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Theorem cdleme3h 34339
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 34340 and cdleme3 34341. (Contributed by NM, 6-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l = (le‘𝐾)
cdleme1.j = (join‘𝐾)
cdleme1.m = (meet‘𝐾)
cdleme1.a 𝐴 = (Atoms‘𝐾)
cdleme1.h 𝐻 = (LHyp‘𝐾)
cdleme1.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme1.f 𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
cdleme3.3 𝑉 = ((𝑃 𝑅) 𝑊)
Assertion
Ref Expression
cdleme3h (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐹𝐴)

Proof of Theorem cdleme3h
StepHypRef Expression
1 cdleme1.l . . 3 = (le‘𝐾)
2 cdleme1.j . . 3 = (join‘𝐾)
3 cdleme1.m . . 3 = (meet‘𝐾)
4 cdleme1.a . . 3 𝐴 = (Atoms‘𝐾)
5 cdleme1.h . . 3 𝐻 = (LHyp‘𝐾)
6 cdleme1.u . . 3 𝑈 = ((𝑃 𝑄) 𝑊)
7 cdleme1.f . . 3 𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
8 cdleme3.3 . . 3 𝑉 = ((𝑃 𝑅) 𝑊)
91, 2, 3, 4, 5, 6, 7, 8cdleme3d 34335 . 2 𝐹 = ((𝑅 𝑈) (𝑄 𝑉))
10 simp1l 1077 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ HL)
11 simp23l 1174 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅𝐴)
12 simp1 1053 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
13 simp21 1086 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
14 simp22l 1172 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑄𝐴)
15 simp3l 1081 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑃𝑄)
161, 2, 3, 4, 5, 6lhpat2 34148 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
1712, 13, 14, 15, 16syl112anc 1321 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑈𝐴)
18 eqid 2605 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
1918, 2, 4hlatjcl 33470 . . . 4 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴) → (𝑅 𝑈) ∈ (Base‘𝐾))
2010, 11, 17, 19syl3anc 1317 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 𝑈) ∈ (Base‘𝐾))
21 simp3r 1082 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑅 (𝑃 𝑄))
2211, 21jca 552 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)))
231, 2, 3, 4, 5, 6, 7, 8cdleme3e 34336 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑉𝐴)
2412, 13, 14, 22, 23syl13anc 1319 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑉𝐴)
2518, 2, 4hlatjcl 33470 . . . 4 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑉𝐴) → (𝑄 𝑉) ∈ (Base‘𝐾))
2610, 14, 24, 25syl3anc 1317 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑄 𝑉) ∈ (Base‘𝐾))
27 hllat 33467 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2810, 27syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ Lat)
29 simp21l 1170 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑃𝐴)
3018, 2, 4hlatjcl 33470 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
3110, 29, 14, 30syl3anc 1317 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃 𝑄) ∈ (Base‘𝐾))
32 simp1r 1078 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑊𝐻)
3318, 5lhpbase 34101 . . . . . . . . 9 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3432, 33syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑊 ∈ (Base‘𝐾))
3518, 1, 3latmle2 16842 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
3628, 31, 34, 35syl3anc 1317 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑊) 𝑊)
376, 36syl5eqbr 4608 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑈 𝑊)
38 simp23r 1175 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑅 𝑊)
39 nbrne2 4593 . . . . . 6 ((𝑈 𝑊 ∧ ¬ 𝑅 𝑊) → 𝑈𝑅)
4037, 38, 39syl2anc 690 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑈𝑅)
4140necomd 2832 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅𝑈)
42 eqid 2605 . . . . 5 (Lines‘𝐾) = (Lines‘𝐾)
43 eqid 2605 . . . . 5 (pmap‘𝐾) = (pmap‘𝐾)
442, 4, 42, 43linepmap 33878 . . . 4 (((𝐾 ∈ Lat ∧ 𝑅𝐴𝑈𝐴) ∧ 𝑅𝑈) → ((pmap‘𝐾)‘(𝑅 𝑈)) ∈ (Lines‘𝐾))
4528, 11, 17, 41, 44syl31anc 1320 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((pmap‘𝐾)‘(𝑅 𝑈)) ∈ (Lines‘𝐾))
4618, 2, 4hlatjcl 33470 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
4710, 29, 11, 46syl3anc 1317 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃 𝑅) ∈ (Base‘𝐾))
4818, 1, 3latmle2 16842 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑊) 𝑊)
4928, 47, 34, 48syl3anc 1317 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑅) 𝑊) 𝑊)
508, 49syl5eqbr 4608 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑉 𝑊)
51 simp22r 1173 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑄 𝑊)
52 nbrne2 4593 . . . . . 6 ((𝑉 𝑊 ∧ ¬ 𝑄 𝑊) → 𝑉𝑄)
5350, 51, 52syl2anc 690 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑉𝑄)
5453necomd 2832 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑄𝑉)
552, 4, 42, 43linepmap 33878 . . . 4 (((𝐾 ∈ Lat ∧ 𝑄𝐴𝑉𝐴) ∧ 𝑄𝑉) → ((pmap‘𝐾)‘(𝑄 𝑉)) ∈ (Lines‘𝐾))
5628, 14, 24, 54, 55syl31anc 1320 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((pmap‘𝐾)‘(𝑄 𝑉)) ∈ (Lines‘𝐾))
571, 2, 4hlatlej1 33478 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴) → 𝑅 (𝑅 𝑈))
5810, 11, 17, 57syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅 (𝑅 𝑈))
59 nbrne2 4593 . . . . . . . . 9 ((𝑉 𝑊 ∧ ¬ 𝑅 𝑊) → 𝑉𝑅)
6050, 38, 59syl2anc 690 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑉𝑅)
6160necomd 2832 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅𝑉)
621, 2, 4hlatexch2 33499 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑉𝐴) ∧ 𝑅𝑉) → (𝑅 (𝑄 𝑉) → 𝑄 (𝑅 𝑉)))
6310, 11, 14, 24, 61, 62syl131anc 1330 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 (𝑄 𝑉) → 𝑄 (𝑅 𝑉)))
648oveq2i 6534 . . . . . . . 8 (𝑅 𝑉) = (𝑅 ((𝑃 𝑅) 𝑊))
651, 2, 4hlatlej2 33479 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → 𝑅 (𝑃 𝑅))
6610, 29, 11, 65syl3anc 1317 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑅))
6718, 1, 3latmle1 16841 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑊) (𝑃 𝑅))
6828, 47, 34, 67syl3anc 1317 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑅) 𝑊) (𝑃 𝑅))
6918, 4atbase 33393 . . . . . . . . . . 11 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
7011, 69syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅 ∈ (Base‘𝐾))
7118, 3latmcl 16817 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾))
7228, 47, 34, 71syl3anc 1317 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾))
7318, 1, 2latjle12 16827 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾) ∧ (𝑃 𝑅) ∈ (Base‘𝐾))) → ((𝑅 (𝑃 𝑅) ∧ ((𝑃 𝑅) 𝑊) (𝑃 𝑅)) ↔ (𝑅 ((𝑃 𝑅) 𝑊)) (𝑃 𝑅)))
7428, 70, 72, 47, 73syl13anc 1319 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑅 (𝑃 𝑅) ∧ ((𝑃 𝑅) 𝑊) (𝑃 𝑅)) ↔ (𝑅 ((𝑃 𝑅) 𝑊)) (𝑃 𝑅)))
7566, 68, 74mpbi2and 957 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 ((𝑃 𝑅) 𝑊)) (𝑃 𝑅))
7664, 75syl5eqbr 4608 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 𝑉) (𝑃 𝑅))
7718, 4atbase 33393 . . . . . . . . 9 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
7814, 77syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑄 ∈ (Base‘𝐾))
7918, 2, 4hlatjcl 33470 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑉𝐴) → (𝑅 𝑉) ∈ (Base‘𝐾))
8010, 11, 24, 79syl3anc 1317 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 𝑉) ∈ (Base‘𝐾))
8118, 1lattr 16821 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑅 𝑉) ∈ (Base‘𝐾) ∧ (𝑃 𝑅) ∈ (Base‘𝐾))) → ((𝑄 (𝑅 𝑉) ∧ (𝑅 𝑉) (𝑃 𝑅)) → 𝑄 (𝑃 𝑅)))
8228, 78, 80, 47, 81syl13anc 1319 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑄 (𝑅 𝑉) ∧ (𝑅 𝑉) (𝑃 𝑅)) → 𝑄 (𝑃 𝑅)))
8376, 82mpan2d 705 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑄 (𝑅 𝑉) → 𝑄 (𝑃 𝑅)))
8415necomd 2832 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑄𝑃)
851, 2, 4hlatexch1 33498 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑅) → 𝑅 (𝑃 𝑄)))
8610, 14, 11, 29, 84, 85syl131anc 1330 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑄 (𝑃 𝑅) → 𝑅 (𝑃 𝑄)))
8763, 83, 863syld 57 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 (𝑄 𝑉) → 𝑅 (𝑃 𝑄)))
8821, 87mtod 187 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑅 (𝑄 𝑉))
89 nbrne1 4592 . . . 4 ((𝑅 (𝑅 𝑈) ∧ ¬ 𝑅 (𝑄 𝑉)) → (𝑅 𝑈) ≠ (𝑄 𝑉))
9058, 88, 89syl2anc 690 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 𝑈) ≠ (𝑄 𝑉))
9114, 15jca 552 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑄𝐴𝑃𝑄))
92 simp23 1088 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
93 eqid 2605 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
941, 2, 3, 4, 5, 6, 7, 93cdleme3c 34334 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐹 ≠ (0.‘𝐾))
9512, 13, 91, 92, 94syl13anc 1319 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐹 ≠ (0.‘𝐾))
969, 95syl5eqner 2852 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑈) (𝑄 𝑉)) ≠ (0.‘𝐾))
9718, 3, 93, 4, 42, 432lnat 33887 . . 3 (((𝐾 ∈ HL ∧ (𝑅 𝑈) ∈ (Base‘𝐾) ∧ (𝑄 𝑉) ∈ (Base‘𝐾)) ∧ (((pmap‘𝐾)‘(𝑅 𝑈)) ∈ (Lines‘𝐾) ∧ ((pmap‘𝐾)‘(𝑄 𝑉)) ∈ (Lines‘𝐾)) ∧ ((𝑅 𝑈) ≠ (𝑄 𝑉) ∧ ((𝑅 𝑈) (𝑄 𝑉)) ≠ (0.‘𝐾))) → ((𝑅 𝑈) (𝑄 𝑉)) ∈ 𝐴)
9810, 20, 26, 45, 56, 90, 96, 97syl322anc 1345 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑈) (𝑄 𝑉)) ∈ 𝐴)
999, 98syl5eqel 2687 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775   class class class wbr 4573  cfv 5786  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  meetcmee 16710  0.cp0 16802  Latclat 16810  Atomscatm 33367  HLchlt 33454  Linesclines 33597  pmapcpmap 33600  LHypclh 34087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-iin 4448  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-preset 16693  df-poset 16711  df-plt 16723  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-p0 16804  df-p1 16805  df-lat 16811  df-clat 16873  df-oposet 33280  df-ol 33282  df-oml 33283  df-covers 33370  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455  df-lines 33604  df-psubsp 33606  df-pmap 33607  df-padd 33899  df-lhyp 34091
This theorem is referenced by:  cdleme3fa  34340
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