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Theorem lvolnle3at 37151
 Description: A lattice plane (or lattice line or atom) cannot majorize a lattice volume. (Contributed by NM, 8-Jul-2012.)
Hypotheses
Ref Expression
lvolnle3at.l = (le‘𝐾)
lvolnle3at.j = (join‘𝐾)
lvolnle3at.a 𝐴 = (Atoms‘𝐾)
lvolnle3at.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
lvolnle3at (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))

Proof of Theorem lvolnle3at
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simplr 769 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑋𝑉)
2 eqid 2759 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2759 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 eqid 2759 . . . . . 6 (LPlanes‘𝐾) = (LPlanes‘𝐾)
5 lvolnle3at.v . . . . . 6 𝑉 = (LVols‘𝐾)
62, 3, 4, 5islvol 37142 . . . . 5 (𝐾 ∈ HL → (𝑋𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
76ad2antrr 726 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑋𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
81, 7mpbid 235 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))
98simprd 500 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)
10 oveq1 7158 . . . . . . . . 9 (𝑃 = 𝑄 → (𝑃 𝑄) = (𝑄 𝑄))
1110oveq1d 7166 . . . . . . . 8 (𝑃 = 𝑄 → ((𝑃 𝑄) 𝑅) = ((𝑄 𝑄) 𝑅))
1211breq2d 5045 . . . . . . 7 (𝑃 = 𝑄 → (𝑋 ((𝑃 𝑄) 𝑅) ↔ 𝑋 ((𝑄 𝑄) 𝑅)))
1312notbid 322 . . . . . 6 (𝑃 = 𝑄 → (¬ 𝑋 ((𝑃 𝑄) 𝑅) ↔ ¬ 𝑋 ((𝑄 𝑄) 𝑅)))
14 simp1l 1195 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
15 simp3l 1199 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LPlanes‘𝐾))
16 simp21 1204 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑃𝐴)
17 simp22 1205 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑄𝐴)
18 lvolnle3at.l . . . . . . . . . . . . 13 = (le‘𝐾)
19 lvolnle3at.j . . . . . . . . . . . . 13 = (join‘𝐾)
20 lvolnle3at.a . . . . . . . . . . . . 13 𝐴 = (Atoms‘𝐾)
2118, 19, 20, 4lplnnle2at 37110 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑃𝐴𝑄𝐴)) → ¬ 𝑦 (𝑃 𝑄))
2214, 15, 16, 17, 21syl13anc 1370 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦 (𝑃 𝑄))
232, 4lplnbase 37103 . . . . . . . . . . . . . . 15 (𝑦 ∈ (LPlanes‘𝐾) → 𝑦 ∈ (Base‘𝐾))
2415, 23syl 17 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (Base‘𝐾))
25 simp1r 1196 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋𝑉)
262, 5lvolbase 37147 . . . . . . . . . . . . . . 15 (𝑋𝑉𝑋 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ (Base‘𝐾))
28 simp3r 1200 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋)
29 eqid 2759 . . . . . . . . . . . . . . 15 (lt‘𝐾) = (lt‘𝐾)
302, 29, 3cvrlt 36839 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
3114, 24, 27, 28, 30syl31anc 1371 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
32 hlpos 36935 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3314, 32syl 17 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
342, 19, 20hlatjcl 36936 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
3514, 16, 17, 34syl3anc 1369 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑃 𝑄) ∈ (Base‘𝐾))
362, 18, 29pltletr 17640 . . . . . . . . . . . . . 14 ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑃 𝑄)) → 𝑦(lt‘𝐾)(𝑃 𝑄)))
3733, 24, 27, 35, 36syl13anc 1370 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑃 𝑄)) → 𝑦(lt‘𝐾)(𝑃 𝑄)))
3831, 37mpand 695 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑃 𝑄) → 𝑦(lt‘𝐾)(𝑃 𝑄)))
3918, 29pltle 17630 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑃 𝑄) → 𝑦 (𝑃 𝑄)))
4014, 15, 35, 39syl3anc 1369 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑦(lt‘𝐾)(𝑃 𝑄) → 𝑦 (𝑃 𝑄)))
4138, 40syld 47 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑃 𝑄) → 𝑦 (𝑃 𝑄)))
4222, 41mtod 201 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑃 𝑄))
4342adantr 485 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ¬ 𝑋 (𝑃 𝑄))
44 simprr 773 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
4514hllatd 36933 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Lat)
46 simp23 1206 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅𝐴)
472, 20atbase 36858 . . . . . . . . . . . . . 14 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
4846, 47syl 17 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾))
492, 18, 19latleeqj2 17733 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑅 (𝑃 𝑄) ↔ ((𝑃 𝑄) 𝑅) = (𝑃 𝑄)))
5045, 48, 35, 49syl3anc 1369 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 (𝑃 𝑄) ↔ ((𝑃 𝑄) 𝑅) = (𝑃 𝑄)))
5150adantr 485 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑃 𝑄) ↔ ((𝑃 𝑄) 𝑅) = (𝑃 𝑄)))
5244, 51mpbid 235 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) = (𝑃 𝑄))
5352breq2d 5045 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → (𝑋 ((𝑃 𝑄) 𝑅) ↔ 𝑋 (𝑃 𝑄)))
5443, 53mtbird 329 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
5554anassrs 472 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃𝑄) ∧ 𝑅 (𝑃 𝑄)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
56 simpl1l 1222 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ HL)
57 simpl3l 1226 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑦 ∈ (LPlanes‘𝐾))
58 simpl2 1190 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃𝐴𝑄𝐴𝑅𝐴))
59 simpr 489 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))
6018, 19, 20, 4lplni2 37106 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
6156, 58, 59, 60syl3anc 1369 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
6229, 4lplnnlt 37134 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾)) → ¬ 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅))
6356, 57, 61, 62syl3anc 1369 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅))
642, 19latjcl 17720 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
6545, 35, 48, 64syl3anc 1369 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
662, 18, 29pltletr 17640 . . . . . . . . . . . 12 ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋𝑋 ((𝑃 𝑄) 𝑅)) → 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅)))
6733, 24, 27, 65, 66syl13anc 1370 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 ((𝑃 𝑄) 𝑅)) → 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅)))
6831, 67mpand 695 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ((𝑃 𝑄) 𝑅) → 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅)))
6968adantr 485 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑋 ((𝑃 𝑄) 𝑅) → 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅)))
7063, 69mtod 201 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
7170anassrs 472 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃𝑄) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
7255, 71pm2.61dan 813 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃𝑄) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
73 eqid 2759 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
7473, 19, 20, 4lplnnle2at 37110 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑄𝐴𝑅𝐴)) → ¬ 𝑦(le‘𝐾)(𝑄 𝑅))
7514, 15, 17, 46, 74syl13anc 1370 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦(le‘𝐾)(𝑄 𝑅))
762, 19, 20hlatjcl 36936 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
7714, 17, 46, 76syl3anc 1369 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 𝑅) ∈ (Base‘𝐾))
782, 18, 29pltletr 17640 . . . . . . . . . . 11 ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
7933, 24, 27, 77, 78syl13anc 1370 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
8031, 79mpand 695 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑄 𝑅) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
8173, 29pltle 17630 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑄 𝑅) → 𝑦(le‘𝐾)(𝑄 𝑅)))
8214, 15, 77, 81syl3anc 1369 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑦(lt‘𝐾)(𝑄 𝑅) → 𝑦(le‘𝐾)(𝑄 𝑅)))
8380, 82syld 47 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑄 𝑅) → 𝑦(le‘𝐾)(𝑄 𝑅)))
8475, 83mtod 201 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑄 𝑅))
8519, 20hlatjidm 36938 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
8614, 17, 85syl2anc 588 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 𝑄) = 𝑄)
8786oveq1d 7166 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑄 𝑄) 𝑅) = (𝑄 𝑅))
8887breq2d 5045 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ((𝑄 𝑄) 𝑅) ↔ 𝑋 (𝑄 𝑅)))
8984, 88mtbird 329 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ((𝑄 𝑄) 𝑅))
9013, 72, 89pm2.61ne 3037 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
91903expia 1119 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 ((𝑃 𝑄) 𝑅)))
9291expd 420 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑦 ∈ (LPlanes‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ((𝑃 𝑄) 𝑅))))
9392rexlimdv 3208 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ((𝑃 𝑄) 𝑅)))
949, 93mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∃wrex 3072   class class class wbr 5033  ‘cfv 6336  (class class class)co 7151  Basecbs 16534  lecple 16623  Posetcpo 17609  ltcplt 17610  joincjn 17613  Latclat 17714   ⋖ ccvr 36831  Atomscatm 36832  HLchlt 36919  LPlanesclpl 37061  LVolsclvol 37062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  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 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-proset 17597  df-poset 17615  df-plt 17627  df-lub 17643  df-glb 17644  df-join 17645  df-meet 17646  df-p0 17708  df-lat 17715  df-clat 17777  df-oposet 36745  df-ol 36747  df-oml 36748  df-covers 36835  df-ats 36836  df-atl 36867  df-cvlat 36891  df-hlat 36920  df-llines 37067  df-lplanes 37068  df-lvols 37069 This theorem is referenced by:  lvolnleat  37152  lvolnlelln  37153  lvolnlelpln  37154  3atnelvolN  37155  4atlem3  37165  dalem39  37280
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