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Theorem lvolnle3at 36270
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 765 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑋𝑉)
2 eqid 2797 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2797 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 eqid 2797 . . . . . 6 (LPlanes‘𝐾) = (LPlanes‘𝐾)
5 lvolnle3at.v . . . . . 6 𝑉 = (LVols‘𝐾)
62, 3, 4, 5islvol 36261 . . . . 5 (𝐾 ∈ HL → (𝑋𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
76ad2antrr 722 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑋𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
81, 7mpbid 233 . . 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 7030 . . . . . . . . 9 (𝑃 = 𝑄 → (𝑃 𝑄) = (𝑄 𝑄))
1110oveq1d 7038 . . . . . . . 8 (𝑃 = 𝑄 → ((𝑃 𝑄) 𝑅) = ((𝑄 𝑄) 𝑅))
1211breq2d 4980 . . . . . . 7 (𝑃 = 𝑄 → (𝑋 ((𝑃 𝑄) 𝑅) ↔ 𝑋 ((𝑄 𝑄) 𝑅)))
1312notbid 319 . . . . . 6 (𝑃 = 𝑄 → (¬ 𝑋 ((𝑃 𝑄) 𝑅) ↔ ¬ 𝑋 ((𝑄 𝑄) 𝑅)))
14 simp1l 1190 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
15 simp3l 1194 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LPlanes‘𝐾))
16 simp21 1199 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑃𝐴)
17 simp22 1200 . . . . . . . . . . . 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 36229 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑃𝐴𝑄𝐴)) → ¬ 𝑦 (𝑃 𝑄))
2214, 15, 16, 17, 21syl13anc 1365 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦 (𝑃 𝑄))
232, 4lplnbase 36222 . . . . . . . . . . . . . . 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 1191 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋𝑉)
262, 5lvolbase 36266 . . . . . . . . . . . . . . 15 (𝑋𝑉𝑋 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ (Base‘𝐾))
28 simp3r 1195 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋)
29 eqid 2797 . . . . . . . . . . . . . . 15 (lt‘𝐾) = (lt‘𝐾)
302, 29, 3cvrlt 35958 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
3114, 24, 27, 28, 30syl31anc 1366 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
32 hlpos 36054 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3314, 32syl 17 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
342, 19, 20hlatjcl 36055 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
3514, 16, 17, 34syl3anc 1364 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑃 𝑄) ∈ (Base‘𝐾))
362, 18, 29pltletr 17414 . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑃 𝑄)) → 𝑦(lt‘𝐾)(𝑃 𝑄)))
3831, 37mpand 691 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑃 𝑄) → 𝑦(lt‘𝐾)(𝑃 𝑄)))
3918, 29pltle 17404 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑃 𝑄) → 𝑦 (𝑃 𝑄)))
4014, 15, 35, 39syl3anc 1364 . . . . . . . . . . . 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 199 . . . . . . . . . 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 769 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
4514hllatd 36052 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Lat)
46 simp23 1201 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅𝐴)
472, 20atbase 35977 . . . . . . . . . . . . . 14 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
4846, 47syl 17 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾))
492, 18, 19latleeqj2 17507 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑅 (𝑃 𝑄) ↔ ((𝑃 𝑄) 𝑅) = (𝑃 𝑄)))
5045, 48, 35, 49syl3anc 1364 . . . . . . . . . . . 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 233 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) = (𝑃 𝑄))
5352breq2d 4980 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → (𝑋 ((𝑃 𝑄) 𝑅) ↔ 𝑋 (𝑃 𝑄)))
5443, 53mtbird 326 . . . . . . . 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 1217 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ HL)
57 simpl3l 1221 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑦 ∈ (LPlanes‘𝐾))
58 simpl2 1185 . . . . . . . . . . 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 36225 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
6156, 58, 59, 60syl3anc 1364 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
6229, 4lplnnlt 36253 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾)) → ¬ 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅))
6356, 57, 61, 62syl3anc 1364 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅))
642, 19latjcl 17494 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
6545, 35, 48, 64syl3anc 1364 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
662, 18, 29pltletr 17414 . . . . . . . . . . . 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 1365 . . . . . . . . . . 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 691 . . . . . . . . . 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 199 . . . . . . . 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 809 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃𝑄) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
73 eqid 2797 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
7473, 19, 20, 4lplnnle2at 36229 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑄𝐴𝑅𝐴)) → ¬ 𝑦(le‘𝐾)(𝑄 𝑅))
7514, 15, 17, 46, 74syl13anc 1365 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦(le‘𝐾)(𝑄 𝑅))
762, 19, 20hlatjcl 36055 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
7714, 17, 46, 76syl3anc 1364 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 𝑅) ∈ (Base‘𝐾))
782, 18, 29pltletr 17414 . . . . . . . . . . 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 1365 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
8031, 79mpand 691 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑄 𝑅) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
8173, 29pltle 17404 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑄 𝑅) → 𝑦(le‘𝐾)(𝑄 𝑅)))
8214, 15, 77, 81syl3anc 1364 . . . . . . . . 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 199 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑄 𝑅))
8519, 20hlatjidm 36057 . . . . . . . . . 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 7038 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑄 𝑄) 𝑅) = (𝑄 𝑅))
8887breq2d 4980 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ((𝑄 𝑄) 𝑅) ↔ 𝑋 (𝑄 𝑅)))
8984, 88mtbird 326 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ((𝑄 𝑄) 𝑅))
9013, 72, 89pm2.61ne 3072 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
91903expia 1114 . . . 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 3248 . 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 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  wne 2986  wrex 3108   class class class wbr 4968  cfv 6232  (class class class)co 7023  Basecbs 16316  lecple 16405  Posetcpo 17383  ltcplt 17384  joincjn 17387  Latclat 17488  ccvr 35950  Atomscatm 35951  HLchlt 36038  LPlanesclpl 36180  LVolsclvol 36181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-proset 17371  df-poset 17389  df-plt 17401  df-lub 17417  df-glb 17418  df-join 17419  df-meet 17420  df-p0 17482  df-lat 17489  df-clat 17551  df-oposet 35864  df-ol 35866  df-oml 35867  df-covers 35954  df-ats 35955  df-atl 35986  df-cvlat 36010  df-hlat 36039  df-llines 36186  df-lplanes 36187  df-lvols 36188
This theorem is referenced by:  lvolnleat  36271  lvolnlelln  36272  lvolnlelpln  36273  3atnelvolN  36274  4atlem3  36284  dalem39  36399
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