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Theorem lvolex3N 38030
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 2737 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 lvolex3.l . . . 4 ≀ = (leβ€˜πΎ)
3 eqid 2737 . . . 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 38028 . . 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 1198 . . . . . . . 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 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†’ π‘Ÿ ∈ 𝐴)
9 simp1rr 1240 . . . . . . . 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 1138 . . . . . . . 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 37961 . . . . . . . 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 1373 . . . . . . 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 1212 . . . . . . . 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 5114 . . . . . . . . . 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 3176 . . . . . . . 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 3157 . . . . 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 3206 . . . 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 492 . . 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 408 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  lecple 17147  joincjn 18207  Atomscatm 37754  HLchlt 37841  LPlanesclpl 37984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-proset 18191  df-poset 18209  df-plt 18226  df-lub 18242  df-glb 18243  df-join 18244  df-meet 18245  df-p0 18321  df-p1 18322  df-lat 18328  df-clat 18395  df-oposet 37667  df-ol 37669  df-oml 37670  df-covers 37757  df-ats 37758  df-atl 37789  df-cvlat 37813  df-hlat 37842  df-llines 37990  df-lplanes 37991
This theorem is referenced by: (None)
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