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Theorem List for Metamath Proof Explorer - 35201-35300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem4atlem3a 35201 Lemma for 4at 35217. Break inequality into 3 cases. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (¬ 𝑄 ((𝑃 𝑈) 𝑉) ∨ ¬ 𝑅 ((𝑃 𝑈) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑈) 𝑉)))
 
Theorem4atlem3b 35202 Lemma for 4at 35217. Break inequality into 2 cases. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)))
 
Theorem4atlem4a 35203 Lemma for 4at 35217. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = (𝑃 ((𝑄 𝑅) 𝑆)))
 
Theorem4atlem4b 35204 Lemma for 4at 35217. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = (𝑄 ((𝑃 𝑅) 𝑆)))
 
Theorem4atlem4c 35205 Lemma for 4at 35217. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = (𝑅 ((𝑃 𝑄) 𝑆)))
 
Theorem4atlem4d 35206 Lemma for 4at 35217. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = (𝑆 ((𝑃 𝑄) 𝑅)))
 
Theorem4atlem9 35207 Lemma for 4at 35217. Substitute 𝑊 for 𝑆. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑊𝐴) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) → (𝑆 ((𝑃 𝑄) (𝑅 𝑊)) ↔ ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑄) (𝑅 𝑊))))
 
Theorem4atlem10a 35208 Lemma for 4at 35217. Substitute 𝑉 for 𝑅. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑉𝐴𝑊𝐴) ∧ ¬ 𝑅 ((𝑃 𝑄) 𝑊)) → (𝑅 ((𝑃 𝑄) (𝑉 𝑊)) ↔ ((𝑃 𝑄) (𝑅 𝑊)) = ((𝑃 𝑄) (𝑉 𝑊))))
 
Theorem4atlem10b 35209 Lemma for 4at 35217. Substitute 𝑉 for 𝑅 (cont.). (Contributed by NM, 10-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑊𝐴 ∧ ¬ 𝑅 ((𝑃 𝑄) 𝑊) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) ∧ (𝑅 ((𝑃 𝑄) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑄) (𝑉 𝑊)))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑄) (𝑉 𝑊)))
 
Theorem4atlem10 35210 Lemma for 4at 35217. Combine both possible cases. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑅𝐴𝑆𝐴) ∧ 𝑉𝐴𝑊𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑅 𝑆) ((𝑃 𝑄) (𝑉 𝑊)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑄) (𝑉 𝑊))))
 
Theorem4atlem11a 35211 Lemma for 4at 35217. Substitute 𝑈 for 𝑄. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) → (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑄) (𝑉 𝑊)) = ((𝑃 𝑈) (𝑉 𝑊))))
 
Theorem4atlem11b 35212 Lemma for 4at 35217. Substitute 𝑈 for 𝑄 (cont.). (Contributed by NM, 10-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑈) (𝑉 𝑊)))
 
Theorem4atlem11 35213 Lemma for 4at 35217. Combine all three possible cases. (Contributed by NM, 10-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑄 (𝑅 𝑆)) ((𝑃 𝑈) (𝑉 𝑊)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑈) (𝑉 𝑊))))
 
Theorem4atlem12a 35214 Lemma for 4at 35217. Substitute 𝑇 for 𝑃. (Contributed by NM, 9-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) → (𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑈) (𝑉 𝑊)) = ((𝑇 𝑈) (𝑉 𝑊))))
 
Theorem4atlem12b 35215 Lemma for 4at 35217. Substitute 𝑇 for 𝑃 (cont.). (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑇 𝑈) (𝑉 𝑊)))
 
Theorem4atlem12 35216 Lemma for 4at 35217. Combine all four possible cases. (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) (𝑅 𝑆)) ((𝑇 𝑈) (𝑉 𝑊)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑇 𝑈) (𝑉 𝑊))))
 
Theorem4at 35217 Four atoms determine a lattice volume uniquely. Three-dimensional analogue of ps-1 35081 and 3at 35094. (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) (𝑅 𝑆)) ((𝑇 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑇 𝑈) (𝑉 𝑊))))
 
Theorem4at2 35218 Four atoms determine a lattice volume uniquely. (Contributed by NM, 11-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((((𝑃 𝑄) 𝑅) 𝑆) (((𝑇 𝑈) 𝑉) 𝑊) ↔ (((𝑃 𝑄) 𝑅) 𝑆) = (((𝑇 𝑈) 𝑉) 𝑊)))
 
Theoremlplncvrlvol2 35219 A lattice line under a lattice plane is covered by it. (Contributed by NM, 12-Jul-2012.)
= (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝑃 = (LPlanes‘𝐾)    &   𝑉 = (LVols‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → 𝑋𝐶𝑌)
 
Theoremlplncvrlvol 35220 An element covering a lattice plane is a lattice volume and vice-versa. (Contributed by NM, 15-Jul-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝑃 = (LPlanes‘𝐾)    &   𝑉 = (LVols‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → (𝑋𝑃𝑌𝑉))
 
Theoremlvolcmp 35221 If two lattice planes are comparable, they are equal. (Contributed by NM, 12-Jul-2012.)
= (le‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑉) → (𝑋 𝑌𝑋 = 𝑌))
 
TheoremlvolnltN 35222 Two lattice volumes cannot satisfy the less than relation. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
< = (lt‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑉) → ¬ 𝑋 < 𝑌)
 
Theorem2lplnja 35223 The join of two different lattice planes in a lattice volume equals the volume (version of 2lplnj 35224 in terms of atoms). (Contributed by NM, 12-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇))) ∧ (((𝑃 𝑄) 𝑅) 𝑊 ∧ ((𝑆 𝑇) 𝑈) 𝑊 ∧ ((𝑃 𝑄) 𝑅) ≠ ((𝑆 𝑇) 𝑈))) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) = 𝑊)
 
Theorem2lplnj 35224 The join of two different lattice planes in a (3-dimensional) lattice volume equals the volume. (Contributed by NM, 12-Jul-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝑃 = (LPlanes‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (𝑋 𝑌) = 𝑊)
 
Theorem2lplnm2N 35225 The meet of two different lattice planes in a lattice volume is a lattice line. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (meet‘𝐾)    &   𝑁 = (LLines‘𝐾)    &   𝑃 = (LPlanes‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (𝑋 𝑌) ∈ 𝑁)
 
Theorem2lplnmj 35226 The meet of two lattice planes is a lattice line iff their join is a lattice volume. (Contributed by NM, 13-Jul-2012.)
= (join‘𝐾)    &    = (meet‘𝐾)    &   𝑁 = (LLines‘𝐾)    &   𝑃 = (LPlanes‘𝐾)    &   𝑉 = (LVols‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑃) → ((𝑋 𝑌) ∈ 𝑁 ↔ (𝑋 𝑌) ∈ 𝑉))
 
Theoremdalemkehl 35227 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝐾 ∈ HL)
 
Theoremdalemkelat 35228 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝐾 ∈ Lat)
 
Theoremdalemkeop 35229 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝐾 ∈ OP)
 
Theoremdalempea 35230 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝑃𝐴)
 
Theoremdalemqea 35231 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝑄𝐴)
 
Theoremdalemrea 35232 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝑅𝐴)
 
Theoremdalemsea 35233 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝑆𝐴)
 
Theoremdalemtea 35234 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝑇𝐴)
 
Theoremdalemuea 35235 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝑈𝐴)
 
Theoremdalemyeo 35236 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝑌𝑂)
 
Theoremdalemzeo 35237 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝑍𝑂)
 
Theoremdalemclpjs 35238 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝐶 (𝑃 𝑆))
 
Theoremdalemclqjt 35239 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝐶 (𝑄 𝑇))
 
Theoremdalemclrju 35240 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑𝐶 (𝑅 𝑈))
 
Theoremdalem-clpjq 35241 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))       (𝜑 → ¬ 𝐶 (𝑃 𝑄))
 
Theoremdalemceb 35242 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &   𝐴 = (Atoms‘𝐾)       (𝜑𝐶 ∈ (Base‘𝐾))
 
Theoremdalempeb 35243 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &   𝐴 = (Atoms‘𝐾)       (𝜑𝑃 ∈ (Base‘𝐾))
 
Theoremdalemqeb 35244 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &   𝐴 = (Atoms‘𝐾)       (𝜑𝑄 ∈ (Base‘𝐾))
 
Theoremdalemreb 35245 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &   𝐴 = (Atoms‘𝐾)       (𝜑𝑅 ∈ (Base‘𝐾))
 
Theoremdalemseb 35246 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &   𝐴 = (Atoms‘𝐾)       (𝜑𝑆 ∈ (Base‘𝐾))
 
Theoremdalemteb 35247 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &   𝐴 = (Atoms‘𝐾)       (𝜑𝑇 ∈ (Base‘𝐾))
 
Theoremdalemueb 35248 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &   𝐴 = (Atoms‘𝐾)       (𝜑𝑈 ∈ (Base‘𝐾))
 
Theoremdalempjqeb 35249 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
 
Theoremdalemsjteb 35250 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
 
Theoremdalemtjueb 35251 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝜑 → (𝑇 𝑈) ∈ (Base‘𝐾))
 
Theoremdalemqrprot 35252 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝜑 → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
 
Theoremdalemyeb 35253 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &   𝑂 = (LPlanes‘𝐾)       (𝜑𝑌 ∈ (Base‘𝐾))
 
Theoremdalemcnes 35254 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝜑𝐶𝑆)
 
Theoremdalempnes 35255 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)       (𝜑𝑃𝑆)
 
Theoremdalemqnet 35256 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)       (𝜑𝑄𝑇)
 
Theoremdalempjsen 35257 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)       (𝜑 → (𝑃 𝑆) ∈ (LLines‘𝐾))
 
Theoremdalemply 35258 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)       (𝜑𝑃 𝑌)
 
Theoremdalemsly 35259 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑍 = ((𝑆 𝑇) 𝑈)       ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
 
Theoremdalemswapyz 35260 Lemma for dath 35340. Swap the role of planes 𝑌 and 𝑍 to allow reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))))
 
Theoremdalemrot 35261 Lemma for dath 35340. Rotate triangles 𝑌 = 𝑃𝑄𝑅 and 𝑍 = 𝑆𝑇𝑈 to allow reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)       (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))
 
Theoremdalemrotyz 35262 Lemma for dath 35340. Rotate triangles 𝑌 = 𝑃𝑄𝑅 and 𝑍 = 𝑆𝑇𝑈 to allow reuse of analogous proofs. (Contributed by NM, 19-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)       ((𝜑𝑌 = 𝑍) → ((𝑄 𝑅) 𝑃) = ((𝑇 𝑈) 𝑆))
 
Theoremdalem1 35263 Lemma for dath 35340. Show the lines 𝑃𝑆 and 𝑄𝑇 are different. (Contributed by NM, 9-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)       (𝜑 → (𝑃 𝑆) ≠ (𝑄 𝑇))
 
Theoremdalemcea 35264 Lemma for dath 35340. Frequently-used utility lemma. Here we show that 𝐶 must be an atom. This is an assumption in most presentations of Desargue's theorem; instead, we assume only the 𝐶 is a lattice element, in order to make later substitutions for 𝐶 easier. (Contributed by NM, 23-Sep-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)       (𝜑𝐶𝐴)
 
Theoremdalem2 35265 Lemma for dath 35340. Show the lines 𝑃𝑄 and 𝑆𝑇 form a plane. (Contributed by NM, 11-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)       (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝑂)
 
Theoremdalemdea 35266 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝐷 = ((𝑃 𝑄) (𝑆 𝑇))       (𝜑𝐷𝐴)
 
Theoremdalemeea 35267 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝐸 = ((𝑄 𝑅) (𝑇 𝑈))       (𝜑𝐸𝐴)
 
Theoremdalem3 35268 Lemma for dalemdnee 35270. (Contributed by NM, 10-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝐷 = ((𝑃 𝑄) (𝑆 𝑇))    &   𝐸 = ((𝑄 𝑅) (𝑇 𝑈))       ((𝜑𝐷𝑄) → 𝐷𝐸)
 
Theoremdalem4 35269 Lemma for dalemdnee 35270. (Contributed by NM, 10-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝐷 = ((𝑃 𝑄) (𝑆 𝑇))    &   𝐸 = ((𝑄 𝑅) (𝑇 𝑈))       ((𝜑𝐷𝑇) → 𝐷𝐸)
 
Theoremdalemdnee 35270 Lemma for dath 35340. Axis of perspectivity points 𝐷 and 𝐸 are different. (Contributed by NM, 10-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝐷 = ((𝑃 𝑄) (𝑆 𝑇))    &   𝐸 = ((𝑄 𝑅) (𝑇 𝑈))       (𝜑𝐷𝐸)
 
Theoremdalem5 35271 Lemma for dath 35340. Atom 𝑈 (in plane 𝑍 = 𝑆𝑇𝑈) belongs to the 3-dimensional volume formed by 𝑌 and 𝐶. (Contributed by NM, 21-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑊 = (𝑌 𝐶)       (𝜑𝑈 𝑊)
 
Theoremdalem6 35272 Lemma for dath 35340. Analogue of dalem5 35271 for 𝑆. (Contributed by NM, 21-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝑊 = (𝑌 𝐶)       (𝜑𝑆 𝑊)
 
Theoremdalem7 35273 Lemma for dath 35340. Analogue of dalem5 35271 for 𝑇. (Contributed by NM, 21-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝑊 = (𝑌 𝐶)       (𝜑𝑇 𝑊)
 
Theoremdalem8 35274 Lemma for dath 35340. Plane 𝑍 belongs to the 3-dimensional space. (Contributed by NM, 21-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝑊 = (𝑌 𝐶)       (𝜑𝑍 𝑊)
 
Theoremdalem-cly 35275 Lemma for dalem9 35276. Center of perspectivity 𝐶 is not in plane 𝑌 (when 𝑌 and 𝑍 are different planes). (Contributed by NM, 13-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)       ((𝜑𝑌𝑍) → ¬ 𝐶 𝑌)
 
Theoremdalem9 35276 Lemma for dath 35340. Since ¬ 𝐶 𝑌, the join 𝑌 𝐶 forms a 3-dimensional space. (Contributed by NM, 20-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑉 = (LVols‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝑊 = (𝑌 𝐶)       ((𝜑𝑌𝑍) → 𝑊𝑉)
 
Theoremdalem10 35277 Lemma for dath 35340. Atom 𝐷 belongs to the axis of perspectivity 𝑋. (Contributed by NM, 19-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝑋 = (𝑌 𝑍)    &   𝐷 = ((𝑃 𝑄) (𝑆 𝑇))       (𝜑𝐷 𝑋)
 
Theoremdalem11 35278 Lemma for dath 35340. Analogue of dalem10 35277 for 𝐸. (Contributed by NM, 23-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝑋 = (𝑌 𝑍)    &   𝐸 = ((𝑄 𝑅) (𝑇 𝑈))       (𝜑𝐸 𝑋)
 
Theoremdalem12 35279 Lemma for dath 35340. Analogue of dalem10 35277 for 𝐹. (Contributed by NM, 11-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝑋 = (𝑌 𝑍)    &   𝐹 = ((𝑅 𝑃) (𝑈 𝑆))       (𝜑𝐹 𝑋)
 
Theoremdalem13 35280 Lemma for dalem14 35281. (Contributed by NM, 21-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝑊 = (𝑌 𝐶)       ((𝜑𝑌𝑍) → (𝑌 𝑍) = 𝑊)
 
Theoremdalem14 35281 Lemma for dath 35340. Planes 𝑌 and 𝑍 form a 3-dimensional space (when they are different). (Contributed by NM, 22-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑉 = (LVols‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝑊 = (𝑌 𝐶)       ((𝜑𝑌𝑍) → (𝑌 𝑍) ∈ 𝑉)
 
Theoremdalem15 35282 Lemma for dath 35340. The axis of perspectivity 𝑋 is a line. (Contributed by NM, 21-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑁 = (LLines‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝑋 = (𝑌 𝑍)       ((𝜑𝑌𝑍) → 𝑋𝑁)
 
Theoremdalem16 35283 Lemma for dath 35340. The atoms 𝐷, 𝐸, and 𝐹 form a line of perspectivity. This is Desargue's Theorem for the special case where planes 𝑌 and 𝑍 are different. (Contributed by NM, 7-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝐷 = ((𝑃 𝑄) (𝑆 𝑇))    &   𝐸 = ((𝑄 𝑅) (𝑇 𝑈))    &   𝐹 = ((𝑅 𝑃) (𝑈 𝑆))       ((𝜑𝑌𝑍) → 𝐹 (𝐷 𝐸))
 
Theoremdalem17 35284 Lemma for dath 35340. When planes 𝑌 and 𝑍 are equal, the center of perspectivity 𝐶 is in 𝑌. (Contributed by NM, 1-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)       ((𝜑𝑌 = 𝑍) → 𝐶 𝑌)
 
Theoremdalem18 35285* Lemma for dath 35340. Show that a dummy atom 𝑐 exists outside of the 𝑌 and 𝑍 planes (when those planes are equal). This requires that the projective space be 3-dimensional. (Desargue's theorem doesn't always hold in 2 dimensions.) (Contributed by NM, 29-Jul-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)       (𝜑 → ∃𝑐𝐴 ¬ 𝑐 𝑌)
 
Theoremdalem19 35286* Lemma for dath 35340. Show that a second dummy atom 𝑑 exists outside of the 𝑌 and 𝑍 planes (when those planes are equal). (Contributed by NM, 15-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)       ((((𝜑𝑌 = 𝑍) ∧ 𝑐𝐴) ∧ ¬ 𝑐 𝑌) → ∃𝑑𝐴 (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑)))
 
Theoremdalemccea 35287 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
(𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))       (𝜓𝑐𝐴)
 
Theoremdalemddea 35288 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
(𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))       (𝜓𝑑𝐴)
 
Theoremdalem-ccly 35289 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
(𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))       (𝜓 → ¬ 𝑐 𝑌)
 
Theoremdalem-ddly 35290 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
(𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))       (𝜓 → ¬ 𝑑 𝑌)
 
Theoremdalemccnedd 35291 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
(𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))       (𝜓𝑐𝑑)
 
Theoremdalemclccjdd 35292 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
(𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))       (𝜓𝐶 (𝑐 𝑑))
 
Theoremdalemcceb 35293 Lemma for dath 35340. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
(𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))    &   𝐴 = (Atoms‘𝐾)       (𝜓𝑐 ∈ (Base‘𝐾))
 
Theoremdalemswapyzps 35294 Lemma for dath 35340. Swap the 𝑌 and 𝑍 planes, along with dummy concurrency (center of perspectivity) atoms 𝑐 and 𝑑, to allow reuse of analogous proofs. (Contributed by NM, 17-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))       ((𝜑𝑌 = 𝑍𝜓) → ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))))
 
Theoremdalemrotps 35295 Lemma for dath 35340. Rotate triangles 𝑌 = 𝑃𝑄𝑅 and 𝑍 = 𝑆𝑇𝑈 to allow reuse of analogous proofs. (Contributed by NM, 15-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))    &   𝑌 = ((𝑃 𝑄) 𝑅)       ((𝜑𝜓) → ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
 
Theoremdalemcjden 35296 Lemma for dath 35340. Show that the dummy atoms form a line. (Contributed by NM, 15-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))       ((𝜑𝜓) → (𝑐 𝑑) ∈ (LLines‘𝐾))
 
Theoremdalem20 35297* Lemma for dath 35340. Show that a second dummy atom 𝑑 exists outside of the 𝑌 and 𝑍 planes (when those planes are equal). (Contributed by NM, 14-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)       ((𝜑𝑌 = 𝑍) → ∃𝑐𝑑𝜓)
 
Theoremdalem21 35298 Lemma for dath 35340. Show that lines 𝑐𝑑 and 𝑃𝑆 intersect at an atom. (Contributed by NM, 2-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)       ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ 𝐴)
 
Theoremdalem22 35299 Lemma for dath 35340. Show that lines 𝑐𝑑 and 𝑃𝑆 determine a plane. (Contributed by NM, 2-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)       ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ 𝑂)
 
Theoremdalem23 35300 Lemma for dath 35340. Show that auxiliary atom 𝐺 is an atom. (Contributed by NM, 2-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))    &    = (meet‘𝐾)    &   𝑂 = (LPlanes‘𝐾)    &   𝑌 = ((𝑃 𝑄) 𝑅)    &   𝑍 = ((𝑆 𝑇) 𝑈)    &   𝐺 = ((𝑐 𝑃) (𝑑 𝑆))       ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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