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Theorem lvolnlelpln 36880
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 1135 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑃) → 𝑌𝑃)
2 eqid 2801 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
3 lvolnlelpln.l . . . . 5 = (le‘𝐾)
4 eqid 2801 . . . . 5 (join‘𝐾) = (join‘𝐾)
5 eqid 2801 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
6 lvolnlelpln.p . . . . 5 𝑃 = (LPlanes‘𝐾)
72, 3, 4, 5, 6islpln2 36831 . . . 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 1130 . . 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 235 . 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 1263 . . . . . . . 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 1264 . . . . . . . 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 1195 . . . . . . . 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 1196 . . . . . . . 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 1197 . . . . . . . 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 36877 . . . . . . . 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 1374 . . . . . . 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 1208 . . . . . . . 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 5045 . . . . . . 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 328 . . . . . 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 1116 . . . . 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 3255 . . . 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 3246 . . 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 494 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wrex 3110   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16479  lecple 16568  joincjn 17550  Atomscatm 36558  HLchlt 36645  LPlanesclpl 36787  LVolsclvol 36788
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-proset 17534  df-poset 17552  df-plt 17564  df-lub 17580  df-glb 17581  df-join 17582  df-meet 17583  df-p0 17645  df-lat 17652  df-clat 17714  df-oposet 36471  df-ol 36473  df-oml 36474  df-covers 36561  df-ats 36562  df-atl 36593  df-cvlat 36617  df-hlat 36646  df-llines 36793  df-lplanes 36794  df-lvols 36795
This theorem is referenced by:  lvolnelpln  36885
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