𝐴)
Theorem	lvolnelln 39571	No lattice volume is a lattice line. (Contributed by NM, 15-Jul-2012.)
⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 ∈ 𝑁)
Theorem	lvolnelpln 39572	No lattice volume is a lattice plane. (Contributed by NM, 19-Jun-2012.)
⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 ∈ 𝑃)
Theorem	lvoln0N 39573	A lattice volume is nonzero. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → 𝑋 ≠ 0 )
Theorem	islvol2aN 39574	The predicate "is a lattice volume". (Contributed by NM, 16-Jul-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((((𝑃 ∨ 𝑄) ∨ 𝑅) ∨ 𝑆) ∈ 𝑉 ↔ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))))
Theorem	4atlem0a 39575	Lemma for 4at 39595. (Contributed by NM, 10-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ¬ 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆))
Theorem	4atlem0ae 39576	Lemma for 4at 39595. (Contributed by NM, 10-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑄 ≤ (𝑃 ∨ 𝑅))
Theorem	4atlem0be 39577	Lemma for 4at 39595. (Contributed by NM, 10-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑅)
Theorem	4atlem3 39578	Lemma for 4at 39595. Break inequality into 4 cases. (Contributed by NM, 8-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ((¬ 𝑃 ≤ ((𝑇 ∨ 𝑈) ∨ 𝑉) ∨ ¬ 𝑄 ≤ ((𝑇 ∨ 𝑈) ∨ 𝑉)) ∨ (¬ 𝑅 ≤ ((𝑇 ∨ 𝑈) ∨ 𝑉) ∨ ¬ 𝑆 ≤ ((𝑇 ∨ 𝑈) ∨ 𝑉))))
Theorem	4atlem3a 39579	Lemma for 4at 39595. Break inequality into 3 cases. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (¬ 𝑄 ≤ ((𝑃 ∨ 𝑈) ∨ 𝑉) ∨ ¬ 𝑅 ≤ ((𝑃 ∨ 𝑈) ∨ 𝑉) ∨ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑈) ∨ 𝑉)))
Theorem	4atlem3b 39580	Lemma for 4at 39595. Break inequality into 2 cases. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (¬ 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑉) ∨ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑉)))
Theorem	4atlem4a 39581	Lemma for 4at 39595. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = (𝑃 ∨ ((𝑄 ∨ 𝑅) ∨ 𝑆)))
Theorem	4atlem4b 39582	Lemma for 4at 39595. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∨ 𝑆)))
Theorem	4atlem4c 39583	Lemma for 4at 39595. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = (𝑅 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑆)))
Theorem	4atlem4d 39584	Lemma for 4at 39595. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = (𝑆 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
Theorem	4atlem9 39585	Lemma for 4at 39595. Substitute 𝑊 for 𝑆. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) → (𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑊)) ↔ ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑊))))
Theorem	4atlem10a 39586	Lemma for 4at 39595. Substitute 𝑉 for 𝑅. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑊)) → (𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊)) ↔ ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑊)) = ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊))))
Theorem	4atlem10b 39587	Lemma for 4at 39595. Substitute 𝑉 for 𝑅 (cont.). (Contributed by NM, 10-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑊) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) ∧ (𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊)) ∧ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊)))) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊)))
Theorem	4atlem10 39588	Lemma for 4at 39595. Combine both possible cases. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ((𝑅 ∨ 𝑆) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊))))
Theorem	4atlem11a 39589	Lemma for 4at 39595. Substitute 𝑈 for 𝑄. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → (𝑄 ≤ ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) ↔ ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊)) = ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊))))
Theorem	4atlem11b 39590	Lemma for 4at 39595. Substitute 𝑈 for 𝑄 (cont.). (Contributed by NM, 10-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) ∧ (𝑄 ≤ ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) ∧ 𝑅 ≤ ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) ∧ 𝑆 ≤ ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)))) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)))
Theorem	4atlem11 39591	Lemma for 4at 39595. Combine all three possible cases. (Contributed by NM, 10-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ((𝑄 ∨ (𝑅 ∨ 𝑆)) ≤ ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊))))
Theorem	4atlem12a 39592	Lemma for 4at 39595. Substitute 𝑇 for 𝑃. (Contributed by NM, 9-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑃 ≤ ((𝑈 ∨ 𝑉) ∨ 𝑊)) → (𝑃 ≤ ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) ↔ ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) = ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊))))
Theorem	4atlem12b 39593	Lemma for 4at 39595. Substitute 𝑇 for 𝑃 (cont.). (Contributed by NM, 11-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑈 ∨ 𝑉) ∨ 𝑊)) ∧ ((𝑃 ≤ ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) ∧ 𝑄 ≤ ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊))) ∧ (𝑅 ≤ ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) ∧ 𝑆 ≤ ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊))))) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)))
Theorem	4atlem12 39594	Lemma for 4at 39595. Combine all four possible cases. (Contributed by NM, 11-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) ≤ ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊))))
Theorem	4at 39595	Four atoms determine a lattice volume uniquely. Three-dimensional analogue of ps-1 39459 and 3at 39472. (Contributed by NM, 11-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) ≤ ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) ↔ ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑇 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊))))
Theorem	4at2 39596	Four atoms determine a lattice volume uniquely. (Contributed by NM, 11-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ((((𝑃 ∨ 𝑄) ∨ 𝑅) ∨ 𝑆) ≤ (((𝑇 ∨ 𝑈) ∨ 𝑉) ∨ 𝑊) ↔ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∨ 𝑆) = (((𝑇 ∨ 𝑈) ∨ 𝑉) ∨ 𝑊)))
Theorem	lplncvrlvol2 39597	A lattice line under a lattice plane is covered by it. (Contributed by NM, 12-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑋𝐶𝑌)
Theorem	lplncvrlvol 39598	An element covering a lattice plane is a lattice volume and vice-versa. (Contributed by NM, 15-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝑃 = (LPlanes‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑉))
Theorem	lvolcmp 39599	If two lattice planes are comparable, they are equal. (Contributed by NM, 12-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
Theorem	lvolnltN 39600	Two lattice volumes cannot satisfy the less than relation. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
⊢ < = (lt‘𝐾)    &   ⊢ 𝑉 = (LVols‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋 < 𝑌)