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Theorem List for Metamath Proof Explorer - 34501-34600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhlatexch2 34501 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 34502* 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 29191 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 34503* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 29200 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 34504* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 29202 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 34505* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 29192, 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 34506* 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 34507* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 29193 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 34508* A consequence of relative atomicity. (chrelat2i 29194 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 34509 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 34510* A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL → ∃𝑝𝐴𝑞𝐴 𝑝𝑞)

Theorematex 34511 At least one atom exists. (Contributed by NM, 15-Jul-2012.)
𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL → 𝐴 ≠ ∅)

TheoremintnatN 34512 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 34513 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 34514 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 34515 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 29184 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 34516 Less-than and covers equivalence in a Hilbert lattice. (chcv2 29185 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 34517* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 34507. (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 34518* 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 34519* 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 34520* 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 34521 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 29204 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 34522 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 34523 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 34524 Lemma for cvrexch 34525. (cvexchlem 29197 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 34525 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 29198 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 34526 Lemma for cvrat 34527. (atcvatlem 29214 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 34527 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 29215 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 34528 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 34529 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 34530 The covers relation is not transitive. (cvntr 29121 analog.) (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))

Theorematcvr0eq 34531 The covers relation is not transitive. (atcv0eq 29208 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 34532 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 34533 Two atoms covering the zero subspace are equal. (atcv1 29209 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 34534 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 29216 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 34535 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 34536 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 34537 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 34538 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 34539 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 34540 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 34541* 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 34542 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 34543 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 34544* 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 34545 Atom exchange property. Version of hlatexch2 34501 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 34546 Atom exchange property. Version of hlatexch2 34501 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 34547 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 29225 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 34548* 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 29226 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 34549* Commuted version of cvrat4 34548. (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 34550 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 34551 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 34552* 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 34553* 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 34554 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 34555 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 34556 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 34557 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 34558 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 34559 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 34560 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 34561* 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 34562* 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 34563 Lemma for 3dim1 34572. (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 34564 Lemma for 3dim1 34572. (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 34565 Lemma for 3dim3 34574. (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 34566 Lemma for 3dim1 34572. (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 34567 Lemma for 3dim1 34572. (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 34568 Lemma for 3dim3 34574. (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 34569 Lemma for 3dim1 34572. (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 34570 Lemma for 3dim1 34572. (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 34571* Lemma for 3dim1 34572. (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 34572* 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 34573* 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 34574* 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 34575* 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 34576* 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 34577 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 34578* 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 34579 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 34580 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 34581 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 34582 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 34583* 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 34584 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 34585 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 34586 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 34587 Variation of projective geometry axiom ps-2 34583. (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 )

Theorem3atlem1 34588 Lemma for 3at 34595. (Contributed by NM, 22-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))

Theorem3atlem2 34589 Lemma for 3at 34595. (Contributed by NM, 22-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))

Theorem3atlem3 34590 Lemma for 3at 34595. (Contributed by NM, 23-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑈 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))

Theorem3atlem4 34591 Lemma for 3at 34595. (Contributed by NM, 23-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑅)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑅))

Theorem3atlem5 34592 Lemma for 3at 34595. (Contributed by NM, 23-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))

Theorem3atlem6 34593 Lemma for 3at 34595. (Contributed by NM, 23-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))

Theorem3atlem7 34594 Lemma for 3at 34595. (Contributed by NM, 23-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))

Theorem3at 34595 Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 34582 for lines and 4at 34718 for volumes. I could not find this proof in the literature on projective geometry (where it is either given as an axiom or stated as an unproved fact), but it is similar to Theorem 15 of Veblen, "The Foundations of Geometry" (1911), p. 18, which uses different axioms. This proof was written before I became aware of Veblen's, and it is possible that a shorter proof could be obtained by using Veblen's proof for hints. (Contributed by NM, 23-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) ↔ ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))

20.22.12  Projective geometries based on Hilbert lattices

Syntaxclln 34596 Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.
class LLines

Syntaxclpl 34597 Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.
class LPlanes

Syntaxclvol 34598 Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.
class LVols

Syntaxclines 34599 Extend class notation with set of all projective lines for a Hilbert lattice.
class Lines

SyntaxcpointsN 34600 Extend class notation with set of all projective points.
class Points

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