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Theorem lvolnle3at 34345
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 791 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑋𝑉)
2 eqid 2621 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2621 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 eqid 2621 . . . . . 6 (LPlanes‘𝐾) = (LPlanes‘𝐾)
5 lvolnle3at.v . . . . . 6 𝑉 = (LVols‘𝐾)
62, 3, 4, 5islvol 34336 . . . . 5 (𝐾 ∈ HL → (𝑋𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
76ad2antrr 761 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑋𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
81, 7mpbid 222 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))
98simprd 479 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)
10 oveq1 6611 . . . . . . . . 9 (𝑃 = 𝑄 → (𝑃 𝑄) = (𝑄 𝑄))
1110oveq1d 6619 . . . . . . . 8 (𝑃 = 𝑄 → ((𝑃 𝑄) 𝑅) = ((𝑄 𝑄) 𝑅))
1211breq2d 4625 . . . . . . 7 (𝑃 = 𝑄 → (𝑋 ((𝑃 𝑄) 𝑅) ↔ 𝑋 ((𝑄 𝑄) 𝑅)))
1312notbid 308 . . . . . 6 (𝑃 = 𝑄 → (¬ 𝑋 ((𝑃 𝑄) 𝑅) ↔ ¬ 𝑋 ((𝑄 𝑄) 𝑅)))
14 simp1l 1083 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
15 simp3l 1087 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LPlanes‘𝐾))
16 simp21 1092 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑃𝐴)
17 simp22 1093 . . . . . . . . . . . 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 34304 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑃𝐴𝑄𝐴)) → ¬ 𝑦 (𝑃 𝑄))
2214, 15, 16, 17, 21syl13anc 1325 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦 (𝑃 𝑄))
232, 4lplnbase 34297 . . . . . . . . . . . . . . 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 1084 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋𝑉)
262, 5lvolbase 34341 . . . . . . . . . . . . . . 15 (𝑋𝑉𝑋 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ (Base‘𝐾))
28 simp3r 1088 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋)
29 eqid 2621 . . . . . . . . . . . . . . 15 (lt‘𝐾) = (lt‘𝐾)
302, 29, 3cvrlt 34034 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
3114, 24, 27, 28, 30syl31anc 1326 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
32 hlpos 34129 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3314, 32syl 17 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
342, 19, 20hlatjcl 34130 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
3514, 16, 17, 34syl3anc 1323 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑃 𝑄) ∈ (Base‘𝐾))
362, 18, 29pltletr 16892 . . . . . . . . . . . . . 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 1325 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑃 𝑄)) → 𝑦(lt‘𝐾)(𝑃 𝑄)))
3831, 37mpand 710 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑃 𝑄) → 𝑦(lt‘𝐾)(𝑃 𝑄)))
3918, 29pltle 16882 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑃 𝑄) → 𝑦 (𝑃 𝑄)))
4014, 15, 35, 39syl3anc 1323 . . . . . . . . . . . 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 189 . . . . . . . . . 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 795 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
45 hllat 34127 . . . . . . . . . . . . . 14 (𝐾 ∈ HL → 𝐾 ∈ Lat)
4614, 45syl 17 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Lat)
47 simp23 1094 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅𝐴)
482, 20atbase 34053 . . . . . . . . . . . . . 14 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
4947, 48syl 17 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾))
502, 18, 19latleeqj2 16985 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑅 (𝑃 𝑄) ↔ ((𝑃 𝑄) 𝑅) = (𝑃 𝑄)))
5146, 49, 35, 50syl3anc 1323 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 (𝑃 𝑄) ↔ ((𝑃 𝑄) 𝑅) = (𝑃 𝑄)))
5251adantr 481 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑃 𝑄) ↔ ((𝑃 𝑄) 𝑅) = (𝑃 𝑄)))
5344, 52mpbid 222 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) = (𝑃 𝑄))
5453breq2d 4625 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → (𝑋 ((𝑃 𝑄) 𝑅) ↔ 𝑋 (𝑃 𝑄)))
5543, 54mtbird 315 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
5655anassrs 679 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃𝑄) ∧ 𝑅 (𝑃 𝑄)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
57 simpl1l 1110 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ HL)
58 simpl3l 1114 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑦 ∈ (LPlanes‘𝐾))
59 simpl2 1063 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃𝐴𝑄𝐴𝑅𝐴))
60 simpr 477 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))
6118, 19, 20, 4lplni2 34300 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
6257, 59, 60, 61syl3anc 1323 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
6329, 4lplnnlt 34328 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾)) → ¬ 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅))
6457, 58, 62, 63syl3anc 1323 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅))
652, 19latjcl 16972 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
6646, 35, 49, 65syl3anc 1323 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
672, 18, 29pltletr 16892 . . . . . . . . . . . 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)))
6833, 24, 27, 66, 67syl13anc 1325 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 ((𝑃 𝑄) 𝑅)) → 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅)))
6931, 68mpand 710 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ((𝑃 𝑄) 𝑅) → 𝑦(lt‘𝐾)((𝑃 𝑄) 𝑅)))
7069adantr 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)))
7164, 70mtod 189 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
7271anassrs 679 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃𝑄) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
7356, 72pm2.61dan 831 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃𝑄) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
74 eqid 2621 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
7574, 19, 20, 4lplnnle2at 34304 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑄𝐴𝑅𝐴)) → ¬ 𝑦(le‘𝐾)(𝑄 𝑅))
7614, 15, 17, 47, 75syl13anc 1325 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦(le‘𝐾)(𝑄 𝑅))
772, 19, 20hlatjcl 34130 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
7814, 17, 47, 77syl3anc 1323 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 𝑅) ∈ (Base‘𝐾))
792, 18, 29pltletr 16892 . . . . . . . . . . 11 ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
8033, 24, 27, 78, 79syl13anc 1325 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
8131, 80mpand 710 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑄 𝑅) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
8274, 29pltle 16882 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑄 𝑅) → 𝑦(le‘𝐾)(𝑄 𝑅)))
8314, 15, 78, 82syl3anc 1323 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑦(lt‘𝐾)(𝑄 𝑅) → 𝑦(le‘𝐾)(𝑄 𝑅)))
8481, 83syld 47 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑄 𝑅) → 𝑦(le‘𝐾)(𝑄 𝑅)))
8576, 84mtod 189 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑄 𝑅))
8619, 20hlatjidm 34132 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
8714, 17, 86syl2anc 692 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 𝑄) = 𝑄)
8887oveq1d 6619 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑄 𝑄) 𝑅) = (𝑄 𝑅))
8988breq2d 4625 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ((𝑄 𝑄) 𝑅) ↔ 𝑋 (𝑄 𝑅)))
9085, 89mtbird 315 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ((𝑄 𝑄) 𝑅))
9113, 73, 90pm2.61ne 2875 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
92913expia 1264 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 ((𝑃 𝑄) 𝑅)))
9392expd 452 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑦 ∈ (LPlanes‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ((𝑃 𝑄) 𝑅))))
9493rexlimdv 3023 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ((𝑃 𝑄) 𝑅)))
959, 94mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ¬ 𝑋 ((𝑃 𝑄) 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  Posetcpo 16861  ltcplt 16862  joincjn 16865  Latclat 16966  ccvr 34026  Atomscatm 34027  HLchlt 34114  LPlanesclpl 34255  LVolsclvol 34256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-clat 17029  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115  df-llines 34261  df-lplanes 34262  df-lvols 34263
This theorem is referenced by:  lvolnleat  34346  lvolnlelln  34347  lvolnlelpln  34348  3atnelvolN  34349  4atlem3  34359  dalem39  34474
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