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Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhlexch2 36401 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 36402 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 36403 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 36404* 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 36405* 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 36406* 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 36407* 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 36408 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 36409 A Hilbert lattice has the exchange property. (atexch 30086 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 36410 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 36411 A version of hlexchb1 36402 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 36412 A version of hlexchb2 36403 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 36413 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 36414 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 36415* 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 30067 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 36416* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 30076 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 36417* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 30078 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 36418* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 30068, 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 36419* 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 36420* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 30069 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 36421* A consequence of relative atomicity. (chrelat2i 30070 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 36422 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 36423* A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL → ∃𝑝𝐴𝑞𝐴 𝑝𝑞)
 
Theorematex 36424 At least one atom exists. (Contributed by NM, 15-Jul-2012.)
𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL → 𝐴 ≠ ∅)
 
TheoremintnatN 36425 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 36426 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 36427 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 36428 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 30060 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 36429 Less-than and covers equivalence in a Hilbert lattice. (chcv2 30061 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 36430* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 36420. (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 36431* 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 36432* 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 36433* 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 36434 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 30080 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 36435 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 36436 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 36437 Lemma for cvrexch 36438. (cvexchlem 30073 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 36438 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 30074 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 36439 Lemma for cvrat 36440. (atcvatlem 30090 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 36440 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 30091 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 36441 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 36442 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 36443 The covers relation is not transitive. (cvntr 29997 analog.) (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))
 
Theorematcvr0eq 36444 The covers relation is not transitive. (atcv0eq 30084 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 36445 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 36446 Two atoms covering the zero subspace are equal. (atcv1 30085 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 36447 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 30092 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 36448 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 36449 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 36450 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 36451 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 36452 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 36453 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 36454* 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 36455 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 36456 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 36457* 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 36458 Atom exchange property. Version of hlatexch2 36414 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 36459 Atom exchange property. Version of hlatexch2 36414 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 36460 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 30101 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 36461* 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 30102 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 36462* Commuted version of cvrat4 36461. (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 36463 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 36464 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 36465* 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 36466* 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 36467 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 36468 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 36469 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 36470 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 36471 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 36472 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 36473 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 36474* 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 36475* 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 36476 Lemma for 3dim1 36485. (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 36477 Lemma for 3dim1 36485. (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 36478 Lemma for 3dim3 36487. (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 36479 Lemma for 3dim1 36485. (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 36480 Lemma for 3dim1 36485. (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)))
 
Theorem3dimlem4a 36481 Lemma for 3dim3 36487. (Contributed by NM, 27-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑃 (𝑄 𝑅) ∧ ¬ 𝑃 ((𝑄 𝑅) 𝑆))) → ¬ 𝑆 ((𝑃 𝑄) 𝑅))
 
Theorem3dimlem4 36482 Lemma for 3dim1 36485. (Contributed by NM, 25-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅)) ∧ ¬ 𝑃 ((𝑄 𝑅) 𝑆)) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)))
 
Theorem3dimlem4OLDN 36483 Lemma for 3dim1 36485. (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 ∧ ¬ 𝑆 (𝑄 𝑅))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅)) ∧ ¬ 𝑃 ((𝑄 𝑅) 𝑆)) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)))
 
Theorem3dim1lem5 36484* Lemma for 3dim1 36485. (Contributed by NM, 26-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝑢𝐴𝑣𝐴𝑤𝐴) ∧ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑤 ((𝑃 𝑢) 𝑣))) → ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑃𝑞 ∧ ¬ 𝑟 (𝑃 𝑞) ∧ ¬ 𝑠 ((𝑃 𝑞) 𝑟)))
 
Theorem3dim1 36485* Construct a 3-dimensional volume (height-4 element) on top of a given atom 𝑃. (Contributed by NM, 25-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴) → ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑃𝑞 ∧ ¬ 𝑟 (𝑃 𝑞) ∧ ¬ 𝑠 ((𝑃 𝑞) 𝑟)))
 
Theorem3dim2 36486* Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ∃𝑟𝐴𝑠𝐴𝑟 (𝑃 𝑄) ∧ ¬ 𝑠 ((𝑃 𝑄) 𝑟)))
 
Theorem3dim3 36487* Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.)
= (join‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ∃𝑠𝐴 ¬ 𝑠 ((𝑃 𝑄) 𝑅))
 
Theorem2dim 36488* Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴) → ∃𝑞𝐴𝑟𝐴 (𝑃𝐶(𝑃 𝑞) ∧ (𝑃 𝑞)𝐶((𝑃 𝑞) 𝑟)))
 
Theorem1dimN 36489* An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴) → ∃𝑞𝐴 𝑃𝐶(𝑃 𝑞))
 
Theorem1cvrco 36490 The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.)
𝐵 = (Base‘𝐾)    &    1 = (1.‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝐶 1 ↔ ( 𝑋) ∈ 𝐴))
 
Theorem1cvratex 36491* There exists an atom less than an element covered by 1. (Contributed by NM, 7-May-2012.) (Revised by Mario Carneiro, 13-Jun-2014.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑋𝐶 1 ) → ∃𝑝𝐴 𝑝 < 𝑋)
 
Theorem1cvratlt 36492 An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑋𝐶 1𝑃 𝑋)) → 𝑃 < 𝑋)
 
Theorem1cvrjat 36493 An element covered by the lattice unit, when joined with an atom not under it, equals the lattice unit. (Contributed by NM, 30-Apr-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋 𝑃) = 1 )
 
Theorem1cvrat 36494 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) ∈ 𝐴)
 
Theoremps-1 36495 The join of two atoms 𝑅 𝑆 (specifying a projective geometry line) is determined uniquely by any two atoms (specifying two points) less than or equal to that join. Part of Lemma 16.4 of [MaedaMaeda] p. 69, showing projective space postulate PS1 in [MaedaMaeda] p. 67. (Contributed by NM, 15-Nov-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) = (𝑅 𝑆)))
 
Theoremps-2 36496* Lattice analogue for the projective geometry axiom, "if a line intersects two sides of a triangle at different points then it also intersects the third side." Projective space condition PS2 in [MaedaMaeda] p. 68 and part of Theorem 16.4 in [MaedaMaeda] p. 69. (Contributed by NM, 1-Dec-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇) ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ∃𝑢𝐴 (𝑢 (𝑃 𝑅) ∧ 𝑢 (𝑆 𝑇)))
 
Theorem2atjlej 36497 Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴 ∧ (𝑃 𝑄) (𝑅 𝑆))) → 𝑅𝑆)
 
Theoremhlatexch3N 36498 Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑃 𝑄) = (𝑄 𝑅))
 
Theoremhlatexch4 36499 Exchange 2 atoms. (Contributed by NM, 13-May-2013.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑅) = (𝑄 𝑆))
 
Theoremps-2b 36500 Variation of projective geometry axiom ps-2 36496. (Contributed by NM, 3-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (¬ 𝑃 (𝑄 𝑅) ∧ 𝑆𝑇 ∧ (𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑄 𝑅)))) → ((𝑃 𝑅) (𝑆 𝑇)) ≠ 0 )
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