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Theorem lvolex3N 35323
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 2756 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 lvolex3.l . . . 4 = (le‘𝐾)
3 eqid 2756 . . . 4 (join‘𝐾) = (join‘𝐾)
4 lvolex3.a . . . 4 𝐴 = (Atoms‘𝐾)
5 lvolex3.p . . . 4 𝑃 = (LPlanes‘𝐾)
61, 2, 3, 4, 5islpln2 35321 . . 3 (𝐾 ∈ HL → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑟𝐴𝑠𝐴𝑡𝐴 (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))))
7 simp1l 1240 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝐾 ∈ HL)
8 simp1rl 1305 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑟𝐴)
9 simp1rr 1306 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑠𝐴)
10 simp2 1132 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑡𝐴)
113, 2, 43dim3 35254 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴𝑡𝐴)) → ∃𝑞𝐴 ¬ 𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
127, 8, 9, 10, 11syl13anc 1479 . . . . . . 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 1254 . . . . . . . 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 4804 . . . . . . . . . 10 (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (𝑞 𝑋𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
1514notbid 307 . . . . . . . . 9 (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (¬ 𝑞 𝑋 ↔ ¬ 𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
1615rexbidv 3186 . . . . . . . 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 247 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞𝐴 ¬ 𝑞 𝑋)
1918rexlimdv3a 3167 . . . . 5 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) → (∃𝑡𝐴 (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)) → ∃𝑞𝐴 ¬ 𝑞 𝑋))
2019rexlimdvva 3172 . . . 4 (𝐾 ∈ HL → (∃𝑟𝐴𝑠𝐴𝑡𝐴 (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)) → ∃𝑞𝐴 ¬ 𝑞 𝑋))
2120adantld 484 . . 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 230 . 2 (𝐾 ∈ HL → (𝑋𝑃 → ∃𝑞𝐴 ¬ 𝑞 𝑋))
2322imp 444 1 ((𝐾 ∈ HL ∧ 𝑋𝑃) → ∃𝑞𝐴 ¬ 𝑞 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1628  wcel 2135  wne 2928  wrex 3047   class class class wbr 4800  cfv 6045  (class class class)co 6809  Basecbs 16055  lecple 16146  joincjn 17141  Atomscatm 35049  HLchlt 35136  LPlanesclpl 35277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-preset 17125  df-poset 17143  df-plt 17155  df-lub 17171  df-glb 17172  df-join 17173  df-meet 17174  df-p0 17236  df-p1 17237  df-lat 17243  df-clat 17305  df-oposet 34962  df-ol 34964  df-oml 34965  df-covers 35052  df-ats 35053  df-atl 35084  df-cvlat 35108  df-hlat 35137  df-llines 35283  df-lplanes 35284
This theorem is referenced by: (None)
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