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Theorem List for Metamath Proof Explorer - 33701-33800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2atneat 33701 The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄)) → ¬ (𝑃 𝑄) ∈ 𝐴)

Theoremllnn0 33702 A lattice line is nonzero. (Contributed by NM, 15-Jul-2012.)
0 = (0.‘𝐾)    &   𝑁 = (LLines‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑁) → 𝑋0 )

Theoremislln2a 33703 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 ∧ 𝑃𝐴𝑄𝐴) → ((𝑃 𝑄) ∈ 𝑁𝑃𝑄))

Theoremllnle 33704* 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 ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ∃𝑦𝑁 𝑦 𝑋)

Theorematcvrlln2 33705 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 ∧ 𝑃𝐴𝑋𝑁) ∧ 𝑃 𝑋) → 𝑃𝐶𝑋)

Theorematcvrlln 33706 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 ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → (𝑋𝐴𝑌𝑁))

TheoremllnexatN 33707* 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 ∧ 𝑋𝑁𝑃𝐴) ∧ 𝑃 𝑋) → ∃𝑞𝐴 (𝑃𝑞𝑋 = (𝑃 𝑞)))

Theoremllncmp 33708 If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.)
= (le‘𝐾)    &   𝑁 = (LLines‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋 𝑌𝑋 = 𝑌))

Theoremllnnlt 33709 Two lattice lines cannot satisfy the less than relation. (Contributed by NM, 26-Jun-2012.)
< = (lt‘𝐾)    &   𝑁 = (LLines‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → ¬ 𝑋 < 𝑌)

Theorem2llnmat 33710 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 ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐴)

Theorem2at0mat0 33711 Special case of 2atmat0 33712 where one atom could be zero. (Contributed by NM, 30-May-2013.)
= (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴 ∧ (𝑆𝐴𝑆 = 0 ) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴 ∨ ((𝑃 𝑄) (𝑅 𝑆)) = 0 ))

Theorem2atmat0 33712 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 ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴 ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴 ∨ ((𝑃 𝑄) (𝑅 𝑆)) = 0 ))

Theorem2atm 33713 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 ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 = ((𝑃 𝑄) (𝑅 𝑆)))

Theoremps-2c 33714 Variation of projective geometry axiom ps-2 33664. (Contributed by NM, 3-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑃 𝑅) ≠ (𝑆 𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ((𝑃 𝑅) (𝑆 𝑇)) ∈ 𝐴)

Theoremlplnset 33715* 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)       (𝐾𝐴𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})

Theoremislpln 33716* The predicate "is a lattice plane". (Contributed by NM, 16-Jun-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝑁 = (LLines‘𝐾)    &   𝑃 = (LPlanes‘𝐾)       (𝐾𝐴 → (𝑋𝑃 ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋)))

Theoremislpln4 33717* The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝑁 = (LLines‘𝐾)    &   𝑃 = (LPlanes‘𝐾)       ((𝐾𝐴𝑋𝐵) → (𝑋𝑃 ↔ ∃𝑦𝑁 𝑦𝐶𝑋))

Theoremlplni 33718 Condition implying a lattice plane. (Contributed by NM, 20-Jun-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝑁 = (LLines‘𝐾)    &   𝑃 = (LPlanes‘𝐾)       (((𝐾𝐷𝑌𝐵𝑋𝑁) ∧ 𝑋𝐶𝑌) → 𝑌𝑃)

Theoremislpln3 33719* 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 ∧ 𝑋𝐵) → (𝑋𝑃 ↔ ∃𝑦𝑁𝑝𝐴𝑝 𝑦𝑋 = (𝑦 𝑝))))

Theoremlplnbase 33720 A lattice plane is a lattice element. (Contributed by NM, 17-Jun-2012.)
𝐵 = (Base‘𝐾)    &   𝑃 = (LPlanes‘𝐾)       (𝑋𝑃𝑋𝐵)

Theoremislpln5 33721* 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 ∧ 𝑋𝐵) → (𝑋𝑃 ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))

Theoremislpln2 33722* 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 → (𝑋𝑃 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))

Theoremlplni2 33723 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 ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ((𝑄 𝑅) 𝑆) ∈ 𝑃)

Theoremlvolex3N 33724* 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 ∧ 𝑋𝑃) → ∃𝑞𝐴 ¬ 𝑞 𝑋)

TheoremllnmlplnN 33725 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 ∧ 𝑋𝑁𝑌𝑃) ∧ (¬ 𝑋 𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐴)

Theoremlplnle 33726* 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 ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴 ∧ ¬ 𝑋𝑁)) → ∃𝑦𝑃 𝑦 𝑋)

Theoremlplnnle2at 33727 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 ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → ¬ 𝑋 (𝑄 𝑅))

Theoremlplnnleat 33728 A lattice plane cannot majorize an atom. (Contributed by NM, 14-Jul-2012.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑃 = (LPlanes‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) → ¬ 𝑋 𝑄)

Theoremlplnnlelln 33729 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 ∧ 𝑋𝑃𝑌𝑁) → ¬ 𝑋 𝑌)

Theorem2atnelpln 33730 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 ∧ 𝑄𝐴𝑅𝐴) → ¬ (𝑄 𝑅) ∈ 𝑃)

Theoremlplnneat 33731 No lattice plane is an atom. (Contributed by NM, 15-Jul-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (LPlanes‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑃) → ¬ 𝑋𝐴)

Theoremlplnnelln 33732 No lattice plane is a lattice line. (Contributed by NM, 19-Jun-2012.)
𝑁 = (LLines‘𝐾)    &   𝑃 = (LPlanes‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑃) → ¬ 𝑋𝑁)

Theoremlplnn0N 33733 A lattice plane is nonzero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
0 = (0.‘𝐾)    &   𝑃 = (LPlanes‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑃) → 𝑋0 )

Theoremislpln2a 33734 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 ∧ (𝑄𝐴𝑅𝐴𝑆𝐴)) → (((𝑄 𝑅) 𝑆) ∈ 𝑃 ↔ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))))

Theoremislpln2ah 33735 The predicate "is a lattice plane" for join of atoms. Version of islpln2a 33734 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 ∧ (𝑄𝐴𝑅𝐴𝑆𝐴)) → (𝑌𝑃 ↔ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))))

TheoremlplnriaN 33736 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 ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑌𝑃) → ¬ 𝑄 (𝑅 𝑆))

TheoremlplnribN 33737 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 ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑌𝑃) → ¬ 𝑅 (𝑄 𝑆))

Theoremlplnric 33738 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 ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑌𝑃) → ¬ 𝑆 (𝑄 𝑅))

Theoremlplnri1 33739 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 ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑌𝑃) → 𝑄𝑅)

Theoremlplnri2N 33740 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 ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑌𝑃) → 𝑄𝑆)

Theoremlplnri3N 33741 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 ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑌𝑃) → 𝑅𝑆)

TheoremlplnllnneN 33742 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 ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑌𝑃) → (𝑄 𝑆) ≠ (𝑅 𝑆))

Theoremllncvrlpln2 33743 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 ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋𝐶𝑌)

Theoremllncvrlpln 33744 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 ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → (𝑋𝑁𝑌𝑃))

Theorem2lplnmN 33745 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 ∧ 𝑋𝑃𝑌𝑃) ∧ 𝑋𝐶(𝑋 𝑌)) → (𝑋 𝑌) ∈ 𝑁)

Theorem2llnmj 33746 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 ∧ 𝑋𝑁𝑌𝑁) → ((𝑋 𝑌) ∈ 𝐴 ↔ (𝑋 𝑌) ∈ 𝑃))

Theorem2atmat 33747 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 ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴)

Theoremlplncmp 33748 If two lattice planes are comparable, they are equal. (Contributed by NM, 24-Jun-2012.)
= (le‘𝐾)    &   𝑃 = (LPlanes‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑃) → (𝑋 𝑌𝑋 = 𝑌))

TheoremlplnexatN 33749* 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 ∧ 𝑋𝑃𝑌𝑁) ∧ 𝑌 𝑋) → ∃𝑞𝐴𝑞 𝑌𝑋 = (𝑌 𝑞)))

TheoremlplnexllnN 33750* 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 ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))

Theoremlplnnlt 33751 Two lattice planes cannot satisfy the less than relation. (Contributed by NM, 7-Jul-2012.)
< = (lt‘𝐾)    &   𝑃 = (LPlanes‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑃) → ¬ 𝑋 < 𝑌)

Theorem2llnjaN 33752 The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 33753 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 ∧ 𝑊𝑃) ∧ (𝑄𝐴𝑅𝐴𝑄𝑅) ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ ((𝑄 𝑅) 𝑊 ∧ (𝑆 𝑇) 𝑊 ∧ (𝑄 𝑅) ≠ (𝑆 𝑇))) → ((𝑄 𝑅) (𝑆 𝑇)) = 𝑊)

Theorem2llnjN 33753 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 ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (𝑋 𝑌) = 𝑊)

Theorem2llnm2N 33754 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 ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (𝑋 𝑌) ∈ 𝐴)

Theorem2llnm3N 33755 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 ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊)) → (𝑋 𝑌) ≠ 0 )

Theorem2llnm4 33756 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 ∧ (𝑃𝐴𝑋𝑁𝑌𝑁) ∧ (𝑃 𝑋𝑃 𝑌)) → (𝑋 𝑌) ≠ 0 )

Theorem2llnmeqat 33757 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 ∧ (𝑋𝑁𝑌𝑁𝑃𝐴) ∧ (𝑋𝑌𝑃 (𝑋 𝑌))) → 𝑃 = (𝑋 𝑌))

Theoremlvolset 33758* 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)       (𝐾𝐴𝑉 = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})

Theoremislvol 33759* 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)       (𝐾𝐴 → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃𝑦𝑃 𝑦𝐶𝑋)))

Theoremislvol4 33760* 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)       ((𝐾𝐴𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑦𝑃 𝑦𝐶𝑋))

Theoremlvoli 33761 Condition implying a 3-dim lattice volume. (Contributed by NM, 1-Jul-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝑃 = (LPlanes‘𝐾)    &   𝑉 = (LVols‘𝐾)       (((𝐾𝐷𝑌𝐵𝑋𝑃) ∧ 𝑋𝐶𝑌) → 𝑌𝑉)

Theoremislvol3 33762* 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 ∧ 𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑦𝑃𝑝𝐴𝑝 𝑦𝑋 = (𝑦 𝑝))))

Theoremlvoli3 33763 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 ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) ∈ 𝑉)

Theoremlvolbase 33764 A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.)
𝐵 = (Base‘𝐾)    &   𝑉 = (LVols‘𝐾)       (𝑋𝑉𝑋𝐵)

Theoremislvol5 33765* 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 ∧ 𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))

Theoremislvol2 33766* 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 → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))))

Theoremlvoli2 33767 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 ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) ∈ 𝑉)

Theoremlvolnle3at 33768 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 ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))

Theoremlvolnleat 33769 An atom cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑉𝑃𝐴) → ¬ 𝑋 𝑃)

Theoremlvolnlelln 33770 A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
= (le‘𝐾)    &   𝑁 = (LLines‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → ¬ 𝑋 𝑌)

Theoremlvolnlelpln 33771 A lattice plane cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
= (le‘𝐾)    &   𝑃 = (LPlanes‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) → ¬ 𝑋 𝑌)

Theorem3atnelvolN 33772 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 ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑉)

Theorem2atnelvolN 33773 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 ∧ 𝑃𝐴𝑄𝐴) → ¬ (𝑃 𝑄) ∈ 𝑉)

TheoremlvolneatN 33774 No lattice volume is an atom. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑉) → ¬ 𝑋𝐴)

Theoremlvolnelln 33775 No lattice volume is a lattice line. (Contributed by NM, 15-Jul-2012.)
𝑁 = (LLines‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑉) → ¬ 𝑋𝑁)

Theoremlvolnelpln 33776 No lattice volume is a lattice plane. (Contributed by NM, 19-Jun-2012.)
𝑃 = (LPlanes‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑉) → ¬ 𝑋𝑃)

Theoremlvoln0N 33777 A lattice volume is nonzero. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
0 = (0.‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑉) → 𝑋0 )

Theoremislvol2aN 33778 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 ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((((𝑃 𝑄) 𝑅) 𝑆) ∈ 𝑉 ↔ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))))

Theorem4atlem0a 33779 Lemma for 4at 33799. (Contributed by NM, 10-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ¬ 𝑅 ((𝑃 𝑄) 𝑆))

Theorem4atlem0ae 33780 Lemma for 4at 33799. (Contributed by NM, 10-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑄 (𝑃 𝑅))

Theorem4atlem0be 33781 Lemma for 4at 33799. (Contributed by NM, 10-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃𝑅)

Theorem4atlem3 33782 Lemma for 4at 33799. Break inequality into 4 cases. (Contributed by NM, 8-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((¬ 𝑃 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑄 ((𝑇 𝑈) 𝑉)) ∨ (¬ 𝑅 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑆 ((𝑇 𝑈) 𝑉))))

Theorem4atlem3a 33783 Lemma for 4at 33799. Break inequality into 3 cases. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (¬ 𝑄 ((𝑃 𝑈) 𝑉) ∨ ¬ 𝑅 ((𝑃 𝑈) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑈) 𝑉)))

Theorem4atlem3b 33784 Lemma for 4at 33799. Break inequality into 2 cases. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)))

Theorem4atlem4a 33785 Lemma for 4at 33799. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = (𝑃 ((𝑄 𝑅) 𝑆)))

Theorem4atlem4b 33786 Lemma for 4at 33799. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = (𝑄 ((𝑃 𝑅) 𝑆)))

Theorem4atlem4c 33787 Lemma for 4at 33799. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = (𝑅 ((𝑃 𝑄) 𝑆)))

Theorem4atlem4d 33788 Lemma for 4at 33799. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = (𝑆 ((𝑃 𝑄) 𝑅)))

Theorem4atlem9 33789 Lemma for 4at 33799. Substitute 𝑊 for 𝑆. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑊𝐴) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) → (𝑆 ((𝑃 𝑄) (𝑅 𝑊)) ↔ ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑄) (𝑅 𝑊))))

Theorem4atlem10a 33790 Lemma for 4at 33799. Substitute 𝑉 for 𝑅. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑉𝐴𝑊𝐴) ∧ ¬ 𝑅 ((𝑃 𝑄) 𝑊)) → (𝑅 ((𝑃 𝑄) (𝑉 𝑊)) ↔ ((𝑃 𝑄) (𝑅 𝑊)) = ((𝑃 𝑄) (𝑉 𝑊))))

Theorem4atlem10b 33791 Lemma for 4at 33799. Substitute 𝑉 for 𝑅 (cont.). (Contributed by NM, 10-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑊𝐴 ∧ ¬ 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)))

Theorem4atlem10 33792 Lemma for 4at 33799. Combine both possible cases. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑅𝐴𝑆𝐴) ∧ 𝑉𝐴𝑊𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑅 𝑆) ((𝑃 𝑄) (𝑉 𝑊)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑄) (𝑉 𝑊))))

Theorem4atlem11a 33793 Lemma for 4at 33799. Substitute 𝑈 for 𝑄. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) → (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑄) (𝑉 𝑊)) = ((𝑃 𝑈) (𝑉 𝑊))))

Theorem4atlem11b 33794 Lemma for 4at 33799. Substitute 𝑈 for 𝑄 (cont.). (Contributed by NM, 10-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 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)))

Theorem4atlem11 33795 Lemma for 4at 33799. Combine all three possible cases. (Contributed by NM, 10-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑄 (𝑅 𝑆)) ((𝑃 𝑈) (𝑉 𝑊)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑈) (𝑉 𝑊))))

Theorem4atlem12a 33796 Lemma for 4at 33799. Substitute 𝑇 for 𝑃. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) → (𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑈) (𝑉 𝑊)) = ((𝑇 𝑈) (𝑉 𝑊))))

Theorem4atlem12b 33797 Lemma for 4at 33799. Substitute 𝑇 for 𝑃 (cont.). (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 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)))

Theorem4atlem12 33798 Lemma for 4at 33799. Combine all four possible cases. (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) (𝑅 𝑆)) ((𝑇 𝑈) (𝑉 𝑊)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑇 𝑈) (𝑉 𝑊))))

Theorem4at 33799 Four atoms determine a lattice volume uniquely. Three-dimensional analogue of ps-1 33663 and 3at 33676. (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) (𝑅 𝑆)) ((𝑇 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑇 𝑈) (𝑉 𝑊))))

Theorem4at2 33800 Four atoms determine a lattice volume uniquely. (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((((𝑃 𝑄) 𝑅) 𝑆) (((𝑇 𝑈) 𝑉) 𝑊) ↔ (((𝑃 𝑄) 𝑅) 𝑆) = (((𝑇 𝑈) 𝑉) 𝑊)))

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