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Theorem dalawlem15 36901
Description: Lemma for dalaw 36902. Swap variable triples 𝑃𝑄𝑅 and 𝑆𝑇𝑈 in dalawlem14 36900, to obtain the elimination of the remaining conditions in dalawlem1 36887. (Contributed by NM, 6-Oct-2012.)
Hypotheses
Ref Expression
dalawlem.l = (le‘𝐾)
dalawlem.j = (join‘𝐾)
dalawlem.m = (meet‘𝐾)
dalawlem.a 𝐴 = (Atoms‘𝐾)
dalawlem2.o 𝑂 = (LPlanes‘𝐾)
Assertion
Ref Expression
dalawlem15 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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))))

Proof of Theorem dalawlem15
StepHypRef Expression
1 simp11 1195 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)
2 simp12 1196 . . . 4 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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) ∈ 𝑂 ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆))))
3 simp21 1198 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → 𝑃𝐴)
4 simp31 1201 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → 𝑆𝐴)
5 dalawlem.j . . . . . . . . . . 11 = (join‘𝐾)
6 dalawlem.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
75, 6hlatjcom 36384 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) = (𝑆 𝑃))
81, 3, 4, 7syl3anc 1363 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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) = (𝑆 𝑃))
9 simp22 1199 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → 𝑄𝐴)
10 simp32 1202 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → 𝑇𝐴)
115, 6hlatjcom 36384 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) = (𝑇 𝑄))
121, 9, 10, 11syl3anc 1363 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → (𝑄 𝑇) = (𝑇 𝑄))
138, 12oveq12d 7163 . . . . . . . 8 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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 𝑃) (𝑇 𝑄)))
1413breq1d 5067 . . . . . . 7 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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) ↔ ((𝑆 𝑃) (𝑇 𝑄)) (𝑆 𝑇)))
1514notbid 319 . . . . . 6 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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) ↔ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑆 𝑇)))
1613breq1d 5067 . . . . . . 7 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) (𝑇 𝑈) ↔ ((𝑆 𝑃) (𝑇 𝑄)) (𝑇 𝑈)))
1716notbid 319 . . . . . 6 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) (𝑇 𝑈) ↔ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑇 𝑈)))
1813breq1d 5067 . . . . . . 7 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) (𝑈 𝑆) ↔ ((𝑆 𝑃) (𝑇 𝑄)) (𝑈 𝑆)))
1918notbid 319 . . . . . 6 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) (𝑈 𝑆) ↔ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑈 𝑆)))
2015, 17, 193anbi123d 1427 . . . . 5 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) ↔ (¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑆 𝑇) ∧ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑇 𝑈) ∧ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑈 𝑆))))
2120anbi2d 628 . . . 4 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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) ∈ 𝑂 ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆))) ↔ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑆 𝑇) ∧ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑇 𝑈) ∧ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑈 𝑆)))))
222, 21mtbid 325 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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) ∈ 𝑂 ∧ (¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑆 𝑇) ∧ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑇 𝑈) ∧ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑈 𝑆))))
23 simp13 1197 . . . 4 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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))
245, 6hlatjcom 36384 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑃𝐴) → (𝑆 𝑃) = (𝑃 𝑆))
251, 4, 3, 24syl3anc 1363 . . . . 5 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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 𝑃) = (𝑃 𝑆))
265, 6hlatjcom 36384 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑄𝐴) → (𝑇 𝑄) = (𝑄 𝑇))
271, 10, 9, 26syl3anc 1363 . . . . 5 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → (𝑇 𝑄) = (𝑄 𝑇))
2825, 27oveq12d 7163 . . . 4 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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 𝑃) (𝑇 𝑄)) = ((𝑃 𝑆) (𝑄 𝑇)))
29 simp33 1203 . . . . 5 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → 𝑈𝐴)
30 simp23 1200 . . . . 5 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → 𝑅𝐴)
315, 6hlatjcom 36384 . . . . 5 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑅𝐴) → (𝑈 𝑅) = (𝑅 𝑈))
321, 29, 30, 31syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → (𝑈 𝑅) = (𝑅 𝑈))
3323, 28, 323brtr4d 5089 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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 𝑃) (𝑇 𝑄)) (𝑈 𝑅))
34 simp3 1130 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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))
35 simp2 1129 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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))
36 dalawlem.l . . . 4 = (le‘𝐾)
37 dalawlem.m . . . 4 = (meet‘𝐾)
38 dalawlem2.o . . . 4 𝑂 = (LPlanes‘𝐾)
3936, 5, 37, 6, 38dalawlem14 36900 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑆 𝑇) ∧ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑇 𝑈) ∧ ¬ ((𝑆 𝑃) (𝑇 𝑄)) (𝑈 𝑆))) ∧ ((𝑆 𝑃) (𝑇 𝑄)) (𝑈 𝑅)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑆 𝑇) (𝑃 𝑄)) (((𝑇 𝑈) (𝑄 𝑅)) ((𝑈 𝑆) (𝑅 𝑃))))
401, 22, 33, 34, 35, 39syl311anc 1376 . 2 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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) (𝑃 𝑄)) (((𝑇 𝑈) (𝑄 𝑅)) ((𝑈 𝑆) (𝑅 𝑃))))
411hllatd 36380 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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 ∈ Lat)
42 eqid 2818 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
4342, 5, 6hlatjcl 36383 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
441, 3, 9, 43syl3anc 1363 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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) ∈ (Base‘𝐾))
4542, 5, 6hlatjcl 36383 . . . 4 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
461, 4, 10, 45syl3anc 1363 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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) ∈ (Base‘𝐾))
4742, 37latmcom 17673 . . 3 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑆 𝑇) (𝑃 𝑄)))
4841, 44, 46, 47syl3anc 1363 . 2 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) = ((𝑆 𝑇) (𝑃 𝑄)))
4942, 5, 6hlatjcl 36383 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
501, 9, 30, 49syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → (𝑄 𝑅) ∈ (Base‘𝐾))
5142, 5, 6hlatjcl 36383 . . . . 5 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
521, 10, 29, 51syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → (𝑇 𝑈) ∈ (Base‘𝐾))
5342, 37latmcom 17673 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → ((𝑄 𝑅) (𝑇 𝑈)) = ((𝑇 𝑈) (𝑄 𝑅)))
5441, 50, 52, 53syl3anc 1363 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → ((𝑄 𝑅) (𝑇 𝑈)) = ((𝑇 𝑈) (𝑄 𝑅)))
5542, 5, 6hlatjcl 36383 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑃𝐴) → (𝑅 𝑃) ∈ (Base‘𝐾))
561, 30, 3, 55syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → (𝑅 𝑃) ∈ (Base‘𝐾))
5742, 5, 6hlatjcl 36383 . . . . 5 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑆𝐴) → (𝑈 𝑆) ∈ (Base‘𝐾))
581, 29, 4, 57syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → (𝑈 𝑆) ∈ (Base‘𝐾))
5942, 37latmcom 17673 . . . 4 ((𝐾 ∈ Lat ∧ (𝑅 𝑃) ∈ (Base‘𝐾) ∧ (𝑈 𝑆) ∈ (Base‘𝐾)) → ((𝑅 𝑃) (𝑈 𝑆)) = ((𝑈 𝑆) (𝑅 𝑃)))
6041, 56, 58, 59syl3anc 1363 . . 3 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → ((𝑅 𝑃) (𝑈 𝑆)) = ((𝑈 𝑆) (𝑅 𝑃)))
6154, 60oveq12d 7163 . 2 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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)) → (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))) = (((𝑇 𝑈) (𝑄 𝑅)) ((𝑈 𝑆) (𝑅 𝑃))))
6240, 48, 613brtr4d 5089 1 (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((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))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  meetcmee 17543  Latclat 17643  Atomscatm 36279  HLchlt 36366  LPlanesclpl 36508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-llines 36514  df-lplanes 36515  df-psubsp 36519  df-pmap 36520  df-padd 36812
This theorem is referenced by:  dalaw  36902
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