Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme22b Structured version   Visualization version   GIF version

Theorem cdleme22b 36150
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 1239 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ HL)
2 simp1r1 1353 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆𝐴)
3 simp1r2 1354 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇𝐴)
4 simp1r3 1355 . . . . . 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 2771 . . . . . . 7 (LLines‘𝐾) = (LLines‘𝐾)
85, 6, 7llni2 35320 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
91, 2, 3, 4, 8syl31anc 1479 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ∈ (LLines‘𝐾))
106, 7llnneat 35322 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → ¬ (𝑆 𝑇) ∈ 𝐴)
111, 9, 10syl2anc 573 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ¬ (𝑆 𝑇) ∈ 𝐴)
12 eqid 2771 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
1312, 7llnn0 35324 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → (𝑆 𝑇) ≠ (0.‘𝐾))
141, 9, 13syl2anc 573 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ≠ (0.‘𝐾))
1511, 14jca 501 . . 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 2944 . . . . 5 ((𝑆 𝑇) ≠ (0.‘𝐾) ↔ ¬ (𝑆 𝑇) = (0.‘𝐾))
1716anbi2i 609 . . . 4 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ (𝑆 𝑇) ≠ (0.‘𝐾)) ↔ (¬ (𝑆 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 𝑇) = (0.‘𝐾)))
18 pm4.56 973 . . . 4 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 𝑇) = (0.‘𝐾)) ↔ ¬ ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
1917, 18bitri 264 . . 3 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ (𝑆 𝑇) ≠ (0.‘𝐾)) ↔ ¬ ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
2015, 19sylib 208 . 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 1366 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 (𝑇 𝑉))
22 simp3l 1243 . . . . . . . 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 35183 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → 𝑇 (𝑇 𝑉))
251, 3, 22, 24syl3anc 1476 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇 (𝑇 𝑉))
26 hllat 35172 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Lat)
271, 26syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ Lat)
28 eqid 2771 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
2928, 6atbase 35098 . . . . . . . . 9 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
302, 29syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 ∈ (Base‘𝐾))
3128, 6atbase 35098 . . . . . . . . 9 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
323, 31syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇 ∈ (Base‘𝐾))
3328, 5, 6hlatjcl 35175 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → (𝑇 𝑉) ∈ (Base‘𝐾))
341, 3, 22, 33syl3anc 1476 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 𝑉) ∈ (Base‘𝐾))
3528, 23, 5latjle12 17270 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑇 𝑉) ∈ (Base‘𝐾))) → ((𝑆 (𝑇 𝑉) ∧ 𝑇 (𝑇 𝑉)) ↔ (𝑆 𝑇) (𝑇 𝑉)))
3627, 30, 32, 34, 35syl13anc 1478 . . . . . . 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)))
3721, 25, 36mpbi2and 691 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) (𝑇 𝑉))
3837adantr 466 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → (𝑆 𝑇) (𝑇 𝑉))
39 simp3r3 1367 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 (𝑃 𝑄))
4039adantr 466 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → 𝑆 (𝑃 𝑄))
41 simpr 471 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → 𝑇 (𝑃 𝑄))
42 simp21 1248 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑃𝐴)
43 simp22 1249 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑄𝐴)
4428, 5, 6hlatjcl 35175 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
451, 42, 43, 44syl3anc 1476 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑃 𝑄) ∈ (Base‘𝐾))
4628, 23, 5latjle12 17270 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ↔ (𝑆 𝑇) (𝑃 𝑄)))
4727, 30, 32, 45, 46syl13anc 1478 . . . . . . 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)))
4847adantr 466 . . . . . 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)))
4940, 41, 48mpbi2and 691 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → (𝑆 𝑇) (𝑃 𝑄))
5028, 5, 6hlatjcl 35175 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
511, 2, 3, 50syl3anc 1476 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ∈ (Base‘𝐾))
52 cdleme22.m . . . . . . . 8 = (meet‘𝐾)
5328, 23, 52latlem12 17286 . . . . . . 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))))
5427, 51, 34, 45, 53syl13anc 1478 . . . . . 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))))
5554adantr 466 . . . . 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))))
5638, 49, 55mpbi2and 691 . . . 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)))
5756ex 397 . . 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))))
58 hlop 35171 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ OP)
591, 58syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ OP)
6059adantr 466 . . . . . 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)
6151adantr 466 . . . . . 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))
62 simprl 754 . . . . . 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)
63 simprr 756 . . . . . 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)))
6428, 23, 12, 6leat3 35104 . . . . . 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)))
6560, 61, 62, 63, 64syl31anc 1479 . . . . 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)))
6665exp32 407 . . . 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)))))
67 breq2 4790 . . . . . . . . 9 (((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) → ((𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)) ↔ (𝑆 𝑇) (0.‘𝐾)))
6867biimpa 462 . . . . . . . 8 ((((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))) → (𝑆 𝑇) (0.‘𝐾))
6928, 23, 12ople0 34996 . . . . . . . . 9 ((𝐾 ∈ OP ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑆 𝑇) (0.‘𝐾) ↔ (𝑆 𝑇) = (0.‘𝐾)))
7059, 51, 69syl2anc 573 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ((𝑆 𝑇) (0.‘𝐾) ↔ (𝑆 𝑇) = (0.‘𝐾)))
7168, 70syl5ib 234 . . . . . . 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)))
7271imp 393 . . . . . 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))
7372olcd 863 . . . . 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)))
7473exp32 407 . . . 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)))))
75 simp3r1 1365 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 𝑉) ≠ (𝑃 𝑄))
765, 52, 12, 62atmat0 35334 . . . . 5 (((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑇 𝑉) ≠ (𝑃 𝑄))) → (((𝑇 𝑉) (𝑃 𝑄)) ∈ 𝐴 ∨ ((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾)))
771, 3, 22, 42, 43, 75, 76syl33anc 1491 . . . 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)))
7866, 74, 77mpjaod 849 . . 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))))
7957, 78syld 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))))
8020, 79mtod 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 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  0.cp0 17245  Latclat 17253  OPcops 34981  Atomscatm 35072  HLchlt 35159  LLinesclln 35299  LHypclh 35792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306
This theorem is referenced by:  cdleme22cN  36151  cdleme27a  36176
  Copyright terms: Public domain W3C validator