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Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhllat 35501 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ Lat)

Theoremhllatd 35502 Deduction form of hllat 35501. A Hilbert lattice is a lattice. (Contributed by BJ, 14-Aug-2022.)
(𝜑𝐾 ∈ HL)       (𝜑𝐾 ∈ Lat)

Theoremhlomcmat 35503 A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))

Theoremhlpos 35504 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ Poset)

Theoremhlatjcl 35505 Closure of join operation. Frequently-used special case of latjcl 17437 for atoms. (Contributed by NM, 15-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 𝑌) ∈ 𝐵)

Theoremhlatjcom 35506 Commutatitivity of join operation. Frequently-used special case of latjcom 17445 for atoms. (Contributed by NM, 15-Jun-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 𝑌) = (𝑌 𝑋))

Theoremhlatjidm 35507 Idempotence of join operation. Frequently-used special case of latjcom 17445 for atoms. (Contributed by NM, 15-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑋 𝑋) = 𝑋)

Theoremhlatjass 35508 Lattice join is associative. Frequently-used special case of latjass 17481 for atoms. (Contributed by NM, 27-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = (𝑃 (𝑄 𝑅)))

Theoremhlatj12 35509 Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 17483 for atoms. (Contributed by NM, 4-Jun-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 (𝑄 𝑅)) = (𝑄 (𝑃 𝑅)))

Theoremhlatj32 35510 Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 17483 for atoms. (Contributed by NM, 21-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑃 𝑅) 𝑄))

Theoremhlatjrot 35511 Rotate lattice join of 3 classes. Frequently-used special case of latjrot 17486 for atoms. (Contributed by NM, 2-Aug-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑅 𝑃) 𝑄))

Theoremhlatj4 35512 Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 17487 for atoms. (Contributed by NM, 9-Aug-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑅) (𝑄 𝑆)))

Theoremhlatlej1 35513 A join's first argument is less than or equal to the join. Special case of latlej1 17446 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))

Theoremhlatlej2 35514 A join's second argument is less than or equal to the join. Special case of latlej2 17447 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))

TheoremglbconN 35515* De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume HL for convenience. (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝐺𝑆) = ( ‘(𝑈‘{𝑥𝐵 ∣ ( 𝑥) ∈ 𝑆})))

TheoremglbconxN 35516* De Morgan's law for GLB and LUB. Index-set version of glbconN 35515, where we read 𝑆 as 𝑆(𝑖). (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})))

Theorematnlej1 35517 If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑄)

Theorematnlej2 35518 If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑅)

Theoremhlsuprexch 35519* A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ((𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))

Theoremhlexch1 35520 A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Theoremhlexch2 35521 A Hilbert lattice has the exchange property. (Contributed by NM, 6-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) → 𝑄 (𝑃 𝑋)))

Theoremhlexchb1 35522 A Hilbert lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Theoremhlexchb2 35523 A Hilbert lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) ↔ (𝑃 𝑋) = (𝑄 𝑋)))

Theoremhlsupr 35524* A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)))

Theoremhlsupr2 35525* A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ∃𝑟𝐴 (𝑃 𝑟) = (𝑄 𝑟))

Theoremhlhgt4 35526* A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ HL → ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))

Theoremhlhgt2 35527* A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ HL → ∃𝑥𝐵 ( 0 < 𝑥𝑥 < 1 ))

Theoremhl0lt1N 35528 Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)
< = (lt‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ HL → 0 < 1 )

Theoremhlexch3 35529 A Hilbert lattice has the exchange property. (atexch 29812 analog.) (Contributed by NM, 15-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Theoremhlexch4N 35530 A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Theoremhlatexchb1 35531 A version of hlexchb1 35522 for atoms. (Contributed by NM, 15-Nov-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑄) ↔ (𝑅 𝑃) = (𝑅 𝑄)))

Theoremhlatexchb2 35532 A version of hlexchb2 35523 for atoms. (Contributed by NM, 7-Feb-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))

Theoremhlatexch1 35533 Atom exchange property. (Contributed by NM, 7-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑄) → 𝑄 (𝑅 𝑃)))

Theoremhlatexch2 35534 Atom exchange property. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) → 𝑄 (𝑃 𝑅)))

TheoremhlatmstcOLDN 35535* An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 29793 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑈‘{𝑦𝐴𝑦 𝑋}) = 𝑋)

Theoremhlatle 35536* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 29802 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌)))

Theoremhlateq 35537* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 29804 analog.) (Contributed by NM, 24-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ 𝑋 = 𝑌))

Theoremhlrelat1 35538* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 29794, with swapped, analog.) (Contributed by NM, 4-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ∃𝑝𝐴𝑝 𝑋𝑝 𝑌)))

Theoremhlrelat5N 35539* An atomistic lattice with 0 is relatively atomic, using the definition in Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝𝐴 (𝑋 < (𝑋 𝑝) ∧ 𝑝 𝑌))

Theoremhlrelat 35540* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 29795 analog.) (Contributed by NM, 4-Feb-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝𝐴 (𝑋 < (𝑋 𝑝) ∧ (𝑋 𝑝) 𝑌))

Theoremhlrelat2 35541* A consequence of relative atomicity. (chrelat2i 29796 analog.) (Contributed by NM, 5-Feb-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))

TheoremexatleN 35542 A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 𝑋𝑅 = 𝑃))

Theoremhl2at 35543* A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL → ∃𝑝𝐴𝑞𝐴 𝑝𝑞)

Theorematex 35544 At least one atom exists. (Contributed by NM, 15-Jul-2012.)
𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL → 𝐴 ≠ ∅)

TheoremintnatN 35545 If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (¬ 𝑌 𝑋 ∧ (𝑋 𝑌) ∈ 𝐴)) → ¬ 𝑌𝐴)

Theorem2llnne2N 35546 Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑅 𝑄)) → (𝑅 𝑃) ≠ (𝑅 𝑄))

Theorem2llnneN 35547 Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 𝑃) ≠ (𝑅 𝑄))

Theoremcvr1 35548 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 29786 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) → (¬ 𝑃 𝑋𝑋𝐶(𝑋 𝑃)))

Theoremcvr2N 35549 Less-than and covers equivalence in a Hilbert lattice. (chcv2 29787 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) → (𝑋 < (𝑋 𝑃) ↔ 𝑋𝐶(𝑋 𝑃)))

Theoremhlrelat3 35550* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 35540. (Contributed by NM, 2-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝𝐴 (𝑋𝐶(𝑋 𝑝) ∧ (𝑋 𝑝) 𝑌))

Theoremcvrval3 35551* Binary relation expressing 𝑌 covers 𝑋. (Contributed by NM, 16-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ∃𝑝𝐴𝑝 𝑋 ∧ (𝑋 𝑝) = 𝑌)))

Theoremcvrval4N 35552* Binary relation expressing 𝑌 covers 𝑋. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)))

Theoremcvrval5 35553* Binary relation expressing 𝑋 covers 𝑋 𝑌. (Contributed by NM, 7-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑋 ↔ ∃𝑝𝐴𝑝 𝑌 ∧ (𝑝 (𝑋 𝑌)) = 𝑋)))

Theoremcvrp 35554 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 29806 analog.) (Contributed by NM, 18-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) → ((𝑋 𝑃) = 0𝑋𝐶(𝑋 𝑃)))

Theorematcvr1 35555 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄𝑃𝐶(𝑃 𝑄)))

Theorematcvr2 35556 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄𝑃𝐶(𝑄 𝑃)))

Theoremcvrexchlem 35557 Lemma for cvrexch 35558. (cvexchlem 29799 analog.) (Contributed by NM, 18-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌𝑋𝐶(𝑋 𝑌)))

Theoremcvrexch 35558 A Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (cvexchi 29800 analog.) (Contributed by NM, 18-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌𝑋𝐶(𝑋 𝑌)))

Theoremcvratlem 35559 Lemma for cvrat 35560. (atcvatlem 29816 analog.) (Contributed by NM, 22-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑋0𝑋 < (𝑃 𝑄))) → (¬ 𝑃(le‘𝐾)𝑋𝑋𝐴))

Theoremcvrat 35560 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 29817 analog.) (Contributed by NM, 22-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑋 < (𝑃 𝑄)) → 𝑋𝐴))

Theoremltltncvr 35561 A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → ¬ 𝑋𝐶𝑍))

Theoremltcvrntr 35562 Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))

Theoremcvrntr 35563 The covers relation is not transitive. (cvntr 29723 analog.) (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))

Theorematcvr0eq 35564 The covers relation is not transitive. (atcv0eq 29810 analog.) (Contributed by NM, 29-Nov-2011.)
= (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))

Theoremlnnat 35565 A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄 ↔ ¬ (𝑃 𝑄) ∈ 𝐴))

Theorematcvrj0 35566 Two atoms covering the zero subspace are equal. (atcv1 29811 analog.) (Contributed by NM, 29-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))

Theoremcvrat2 35567 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 29818 analog.) (Contributed by NM, 30-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄𝑋𝐶(𝑃 𝑄))) → 𝑋𝐴)

TheorematcvrneN 35568 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄𝑅)

Theorematcvrj1 35569 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))

Theorematcvrj2b 35570 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) ↔ 𝑃𝐶(𝑄 𝑅)))

Theorematcvrj2 35571 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))

TheorematleneN 35572 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑅𝑃 (𝑄 𝑅))) → 𝑄𝑅)

Theorematltcvr 35573 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)
< = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 < (𝑄 𝑅) ↔ 𝑃𝐶(𝑄 𝑅)))

Theorematle 35574* Any nonzero element has an atom under it. (Contributed by NM, 28-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑋0 ) → ∃𝑝𝐴 𝑝 𝑋)

Theorematlt 35575 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)
< = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 < (𝑃 𝑄) ↔ 𝑃𝑄))

Theorematlelt 35576 Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 < 𝑋)

Theorem2atlt 35577* Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞𝐴 (𝑞𝑃𝑞 < 𝑋))

TheorematexchcvrN 35578 Atom exchange property. Version of hlatexch2 35534 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃𝐶(𝑄 𝑅) → 𝑄𝐶(𝑃 𝑅)))

TheorematexchltN 35579 Atom exchange property. Version of hlatexch2 35534 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
< = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 < (𝑄 𝑅) → 𝑄 < (𝑃 𝑅)))

Theoremcvrat3 35580 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 29827 analog.) (Contributed by NM, 30-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))

Theoremcvrat4 35581* A condition implying existence of an atom with the properties shown. Lemma 3.2.20 in [PtakPulmannova] p. 68. Also Lemma 9.2(delta) in [MaedaMaeda] p. 41. (atcvat4i 29828 analog.) (Contributed by NM, 30-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟))))

Theoremcvrat42 35582* Commuted version of cvrat4 35581. (Contributed by NM, 28-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑟 𝑄))))

Theorem2atjm 35583 The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) = 𝑃)

Theorematbtwn 35584 Property of a 3rd atom 𝑅 on a line 𝑃 𝑄 intersecting element 𝑋 at 𝑃. (Contributed by NM, 30-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅𝑃 ↔ ¬ 𝑅 𝑋))

TheorematbtwnexOLDN 35585* There exists a 3rd atom 𝑟 on a line 𝑃 𝑄 intersecting element 𝑋 at 𝑃, such that 𝑟 is different from 𝑄 and not in 𝑋. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑟𝐴 (𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 (𝑃 𝑄)))

Theorematbtwnex 35586* Given atoms 𝑃 in 𝑋 and 𝑄 not in 𝑋, there exists an atom 𝑟 not in 𝑋 such that the line 𝑄 𝑟 intersects 𝑋 at 𝑃. (Contributed by NM, 1-Aug-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑟𝐴 (𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑃 (𝑄 𝑟)))

Theorem3noncolr2 35587 Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑄𝑅 ∧ ¬ 𝑃 (𝑄 𝑅)))

Theorem3noncolr1N 35588 Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅𝑃 ∧ ¬ 𝑄 (𝑅 𝑃)))

Theoremhlatcon3 35589 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑃 (𝑄 𝑅))

Theoremhlatcon2 35590 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑃 (𝑅 𝑄))

Theorem4noncolr3 35591 A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑃 ((𝑄 𝑅) 𝑆)))

Theorem4noncolr2 35592 A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑅𝑆 ∧ ¬ 𝑃 (𝑅 𝑆) ∧ ¬ 𝑄 ((𝑅 𝑆) 𝑃)))

Theorem4noncolr1 35593 A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑆𝑃 ∧ ¬ 𝑄 (𝑆 𝑃) ∧ ¬ 𝑅 ((𝑆 𝑃) 𝑄)))

Theoremathgt 35594* A Hilbert lattice, whose height is at least 4, has a chain of 4 successively covering atom joins. (Contributed by NM, 3-May-2012.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL → ∃𝑝𝐴𝑞𝐴 (𝑝𝐶(𝑝 𝑞) ∧ ∃𝑟𝐴 ((𝑝 𝑞)𝐶((𝑝 𝑞) 𝑟) ∧ ∃𝑠𝐴 ((𝑝 𝑞) 𝑟)𝐶(((𝑝 𝑞) 𝑟) 𝑠))))

Theorem3dim0 35595* There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL → ∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))

Theorem3dimlem1 35596 Lemma for 3dim1 35605. (Contributed by NM, 25-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆)) ∧ 𝑃 = 𝑄) → (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅) ∧ ¬ 𝑇 ((𝑃 𝑅) 𝑆)))

Theorem3dimlem2 35597 Lemma for 3dim1 35605. (Contributed by NM, 25-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆)) ∧ (𝑃𝑄𝑃 (𝑄 𝑅))) → (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 ((𝑃 𝑄) 𝑆)))

Theorem3dimlem3a 35598 Lemma for 3dim3 35607. (Contributed by NM, 27-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (¬ 𝑇 ((𝑄 𝑅) 𝑆) ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → ¬ 𝑇 ((𝑃 𝑄) 𝑅))

Theorem3dimlem3 35599 Lemma for 3dim1 35605. (Contributed by NM, 25-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑇 ((𝑃 𝑄) 𝑅)))

Theorem3dimlem3OLDN 35600 Lemma for 3dim1 35605. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑇 ((𝑃 𝑄) 𝑅)))

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