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Theorem lvolex3N 36710
 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 2820 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 lvolex3.l . . . 4 = (le‘𝐾)
3 eqid 2820 . . . 4 (join‘𝐾) = (join‘𝐾)
4 lvolex3.a . . . 4 𝐴 = (Atoms‘𝐾)
5 lvolex3.p . . . 4 𝑃 = (LPlanes‘𝐾)
61, 2, 3, 4, 5islpln2 36708 . . 3 (𝐾 ∈ HL → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑟𝐴𝑠𝐴𝑡𝐴 (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))))
7 simp1l 1193 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝐾 ∈ HL)
8 simp1rl 1234 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑟𝐴)
9 simp1rr 1235 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑠𝐴)
10 simp2 1133 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑡𝐴)
113, 2, 43dim3 36641 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴𝑡𝐴)) → ∃𝑞𝐴 ¬ 𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
127, 8, 9, 10, 11syl13anc 1368 . . . . . . 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 1207 . . . . . . . 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 5046 . . . . . . . . . 10 (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (𝑞 𝑋𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
1514notbid 320 . . . . . . . . 9 (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (¬ 𝑞 𝑋 ↔ ¬ 𝑞 ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
1615rexbidv 3284 . . . . . . . 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 259 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑡𝐴 ∧ (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞𝐴 ¬ 𝑞 𝑋)
1918rexlimdv3a 3273 . . . . 5 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑠𝐴)) → (∃𝑡𝐴 (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)) → ∃𝑞𝐴 ¬ 𝑞 𝑋))
2019rexlimdvva 3281 . . . 4 (𝐾 ∈ HL → (∃𝑟𝐴𝑠𝐴𝑡𝐴 (𝑟𝑠 ∧ ¬ 𝑡 (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)) → ∃𝑞𝐴 ¬ 𝑞 𝑋))
2120adantld 493 . . 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 242 . 2 (𝐾 ∈ HL → (𝑋𝑃 → ∃𝑞𝐴 ¬ 𝑞 𝑋))
2322imp 409 1 ((𝐾 ∈ HL ∧ 𝑋𝑃) → ∃𝑞𝐴 ¬ 𝑞 𝑋)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3006  ∃wrex 3126   class class class wbr 5042  ‘cfv 6331  (class class class)co 7133  Basecbs 16462  lecple 16551  joincjn 17533  Atomscatm 36435  HLchlt 36522  LPlanesclpl 36664 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-proset 17517  df-poset 17535  df-plt 17547  df-lub 17563  df-glb 17564  df-join 17565  df-meet 17566  df-p0 17628  df-p1 17629  df-lat 17635  df-clat 17697  df-oposet 36348  df-ol 36350  df-oml 36351  df-covers 36438  df-ats 36439  df-atl 36470  df-cvlat 36494  df-hlat 36523  df-llines 36670  df-lplanes 36671 This theorem is referenced by: (None)
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