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Theorem List for Metamath Proof Explorer - 34201-34300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlhpexle1lem 34201* Lemma for lhpexle1 34202 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 34202* 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 34203* Lemma for lhpexle2 34204. (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 34204* 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 34205* 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 34206* 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 34207* 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 34208 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 34209 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 34210 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 34211 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 34212 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 34213 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 34214 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 34215 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 34216 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 34217 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 34218* 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 34219 Specialization of lhpmcvr2 34218. 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 34220 Specialization of lhpmcvr2 34218. (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 34221* Specialization of lhpmcvr2 34218. (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 34222* Specialization of lhpmcvr2 34218. (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 34223 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 34224 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 34225 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 34226 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 34227 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 34228 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 34229 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 34230 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 34231 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 34224 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 34232 Modular law for hyperplanes analogous to atmod2i2 34056 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 34233 Modular law for hyperplanes analogous to complement of atmod2i1 34055 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 34234* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 33606. (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 34235* 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 34236 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 34237 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 34238 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 34239 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 34240 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 34241 Lemma for 4atexlem7 34269. (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 34242 Lemma for 4atexlem7 34269. (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 34243 Lemma for 4atexlem7 34269. (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 34244 Lemma for 4atexlem7 34269. (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 34245 Lemma for 4atexlem7 34269. (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 34246 Lemma for 4atexlem7 34269. (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 34247 Lemma for 4atexlem7 34269. (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 34248 Lemma for 4atexlem7 34269. (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 34249 Lemma for 4atexlem7 34269. (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 34250 Lemma for 4atexlem7 34269. (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 34251 Lemma for 4atexlem7 34269. (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 34252 Lemma for 4atexlem7 34269. (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 34253 Lemma for 4atexlem7 34269. (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 34254 Lemma for 4atexlem7 34269. (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 34255 Lemma for 4atexlem7 34269. (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 34256 Lemma for 4atexlem7 34269. (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 34257 Lemma for 4atexlem7 34269. 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 34258 Lemma for 4atexlem7 34269. (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 34259 Lemma for 4atexlem7 34269. (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 34260 Lemma for 4atexlem7 34269. (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 34261 Lemma for 4atexlem7 34269. (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 34262 Lemma for 4atexlem7 34269. (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 34263 Lemma for 4atexlem7 34269. (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 34264 Lemma for 4atexlem7 34269. (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 34265* Lemma for 4atexlem7 34269. 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 34266 Lemma for 4atexlem7 34269. (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 34267* Lemma for 4atexlem7 34269. 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 34268* Lemma for 4atexlem7 34269. (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 34269* 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 33538, 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 34270), 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 34270* 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 33538, 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 34271* More general version of 4atex 34270 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 34272* Same as 4atex2 34271 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 34273* Same as 4atex2 34271 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 34274* Same as 4atex2 34271 except that 𝑆 and 𝑇 are zero. TODO: do we need this one or 4atex2-0aOLDN 34272 or 4atex2-0bOLDN 34273? (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 34275* More general version of 4atex 34270 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 34276* 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 34277* 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 34278 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 34279 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 34280 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 34281 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 34282 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 34283 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 34284 The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
𝐼 = (LAut‘𝐾)       ((𝐾𝑉𝐹𝐼) → 𝐹𝐼)

Theoremlautlt 34285 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 34286 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 34287 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 34288 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 34289* 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 34290 The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐼 = (LAut‘𝐾)       (𝐾𝐴 → ( I ↾ 𝐵) ∈ 𝐼)

Theoremlautco 34291 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
𝐼 = (LAut‘𝐾)       ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺) ∈ 𝐼)

TheorempautsetN 34292* The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &   𝑀 = (PAut‘𝐾)       (𝐾𝐵𝑀 = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})

TheoremispautN 34293* The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &   𝑀 = (PAut‘𝐾)       (𝐾𝐵 → (𝐹𝑀 ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))))

Syntaxcldil 34294 Extend class notation with set of all lattice dilations.
class LDil

Syntaxcltrn 34295 Extend class notation with set of all lattice translations.
class LTrn

SyntaxcdilN 34296 Extend class notation with set of all dilations.
class Dil

SyntaxctrnN 34297 Extend class notation with set of all translations.
class Trn

Definitiondf-ldil 34298* Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)}))

Definitiondf-ltrn 34299* Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
LTrn = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ ((LDil‘𝑘)‘𝑤) ∣ ∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) → ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓𝑞))(meet‘𝑘)𝑤))}))

Definitiondf-dilN 34300* Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)
Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)}))

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