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Theorem cdleme22b 36300
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 1254 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ HL)
2 simp1r1 1368 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆𝐴)
3 simp1r2 1369 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇𝐴)
4 simp1r3 1370 . . . . . 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 2765 . . . . . . 7 (LLines‘𝐾) = (LLines‘𝐾)
85, 6, 7llni2 35471 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
91, 2, 3, 4, 8syl31anc 1492 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ∈ (LLines‘𝐾))
106, 7llnneat 35473 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → ¬ (𝑆 𝑇) ∈ 𝐴)
111, 9, 10syl2anc 579 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ¬ (𝑆 𝑇) ∈ 𝐴)
12 eqid 2765 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
1312, 7llnn0 35475 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → (𝑆 𝑇) ≠ (0.‘𝐾))
141, 9, 13syl2anc 579 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ≠ (0.‘𝐾))
1511, 14jca 507 . . 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 2938 . . . . 5 ((𝑆 𝑇) ≠ (0.‘𝐾) ↔ ¬ (𝑆 𝑇) = (0.‘𝐾))
1716anbi2i 616 . . . 4 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ (𝑆 𝑇) ≠ (0.‘𝐾)) ↔ (¬ (𝑆 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 𝑇) = (0.‘𝐾)))
18 pm4.56 1011 . . . 4 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 𝑇) = (0.‘𝐾)) ↔ ¬ ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
1917, 18bitri 266 . . 3 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ (𝑆 𝑇) ≠ (0.‘𝐾)) ↔ ¬ ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
2015, 19sylib 209 . 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 1381 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 (𝑇 𝑉))
22 simp3l 1258 . . . . . . . 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 35334 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → 𝑇 (𝑇 𝑉))
251, 3, 22, 24syl3anc 1490 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇 (𝑇 𝑉))
261hllatd 35323 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ Lat)
27 eqid 2765 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
2827, 6atbase 35248 . . . . . . . . 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 35248 . . . . . . . . 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 35326 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → (𝑇 𝑉) ∈ (Base‘𝐾))
331, 3, 22, 32syl3anc 1490 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 𝑉) ∈ (Base‘𝐾))
3427, 23, 5latjle12 17331 . . . . . . . 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 1491 . . . . . . 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 703 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) (𝑇 𝑉))
3736adantr 472 . . . . 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 1382 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 (𝑃 𝑄))
3938adantr 472 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → 𝑆 (𝑃 𝑄))
40 simpr 477 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → 𝑇 (𝑃 𝑄))
41 simp21 1263 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑃𝐴)
42 simp22 1264 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑄𝐴)
4327, 5, 6hlatjcl 35326 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
441, 41, 42, 43syl3anc 1490 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑃 𝑄) ∈ (Base‘𝐾))
4527, 23, 5latjle12 17331 . . . . . . . 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 1491 . . . . . . 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 472 . . . . . 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 703 . . . . 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 35326 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
501, 2, 3, 49syl3anc 1490 . . . . . . 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 17347 . . . . . . 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 1491 . . . . . 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 472 . . . . 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 703 . . . 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 401 . . 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 35321 . . . . . . . 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 472 . . . . . 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 472 . . . . . 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 787 . . . . . 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 789 . . . . . 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 35254 . . . . . 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 1492 . . . . 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 411 . . . 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 4815 . . . . . . . . 9 (((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) → ((𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)) ↔ (𝑆 𝑇) (0.‘𝐾)))
6766biimpa 468 . . . . . . . 8 ((((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))) → (𝑆 𝑇) (0.‘𝐾))
6827, 23, 12ople0 35146 . . . . . . . . 9 ((𝐾 ∈ OP ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑆 𝑇) (0.‘𝐾) ↔ (𝑆 𝑇) = (0.‘𝐾)))
6958, 50, 68syl2anc 579 . . . . . . . 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, 69syl5ib 235 . . . . . . 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 395 . . . . . 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 900 . . . . 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 411 . . . 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 1380 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 𝑉) ≠ (𝑃 𝑄))
755, 51, 12, 62atmat0 35485 . . . . 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 1504 . . . 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 886 . . 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 189 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4811  cfv 6070  (class class class)co 6844  Basecbs 16133  lecple 16224  joincjn 17213  meetcmee 17214  0.cp0 17306  Latclat 17314  OPcops 35131  Atomscatm 35222  HLchlt 35309  LLinesclln 35450  LHypclh 35943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-proset 17197  df-poset 17215  df-plt 17227  df-lub 17243  df-glb 17244  df-join 17245  df-meet 17246  df-p0 17308  df-p1 17309  df-lat 17315  df-clat 17377  df-oposet 35135  df-ol 35137  df-oml 35138  df-covers 35225  df-ats 35226  df-atl 35257  df-cvlat 35281  df-hlat 35310  df-llines 35457
This theorem is referenced by:  cdleme22cN  36301  cdleme27a  36326
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