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Theorem cdleme22b 39668
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 5th line on p. 115. Show that t v =/= p q and s p q implies ¬ t p q. (Contributed by NM, 2-Dec-2012.)
Hypotheses
Ref Expression
cdleme22.l = (le‘𝐾)
cdleme22.j = (join‘𝐾)
cdleme22.m = (meet‘𝐾)
cdleme22.a 𝐴 = (Atoms‘𝐾)
cdleme22.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
cdleme22b (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ¬ 𝑇 (𝑃 𝑄))

Proof of Theorem cdleme22b
StepHypRef Expression
1 simp1l 1194 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ HL)
2 simp1r1 1266 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆𝐴)
3 simp1r2 1267 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇𝐴)
4 simp1r3 1268 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆𝑇)
5 cdleme22.j . . . . . . 7 = (join‘𝐾)
6 cdleme22.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
7 eqid 2724 . . . . . . 7 (LLines‘𝐾) = (LLines‘𝐾)
85, 6, 7llni2 38839 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
91, 2, 3, 4, 8syl31anc 1370 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ∈ (LLines‘𝐾))
106, 7llnneat 38841 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → ¬ (𝑆 𝑇) ∈ 𝐴)
111, 9, 10syl2anc 583 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ¬ (𝑆 𝑇) ∈ 𝐴)
12 eqid 2724 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
1312, 7llnn0 38843 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → (𝑆 𝑇) ≠ (0.‘𝐾))
141, 9, 13syl2anc 583 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ≠ (0.‘𝐾))
1511, 14jca 511 . . 3 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (¬ (𝑆 𝑇) ∈ 𝐴 ∧ (𝑆 𝑇) ≠ (0.‘𝐾)))
16 df-ne 2933 . . . . 5 ((𝑆 𝑇) ≠ (0.‘𝐾) ↔ ¬ (𝑆 𝑇) = (0.‘𝐾))
1716anbi2i 622 . . . 4 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ (𝑆 𝑇) ≠ (0.‘𝐾)) ↔ (¬ (𝑆 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 𝑇) = (0.‘𝐾)))
18 pm4.56 985 . . . 4 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 𝑇) = (0.‘𝐾)) ↔ ¬ ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
1917, 18bitri 275 . . 3 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ (𝑆 𝑇) ≠ (0.‘𝐾)) ↔ ¬ ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
2015, 19sylib 217 . 2 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ¬ ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
21 simp3r2 1279 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 (𝑇 𝑉))
22 simp3l 1198 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑉𝐴)
23 cdleme22.l . . . . . . . . 9 = (le‘𝐾)
2423, 5, 6hlatlej1 38701 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → 𝑇 (𝑇 𝑉))
251, 3, 22, 24syl3anc 1368 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇 (𝑇 𝑉))
261hllatd 38690 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ Lat)
27 eqid 2724 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
2827, 6atbase 38615 . . . . . . . . 9 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
292, 28syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 ∈ (Base‘𝐾))
3027, 6atbase 38615 . . . . . . . . 9 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
313, 30syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇 ∈ (Base‘𝐾))
3227, 5, 6hlatjcl 38693 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → (𝑇 𝑉) ∈ (Base‘𝐾))
331, 3, 22, 32syl3anc 1368 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 𝑉) ∈ (Base‘𝐾))
3427, 23, 5latjle12 18404 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑇 𝑉) ∈ (Base‘𝐾))) → ((𝑆 (𝑇 𝑉) ∧ 𝑇 (𝑇 𝑉)) ↔ (𝑆 𝑇) (𝑇 𝑉)))
3526, 29, 31, 33, 34syl13anc 1369 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ((𝑆 (𝑇 𝑉) ∧ 𝑇 (𝑇 𝑉)) ↔ (𝑆 𝑇) (𝑇 𝑉)))
3621, 25, 35mpbi2and 709 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) (𝑇 𝑉))
3736adantr 480 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → (𝑆 𝑇) (𝑇 𝑉))
38 simp3r3 1280 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 (𝑃 𝑄))
3938adantr 480 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → 𝑆 (𝑃 𝑄))
40 simpr 484 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → 𝑇 (𝑃 𝑄))
41 simp21 1203 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑃𝐴)
42 simp22 1204 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑄𝐴)
4327, 5, 6hlatjcl 38693 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
441, 41, 42, 43syl3anc 1368 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑃 𝑄) ∈ (Base‘𝐾))
4527, 23, 5latjle12 18404 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ↔ (𝑆 𝑇) (𝑃 𝑄)))
4626, 29, 31, 44, 45syl13anc 1369 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ((𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ↔ (𝑆 𝑇) (𝑃 𝑄)))
4746adantr 480 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → ((𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ↔ (𝑆 𝑇) (𝑃 𝑄)))
4839, 40, 47mpbi2and 709 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → (𝑆 𝑇) (𝑃 𝑄))
4927, 5, 6hlatjcl 38693 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
501, 2, 3, 49syl3anc 1368 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ∈ (Base‘𝐾))
51 cdleme22.m . . . . . . . 8 = (meet‘𝐾)
5227, 23, 51latlem12 18420 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝑆 𝑇) ∈ (Base‘𝐾) ∧ (𝑇 𝑉) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → (((𝑆 𝑇) (𝑇 𝑉) ∧ (𝑆 𝑇) (𝑃 𝑄)) ↔ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))))
5326, 50, 33, 44, 52syl13anc 1369 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (((𝑆 𝑇) (𝑇 𝑉) ∧ (𝑆 𝑇) (𝑃 𝑄)) ↔ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))))
5453adantr 480 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → (((𝑆 𝑇) (𝑇 𝑉) ∧ (𝑆 𝑇) (𝑃 𝑄)) ↔ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))))
5537, 48, 54mpbi2and 709 . . . 4 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)))
5655ex 412 . . 3 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 (𝑃 𝑄) → (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))))
57 hlop 38688 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ OP)
581, 57syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ OP)
5958adantr 480 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ (((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴 ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)))) → 𝐾 ∈ OP)
6050adantr 480 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ (((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴 ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)))) → (𝑆 𝑇) ∈ (Base‘𝐾))
61 simprl 768 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ (((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴 ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)))) → ((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴)
62 simprr 770 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ (((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴 ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)))) → (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)))
6327, 23, 12, 6leat3 38621 . . . . . 6 (((𝐾 ∈ OP ∧ (𝑆 𝑇) ∈ (Base‘𝐾) ∧ ((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴) ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))) → ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
6459, 60, 61, 62, 63syl31anc 1370 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ (((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴 ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)))) → ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
6564exp32 420 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴 → ((𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)) → ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))))
66 breq2 5142 . . . . . . . . 9 (((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) → ((𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)) ↔ (𝑆 𝑇) (0.‘𝐾)))
6766biimpa 476 . . . . . . . 8 ((((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))) → (𝑆 𝑇) (0.‘𝐾))
6827, 23, 12ople0 38513 . . . . . . . . 9 ((𝐾 ∈ OP ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑆 𝑇) (0.‘𝐾) ↔ (𝑆 𝑇) = (0.‘𝐾)))
6958, 50, 68syl2anc 583 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ((𝑆 𝑇) (0.‘𝐾) ↔ (𝑆 𝑇) = (0.‘𝐾)))
7067, 69imbitrid 243 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ((((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))) → (𝑆 𝑇) = (0.‘𝐾)))
7170imp 406 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ (((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)))) → (𝑆 𝑇) = (0.‘𝐾))
7271olcd 871 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ (((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)))) → ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
7372exp32 420 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) → ((𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)) → ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))))
74 simp3r1 1278 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 𝑉) ≠ (𝑃 𝑄))
755, 51, 12, 62atmat0 38853 . . . . 5 (((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑇 𝑉) ≠ (𝑃 𝑄))) → (((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴 ∨ ((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾)))
761, 3, 22, 41, 42, 74, 75syl33anc 1382 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴 ∨ ((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾)))
7765, 73, 76mpjaod 857 . . 3 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ((𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)) → ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾))))
7856, 77syld 47 . 2 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 (𝑃 𝑄) → ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾))))
7920, 78mtod 197 1 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ¬ 𝑇 (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844  w3a 1084   = wceq 1533  wcel 2098  wne 2932   class class class wbr 5138  cfv 6533  (class class class)co 7401  Basecbs 17142  lecple 17202  joincjn 18265  meetcmee 18266  0.cp0 18377  Latclat 18385  OPcops 38498  Atomscatm 38589  HLchlt 38676  LLinesclln 38818  LHypclh 39311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-proset 18249  df-poset 18267  df-plt 18284  df-lub 18300  df-glb 18301  df-join 18302  df-meet 18303  df-p0 18379  df-p1 18380  df-lat 18386  df-clat 18453  df-oposet 38502  df-ol 38504  df-oml 38505  df-covers 38592  df-ats 38593  df-atl 38624  df-cvlat 38648  df-hlat 38677  df-llines 38825
This theorem is referenced by:  cdleme22cN  39669  cdleme27a  39694
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