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Theorem lvolnlelln 37985
Description: A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
Hypotheses
Ref Expression
lvolnlelln.l = (le‘𝐾)
lvolnlelln.n 𝑁 = (LLines‘𝐾)
lvolnlelln.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
lvolnlelln ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → ¬ 𝑋 𝑌)

Proof of Theorem lvolnlelln
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1138 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → 𝑌𝑁)
2 eqid 2737 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2737 . . . . 5 (join‘𝐾) = (join‘𝐾)
4 eqid 2737 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
5 lvolnlelln.n . . . . 5 𝑁 = (LLines‘𝐾)
62, 3, 4, 5islln2 37912 . . . 4 (𝐾 ∈ HL → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞)))))
763ad2ant1 1133 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞)))))
81, 7mpbid 231 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))))
9 simp11 1203 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝐾 ∈ HL)
10 simp12 1204 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑋𝑉)
11 simp2l 1199 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑝 ∈ (Atoms‘𝐾))
12 simp2r 1200 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑞 ∈ (Atoms‘𝐾))
13 lvolnlelln.l . . . . . . . 8 = (le‘𝐾)
14 lvolnlelln.v . . . . . . . 8 𝑉 = (LVols‘𝐾)
1513, 3, 4, 14lvolnle3at 37983 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ¬ 𝑋 ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
169, 10, 11, 11, 12, 15syl23anc 1377 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
17 simp3r 1202 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑌 = (𝑝(join‘𝐾)𝑞))
183, 4hlatjidm 37769 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑝) = 𝑝)
199, 11, 18syl2anc 584 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → (𝑝(join‘𝐾)𝑝) = 𝑝)
2019oveq1d 7366 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞) = (𝑝(join‘𝐾)𝑞))
2117, 20eqtr4d 2780 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑌 = ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
2221breq2d 5115 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → (𝑋 𝑌𝑋 ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞)))
2316, 22mtbird 324 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 𝑌)
24233exp 1119 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞)) → ¬ 𝑋 𝑌)))
2524rexlimdvv 3202 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞)) → ¬ 𝑋 𝑌))
2625adantld 491 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 𝑌))
278, 26mpd 15 1 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → ¬ 𝑋 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wrex 3071   class class class wbr 5103  cfv 6493  (class class class)co 7351  Basecbs 17043  lecple 17100  joincjn 18160  Atomscatm 37663  HLchlt 37750  LLinesclln 37892  LVolsclvol 37894
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-proset 18144  df-poset 18162  df-plt 18179  df-lub 18195  df-glb 18196  df-join 18197  df-meet 18198  df-p0 18274  df-lat 18281  df-clat 18348  df-oposet 37576  df-ol 37578  df-oml 37579  df-covers 37666  df-ats 37667  df-atl 37698  df-cvlat 37722  df-hlat 37751  df-llines 37899  df-lplanes 37900  df-lvols 37901
This theorem is referenced by:  lvolnelln  37990
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