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Theorem List for Metamath Proof Explorer - 35601-35700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremislhp2 35601 The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 18-May-2012.)
𝐵 = (Base‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾𝐴𝑊𝐵) → (𝑊𝐻𝑊𝐶 1 ))
 
Theoremlhpbase 35602 A co-atom is a member of the lattice base set (i.e. a lattice element). (Contributed by NM, 18-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝑊𝐻𝑊𝐵)
 
Theoremlhp1cvr 35603 The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)
1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾𝐴𝑊𝐻) → 𝑊𝐶 1 )
 
Theoremlhplt 35604 An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
= (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑃 𝑊)) → 𝑃 < 𝑊)
 
Theoremlhp2lt 35605 The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)
= (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑃 𝑊) ∧ (𝑄𝐴𝑄 𝑊)) → (𝑃 𝑄) < 𝑊)
 
Theoremlhpexlt 35606* There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
< = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 𝑝 < 𝑊)
 
Theoremlhp0lt 35607 A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
< = (lt‘𝐾)    &    0 = (0.‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 < 𝑊)
 
Theoremlhpn0 35608 A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)
0 = (0.‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑊0 )
 
Theoremlhpexle 35609* There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 𝑝 𝑊)
 
Theoremlhpexnle 35610* There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 ¬ 𝑝 𝑊)
 
Theoremlhpexle1lem 35611* Lemma for lhpexle1 35612 and others that eliminates restrictions on 𝑋. (Contributed by NM, 24-Jul-2013.)
(𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓))    &   ((𝜑 ∧ (𝑋𝐴𝑋 𝑊)) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))       (𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
 
Theoremlhpexle1 35612* There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 (𝑝 𝑊𝑝𝑋))
 
Theoremlhpexle2lem 35613* Lemma for lhpexle2 35614. (Contributed by NM, 19-Jun-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐴𝑋 𝑊) ∧ (𝑌𝐴𝑌 𝑊)) → ∃𝑝𝐴 (𝑝 𝑊𝑝𝑋𝑝𝑌))
 
Theoremlhpexle2 35614* There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 (𝑝 𝑊𝑝𝑋𝑝𝑌))
 
Theoremlhpexle3lem 35615* There exists atom under a co-atom different from any 3 other atoms. TODO: study if adant*,simp* usage can be improved. (Contributed by NM, 9-Jul-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑋 𝑊𝑌 𝑊𝑍 𝑊)) → ∃𝑝𝐴 (𝑝 𝑊 ∧ (𝑝𝑋𝑝𝑌𝑝𝑍)))
 
Theoremlhpexle3 35616* There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 (𝑝 𝑊 ∧ (𝑝𝑋𝑝𝑌𝑝𝑍)))
 
Theoremlhpex2leN 35617* There exist at least two different atoms under a co-atom. This allows us to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴𝑞𝐴 (𝑝 𝑊𝑞 𝑊𝑝𝑞))
 
Theoremlhpoc 35618 The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐵) → (𝑊𝐻 ↔ ( 𝑊) ∈ 𝐴))
 
Theoremlhpoc2N 35619 The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐵) → (𝑊𝐴 ↔ ( 𝑊) ∈ 𝐻))
 
Theoremlhpocnle 35620 The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)
= (le‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ¬ ( 𝑊) 𝑊)
 
Theoremlhpocat 35621 The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)
= (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( 𝑊) ∈ 𝐴)
 
Theoremlhpocnel 35622 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 25-May-2012.)
= (le‘𝐾)    &    = (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (( 𝑊) ∈ 𝐴 ∧ ¬ ( 𝑊) 𝑊))
 
Theoremlhpocnel2 35623 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
 
Theoremlhpjat1 35624 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = 1 )
 
Theoremlhpjat2 35625 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 4-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = 1 )
 
Theoremlhpj1 35626 The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝑊 𝑋) = 1 )
 
Theoremlhpmcvr 35627 The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝑋 𝑊)𝐶𝑋)
 
Theoremlhpmcvr2 35628* Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑝𝐴𝑝 𝑊 ∧ (𝑝 (𝑋 𝑊)) = 𝑋))
 
Theoremlhpmcvr3 35629 Specialization of lhpmcvr2 35628. TODO: Use this to simplify many uses of (𝑃 (𝑋 𝑊)) = 𝑋 to become 𝑃 𝑋. (Contributed by NM, 6-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑋 ↔ (𝑃 (𝑋 𝑊)) = 𝑋))
 
Theoremlhpmcvr4N 35630 Specialization of lhpmcvr2 35628. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑃 𝑋)) → ¬ 𝑃 𝑌)
 
Theoremlhpmcvr5N 35631* Specialization of lhpmcvr2 35628. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊)) → ∃𝑝𝐴𝑝 𝑊 ∧ ¬ 𝑝 𝑌 ∧ (𝑝 (𝑋 𝑊)) = 𝑋))
 
Theoremlhpmcvr6N 35632* Specialization of lhpmcvr2 35628. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊)) → ∃𝑝𝐴𝑝 𝑊 ∧ ¬ 𝑝 𝑌𝑝 𝑋))
 
Theoremlhpm0atN 35633 If the meet of a lattice hyperplane with a nonzero element is zero, the element is an atom. (Contributed by NM, 28-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑋𝐴)
 
Theoremlhpmat 35634 An element covered by the lattice unit, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.)
= (le‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = 0 )
 
Theoremlhpmatb 35635 An element covered by the lattice unit, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.)
= (le‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴) → (¬ 𝑃 𝑊 ↔ (𝑃 𝑊) = 0 ))
 
Theoremlhp2at0 35636 Join and meet with different atoms under co-atom 𝑊. (Contributed by NM, 15-Jun-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑈𝑉) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) → ((𝑃 𝑈) 𝑉) = 0 )
 
Theoremlhp2atnle 35637 Inequality for 2 different atoms under co-atom 𝑊. (Contributed by NM, 17-Jun-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑈𝑉) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) → ¬ 𝑉 (𝑃 𝑈))
 
Theoremlhp2atne 35638 Inequality for joins with 2 different atoms under co-atom 𝑊. (Contributed by NM, 22-Jul-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ ((𝑈𝐴𝑈 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ 𝑈𝑉) → (𝑃 𝑈) ≠ (𝑄 𝑉))
 
Theoremlhp2at0nle 35639 Inequality for 2 different atoms (or an atom and zero) under co-atom 𝑊. (Contributed by NM, 28-Jul-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑈𝑉) ∧ ((𝑈𝐴𝑈 = 0 ) ∧ 𝑈 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) → ¬ 𝑉 (𝑃 𝑈))
 
Theoremlhp2at0ne 35640 Inequality for joins with 2 different atoms (or an atom and zero) under co-atom 𝑊. (Contributed by NM, 28-Jul-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (((𝑈𝐴𝑈 = 0 ) ∧ 𝑈 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ 𝑈𝑉) → (𝑃 𝑈) ≠ (𝑄 𝑉))
 
Theoremlhpelim 35641 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 35634 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) → ((𝑃 (𝑋 𝑊)) 𝑊) = (𝑋 𝑊))
 
Theoremlhpmod2i2 35642 Modular law for hyperplanes analogous to atmod2i2 35466 for atoms. (Contributed by NM, 9-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → ((𝑋 𝑊) 𝑌) = (𝑋 (𝑊 𝑌)))
 
Theoremlhpmod6i1 35643 Modular law for hyperplanes analogous to complement of atmod2i1 35465 for atoms. (Contributed by NM, 1-Jun-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑊) → (𝑋 (𝑌 𝑊)) = ((𝑋 𝑌) 𝑊))
 
Theoremlhprelat3N 35644* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 35016. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
 
Theoremcdlemb2 35645* Given two atoms not under the fiducial (reference) co-atom 𝑊, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 30-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑟𝐴𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄)))
 
Theoremlhple 35646 Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝑃 𝑋) 𝑊) = 𝑋)
 
Theoremlhpat 35647 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → ((𝑃 𝑄) 𝑊) ∈ 𝐴)
 
Theoremlhpat4N 35648 Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑈𝐴𝑈 𝑊)) → ((𝑃 𝑈) 𝑊) = 𝑈)
 
Theoremlhpat2 35649 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑅 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑅𝐴)
 
Theoremlhpat3 35650 There is only one atom under both 𝑃 𝑄 and co-atom 𝑊. (Contributed by NM, 21-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑅 = ((𝑃 𝑄) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑄𝐴𝑆𝐴) ∧ (𝑃𝑄𝑆 (𝑃 𝑄))) → (¬ 𝑆 𝑊𝑆𝑅))
 
Theorem4atexlemk 35651 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑𝐾 ∈ HL)
 
Theorem4atexlemw 35652 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑𝑊𝐻)
 
Theorem4atexlempw 35653 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
 
Theorem4atexlemp 35654 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑𝑃𝐴)
 
Theorem4atexlemq 35655 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑𝑄𝐴)
 
Theorem4atexlems 35656 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑𝑆𝐴)
 
Theorem4atexlemt 35657 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑𝑇𝐴)
 
Theorem4atexlemutvt 35658 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
 
Theorem4atexlempnq 35659 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑𝑃𝑄)
 
Theorem4atexlemnslpq 35660 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑 → ¬ 𝑆 (𝑃 𝑄))
 
Theorem4atexlemkl 35661 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑𝐾 ∈ Lat)
 
Theorem4atexlemkc 35662 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))       (𝜑𝐾 ∈ CvLat)
 
Theorem4atexlemwb 35663 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &   𝐻 = (LHyp‘𝐾)       (𝜑𝑊 ∈ (Base‘𝐾))
 
Theorem4atexlempsb 35664 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
 
Theorem4atexlemqtb 35665 Lemma for 4atexlem7 35679. (Contributed by NM, 24-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
 
Theorem4atexlempns 35666 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝜑𝑃𝑆)
 
Theorem4atexlemswapqr 35667 Lemma for 4atexlem7 35679. Swap 𝑄 and 𝑅, so that theorems involving 𝐶 can be reused for 𝐷. Note that 𝑈 must be expanded because it involves 𝑄. (Contributed by NM, 25-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (𝜑 → (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑆𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)) ∧ (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅))))
 
Theorem4atexlemu 35668 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (𝜑𝑈𝐴)
 
Theorem4atexlemv 35669 Lemma for 4atexlem7 35679. (Contributed by NM, 23-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑆) 𝑊)       (𝜑𝑉𝐴)
 
Theorem4atexlemunv 35670 Lemma for 4atexlem7 35679. (Contributed by NM, 21-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑆) 𝑊)       (𝜑𝑈𝑉)
 
Theorem4atexlemtlw 35671 Lemma for 4atexlem7 35679. (Contributed by NM, 24-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑆) 𝑊)       (𝜑𝑇 𝑊)
 
Theorem4atexlemntlpq 35672 Lemma for 4atexlem7 35679. (Contributed by NM, 24-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑆) 𝑊)       (𝜑 → ¬ 𝑇 (𝑃 𝑄))
 
Theorem4atexlemc 35673 Lemma for 4atexlem7 35679. (Contributed by NM, 24-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑆) 𝑊)    &   𝐶 = ((𝑄 𝑇) (𝑃 𝑆))       (𝜑𝐶𝐴)
 
Theorem4atexlemnclw 35674 Lemma for 4atexlem7 35679. (Contributed by NM, 24-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑆) 𝑊)    &   𝐶 = ((𝑄 𝑇) (𝑃 𝑆))       (𝜑 → ¬ 𝐶 𝑊)
 
Theorem4atexlemex2 35675* Lemma for 4atexlem7 35679. Show that when 𝐶𝑆, 𝐶 satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑆) 𝑊)    &   𝐶 = ((𝑄 𝑇) (𝑃 𝑆))       ((𝜑𝐶𝑆) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
 
Theorem4atexlemcnd 35676 Lemma for 4atexlem7 35679. (Contributed by NM, 24-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑆) 𝑊)    &   𝐶 = ((𝑄 𝑇) (𝑃 𝑆))    &   𝐷 = ((𝑅 𝑇) (𝑃 𝑆))       (𝜑𝐶𝐷)
 
Theorem4atexlemex4 35677* Lemma for 4atexlem7 35679. Show that when 𝐶 = 𝑆, 𝐷 satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑆) 𝑊)    &   𝐶 = ((𝑄 𝑇) (𝑃 𝑆))    &   𝐷 = ((𝑅 𝑇) (𝑃 𝑆))       ((𝜑𝐶 = 𝑆) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
 
Theorem4atexlemex6 35678* Lemma for 4atexlem7 35679. (Contributed by NM, 25-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
 
Theorem4atexlem7 35679* Whenever there are at least 4 atoms under 𝑃 𝑄 (specifically, 𝑃, 𝑄, 𝑟, and (𝑃 𝑄) 𝑊), there are also at least 4 atoms under 𝑃 𝑆. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 34948, our (𝑃 𝑟) = (𝑄 𝑟) is a shorter way to express 𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄). With a longer proof, the condition ¬ 𝑆 (𝑃 𝑄) could be eliminated (see 4atex 35680), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
 
Theorem4atex 35680* Whenever there are at least 4 atoms under 𝑃 𝑄 (specifically, 𝑃, 𝑄, 𝑟, and (𝑃 𝑄) 𝑊), there are also at least 4 atoms under 𝑃 𝑆. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 34948, our (𝑃 𝑟) = (𝑄 𝑟) is a shorter way to express 𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄). (Contributed by NM, 27-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
 
Theorem4atex2 35681* More general version of 4atex 35680 for a line 𝑆 𝑇 not necessarily connected to 𝑃 𝑄. (Contributed by NM, 27-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑃𝑄𝑇𝐴 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)))
 
Theorem4atex2-0aOLDN 35682* Same as 4atex2 35681 except that 𝑆 is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)))
 
Theorem4atex2-0bOLDN 35683* Same as 4atex2 35681 except that 𝑇 is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)))
 
Theorem4atex2-0cOLDN 35684* Same as 4atex2 35681 except that 𝑆 and 𝑇 are zero. TODO: do we need this one or 4atex2-0aOLDN 35682 or 4atex2-0bOLDN 35683? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)))
 
Theorem4atex3 35685* More general version of 4atex 35680 for a line 𝑆 𝑇 not necessarily connected to 𝑃 𝑄. (Contributed by NM, 29-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑃𝑄 ∧ (𝑇𝐴𝑆𝑇) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑧𝑆𝑧𝑇𝑧 (𝑆 𝑇))))
 
Theoremlautset 35686* The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐼 = (LAut‘𝐾)       (𝐾𝐴𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
 
Theoremislaut 35687* The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐼 = (LAut‘𝐾)       (𝐾𝐴 → (𝐹𝐼 ↔ (𝐹:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))))
 
Theoremlautle 35688 Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐼 = (LAut‘𝐾)       (((𝐾𝑉𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌)))
 
Theoremlaut1o 35689 A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐼 = (LAut‘𝐾)       ((𝐾𝐴𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
 
Theoremlaut11 35690 One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐼 = (LAut‘𝐾)       (((𝐾𝑉𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
 
Theoremlautcl 35691 A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐼 = (LAut‘𝐾)       (((𝐾𝑉𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
 
TheoremlautcnvclN 35692 Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐼 = (LAut‘𝐾)       (((𝐾𝑉𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
 
Theoremlautcnvle 35693 Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐼 = (LAut‘𝐾)       (((𝐾𝑉𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌)))
 
Theoremlautcnv 35694 The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
𝐼 = (LAut‘𝐾)       ((𝐾𝑉𝐹𝐼) → 𝐹𝐼)
 
Theoremlautlt 35695 Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐼 = (LAut‘𝐾)       ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝐹𝑋) < (𝐹𝑌)))
 
Theoremlautcvr 35696 Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐼 = (LAut‘𝐾)       ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋𝐶𝑌 ↔ (𝐹𝑋)𝐶(𝐹𝑌)))
 
Theoremlautj 35697 Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐼 = (LAut‘𝐾)       ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
 
Theoremlautm 35698 Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &   𝐼 = (LAut‘𝐾)       ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
 
Theoremlauteq 35699* A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐼 = (LAut‘𝐾)       (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (𝐹𝑋) = 𝑋)
 
Theoremidlaut 35700 The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐼 = (LAut‘𝐾)       (𝐾𝐴 → ( I ↾ 𝐵) ∈ 𝐼)
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