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Theorem lvolnlelpln 34351
Description: A lattice plane cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
Hypotheses
Ref Expression
lvolnlelpln.l = (le‘𝐾)
lvolnlelpln.p 𝑃 = (LPlanes‘𝐾)
lvolnlelpln.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
lvolnlelpln ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) → ¬ 𝑋 𝑌)

Proof of Theorem lvolnlelpln
Dummy variables 𝑟 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1061 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) → 𝑌𝑃)
2 eqid 2621 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
3 lvolnlelpln.l . . . . 5 = (le‘𝐾)
4 eqid 2621 . . . . 5 (join‘𝐾) = (join‘𝐾)
5 eqid 2621 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
6 lvolnlelpln.p . . . . 5 𝑃 = (LPlanes‘𝐾)
72, 3, 4, 5, 6islpln2 34302 . . . 4 (𝐾 ∈ HL → (𝑌𝑃 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)))))
873ad2ant1 1080 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) → (𝑌𝑃 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)))))
91, 8mpbid 222 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) → (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))))
10 simp1l1 1152 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝐾 ∈ HL)
11 simp1l2 1153 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝑋𝑉)
12 simp1r 1084 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝑞 ∈ (Atoms‘𝐾))
13 simp2l 1085 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝑟 ∈ (Atoms‘𝐾))
14 simp2r 1086 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝑠 ∈ (Atoms‘𝐾))
15 lvolnlelpln.v . . . . . . . . 9 𝑉 = (LVols‘𝐾)
163, 4, 5, 15lvolnle3at 34348 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) → ¬ 𝑋 ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))
1710, 11, 12, 13, 14, 16syl23anc 1330 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → ¬ 𝑋 ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))
18 simp33 1097 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))
1918breq2d 4625 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → (𝑋 𝑌𝑋 ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)))
2017, 19mtbird 315 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → ¬ 𝑋 𝑌)
21203exp 1261 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) → ((𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)) → ¬ 𝑋 𝑌)))
2221rexlimdvv 3030 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)) → ¬ 𝑋 𝑌))
2322rexlimdva 3024 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)) → ¬ 𝑋 𝑌))
2423adantld 483 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → ¬ 𝑋 𝑌))
259, 24mpd 15 1 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) → ¬ 𝑋 𝑌)
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  joincjn 16865  Atomscatm 34030  HLchlt 34117  LPlanesclpl 34258  LVolsclvol 34259
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 33943  df-ol 33945  df-oml 33946  df-covers 34033  df-ats 34034  df-atl 34065  df-cvlat 34089  df-hlat 34118  df-llines 34264  df-lplanes 34265  df-lvols 34266
This theorem is referenced by:  lvolnelpln  34356
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