Theorem lvolex3N 40340

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 ∧ 𝑋 ∈ 𝑃) → ∃𝑞 ∈ 𝐴 ¬ 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.

Step	Hyp	Ref	Expression
1		eqid 2762	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lvolex3.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		eqid 2762	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
4		lvolex3.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5		lvolex3.p	. . . 4 ⊢ 𝑃 = (LPlanes‘𝐾)
6	1, 2, 3, 4, 5	islpln2 40338	. . 3 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))))
7		simp1l 1215	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝐾 ∈ HL)
8		simp1rl 1256	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑟 ∈ 𝐴)
9		simp1rr 1257	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑠 ∈ 𝐴)
10		simp2 1154	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑡 ∈ 𝐴)
11	3, 2, 4	3dim3 40271	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴)) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
12	7, 8, 9, 10, 11	syl13anc 1398	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
13		simp33 1229	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
14		breq2 5112	. . . . . . . . . 10 ⊢ (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (𝑞 ≤ 𝑋 ↔ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
15	14	notbid 321	. . . . . . . . 9 ⊢ (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (¬ 𝑞 ≤ 𝑋 ↔ ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
16	15	rexbidv 3188	. . . . . . . 8 ⊢ (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋 ↔ ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
17	13, 16	syl 18	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → (∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋 ↔ ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
18	12, 17	mpbird 260	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋)
19	18	rexlimdv3a 3169	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) → (∃𝑡 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋))
20	19	rexlimdvva 3221	. . . 4 ⊢ (𝐾 ∈ HL → (∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋))
21	20	adantld 495	. . 3 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋))
22	6, 21	sylbid 243	. 2 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋))
23	22	imp 411	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3088   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  lecple 17323  joincjn 18373  Atomscatm 40065  HLchlt 40152  LPlanesclpl 40294

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-proset 18356  df-poset 18375  df-plt 18390  df-lub 18406  df-glb 18407  df-join 18408  df-meet 18409  df-p0 18485  df-p1 18486  df-lat 18494  df-clat 18561  df-oposet 39978  df-ol 39980  df-oml 39981  df-covers 40068  df-ats 40069  df-atl 40100  df-cvlat 40124  df-hlat 40153  df-llines 40300  df-lplanes 40301

This theorem is used by: (None)