Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalaw Structured version   Visualization version   GIF version

Theorem dalaw 37179
Description: Desargues's law, derived from Desargues's theorem dath 37029 and with no conditions on the atoms. If triples 𝑃, 𝑄, 𝑅 and 𝑆, 𝑇, 𝑈 are centrally perspective, i.e. ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈), then they are axially perspective. Theorem 13.3 of [Crawley] p. 110. (Contributed by NM, 7-Oct-2012.)
Hypotheses
Ref Expression
dalaw.l = (le‘𝐾)
dalaw.j = (join‘𝐾)
dalaw.m = (meet‘𝐾)
dalaw.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
dalaw ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))

Proof of Theorem dalaw
StepHypRef Expression
1 dalaw.l . . . . . . . . 9 = (le‘𝐾)
2 dalaw.j . . . . . . . . 9 = (join‘𝐾)
3 dalaw.m . . . . . . . . 9 = (meet‘𝐾)
4 dalaw.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
5 eqid 2798 . . . . . . . . 9 (LPlanes‘𝐾) = (LPlanes‘𝐾)
61, 2, 3, 4, 5dalawlem14 37177 . . . . . . . 8 (((𝐾 ∈ HL ∧ ¬ (((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
763expib 1119 . . . . . . 7 ((𝐾 ∈ HL ∧ ¬ (((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
873exp 1116 . . . . . 6 (𝐾 ∈ HL → (¬ (((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
91, 2, 3, 4, 5dalawlem15 37178 . . . . . . . 8 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
1093expib 1119 . . . . . . 7 ((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
11103exp 1116 . . . . . 6 (𝐾 ∈ HL → (¬ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆))) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
12 simp11 1200 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
13 simp2 1134 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃𝐴𝑄𝐴𝑅𝐴))
14 simp3 1135 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆𝐴𝑇𝐴𝑈𝐴))
15 simp2ll 1237 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
16153ad2ant1 1130 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
17 simp2rl 1239 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → ((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾))
18173ad2ant1 1130 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾))
19 simp2lr 1238 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)))
20193ad2ant1 1130 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)))
21 simp2rr 1240 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))
22213ad2ant1 1130 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))
23 simp13 1202 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))
241, 2, 3, 4, 5dalawlem1 37164 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ ((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾)) ∧ ((¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
2512, 13, 14, 16, 18, 20, 22, 23, 24syl323anc 1397 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
26253expib 1119 . . . . . . 7 ((𝐾 ∈ HL ∧ ((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
27263exp 1116 . . . . . 6 (𝐾 ∈ HL → (((((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ (((𝑆 𝑇) 𝑈) ∈ (LPlanes‘𝐾) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)))) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
288, 11, 27ecased 1031 . . . . 5 (𝐾 ∈ HL → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))))
2928exp4a 435 . . . 4 (𝐾 ∈ HL → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → ((𝑃𝐴𝑄𝐴𝑅𝐴) → ((𝑆𝐴𝑇𝐴𝑈𝐴) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
3029com34 91 . . 3 (𝐾 ∈ HL → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → ((𝑆𝐴𝑇𝐴𝑈𝐴) → ((𝑃𝐴𝑄𝐴𝑅𝐴) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
3130com24 95 . 2 (𝐾 ∈ HL → ((𝑃𝐴𝑄𝐴𝑅𝐴) → ((𝑆𝐴𝑇𝐴𝑈𝐴) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
32313imp 1108 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111   class class class wbr 5030  cfv 6324  (class class class)co 7135  lecple 16564  joincjn 17546  meetcmee 17547  Atomscatm 36556  HLchlt 36643  LPlanesclpl 36785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36469  df-ol 36471  df-oml 36472  df-covers 36559  df-ats 36560  df-atl 36591  df-cvlat 36615  df-hlat 36644  df-llines 36791  df-lplanes 36792  df-lvols 36793  df-psubsp 36796  df-pmap 36797  df-padd 37089
This theorem is referenced by:  cdleme14  37566  cdleme20f  37607  cdlemg9  37927  cdlemg12c  37938  cdlemk6  38130  cdlemk6u  38155
  Copyright terms: Public domain W3C validator