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Theorem cdleme22b 35946
 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 1105 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝐾 ∈ HL)
2 simp1r1 1177 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆𝐴)
3 simp1r2 1178 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇𝐴)
4 simp1r3 1179 . . . . . 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 2651 . . . . . . 7 (LLines‘𝐾) = (LLines‘𝐾)
85, 6, 7llni2 35116 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
91, 2, 3, 4, 8syl31anc 1369 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ∈ (LLines‘𝐾))
106, 7llnneat 35118 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → ¬ (𝑆 𝑇) ∈ 𝐴)
111, 9, 10syl2anc 694 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ¬ (𝑆 𝑇) ∈ 𝐴)
12 eqid 2651 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
1312, 7llnn0 35120 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → (𝑆 𝑇) ≠ (0.‘𝐾))
141, 9, 13syl2anc 694 . . . 4 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) ≠ (0.‘𝐾))
1511, 14jca 553 . . 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 2824 . . . . 5 ((𝑆 𝑇) ≠ (0.‘𝐾) ↔ ¬ (𝑆 𝑇) = (0.‘𝐾))
1716anbi2i 730 . . . 4 ((¬ (𝑆 𝑇) ∈ 𝐴 ∧ (𝑆 𝑇) ≠ (0.‘𝐾)) ↔ (¬ (𝑆 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 𝑇) = (0.‘𝐾)))
18 pm4.56 515 . . . 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 1190 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 (𝑇 𝑉))
22 simp3l 1109 . . . . . . . 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 34979 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → 𝑇 (𝑇 𝑉))
251, 3, 22, 24syl3anc 1366 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑇 (𝑇 𝑉))
26 hllat 34968 . . . . . . . . 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 2651 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
2928, 6atbase 34894 . . . . . . . . 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 34894 . . . . . . . . 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 34971 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → (𝑇 𝑉) ∈ (Base‘𝐾))
341, 3, 22, 33syl3anc 1366 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 𝑉) ∈ (Base‘𝐾))
3528, 23, 5latjle12 17109 . . . . . . . 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 1368 . . . . . . 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 976 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑆 𝑇) (𝑇 𝑉))
3837adantr 480 . . . . 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 1191 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑆 (𝑃 𝑄))
4039adantr 480 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → 𝑆 (𝑃 𝑄))
41 simpr 476 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) ∧ 𝑇 (𝑃 𝑄)) → 𝑇 (𝑃 𝑄))
42 simp21 1114 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑃𝐴)
43 simp22 1115 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → 𝑄𝐴)
4428, 5, 6hlatjcl 34971 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
451, 42, 43, 44syl3anc 1366 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑃 𝑄) ∈ (Base‘𝐾))
4628, 23, 5latjle12 17109 . . . . . . . 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 1368 . . . . . . 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 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)))
4940, 41, 48mpbi2and 976 . . . . 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 34971 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
511, 2, 3, 50syl3anc 1366 . . . . . . 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 17125 . . . . . . 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 1368 . . . . . 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 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))))
5638, 49, 55mpbi2and 976 . . . 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 449 . . 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 34967 . . . . . . . 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 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)
6151adantr 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))
62 simprl 809 . . . . . 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 811 . . . . . 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 34900 . . . . . 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 1369 . . . . 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 630 . . . 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 4689 . . . . . . . . 9 (((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) → ((𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄)) ↔ (𝑆 𝑇) (0.‘𝐾)))
6867biimpa 500 . . . . . . . 8 ((((𝑇 𝑉) (𝑃 𝑄)) = (0.‘𝐾) ∧ (𝑆 𝑇) ((𝑇 𝑉) (𝑃 𝑄))) → (𝑆 𝑇) (0.‘𝐾))
6928, 23, 12ople0 34792 . . . . . . . . 9 ((𝐾 ∈ OP ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑆 𝑇) (0.‘𝐾) ↔ (𝑆 𝑇) = (0.‘𝐾)))
7059, 51, 69syl2anc 694 . . . . . . . 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 444 . . . . . 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 407 . . . . 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 630 . . . 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 1189 . . . . 5 (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → (𝑇 𝑉) ≠ (𝑃 𝑄))
765, 52, 12, 62atmat0 35130 . . . . 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 1381 . . . 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 395 . . 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   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  0.cp0 17084  Latclat 17092  OPcops 34777  Atomscatm 34868  HLchlt 34955  LLinesclln 35095  LHypclh 35588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102 This theorem is referenced by:  cdleme22cN  35947  cdleme27a  35972
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