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Theorem cdleme3h 35840
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 35841 and cdleme3 35842. (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 35836 . 2 𝐹 = ((𝑅 𝑈) (𝑄 𝑉))
10 simp1l 1105 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ HL)
11 simp23l 1202 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅𝐴)
12 simp1 1081 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
13 simp21 1114 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
14 simp22l 1200 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑄𝐴)
15 simp3l 1109 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑃𝑄)
161, 2, 3, 4, 5, 6lhpat2 35649 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
1712, 13, 14, 15, 16syl112anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑈𝐴)
18 eqid 2651 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
1918, 2, 4hlatjcl 34971 . . . 4 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴) → (𝑅 𝑈) ∈ (Base‘𝐾))
2010, 11, 17, 19syl3anc 1366 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 𝑈) ∈ (Base‘𝐾))
21 simp3r 1110 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑅 (𝑃 𝑄))
2211, 21jca 553 . . . . 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 35837 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑉𝐴)
2412, 13, 14, 22, 23syl13anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑉𝐴)
2518, 2, 4hlatjcl 34971 . . . 4 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑉𝐴) → (𝑄 𝑉) ∈ (Base‘𝐾))
2610, 14, 24, 25syl3anc 1366 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑄 𝑉) ∈ (Base‘𝐾))
27 hllat 34968 . . . . 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 1198 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑃𝐴)
3018, 2, 4hlatjcl 34971 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
3110, 29, 14, 30syl3anc 1366 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃 𝑄) ∈ (Base‘𝐾))
32 simp1r 1106 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑊𝐻)
3318, 5lhpbase 35602 . . . . . . . . 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 17124 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
3628, 31, 34, 35syl3anc 1366 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑊) 𝑊)
376, 36syl5eqbr 4720 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑈 𝑊)
38 simp23r 1203 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑅 𝑊)
39 nbrne2 4705 . . . . . 6 ((𝑈 𝑊 ∧ ¬ 𝑅 𝑊) → 𝑈𝑅)
4037, 38, 39syl2anc 694 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑈𝑅)
4140necomd 2878 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅𝑈)
42 eqid 2651 . . . . 5 (Lines‘𝐾) = (Lines‘𝐾)
43 eqid 2651 . . . . 5 (pmap‘𝐾) = (pmap‘𝐾)
442, 4, 42, 43linepmap 35379 . . . 4 (((𝐾 ∈ Lat ∧ 𝑅𝐴𝑈𝐴) ∧ 𝑅𝑈) → ((pmap‘𝐾)‘(𝑅 𝑈)) ∈ (Lines‘𝐾))
4528, 11, 17, 41, 44syl31anc 1369 . . 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 34971 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
4710, 29, 11, 46syl3anc 1366 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃 𝑅) ∈ (Base‘𝐾))
4818, 1, 3latmle2 17124 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑊) 𝑊)
4928, 47, 34, 48syl3anc 1366 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑅) 𝑊) 𝑊)
508, 49syl5eqbr 4720 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑉 𝑊)
51 simp22r 1201 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑄 𝑊)
52 nbrne2 4705 . . . . . 6 ((𝑉 𝑊 ∧ ¬ 𝑄 𝑊) → 𝑉𝑄)
5350, 51, 52syl2anc 694 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑉𝑄)
5453necomd 2878 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑄𝑉)
552, 4, 42, 43linepmap 35379 . . . 4 (((𝐾 ∈ Lat ∧ 𝑄𝐴𝑉𝐴) ∧ 𝑄𝑉) → ((pmap‘𝐾)‘(𝑄 𝑉)) ∈ (Lines‘𝐾))
5628, 14, 24, 54, 55syl31anc 1369 . . 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 34979 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴) → 𝑅 (𝑅 𝑈))
5810, 11, 17, 57syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅 (𝑅 𝑈))
59 nbrne2 4705 . . . . . . . . 9 ((𝑉 𝑊 ∧ ¬ 𝑅 𝑊) → 𝑉𝑅)
6050, 38, 59syl2anc 694 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑉𝑅)
6160necomd 2878 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅𝑉)
621, 2, 4hlatexch2 35000 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑉𝐴) ∧ 𝑅𝑉) → (𝑅 (𝑄 𝑉) → 𝑄 (𝑅 𝑉)))
6310, 11, 14, 24, 61, 62syl131anc 1379 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 (𝑄 𝑉) → 𝑄 (𝑅 𝑉)))
648oveq2i 6701 . . . . . . . 8 (𝑅 𝑉) = (𝑅 ((𝑃 𝑅) 𝑊))
651, 2, 4hlatlej2 34980 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → 𝑅 (𝑃 𝑅))
6610, 29, 11, 65syl3anc 1366 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑅))
6718, 1, 3latmle1 17123 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑊) (𝑃 𝑅))
6828, 47, 34, 67syl3anc 1366 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑅) 𝑊) (𝑃 𝑅))
6918, 4atbase 34894 . . . . . . . . . . 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 17099 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾))
7228, 47, 34, 71syl3anc 1366 . . . . . . . . . 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 17109 . . . . . . . . . 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 1368 . . . . . . . . 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 976 . . . . . . . 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 4720 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 𝑉) (𝑃 𝑅))
7718, 4atbase 34894 . . . . . . . . 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 34971 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑉𝐴) → (𝑅 𝑉) ∈ (Base‘𝐾))
8010, 11, 24, 79syl3anc 1366 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 𝑉) ∈ (Base‘𝐾))
8118, 1lattr 17103 . . . . . . . 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 1368 . . . . . . 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 710 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑄 (𝑅 𝑉) → 𝑄 (𝑃 𝑅)))
8415necomd 2878 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑄𝑃)
851, 2, 4hlatexch1 34999 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑅) → 𝑅 (𝑃 𝑄)))
8610, 14, 11, 29, 84, 85syl131anc 1379 . . . . . 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 60 . . . . 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 189 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑅 (𝑄 𝑉))
89 nbrne1 4704 . . . 4 ((𝑅 (𝑅 𝑈) ∧ ¬ 𝑅 (𝑄 𝑉)) → (𝑅 𝑈) ≠ (𝑄 𝑉))
9058, 88, 89syl2anc 694 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 𝑈) ≠ (𝑄 𝑉))
9114, 15jca 553 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑄𝐴𝑃𝑄))
92 simp23 1116 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
93 eqid 2651 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
941, 2, 3, 4, 5, 6, 7, 93cdleme3c 35835 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐹 ≠ (0.‘𝐾))
9512, 13, 91, 92, 94syl13anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐹 ≠ (0.‘𝐾))
969, 95syl5eqner 2898 . . 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 35388 . . 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 1394 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑈) (𝑄 𝑉)) ∈ 𝐴)
999, 98syl5eqel 2734 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 196  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  Atomscatm 34868  HLchlt 34955  Linesclines 35098  pmapcpmap 35101  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-iin 4555  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-mpt2 6695  df-1st 7210  df-2nd 7211  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-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592
This theorem is referenced by:  cdleme3fa  35841
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