Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lvolex3N Structured version   Visualization version   GIF version
Theorem lvolex3N 38397
Description: There is an atom outside of a lattice plane i.e. a 3-dimensional lattice volume exists. (Contributed by NM, 28-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lvolex3.l ≀ = (leβ€˜πΎ)
lvolex3.a 𝐴 = (Atomsβ€˜πΎ)
lvolex3.p 𝑃 = (LPlanesβ€˜πΎ)
Assertion
Ref Expression
lvolex3N ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) β†’ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ 𝑋)
Distinct variable groups:   𝐴,π‘ž   𝐾,π‘ž   ≀ ,π‘ž   𝑋,π‘ž
Allowed substitution hint:   𝑃(π‘ž)
Proof of Theorem lvolex3N
Dummy variables π‘Ÿ 𝑠 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 lvolex3.l . . . 4 ≀ = (leβ€˜πΎ)
3 eqid 2732 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
4 lvolex3.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
5 lvolex3.p . . . 4 𝑃 = (LPlanesβ€˜πΎ)
61, 2, 3, 4, 5islpln2 38395 . . 3 (𝐾 ∈ HL β†’ (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 βˆƒπ‘  ∈ 𝐴 βˆƒπ‘‘ ∈ 𝐴 (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑)))))
7 simp1l 1197 . . . . . . . 8 (((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑑 ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))) β†’ 𝐾 ∈ HL)
8 simp1rl 1238 . . . . . . . 8 (((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑑 ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))) β†’ π‘Ÿ ∈ 𝐴)
9 simp1rr 1239 . . . . . . . 8 (((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑑 ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))) β†’ 𝑠 ∈ 𝐴)
10 simp2 1137 . . . . . . . 8 (((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑑 ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))) β†’ 𝑑 ∈ 𝐴)
113, 2, 43dim3 38328 . . . . . . . 8 ((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) β†’ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))
127, 8, 9, 10, 11syl13anc 1372 . . . . . . 7 (((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑑 ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))) β†’ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))
13 simp33 1211 . . . . . . . 8 (((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑑 ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))) β†’ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))
14 breq2 5151 . . . . . . . . . 10 (𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑) β†’ (π‘ž ≀ 𝑋 ↔ π‘ž ≀ ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑)))
1514notbid 317 . . . . . . . . 9 (𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑) β†’ (Β¬ π‘ž ≀ 𝑋 ↔ Β¬ π‘ž ≀ ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑)))
1615rexbidv 3178 . . . . . . . 8 (𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑) β†’ (βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ 𝑋 ↔ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑)))
1713, 16syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑑 ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))) β†’ (βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ 𝑋 ↔ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑)))
1812, 17mpbird 256 . . . . . 6 (((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑑 ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))) β†’ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ 𝑋)
1918rexlimdv3a 3159 . . . . 5 ((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) β†’ (βˆƒπ‘‘ ∈ 𝐴 (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ 𝑋))
2019rexlimdvva 3211 . . . 4 (𝐾 ∈ HL β†’ (βˆƒπ‘Ÿ ∈ 𝐴 βˆƒπ‘  ∈ 𝐴 βˆƒπ‘‘ ∈ 𝐴 (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ 𝑋))
2120adantld 491 . . 3 (𝐾 ∈ HL β†’ ((𝑋 ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 βˆƒπ‘  ∈ 𝐴 βˆƒπ‘‘ ∈ 𝐴 (π‘Ÿ β‰  𝑠 ∧ Β¬ 𝑑 ≀ (π‘Ÿ(joinβ€˜πΎ)𝑠) ∧ 𝑋 = ((π‘Ÿ(joinβ€˜πΎ)𝑠)(joinβ€˜πΎ)𝑑))) β†’ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ 𝑋))
226, 21sylbid 239 . 2 (𝐾 ∈ HL β†’ (𝑋 ∈ 𝑃 β†’ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ 𝑋))
2322imp 407 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) β†’ βˆƒπ‘ž ∈ 𝐴 Β¬ π‘ž ≀ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  Atomscatm 38121  HLchlt 38208  LPlanesclpl 38351
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-p1 18375  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-llines 38357  df-lplanes 38358
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator