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Theorem lvolex3N 37164
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 = (le‘𝐾)
lvolex3.a 𝐴 = (Atoms‘𝐾)
lvolex3.p 𝑃 = (LPlanes‘𝐾)
Assertion
Ref Expression
lvolex3N ((𝐾 ∈ HL ∧ 𝑋𝑃) → ∃𝑞𝐴 ¬ 𝑞 𝑋)
Distinct variable groups:   𝐴,𝑞   𝐾,𝑞   ,𝑞   𝑋,𝑞
Allowed substitution hint:   𝑃(𝑞)

Proof of Theorem lvolex3N
Dummy variables 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 lvolex3.l . . . 4 = (le‘𝐾)
3 eqid 2738 . . . 4 (join‘𝐾) = (join‘𝐾)
4 lvolex3.a . . . 4 𝐴 = (Atoms‘𝐾)
5 lvolex3.p . . . 4 𝑃 = (LPlanes‘𝐾)
61, 2, 3, 4, 5islpln2 37162 . . 3 (𝐾 ∈ HL → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑟𝐴𝑠𝐴𝑡𝐴 (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))))
7 simp1l 1198 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝐾 ∈ HL)
8 simp1rl 1239 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑟𝐴)
9 simp1rr 1240 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑠𝐴)
10 simp2 1138 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑡𝐴)
113, 2, 43dim3 37095 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴𝑡𝐴)) → ∃𝑞𝐴 ¬ 𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
127, 8, 9, 10, 11syl13anc 1373 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞𝐴 ¬ 𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
13 simp33 1212 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
14 breq2 5031 . . . . . . . . . 10 (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (𝑞 𝑋𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
1514notbid 321 . . . . . . . . 9 (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (¬ 𝑞 𝑋 ↔ ¬ 𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
1615rexbidv 3206 . . . . . . . 8 (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (∃𝑞𝐴 ¬ 𝑞 𝑋 ↔ ∃𝑞𝐴 ¬ 𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
1713, 16syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → (∃𝑞𝐴 ¬ 𝑞 𝑋 ↔ ∃𝑞𝐴 ¬ 𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
1812, 17mpbird 260 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞𝐴 ¬ 𝑞 𝑋)
1918rexlimdv3a 3195 . . . . 5 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) → (∃𝑡𝐴 (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)) → ∃𝑞𝐴 ¬ 𝑞 𝑋))
2019rexlimdvva 3203 . . . 4 (𝐾 ∈ HL → (∃𝑟𝐴𝑠𝐴𝑡𝐴 (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)) → ∃𝑞𝐴 ¬ 𝑞 𝑋))
2120adantld 494 . . 3 (𝐾 ∈ HL → ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑟𝐴𝑠𝐴𝑡𝐴 (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞𝐴 ¬ 𝑞 𝑋))
226, 21sylbid 243 . 2 (𝐾 ∈ HL → (𝑋𝑃 → ∃𝑞𝐴 ¬ 𝑞 𝑋))
2322imp 410 1 ((𝐾 ∈ HL ∧ 𝑋𝑃) → ∃𝑞𝐴 ¬ 𝑞 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wne 2934  wrex 3054   class class class wbr 5027  cfv 6333  (class class class)co 7164  Basecbs 16579  lecple 16668  joincjn 17663  Atomscatm 36889  HLchlt 36976  LPlanesclpl 37118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-proset 17647  df-poset 17665  df-plt 17677  df-lub 17693  df-glb 17694  df-join 17695  df-meet 17696  df-p0 17758  df-p1 17759  df-lat 17765  df-clat 17827  df-oposet 36802  df-ol 36804  df-oml 36805  df-covers 36892  df-ats 36893  df-atl 36924  df-cvlat 36948  df-hlat 36977  df-llines 37124  df-lplanes 37125
This theorem is referenced by: (None)
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