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Theorem lvolnlelpln 38760
Description: A lattice plane cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
Hypotheses
Ref Expression
lvolnlelpln.l ≀ = (leβ€˜πΎ)
lvolnlelpln.p 𝑃 = (LPlanesβ€˜πΎ)
lvolnlelpln.v 𝑉 = (LVolsβ€˜πΎ)
Assertion
Ref Expression
lvolnlelpln ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) β†’ Β¬ 𝑋 ≀ π‘Œ)
Proof of Theorem lvolnlelpln
Dummy variables π‘Ÿ π‘ž 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1137 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) β†’ π‘Œ ∈ 𝑃)
2 eqid 2731 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3 lvolnlelpln.l . . . . 5 ≀ = (leβ€˜πΎ)
4 eqid 2731 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
5 eqid 2731 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
6 lvolnlelpln.p . . . . 5 𝑃 = (LPlanesβ€˜πΎ)
72, 3, 4, 5, 6islpln2 38711 . . . 4 (𝐾 ∈ HL β†’ (π‘Œ ∈ 𝑃 ↔ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)βˆƒπ‘  ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠)))))
873ad2ant1 1132 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) β†’ (π‘Œ ∈ 𝑃 ↔ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)βˆƒπ‘  ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠)))))
91, 8mpbid 231 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) β†’ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)βˆƒπ‘  ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))))
10 simp1l1 1265 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))) β†’ 𝐾 ∈ HL)
11 simp1l2 1266 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))) β†’ 𝑋 ∈ 𝑉)
12 simp1r 1197 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))) β†’ π‘ž ∈ (Atomsβ€˜πΎ))
13 simp2l 1198 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))) β†’ π‘Ÿ ∈ (Atomsβ€˜πΎ))
14 simp2r 1199 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))) β†’ 𝑠 ∈ (Atomsβ€˜πΎ))
15 lvolnlelpln.v . . . . . . . . 9 𝑉 = (LVolsβ€˜πΎ)
163, 4, 5, 15lvolnle3at 38757 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ))) β†’ Β¬ 𝑋 ≀ ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))
1710, 11, 12, 13, 14, 16syl23anc 1376 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))) β†’ Β¬ 𝑋 ≀ ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))
18 simp33 1210 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))) β†’ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))
1918breq2d 5160 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 ≀ ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠)))
2017, 19mtbird 325 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))) β†’ Β¬ 𝑋 ≀ π‘Œ)
21203exp 1118 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) β†’ ((π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) β†’ ((π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠)) β†’ Β¬ 𝑋 ≀ π‘Œ)))
2221rexlimdvv 3209 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) β†’ (βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)βˆƒπ‘  ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠)) β†’ Β¬ 𝑋 ≀ π‘Œ))
2322rexlimdva 3154 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) β†’ (βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)βˆƒπ‘  ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠)) β†’ Β¬ 𝑋 ≀ π‘Œ))
2423adantld 490 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) β†’ ((π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)βˆƒπ‘  ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ Β¬ 𝑠 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ) ∧ π‘Œ = ((π‘ž(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))) β†’ Β¬ 𝑋 ≀ π‘Œ))
259, 24mpd 15 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑃) β†’ Β¬ 𝑋 ≀ π‘Œ)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆƒwrex 3069   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  lecple 17209  joincjn 18269  Atomscatm 38437  HLchlt 38524  LPlanesclpl 38667  LVolsclvol 38668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-llines 38673  df-lplanes 38674  df-lvols 38675
This theorem is referenced by:  lvolnelpln  38765
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