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 37978
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 1198 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ HL)
2 simp1r1 1270 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆𝐴)
3 simp1r2 1271 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇𝐴)
4 simp1r3 1272 . . . . . 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 2738 . . . . . . 7 (LLines‘𝐾) = (LLines‘𝐾)
85, 6, 7llni2 37149 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
91, 2, 3, 4, 8syl31anc 1374 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ∈ (LLines‘𝐾))
106, 7llnneat 37151 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → ¬ (𝑆 𝑇) ∈ 𝐴)
111, 9, 10syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ¬ (𝑆 𝑇) ∈ 𝐴)
12 eqid 2738 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
1312, 7llnn0 37153 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → (𝑆 𝑇) ≠ (0.‘𝐾))
141, 9, 13syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ≠ (0.‘𝐾))
1511, 14jca 515 . . 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 2935 . . . . 5 ((𝑆 𝑇) ≠ (0.‘𝐾) ↔ ¬ (𝑆 𝑇) = (0.‘𝐾))
1716anbi2i 626 . . . 4 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ (𝑆 𝑇) ≠ (0.‘𝐾)) ↔ (¬ (𝑆 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 𝑇) = (0.‘𝐾)))
18 pm4.56 988 . . . 4 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 𝑇) = (0.‘𝐾)) ↔ ¬ ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
1917, 18bitri 278 . . 3 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ (𝑆 𝑇) ≠ (0.‘𝐾)) ↔ ¬ ((𝑆 𝑇) ∈ 𝐴 ∨ (𝑆 𝑇) = (0.‘𝐾)))
2015, 19sylib 221 . 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 1283 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 (𝑇 𝑉))
22 simp3l 1202 . . . . . . . 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 37012 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → 𝑇 (𝑇 𝑉))
251, 3, 22, 24syl3anc 1372 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇 (𝑇 𝑉))
261hllatd 37001 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ Lat)
27 eqid 2738 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
2827, 6atbase 36926 . . . . . . . . 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 36926 . . . . . . . . 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 37004 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → (𝑇 𝑉) ∈ (Base‘𝐾))
331, 3, 22, 32syl3anc 1372 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 𝑉) ∈ (Base‘𝐾))
3427, 23, 5latjle12 17788 . . . . . . . 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 1373 . . . . . . 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 712 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) (𝑇 𝑉))
3736adantr 484 . . . . 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 1284 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 (𝑃 𝑄))
3938adantr 484 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → 𝑆 (𝑃 𝑄))
40 simpr 488 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → 𝑇 (𝑃 𝑄))
41 simp21 1207 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑃𝐴)
42 simp22 1208 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑄𝐴)
4327, 5, 6hlatjcl 37004 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
441, 41, 42, 43syl3anc 1372 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑃 𝑄) ∈ (Base‘𝐾))
4527, 23, 5latjle12 17788 . . . . . . . 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 1373 . . . . . . 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 484 . . . . . 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 712 . . . . 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 37004 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
501, 2, 3, 49syl3anc 1372 . . . . . . 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 17804 . . . . . . 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 1373 . . . . . 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 484 . . . . 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 712 . . . 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 416 . . 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 36999 . . . . . . . 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 484 . . . . . 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 484 . . . . . 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 771 . . . . . 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 773 . . . . . 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 36932 . . . . . 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 1374 . . . . 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 424 . . . 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 5034 . . . . . . . . 9 (((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) → ((𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)) ↔ (𝑆 𝑇) (0.‘𝐾)))
6766biimpa 480 . . . . . . . 8 ((((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))) → (𝑆 𝑇) (0.‘𝐾))
6827, 23, 12ople0 36824 . . . . . . . . 9 ((𝐾 ∈ OP ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑆 𝑇) (0.‘𝐾) ↔ (𝑆 𝑇) = (0.‘𝐾)))
6958, 50, 68syl2anc 587 . . . . . . . 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 247 . . . . . . 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 410 . . . . . 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 873 . . . . 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 424 . . . 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 1282 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 𝑉) ≠ (𝑃 𝑄))
755, 51, 12, 62atmat0 37163 . . . . 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 1386 . . . 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 859 . . 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 201 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 209  wa 399  wo 846  w3a 1088   = wceq 1542  wcel 2114  wne 2934   class class class wbr 5030  cfv 6339  (class class class)co 7170  Basecbs 16586  lecple 16675  joincjn 17670  meetcmee 17671  0.cp0 17763  Latclat 17771  OPcops 36809  Atomscatm 36900  HLchlt 36987  LLinesclln 37128  LHypclh 37621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-glb 17701  df-join 17702  df-meet 17703  df-p0 17765  df-p1 17766  df-lat 17772  df-clat 17834  df-oposet 36813  df-ol 36815  df-oml 36816  df-covers 36903  df-ats 36904  df-atl 36935  df-cvlat 36959  df-hlat 36988  df-llines 37135
This theorem is referenced by:  cdleme22cN  37979  cdleme27a  38004
  Copyright terms: Public domain W3C validator