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Theorem lvolex3N 38922
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 2726 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 lvolex3.l . . . 4 ≀ = (leβ€˜πΎ)
3 eqid 2726 . . . 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 38920 . . 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 1194 . . . . . . . 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 1235 . . . . . . . 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 1236 . . . . . . . 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 1134 . . . . . . . 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 38853 . . . . . . . 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 1369 . . . . . . 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 1208 . . . . . . . 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 5145 . . . . . . . . . 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 318 . . . . . . . . 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 3172 . . . . . . . 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 257 . . . . . 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 3153 . . . . 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 3205 . . . 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 490 . . 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 406 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  Atomscatm 38646  HLchlt 38733  LPlanesclpl 38876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-llines 38882  df-lplanes 38883
This theorem is referenced by: (None)
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