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Theorem lvolnle3at 38045
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 767 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑋𝑉)
2 eqid 2736 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2736 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 eqid 2736 . . . . . 6 (LPlanes‘𝐾) = (LPlanes‘𝐾)
5 lvolnle3at.v . . . . . 6 𝑉 = (LVols‘𝐾)
62, 3, 4, 5islvol 38036 . . . . 5 (𝐾 ∈ HL → (𝑋𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
76ad2antrr 724 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑋𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
81, 7mpbid 231 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))
98simprd 496 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)
10 oveq1 7364 . . . . . . . . 9 (𝑃 = 𝑄 → (𝑃 𝑄) = (𝑄 𝑄))
1110oveq1d 7372 . . . . . . . 8 (𝑃 = 𝑄 → ((𝑃 𝑄) 𝑅) = ((𝑄 𝑄) 𝑅))
1211breq2d 5117 . . . . . . 7 (𝑃 = 𝑄 → (𝑋 ((𝑃 𝑄) 𝑅) ↔ 𝑋 ((𝑄 𝑄) 𝑅)))
1312notbid 317 . . . . . 6 (𝑃 = 𝑄 → (¬ 𝑋 ((𝑃 𝑄) 𝑅) ↔ ¬ 𝑋 ((𝑄 𝑄) 𝑅)))
14 simp1l 1197 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
15 simp3l 1201 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LPlanes‘𝐾))
16 simp21 1206 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑃𝐴)
17 simp22 1207 . . . . . . . . . . . 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 38004 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑃𝐴𝑄𝐴)) → ¬ 𝑦 (𝑃 𝑄))
2214, 15, 16, 17, 21syl13anc 1372 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦 (𝑃 𝑄))
232, 4lplnbase 37997 . . . . . . . . . . . . . . 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 1198 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋𝑉)
262, 5lvolbase 38041 . . . . . . . . . . . . . . 15 (𝑋𝑉𝑋 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ (Base‘𝐾))
28 simp3r 1202 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋)
29 eqid 2736 . . . . . . . . . . . . . . 15 (lt‘𝐾) = (lt‘𝐾)
302, 29, 3cvrlt 37732 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
3114, 24, 27, 28, 30syl31anc 1373 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
32 hlpos 37828 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3314, 32syl 17 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
342, 19, 20hlatjcl 37829 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
3514, 16, 17, 34syl3anc 1371 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑃 𝑄) ∈ (Base‘𝐾))
362, 18, 29pltletr 18232 . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑃 𝑄)) → 𝑦(lt‘𝐾)(𝑃 𝑄)))
3831, 37mpand 693 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑃 𝑄) → 𝑦(lt‘𝐾)(𝑃 𝑄)))
3918, 29pltle 18222 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑃 𝑄) → 𝑦 (𝑃 𝑄)))
4014, 15, 35, 39syl3anc 1371 . . . . . . . . . . . 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 197 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑃 𝑄))
4342adantr 481 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ¬ 𝑋 (𝑃 𝑄))
44 simprr 771 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
4514hllatd 37826 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Lat)
46 simp23 1208 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅𝐴)
472, 20atbase 37751 . . . . . . . . . . . . . 14 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
4846, 47syl 17 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾))
492, 18, 19latleeqj2 18341 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑅 (𝑃 𝑄) ↔ ((𝑃 𝑄) 𝑅) = (𝑃 𝑄)))
5045, 48, 35, 49syl3anc 1371 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 (𝑃 𝑄) ↔ ((𝑃 𝑄) 𝑅) = (𝑃 𝑄)))
5150adantr 481 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑃 𝑄) ↔ ((𝑃 𝑄) 𝑅) = (𝑃 𝑄)))
5244, 51mpbid 231 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) = (𝑃 𝑄))
5352breq2d 5117 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → (𝑋 ((𝑃 𝑄) 𝑅) ↔ 𝑋 (𝑃 𝑄)))
5443, 53mtbird 324 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
5554anassrs 468 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃𝑄) ∧ 𝑅 (𝑃 𝑄)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
56 simpl1l 1224 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ HL)
57 simpl3l 1228 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑦 ∈ (LPlanes‘𝐾))
58 simpl2 1192 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃𝐴𝑄𝐴𝑅𝐴))
59 simpr 485 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))
6018, 19, 20, 4lplni2 38000 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
6156, 58, 59, 60syl3anc 1371 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
6229, 4lplnnlt 38028 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾)) → ¬ 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅))
6356, 57, 61, 62syl3anc 1371 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅))
642, 19latjcl 18328 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
6545, 35, 48, 64syl3anc 1371 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
662, 18, 29pltletr 18232 . . . . . . . . . . . 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 1372 . . . . . . . . . . 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 693 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ((𝑃 𝑄) 𝑅) → 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅)))
6968adantr 481 . . . . . . . . 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 197 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
7170anassrs 468 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃𝑄) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
7255, 71pm2.61dan 811 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃𝑄) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
73 eqid 2736 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
7473, 19, 20, 4lplnnle2at 38004 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑄𝐴𝑅𝐴)) → ¬ 𝑦(le‘𝐾)(𝑄 𝑅))
7514, 15, 17, 46, 74syl13anc 1372 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦(le‘𝐾)(𝑄 𝑅))
762, 19, 20hlatjcl 37829 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
7714, 17, 46, 76syl3anc 1371 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 𝑅) ∈ (Base‘𝐾))
782, 18, 29pltletr 18232 . . . . . . . . . . 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 1372 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
8031, 79mpand 693 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑄 𝑅) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
8173, 29pltle 18222 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑄 𝑅) → 𝑦(le‘𝐾)(𝑄 𝑅)))
8214, 15, 77, 81syl3anc 1371 . . . . . . . . 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 197 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑄 𝑅))
8519, 20hlatjidm 37831 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
8614, 17, 85syl2anc 584 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 𝑄) = 𝑄)
8786oveq1d 7372 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑄 𝑄) 𝑅) = (𝑄 𝑅))
8887breq2d 5117 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ((𝑄 𝑄) 𝑅) ↔ 𝑋 (𝑄 𝑅)))
8984, 88mtbird 324 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ((𝑄 𝑄) 𝑅))
9013, 72, 89pm2.61ne 3030 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
91903expia 1121 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 ((𝑃 𝑄) 𝑅)))
9291expd 416 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑦 ∈ (LPlanes‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ((𝑃 𝑄) 𝑅))))
9392rexlimdv 3150 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  Posetcpo 18196  ltcplt 18197  joincjn 18200  Latclat 18320  ccvr 37724  Atomscatm 37725  HLchlt 37812  LPlanesclpl 37955  LVolsclvol 37956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963
This theorem is referenced by:  lvolnleat  38046  lvolnlelln  38047  lvolnlelpln  38048  3atnelvolN  38049  4atlem3  38059  dalem39  38174
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